US20010019630A1 - Method for transferring and displaying compressed images - Google Patents

Abstract

An image of certain resolution higher than possible in a single transmission over a finite bandwidth channel is obtained by transferring a progressively-rendered, compressed image. Initially, a low quality image is compressed and transmitted over a finite bandwidth channel. Then, a successively higher resolution image information is compressed at a source and is transmitted. The successively higher resolution image information received at the destination end is used to display a higher resolution image at the destination end.

Description

BACKGROUND OF THE INVENTION
• 1. Field of the Invention[0001]
• This invention relates to the compression and decompression of digital data and, more particularly, to the reduction in the amount of digital data necessary to store and transmit images.[0002]
• 2. Background of the Invention[0003]
• Image compression systems are commonly used in computers to reduce the storage space and transmittal times associated with storing, transferring and retrieving images. Due to increased use of images in computer applications, and the increase in the transfer of images, a variety of image compression techniques have attempted to solve the problems associated with the large amounts of storage space (i.e., hard disks, tapes or other devices) needed to store images.[0004]
• Conventional devices store an image as a two-dimensional array of picture elements, or pixels. The number of pixels determines the resolution of an image. Typically the resolution is measured by stating the number of horizontal and vertical pixels contained in the two dimensional image array. For example, a 640 by 480 image has 640 pixels across and 480 from top to bottom to total 307,200 pixels.[0005]
• While the number of pixels represents the image resolution, the number of bits assigned to each pixel represents the number of available intensity levels of each pixel. For example, if a pixel is only assigned one bit, the pixel can represent a maximum of two values. Thus the range of colors which can be assigned to that pixel is limited to two (typically black and white). In color images, the bits assigned to each pixel represent the intensity values of the three primary colors of red, green and blue. In present “true color” applications, each pixel is normally represented by 24 bits where 8 bits are assigned to each primary color allowing the encoding of 16.8 million (2[0006]⁸×2⁸×2⁸) different colors.
• Consequently, color images require large amounts of storage capacity. For example, a typical color (24 bits per 5 pixel) image with a resolution of 640 by 480 requires approximately 922,000 bytes of storage. A larger 24-bit color image with a 2000 by 2000 pixel resolution requires approximately twelve million bytes of storage. As a result, image-based applications such as interactive shopping, multimedia products, electronic games and other image-based presentations require large amounts of storage space to display high quality color images.[0007]
• In order to reduce storage requirements, an image is compressed (encoded) and stored as a smaller file which requires less storage space. In order to retrieve and view the compressed image, the compressed image file is expanded (decoded) to its original size. The decoded (or “reconstructed”) image is usually an imperfect or “lossy” representation of the original image because some information may be lost in the compression process. Normally, the greater the amount of compression the greater the divergence between the original image and the reconstructed image. The amount of compression is often referred to as the compression ratio. The compression ratio is the amount of storage space needed to store the original (uncompressed) digitized image file divided by the amount of storage space needed to store the corresponding compressed image file.[0008]
• By reducing the amount of storage space needed to store an image, compression is also used to reduce the time needed to transfer and communicate images to other locations. In order to transfer an image, the data bits that represent the image are sent via a data channel to another location. The sequence of transmitted bytes is called the data stream. Generally, the image data is encoded and the compressed image data stream is sent over a data channel and when received, the compressed image data is decoded to recreate the original image. Thus, compression speeds the transmission of image files by reducing their size.[0009]
• Several processes have been developed for compressing the data required to represent an image. Generally, the processes rely on two methods: 1) spatial or time domain compression, and 2) frequency domain compression. In frequency domain compression, the binary data representing each pixel in the space or time domain are mapped into a new coordinate system in the frequency domain.[0010]
• In general, the mathematical transforms, such as the discrete cosine transform (DCT), are chosen so that the signal energy of the original image is preserved, but the energy is concentrated in a relatively few transform coefficients. Once transformed, the data is compressed by quantization and encoding of the transform coefficients.[0011]
• Optimization of the process of compressing an image includes increasing the compression ratio while maintaining the quality of the original image, reducing the time to encode an image, and reducing the time to decode a compressed image. In general, a process that increases the compression ratio or decreases the time to compress an image results in a loss of image quality. A process that increases the compression ratio and maintains a high quality image often results in longer encoding and decoding times. Accordingly, it would be advantageous to increase the compression ratio and reduce the time needed to encode and decode an image while maintaining a high quality image.[0012]
• It is well known that image encoders can be optimized for specific image types. For example, different types of images may include graphical, photographic, or typographic information or combinations thereof. As discussed in more detail below, the encoding of an image can be viewed as a multi-step process that uses a variety of compression methods which include filters, mathematical transformations, quantization techniques, etc. In general each compression method will compress different image types with varying comparative efficiency. These compression methods can be selectively applied to optimize an encoder with respect to a certain type of image. In addition to selectively applying various compression methods, it is also possible to optimize an encoder by varying the parameters (e.g., quantization tables) of a particular compression method.[0013]
• Broadly speaking, however, the prior art does not provide an adaptive encoder that automatically decomposes a source image, classifies its parts, and selects the optimal compression methods and the optimal parameters of the selected compression methods resulting in an optimized encoder that increases relative compression rates.[0014]
• Once an image is optimally compressed with an encoder, the set of compressed data are stored in a file. The structure of the compressed file is referred to as the file format. The file format can be fairly simple and common, or the format can be quite complex and include a particular sequence of compressed data or various types of control instructions and codes.[0015]
• The file format (the structure of the data in the file) is especially important when compressed data in the file will be read and processed sequentially and when the user desires to view or transmit only part of a compressed image file. Accordingly, it would be advantageous to provide a file format that “layers” the compressed image components, arranging those of greatest visual importance first, those of secondary visual importance second, and so on. Layering the compressed file format in such a way allows the first segment of the compressed image file to be decoded prior to the remainder of the file being received or read by the decoder. The decoder can display the first segment (layer) as a miniature version of the entire image or can enlarge the miniature to display a coarse or “splash” quality rendition of the original image. As each successive file segment or layer is received, the decoder enhances the quality of the displayed picture by selectively adding detail and correcting pixel values.[0016]
• Like the encoding process, the decoding of an image can be viewed as a multi-step process that uses a variety of decoding methods which include inverse mathematical transformations, inverse quantization techniques, etc. Conventional decoders are designed to have an inverse function relative to the encoding system. These inverse decoding methods must match the encoding process used to encode the image. In addition, where an encoder makes content-sensitive adaptations to the compression algorithm, the decoder must apply a matching content-sensitive decoding process.[0017]
• Generally, a decoder is designed to match a specific encoding process. Prior art compression systems exist that allow the decoder to adjust particular parameters, but the prior art encoders must also transmit accompanying tables and other information. In addition, many conventional decoders are limited to specific decoding methods that do not accommodate content-sensitive adaptations.[0018]
SUMMARY OF THE INVENTION
• The problems outlined above are solved by the method and apparatus of the present invention. That is, the computer-based image compression system of the present invention includes a unique encoder which compresses images and a unique decoder which decompresses images. The unique compression system obtains high compression ratios at all image quality levels while achieving relatively quick encoding and decoding times.[0019]
• A high compression ratio enables faster image transmission and reduces the amount of storage space required to store an image. When compared with conventional compression techniques, such as the Joint Photographic Experts Group (JPEG), the present invention significantly increases the compression ratio for color images which, when decompressed, are of comparable quality to the JPEG images. The exact improvement over JPEG will depend on image content, resolution, and other factors.[0020]
• Smaller image files translate into direct storage and transmission time savings. In addition, the present invention reduces the number of operations to encode and decode an image when compared to JPEG and other compression methods of a similar nature. Reducing the number of operations reduces the amount of time and computing resources needed to encode and decode an image, and thus improves computer system response times.[0021]
• Furthermore, the image compression system of the present invention optimizes the encoding process to accommodate different image types. As explained below, the present invention uses fuzzy logic techniques to automatically analyze and decompose a source image, classify its components, select the optimal compression method for each component, and determine the optimal content-sensitive parameters of the selected compression methods. The encoder does not need prior information regarding the type of image or information regarding which compression methods to apply. Thus, a user does not need to provide compression system customization or need to set the parameters of the compression methods.[0022]
• The present invention is designed with the goal of providing an image compression system that reliably compresses any type of image with the highest achievable efficiency, while maintaining a consistent range of viewing qualities. Automating the system's adaptivity to varied image types allows for a minimum of human intervention in the encoding process and results in a system where the compression and decompression process are virtually transparent to the users.[0023]
• The encoder and decoder of the present invention contain a library of encoding methods that are treated as a “toolbox.” The toolbox allows the encoder to selectively apply particular encoding methods or tools that optimize the compression ratio for a particular image component. The toolbox approach allows the encoder to support many different encoding methods in one program, and accommodates the invention of new encoding methods without invalidating existing decoders. The toolbox approach thus allows upgradeability for future improvements in compression methods and adaptation to new technologies.[0024]
• A further feature of the present invention is that the encoder creates a file format that segments or “layers” the compressed image. The layering of the compressed image allows the decoder to display image file segments, beginning with the data at the front of the file, in a coherent sequence which begins with the decoding and display of the information that constitutes the core of the image as defined by human perception. This core information can appear as a good quality miniature of the image and/or as a full sized “splash” or coarse quality version of the image. Both the miniature and splash image enable the user to view the essence of an image from a relatively small amount of encoded data. In applications where the image file is being transmitted over a data channel, such as a telephone line or limited bandwidth wireless channel, display of the miniature and/or splash image occurs as soon as the first segment or layer of the file is received. This allows users to view the image quickly and to see detail being added to the image as subsequent layers are received, decoded, and added to the core image.[0025]
• The decoder decompresses the miniature and the full sized splash quality image from the same information. User specified preferences and the application determine whether the miniature and/or the full sized splash quality image are displayed for any given image.[0026]
• Whether the first layer is displayed as a miniature or a splash quality full size image, the receipt of each successive layer allows the decoder to add additional image detail and sharpness. Information from the previous layer is supplemented, not discarded, so that the image is built layer by layer. Thus a single compressed file with a layered file format can store both a thumbnail and a full size version of the image and can store the full size version at various quality levels without storing any redundant information.[0027]
• The layered approach of the present invention allows the transmission or decoding of only the part of the compressed file which is necessary to display a desired image quality. Thus, a single compressed file can generate a thumbnail and different quality full size images without the need to recompress the file to a smaller size and lesser quality, or store multiple files compressed to different file sizes and quality levels.[0028]
• This feature is particularly advantageous for on line service applications, such as shopping or other applications where the user or the application developer may want several thumbnail images downloaded and presented before the user chooses to receive the entire full size, high quality image. In addition to conserving the time and transmission costs associated with viewing a variety of high quality images that may not be of interest, the user need only subsequently download the remainder of each image file to view the higher detail versions of the image.[0029]
• The layered format also allows the storage of different layers of the compressed data file separate from one another. Thus, the core image data (miniature) can be stored locally (e.g., in fast RAM memory for fast access), and the higher quality “enhancement” layers can be stored remotely in lower cost bulk storage.[0030]
• A further feature of the layered file format of the present invention allows the addition of other compressed data information. The layered and segmented file format is extendable so that new layers of compressed information such as sound, text and video can be added to the compressed image data file. The extendable file format allows the compression system to adapt to new image types and to combine compressed image data with sound, text and video.[0031]
• Like the encoder, the decoder of the present invention includes a toolbox of decoding methods. The decoding process can begin with the decoder first determining the encoding methods used to encode each data segment. The decoder determines the encoding methods from instructions the encoder inserts into the compressed data file.[0032]
• Adding decoder instructions to the compressed image data provides several advantages. A decoder that recognizes the instructions can decode files from a variety of different encoders, accommodate content-sensitive encoding methods, and adjust to user specific needs. The decoder of the present invention also skips parts of the data stream that contain data that are unnecessary for a given rendition of the image, or ignore parts of the data stream that are in an unknown format. The ability to ignore unknown formats allows future file layers to be added while maintaining compatibility with older decoders.[0033]
• In a preferred embodiment of the present invention, the encoder compresses an image using a first Reed Spline Filter, an image classifier, a discrete cosine transform, a second and third Reed Spline Filter, a differential pulse code modulator, an enhancement analyzer, and an adaptive vector quantizer to generate a plurality of data segments that contain the compressed image. The plurality of data segments are further compressed with a channel encoder.[0034]
• The Reed Spline Filter includes a color space conversion transform, a decimation step and a least mean squared error (LMSE) spline fitting step. The output of the first Reed Spline Filter is then analyzed to determine an image type for optimal compression. The first Reed Spline Filter outputs three components which are analyzed by the image classifier. The image classifier uses fuzzy logic techniques to classify the image type. Once the image type is determined, the first component is separated from the second and third components and further compressed with an optimized discrete cosine transform and an adaptive vector quantizer. The second and third components are further compressed with a second and third Reed Spline Filter, the adaptive vector quantizer, and a differential pulse code modulator.[0035]
• The enhancement analyzer enhances areas of an image determined to be the most visually important, such as text or edges. The enhancement analyzer determines the visual priority of pixel blocks. The pixel block dimensions typically correspond to 16×16 pixel blocks in the source image. In addition, the enhancement analyzer prioritizes each pixel block so that the most important enhancement information is placed in the earliest enhancement layers so that it can be decoded first. The output of the enhancement analyzer is compressed with the adaptive vector quantizer.[0036]
• A user may set the encoder to compute a color palette optimized to the color image. The color palette is combined with the output of the discrete cosine transform, the adaptive vector quantizer, the differential pulse code modulator, and the enhancement analyzer to create a plurality of data segments. The channel encoder then interleaves and compresses the plurality of data segments.[0037]
BRIEF DESCRIPTION OF THE DRAWINGS
• These and other aspects, advantages, and novel features of the invention will become apparent upon reading the following detailed description and upon reference to accompanying drawings in which:[0038]
• FIG. 1 is a block diagram of an image compression system that encodes, transfers and decodes an image and includes a source image, an encoder, a compressed file, a first storage device, a data channel, a data stream, a decoder, a display, a second storage device, and a printer;[0039]
• FIG. 2 illustrates the multi-step decoding process and includes the source image, the encoder, the compressed file, the data channel, the data stream, the decoder, a thumbnail image, a splash image, a panellized standard image, and the final representation of the source image;[0040]
• FIG. 3 is a block diagram of the encoder showing the four stages of the encoding process;[0041]
• FIG. 4 is a block diagram of the encoder showing a first Reed Spline Filter, a color space conversion transform, a Y miniature, a U miniature, an X miniature, an image classifier, an optimized discrete cosine transform, a discrete cosine transform residual calculator, an adaptive vector quantizer, a second and third Reed Spline Filter, a Reed Spline residual calculator, a differential pulse coder modulator, an enhancement analyzer, a high resolution residual calculator, a palette selector, a plurality of data segments and a channel encoder;[0042]
• FIG. 5 is a block diagram of the image formatter;[0043]
• FIG. 6 is a block diagram of the Reed Spline Filter;[0044]
• FIG. 7 is a block diagram of the color space conversion transform;[0045]
• FIG. 8 is a block diagram of the image classifier;[0046]
• FIG. 9 is a block diagram of the optimized discrete cosine transform;[0047]
• FIG. 10 is a block diagram of the DCT residual calculator;[0048]
• FIG. 11 is a block diagram of the adaptive vector quantizer;[0049]
• FIG. 12 is a block diagram of the second and third Reed Spline Filters;[0050]
• FIG. 13 is a block diagram of the Reed Spline residual calculator;[0051]
• FIG. 14 is a block diagram of the differential pulse code modulator;[0052]
• FIG. 15 is a block diagram of the enhancement analyzer;[0053]
• FIG. 16 is a block diagram of the high resolution residual calculator;[0054]
• FIG. 17 is the block diagram of the palette selector;[0055]
• FIG. 18 is the block diagram of the channel encoder;[0056]
• FIG. 19 is a block diagram of the vector quantization process;[0057]
• FIGS. 20[0058]aand20bshow the segmented architecture of the data stream;
• FIG. 21 illustrates the normal segment;[0059]
• FIG. 22[0060]a,22b,22cand22dillustrate the layering and interleaving of the plurality of data segments;
• FIG. 23 is a block diagram of the decoder of the present invention;[0061]
• FIG. 24 illustrates the multi-step decoding process and includes a Ym miniature, a Um miniature, an Xm miniature, the thumbnail miniature, the splash image and the standard image, and the enhanced image;[0062]
• FIG. 25 is a block diagram of the decoder and includes an inverse Huffman encoder, an inverse DPCM, a dequantizer, a combiner, an inverse DCT, a demultiplexer, and an adder;[0063]
• FIG. 26 is a block diagram of the decoder and includes the interpolator, interpolation factors, a scaler, scale factors, a replicator, and an inverse color converter;[0064]
• FIG. 27 is a block diagram of the decoder that includes the inverse Huffman encoder, the combiner, the dequantizer, the inverse DCT, a pattern matcher, the adder, the interpolator, and an enhancement overlay builder;[0065]
• FIG. 28 is block diagram of the scaler with an input to output ratio of five-to-three in the one dimensional case;[0066]
• FIG. 29 illustrates the process of bilinear interpolation;[0067]
• FIG. 30 is a block diagram of the process of optimizing the compression methods with the image classifier, the enhancement analyzer, the optimized DCT, the AVQ, and the channel encoder;[0068]
• FIG. 31 is a block diagram of the image classifier;[0069]
• FIG. 32 is a flow chart of the process of creating an adaptive uniform DCT quantization table;[0070]
• FIG. 33 illustrates a table of several examples showing the mapping from input measurements to input sets to output sets;[0071]
• FIG. 34 is a block diagram of image data compression;[0072]
• FIG. 35 is a block diagram of a spline decimation/interpolation filter;[0073]
• FIG. 36 is a block diagram of an optimal spline filter;[0074]
• FIG. 37 is a vector representation of the image, processed image, and residual image;[0075]
• FIG. 38 is a block diagram showing a basic optimization block of the present invention;[0076]
• FIG. 39 is a graphical illustration of a one-dimensional bi-linear spline projection;[0077]
• FIG. 40 is a schematic view showing periodic replication of a two-dimensional image;[0078]
• FIGS. 41[0079]a,41band41care perspective and plan views of a two-dimensional planar spline basis;
• FIG. 42 is a diagram showing representations of the hexagonal tent function;[0080]
• FIG. 43 is a flow diagram of compression and reconstruction of image data;[0081]
• FIG. 44 is a graphical representation of a normalized frequency response of a one-dimensional bi-linear spline basis;[0082]
• FIG. 45 is a graphical representation of a one-dimensional eigenfilter frequency response;[0083]
• FIG. 46 is a perspective view of a two-dimensional eigenfilter frequency response;[0084]
• FIG. 47 is a plot of standard error as a function of frequency for a one-dimensional cosinusoidal image;[0085]
• FIG. 48 is a plot of original and reconstructed one-dimensional images and a plot of standard error;[0086]
• FIG. 49 is a first two-dimensional image reconstruction for different compression factors;[0087]
• FIG. 50 is a second two-dimensional image reconstruction for different compression factors;[0088]
• FIG. 51 is plots of standard error for[0089]representative images1 and2;
• FIG. 52 is a compressed two- miniature using the optimized decomposition weights;[0090]
• FIG. 53 is a block diagram of a preferred adaptive compression scheme in which the method of the present invention is particularly suited;[0091]
• FIG. 54 is a block diagram showing a combined sublevel and optimal-spline compression arrangement;[0092]
• FIG. 55 is a block diagram showing a combined sublevel and optimal-spline reconstruction arrangement;[0093]
• FIG. 56 is a block diagram showing a multi-resolution optimized interpolation arrangement; and[0094]
• FIG. 57 is a block diagram showing an embodiment of the optimizing process in the image domain.[0095]
DETAILED DESCRIPTION OF THE INVENTION
• FIG. 1 illustrates a block diagram of an image compression system that includes a[0096]source image100, anencoder102, acompressed file104, afirst storage device106, acommunication data channel108, adecoder110, adisplay112, asecond storage device114, and aprinter116. Thesource image100 is represented as a two-dimensional image array of picture elements, or pixels. The number of pixels determines the resolution of thesource image100, which is typically measured by the number of horizontal and vertical pixels contained in the two-dimensional image array.
• Each pixel is assigned a number of bits that represent the intensity level of the three primary colors: red, green, and blue. In the preferred embodiment, the full-[0097]color source image100 is represented with 24 bits, where 8 bits are assigned to each primary color. Thus, the total storage required for an uncompressed image is computed as the number of pixels in the image times the number of bits used to represent each pixel (referred to as bits per pixel).
• As discussed in more detail below, the[0098]encoder102 uses decimation, filtering, mathematical transforms, and quantization techniques to concentrate the image into fewer data samples representing the image with fewer bits per pixel than the original format. Once thesource image100 is compressed with theencoder102, the set of compressed data are assembled in thecompressed file104. Thecompressed file104 is stored in thefirst storage device106 or transmitted to another location via thedata channel108. If thecompressed file104 is transmitted to another location, the data stored in thecompressed file104 is transmitted sequentially via thedata channel108. The sequence of bits in thecompressed file104 that are transmitted via thedata channel108 is referred to as adata stream118.
• The[0099]decoder110 expands thecompressed file104 to the original source image size. During the process of decoding thecompressed file104, thedecoder110 displays the expandedsource image100 on thedisplay112. In addition, thedecoder110 may store the expandedcompressed file104 in thesecond storage device114 or print the expandedcompressed file104 on theprinter116.
• For example, if the[0100]source image100 comprises a 640×480, 24-bit color image, the amount of memory needed to store and display thesource image100 is approximately 922,000 bytes. In the preferred embodiment, theencoder102 computes the highest compression ratio for a given decoding quality and playback model. The playback model allows a user to select the decoding mode as is discussed in more detail below. The compressed data are then assembled in thecompressed file104 for transmittal via thedata channel108 or stored in thefirst storage device106. For example, at a 92-to-1 compression ratio, the 922,000 bytes that represent thesource image100 are compressed into approximately 10,000 bytes. In addition, theencoder102 arranges the compressed data into layers in thecompressed file104.
• Referring to FIG. 2, it can be seen that the layering of the[0101]compressed file104 allows thedecoder110 to display a thumbnail image and progressively improving quality versions of thesource image100 before thedecoder110 receives the entirecompressed file104. The first data expanded by thedecoder110 can be viewed as athumbnail miniature120 of the original image or as a coarse quality “splash”image122 with the same dimensions as the original image. Thesplash image122 is a result of interpolating the thumbnail miniature to the dimensions of the original image. As thedecoder110 continues to receive data from thedata stream118, thedecoder110 creates astandard image124 by decoding the second layer of information and adding it to thesplash image122 data to create a higher quality image. Theencoder102 can create a user-specified number of layers in which each layer is decoded and added to the displayed image as data is received. Upon receiving the entirecompressed file104 via thedata stream118, thedecoder110 displays an enhanced image105 that is the highest quality reconstructed image that can be obtained from thecompressed data stream118.
• FIG. 3 illustrates a block diagram of the[0102]encoder102 constructed in accordance with the present invention. Theencoder102 compresses thesource image100 in four main stages. In afirst stage126, thesource image100 is formatted, processed by a Reed Spline Filter and color converted. In asecond stage128, theencoder102 classifies thesource image100 in blocks. In athird stage130, theencoder102 selectively applies particular encoding methods that optimize the compression ratio. Finally, the compressed data are interleaved and channel encoded in afourth stage132.
• The[0103]encoder102 contains a library of encoding methods that are treated as a toolbox. The toolbox allows theencoder102 to selectively apply particular encoding methods that optimize the compression ratio for a particular image type. In the preferred embodiment, theencoder102 includes at least one of the following: an adaptive vector quantizer (AVQ134), an optimized discrete cosine transform (optimized DCT136), a Reed Spline Filter138 (RSF), a differential pulse code modulator (DPCM140), a run length encoder (RLE142), and anenhancement analyzer144.
• FIG. 4 illustrates a more detailed block diagram of the[0104]encoder102. Thefirst stage126 of theencoder102 includes aformatter146, a firstReed Spline Filter148 and acolor space converter150 which producesY data186, and U andX data188. Thesecond stage128 includes animage classifier152. The third stage includes an optimized discrete cosine transform and adaptive DCT quantization (optimized DCT136), a DCTresidual calculator154, the adaptive vector quantizer (AVQ134), a second and a thirdReed Spline Filter156, a Reed Splineresidual calculator158, the differential pulse code modulator (DPCM140), aresource file160, theenhancement analyzer144, a high resolutionresidual calculator162, and apalette selector164. The fourth stage includes a plurality ofdata segments166 and achannel encoder168. The output of thechannel encoder168 is stored in thecompressed file104.
• The[0105]formatter146, as shown in more detail in FIG. 5, converts thesource image100 from its native format to a 24-bit red, green and blue pixel array. For example, if thesource image100 is an 8-bit palletized image, the formatter converts the 8-bit palletized image to a 24-bit red, green, and blue equivalent.
• The first[0106]Reed Spline Filter148, illustrated in more detail in FIG. 6, uses a two-step process to compress the formattedsource image100. The two-step process comprises a decimation step performed inblock170 and a spline fitting step performed in ablock172. As explained in more detail below, the decimation step in theblock170 decimates each color component of red, green, and blue by a factor of two along the vertical and horizontal dimensions using a Reed Spline decimation kernal. The decimation factor is called “tau.” The R_tau2′ decimateddata174 corresponds to the red component decimated by a factor of 2. The G_tau2′ decimateddata176 corresponds to the green component decimated by a factor of 2. The B_tau2′ decimateddata178 corresponds to the blue component decimated by a factor of 2.
• In the spline fitting step in[0107]block172, the firstReed Spline Filter148 partially restores the source image detail lost by the decimation inblock170. The spline fitting step inblock172 processes the R_tau2′ decimateddata172, the G_tau2′ decimated data, and the B_tau2′ decimated data to calculate optimal reconstruction weights.
• As explained in more detail below, the[0108]decoder110 will interpolate the decimated data into a full sized image. In this interpolation, thedecoder110 uses the reconstruction weights which have been calculated by the Reed Spline Filter in such a way as to minimize the mean squared error between the original image components and the interpolated image components. Accordingly theReed Spline Filter148 causes the interpolated image to match the original image more closely and increases the overall sharpness of the interpolated picture. In addition, reducing the error arising from the decimation step inblock170 reduces the amount of data needed to represent the residual image. The residual image is the difference between the reconstructed image and the original image.
• The reconstruction weights output from the[0109]Reed Spline Filter148 form a “miniature” of theoriginal source image100 for each primary color of red, green, and blue, wherein each red, green, and blue miniature is one-quarter the resolution of theoriginal source image100 when a tau of 2 is used.
• More specifically, the preferred[0110]color space converter150 transforms theR_tau2 miniature180, theG_tau2 miniature182 and the B_tau2 miniature184 output by the firstReed Spline Filter148 into a different color coordinate system in which one component is theluminance Y data186 and the other two components are related to the chrominance U andX data188. Thecolor space converter150 transforms the RGB to the YUX color space according to the following formulas:

  	Y	=	0.29900R + 0.58700G + 0.11400B
  	U	=	0.16870R + 0.33120G + 0.50000B
  	X	=	0.50000R − 1.08216G + 0.91869B

• Referring to FIG. 6, it can be seen that a[0111]R_tau2 miniature180 corresponds to a miniature that is decimated and spline fitted by a factor of 2. AG_tau2 miniature182 corresponds to a green miniature that is decimated and spline fitted by a factor of 2. AB_tau2 miniature184 corresponds to a blue miniature that is decimated and spline fitted by a factor of 2.
• FIG. 7 illustrates the[0112]color space converter150 of FIG. 4. Thecolor space converter150 transforms theR_tau2 miniature180, theG_tau2 miniature182 and the B_tau2 miniature184 output by the firstReed Spline Filter148 into a different color coordinate system in which one component is theluminance Y data186 and the other two components are related to the chrominance U andX data188 as shown in FIG. 4. Thus thecolor space converter150 transforms theR_tau2 miniature180, theG_tau2 miniature182 and the B_tau2 miniature184 into aY_tau2 miniature190, aU_tau2 miniature192 and anX_tau2 miniature194.
• Referring to FIG. 8, it can be seen that the[0113]second stage128 of theencoder102 includes animage classifier152 that determines the image type by analyzing theY_tau2 miniature190, theU_tau2 miniature192 and theX_tau2 miniature194. Theimage classifier152 uses a fuzzy logic rule base to classify an image into one or more of its known classes. In the preferred embodiment, these classes include gray scale, graphics, text, photographs, high activity and low activity images. Theimage classifier152 also decomposes thesource image100 into block units and classifies each block. Since thesource image100 includes a combination of different image types, theimage classifier152 sub-divides thesource image100 into distinct regions. Theimage classifier152 then outputs thecontrol script196 that specifies the correct compression methods for each region. Thecontrol script196 specifies which compression methods to apply in thethird stage130, and specifies the channel encoding methods to apply in thefourth stage132.
• As shown in FIG. 4, during the[0114]third stage130, theencoder102 uses thecontrol script196 to select the optimal compression methods from its compression toolbox. Theencoder102 separates theY data186 from the U andX data188. Thus, theencoder102 separates the Y_tau2 miniature190 from theU_tau2 miniature192 and theX_tau2 miniature194, and passes the Y_tau2 miniature190 to the optimizedDCT136, and passes theU_tau2 miniature192 and the X_tau2 miniature194 to a second and thirdReed Spline Filter156.
• As illustrated in FIG. 9, the optimized[0115]DCT136 subdivides the Y_tau2 miniature190 into a set of 8×8 pixel blocks and transforms each 8×8 pixel block into sixty-fourDCT coefficients198. The DCT coefficients include theAC terms200 and the DC terms201. The DCT coefficients198 are analyzed by the optimizedDCT136 to determine optimal quantization step sizes and reconstruction values. The optimizedDCT136 stores the optimal quantization step sizes (uniform or non-uniform) in aquantization table Q202 and outputs the reconstruction values to theCS data segment204. The optimizedDCT136 then quantizes theDCT coefficients198 according to thequantization table Q202. Once quantized, the optimizedDCT136 outputs the DCT quantized values206 to theDCT data segment208.
• In order to preserve the image information lost by the optimized[0116]DCT136, the DCT residual calculator154 (shown in FIG. 10) computes and compresses the DCT residual. The DCTresidual calculator154 dequantizes in adequantizer209 the DCT quantized values206 stored in theDCT data segment208 by multiplying the reconstruction values in theCS data segment204 with the DCT quantized values206. The DCTresidual calculator154 then reconstructs the dequantized DCT components with aninverse DCT210 to generate a reconstructeddY_tau2 miniature211. The reconstructeddY_tau2 miniature211 is subtracted from the original Y_tau2 miniature190 to create an rY_tau2 residual212.
• Referring to FIG. 11, it can be seen that the rY_tau2 residual[0117]212 is further compressed with theAVQ134. The technique of vector quantization is used to represent a block of information as a single index that requires fewer bits of storage. As explained in more detail below, theAVQ134 maintains a group of commonly occurring block patterns in a set ofcodebooks214 stored in theresource file160. The index references a particular block pattern within aparticular codebook214. TheAVQ134 compares the input block with the block patterns in the set ofcodebooks214. If a block pattern in the set ofcodebooks214 matches or closely approximates the input block, theAVQ134 replaces the input block pattern with the index.
• Thus, the[0118]AVQ134 compresses the input block information into a list of indexes. The indexes are decompressed by replacing each index with the block pattern each index references in the set ofcodebooks214. Thedecoder110, as explained in more detail below, also has a set of thecodebooks214. During the decoding process thedecoder110 uses the list of indexes to reference block patterns stored in aparticular codebook214. The original source cannot be precisely recovered from the compressed representation since the indexed patterns in the codebook will not match the input block exactly. The degree of loss will depend on how well the codebook matches the input block.
• As shown in FIG. 11, the[0119]AVQ134 compresses the rY_tau2 residual212, by sub-dividing the rY_tau2 residual212 into 4×4 residual blocks and comparing the residual blocks with codebook patterns as explained above. TheAVQ134 replaces the residual blocks with the codebook indexes that minimize the squared error. TheAVQ134 outputs the list of codebook indexes to theVQ1 data segment224. Thus, theVQ1 data segment224 is a list of codebook indexes that identify block patterns in the codebook. As explained in more detail below, theAVQ134 of the preferred embodiment also generates new codebook patterns that theAVQ134 outputs to the set ofcodebooks214. The added codebook patterns are stored in theVQCB data segment223.
• FIG. 12 illustrates a block diagram of the second Reed Spline Filter[0120]225 and thirdReed Spline Filter227. Once theimage classifier152 determines the particular image type, theU_tau2 miniature192 and the X_tau2 miniature194 are further decimated and filtered by the second Reed Spline Filter225. Like the firstReed Spline Filter148 shown in FIG. 6, the second Reed Spline Filter225 compresses theU_tau2 miniature192 and the X_tau2 miniature194 in a two-step process. First, theU_tau2 miniature192 and the X_tau2 miniature194 are vertically and horizontally decimated by a factor of two. The decimated data are then spline fitted to determine optimal reconstruction weights that will minimize the mean square error of the reconstructed decimated miniatures. Once complete, the second Reed Spline Filter225 outputs the optimal reconstruction values to create aU_tau4 miniature226 and anX_tau4 miniature228.
• The third[0121]Reed Spline Filter227 decimates theU_tau4 miniature226 and the X_tau4 miniature228 vertically and horizontally by a factor of four. The decimated image data are again spline fitted to create aU_tau16 miniature230 and anX_tau16 miniature232.
• In FIG. 13 the Reed Spline[0122]residual calculator158 preserves the image information lost by the second Reed Spline Filter225 and the thirdReed Spline Filter227 by computing and compressing the Reed Spline Filter residual. The Reed Splineresidual calculator158 reconstructs theU_tau4 miniature226 and X_tau4 miniature228 by interpolating theU_tau16 miniature230 and theX_tau16 miniature232. The interpolatedU_tau16 miniature230 is referred to as a dU_tau4 miniature234. The interpolatedX_tau16 miniature232 is referred to as adX_tau4 miniature236. The dU_tau4 miniature234 and dX_tau4 miniature236 are subtracted from the actual U_tau4 miniature226 and X_tau4 miniature228 to create an rU_tau4 residual238 and an rX_tau4 residual240.
• As illustrated in FIG. 11, the rU_tau4 residual[0123]238 and the rX_tau4 residual240 are further compressed with theAVQ134. TheAVQ134 subdivides the rU_tau4 residual238 and the rX_tau4 residual240 into 4×4 residual blocks. The residual blocks are compared with blocks in the set ofcodebooks214 to find the codebook patterns that minimize the squared error. TheAVQ134 compresses the residual block by assigning an index that identifies the corresponding block pattern in the set ofcodebooks214. Once complete, theAVQ134 outputs the compressed residual as theVQ3 data segment242 and theVQ4 data segment244.
• The[0124]U_tau16 miniature230 and the X_tau16 miniature232 are also compressed with theDPCM140 as shown in FIG. 14. TheDPCM140 outputs the low-detail color components as theURCA data segment246 and theXRCA data segment248. TheURCA data segment246 and theXRCA data segment248 form the low-detail color components that thedecoder110 uses to create thecolor thumbnail miniature120 if this is included as a playback option in thecompressed data stream118.
• FIG. 15 illustrates the[0125]enhancement analyzer144 of the preferred embodiment. TheY_tau2 miniature190, theU_tau4 miniature226, and the X_tau4 miniature228 are analyzed to determine anenhancement list250 that specifies the visual priority of every 16×16 image block. Theenhancement analyzer144 determines the visual priority of each 16×16 image block by convolving theY_tau2 miniature190, theU_tau4 miniature226, and theX_tau4 miniature228 and comparing the result of the convolution to athreshold value E252. Thethreshold value E252 is user defined. The user can set thethreshold value E252 from zero to200. Thethreshold value E252 determines how much enhancement information theencoder102 adds to thecompressed file104. Thus, setting thethreshold value E252 to zero will suppress any image enhancement information.
• If the result of convolving a particular 16×16 high resolution block is greater than the[0126]threshold value E252, the 16×16 high-resolution block is prioritized and added to theenhancement list250. Thus theenhancement list250 identifies which 16×16 blocks are coded and prioritizes how the 16×16 coded blocks are listed.
• The high resolution[0127]residual calculator162, as shown in FIG. 16, determines the high resolution residual for each 16×16 high resolution block identified in theenhancement list250. The high resolutionresidual calculator162 translates theVQ1 data segment224 from theAVQ134 into a reconstructed rY_tau2 residual212 by mapping the indexes in theVQ1 data segment224 to the patterns in the codebook. The reconstructed rY_tau2 residual is added to the dY_tau2 miniature254 (dequantized DCT components). The result is interpolated by a factor of two in the vertical and horizontal dimensions and is subtracted from theoriginal Y_tau2190 miniature to form the high resolution residual.
• The high resolution[0128]residual calculator162 then extractshigh resolution 16×16 blocks from the high resolution residual according to the priorities in theenhancement list250. As will be explained in more detail below, the high resolutionresidual calculator162 outputs the highest priority blocks in the first enhancement layer, the next-highest priority blocks in the second enhancement layer, etc. The high resolution residual blocks are referred to as the xr_Y residual256.
• The xr_Y residual[0129]256 is further compressed with theAVQ134. TheAVQ134 subdivides the xr_Y residual256 into 4×4 residual blocks. The residual blocks are compared with blocks in the codebook. If a residual block corresponds to a block pattern in the codebook, theAVQ134 compresses the 4×4 residual block by assigning an index that identifies the corresponding block pattern in the codebook. Once complete, theAVQ134 outputs the compressed high resolution residual to theVQ2 data segment258.
• FIG. 17 illustrates a block diagram of the[0130]palette selector164. Thepalette selector164 computes a “best-fit” 24-bit color palette260 for thedecoder110. Thepalette selector164 is optional and is user defined. Thepalette selector164 computes thecolor palette260 from theY_tau2 miniature190, theU_tau2 miniature192 and theX_tau2 miniature194. The user can select a number ofpalette entries N262 to range from 0 to 255 entries. If the user selects a zero, no palette is computed. If enabled, thepalette selector164 adds thecolor palette260 to a plurality ofdata segments166.
• The[0131]channel encoder168, as shown in FIG. 18, interleaves and channel encodes the plurality ofdata segments166. Based on the user definedplayback model261, the plurality ofdata segments166 are interleaved as follows: 1) as a single layer, single-pass comprising the entire image, 2) as two layers comprising thethumbnail miniature120 and the remainder of theimage122 with enhancement information interleaved into each data block (panel) in the second layer, and 3) as multiple layers comprising thethumbnail miniature120, thestandard image124, the sharp image105, and additional layers as specified by the user. For each playback model an option exists to interleave the data for panellized or non-panellized display. The user definedplayback model261 is described in more detail below.
• After interleaving the plurality of[0132]data segments166, thechannel encoder168 compresses the plurality ofdata segments166 in response to thecontrol script196. In the preferred embodiment, thechannel encoder168 compresses the plurality ofdata segments166 with: 1) a Huffman encoding process that uses fixed tables, 2) a Huffman process that uses adaptive tables, 3) a conventional LZ1 coding technique or 4) a run-length encoding process. Thechannel encoder168 chooses the optimal compression method based on the image type identified in thecontrol script196.
• The Adaptive Vector Ouantizer[0133]
• The preferred embodiment of the[0134]AVQ134 is illustrated in FIG. 19. More specifically, theAVQ134 optimizes the vector quantization techniques described above. TheAVQ134 sub-divides the image data into a set of 4×4 pixel blocks216. The 4×4 pixel blocks216 include sixteen (16) elements X₁,X₂,X₃. . .X₁₆218, that start at the upper left-hand corner and move left to right on every row to the bottom right-hand corner.
• The[0135]codebook214 of the present invention comprises M predetermined sixteen-element vectors, P₁,P₂,P₃, . . .P_M220, that correspond to common patterns found in the population of images. The indexes I₁,I₂,I₃, . . . I_M222 refer respectively to the patterns P₁,P₂,P₃, . . . ,P_M220.
• Finding a best-fit pattern from the[0136]codebook214 requires comparing each input block with every pattern in thecodebook214 and selecting the index that corresponds to the pattern with the minimum squared error summed over the 16 elements in the 4×4 block. The optimal code, C, for an input vector, X, is the index j such that pattern P_jsatisfies:$\sum _{i=0}^{15}\left[\frac{{\left(X_i-P_{\mathrm{ij}}\right)}^2}{16}\right]=\underset{P_k\in P}{\min }\sum _{i=0}^{15}\left[\frac{{\left(X_i-P_{\mathrm{ik}}\right)}^2}{16}\right]$

  
• where: X[0137]_iis the ith element of the input vector, X and P_ikis the ith element of the VQ pattern P_k.
• The comparison equation finds the best match by selecting the minimum error term that results from comparing the input block with the codebook patterns. In other words, the[0138]AVQ134 calculates the mean squared error term associated with each pattern in thecodebook214 in order to determine which pattern in thecodebook214 has the minimum squared error (also referred to as the minimum error). The error term is the mean square error produced by subtracting the pattern element P_ikfrom the input block element X_i, squaring the result and dividing by sixteen (16).
• The process of searching for a matching pattern in the[0139]codebook214 is time-consuming. TheAVQ134 of the preferred embodiment accelerates the pattern matching process with a variety of techniques.
• First, in order to find the optimal codebook pattern, the[0140]AVQ134 compares each input block term X_ito the corresponding term in the codebook pattern P_jbeing tested and calculates the total squared error for the first codebook pattern. This value is stored as the initial minimum error. For each of the other patterns P_j=P₂,P₃, . . . , P_M, theAVQ134 subtracts the X₁and P_1jterms and squares the result. TheAVQ134 compares the resulting squared error to the minimum error. If the squared error value is less than the minimum error, theAVQ134 continues with the next input term X₂and computes the squared error associated with X₂and P_2j. TheAVQ134 adds the result to the squared error of the first two terms. TheAVQ134 then compares the accumulated squared error for X₁and X₂to the minimum error. If the accumulated squared error is less than the minimum error the squared error calculation continues until theAVQ134 has evaluated all 16 terms.
• If at any time in the comparison, the accumulated squared error for the new pattern is greater than the minimum squared error, the current pattern is immediately rejected and the[0141]AVQ134 discontinues calculating the squared error for the remaining input block terms for that pattern. If the total squared error for the new pattern is less than the minimum error, theAVQ134 replaces the minimum error with the squared error from the new pattern before making the comparisons for the remaining patterns.
• Also, if the accumulated squared error for a particular codebook pattern is less than a pre-determined threshold, the codebook pattern is immediately accepted and the[0142]AVQ134 quits testing other codebook patterns. Furthermore, the codebook patterns in the present invention are ordered according to the frequency of matches. Thus, theAVQ134 begins by comparing the input block with patterns in thecodebook214 that are most likely to match. Still further, the codebook patterns are grouped by the sum of their squared amplitudes. Thus theAVQ134 selects a group of similar codebook patterns by summing the squared amplitude of an input block in order to determine which group of codebook patterns to search.
• Besides improving the time it takes for the[0143]AVQ134 to find an optimal codebook pattern, theAVQ134 includes a set ofcodebooks214 that are adapted to the input blocks (i.e.,codebooks214 that are optimized for input blocks that contain DCT residual values, high resolution residual values, etc.). Finally, theAVQ134 of the preferred embodiment, adapts acodebook214 to thesource image100 by devising a set of new patterns to add to acodebook214.
• Therefore, the[0144]AVQ134 of the preferred embodiment has three modes of operation: 1) theAVQ134 uses a specifiedcodebook214, 2) theAVQ134 selects the best-fit codebook214, or 3) theAVQ134 uses a combination of existingcodebooks214, and new patterns that theAVQ134 creates. If theAVQ134 creates new patterns, theAVQ134 stores the new patterns in theVQCB data segment223.
• The Compressed File Format[0145]
• FIGS. 20[0146]aand20billustrate the segmented architecture of thedata stream118 that results from transmitting thecompressed file104. The segmented architecture of thecompressed file104 in the preferred embodiment allows layering of the compressed image data. Referring to FIG. 2, the layering of thecompressed file104 allows thedecoder110 to display thethumbnail miniature120, thesplash image122 and thestandard image124 before the entirecompressed file104 is transferred. As thedecoder110 receives each successive layer of components, thedecoder110 adds additional detail to the displayed image.
• In addition to layering the compressed data, the segmented architecture allows the[0147]decoder110 of the preferred embodiment: 1) to move from one segment to the next in the stream without fully decoding segments of data, 2) to skip parts of thedata stream118 that contain data that is unnecessary for a given rendition of the image, 3) to ignore parts of thedata stream118 that are in an unknown format, 4) to process the data in an order that is configurable on the fly if theentire data stream118 is stored locally, and 5) to store different layers of thecompressed file104 separately from one another.
• As shown in FIG. 20[0148]a, the byte arrangement of thedata stream118 and thecompressed file104 includes aheader segment400 and anormal segment402. Theheader segment400 contains header information, and thenormal segment402 contains data. Theheader segment400 is the first segment in thecompressed file104 and is the first segment transmitted with thedata stream118. In the preferred embodiment, theheader segment400 is eight bytes long.
• As shown in FIG. 20[0149]b, the byte arrangement of theheader segment400 includes abyte0406 and abyte1408 of theheader segment400.Byte0406 andbyte1408 of theheader segment400 identify thedata stream118.Byte1408 also indicates if thedata stream118 contains image data (indicated by a “G”) or if it contains resource data (indicated by a “C”). Resource data includes color lookup tables, font information, and vector quantization tables.
• [0150]Byte2410,byte3412,byte4414,byte5416,byte6418 andbyte7420 of theheader segment400 specify which encoder102 created thedata stream118. As new encoding methods are added to theencoder102, new versions of theencoder102 will be sold and distributed to decode the data encoded by the new methods. Thus, to remain compatible withprior encoders102, thedecoder110 needs to identify which encoder102 generated the compressed data. In the preferred embodiment,byte7420 identifies theencoder102 andbyte2410,byte3412,byte4414,byte5416, andbyte6418 are reserved for future enhancements to theencoder102.
• FIG. 21 illustrates the[0151]normal segment402 as a sequence of bytes that are logically separated into two sections: anidentifier section422 and adata section424. Theidentifier section422 precedes thedata section424. Theidentifier section422 specifies the size of thenormal segment402, and identifies a segment type. Thedata section424 contains information about thesource image100.
• The[0152]identification section422 is a sequence of one, two, or three bytes that identifies the length of thenormal segment402 and the segment type. The segment type is an integer number that specifies the method of data encoding. Thecompressed file104 contains256 possible segment types. The data in thenormal segment402 is formatted according to the segment type. In the preferred embodiment, thenormal segments402 are optimally formatted for the color palette, the Huffman bitstreams, the Huffman tables, the image panels, the codebook information, the vector dequantization tables, etc.
• For example, the file format of the preferred embodiment allows the use of different Huffman bitstreams such as an 8-bit Huffman stream, a 10-bit Huffman stream, and a DCT Huffman stream. The[0153]encoder102 uses each Huffman bitstream to optimize thecompressed file104 in response to different image types. Theidentification section422 identifies which Huffman encoder was used and thenormal segment402 contains the compressed data.
• FIGS. 22[0154]a,22b,22c, and22dillustrate the layering and interleaving of the plurality ofdata segments166 in thecompressed file104 of the preferred embodiment. The plurality ofdata segments166 in thecompressed file104 are interleaved based on the user definedplayback model261 as follows: 1) as a single-pass, non-panellized image (FIG. 22a), 2) as a single-pass, panellized image (FIG. 22b), 3) as two layers comprising thethumbnail miniature120, and the sharp image125 (FIG. 22c) and 4) as multiple layers comprising thethumbnail miniature120, thestandard image124, and the sharp image125 (FIG. 22d).
• Block diagram[0155]426 in FIG. 22ashows the compressed file format for the single-pass, non-panellized image. Thecompressed file104 begins with the header, the optional color palette and the resource data such as the tables and Huffman encoding information. The plurality ofdata segments166 are not interleaved or layered. Thus, thedecoder110 must receive the entirecompressed file104 before any part of thesource image100 can be displayed.
• Block diagram[0156]428 in FIG. 22bshows thecompressed file104 for the single-pass, panellized image. The plurality ofdata segments166 are interleaved panel-by-panel, so that all of the segments for each panel are contiguously transmitted. Thedecoder110 can expand and display a panel at a time until the entirecompressed file104 is expanded.
• Block diagram[0157]430 in FIG. 22cshows the compressed file format of thethumbnail miniature120, thesplash image122 and the final orsharp image125. The plurality ofdata segments166 are interleaved panel-by-panel and the resolution components for thethumbnail miniature120 andsplash image122 exist in the first layer, the panels for the final image exist in the second layer. The first layer includes selected portions of the plurality ofdata segments166 that are needed to decode the panels of thethumbnail miniature120 andsplash image122. Thus, thecompressed file104 only stores the low detail color components (URCA data segment246, the XRCA data segment248), theDC terms201 and as many as the first fiveAC terms200 in the first layer. The number ofAC terms200 depends on the user-selected quality of thethumbnail miniature120.
• The plurality of[0158]data segments166 in the first layer are also interleaved panel-by-panel to allow thethumbnail miniature120 andsplash image122 to be decoded a panel at a time. The second layer contains the remaining plurality ofdata segments166 needed to expand thecompressed file104 into the final image. The plurality ofdata segments166 in the second layer are also interleaved panel-by-panel.
• [0159]Block432 in FIG. 22dshows the compressed file format of thethumbnail image120, thesplash image122, the layeredstandard image124, and thesharp image125. Thethumbnail miniature120 andsplash image122 are arranged in the first layer as described above. The remainingdata segments166 are layered at different quality levels. The multi-layering is accomplished by layering and interleaving panel information associated with the VQ2 data segment258 (high resolution residual). The multiple layers allow the display of all the panels at a particular level of detail before decoding the panels in the next layer.
• The Decoder[0160]
• FIG. 23 illustrates the[0161]decoder110 of the present invention. Thedecoder110 takes as input thecompressed data stream118 and expands or decodes it into an image for viewing on thedisplay112. As explained above, thecompressed file104 and the transmitteddata stream118 include image components that are layered with a plurality ofpanels433. Thedecoder110 expands the plurality ofpanels433 one at a time.
• As illustrated in FIG. 24, the[0162]decoder110 expands thecompressed file104 in four steps. In afirst step434, thedecoder110 expands the first layer of image data in thecompressed file104 or thedata stream118 into aYm miniature436, aUm miniature438, and anXm miniature440. In asecond step442, thedecoder110 uses theYm miniature436, theUm miniature438, and an Xm miniature440 to generate thethumbnail miniature120, and thesplash image122. In athird step444, thedecoder110 receives a second layer of image data and generates the higher detail panels445 needed to expand thethumbnail miniature120 into astandard image124, afourth step446 thedecoder110 receives a third layer of image data to generate higher detail panels to enhance the detail of the standard image in order to create an enhanced image105 that corresponds to thesource image100.
• FIG. 25 illustrates the elements of the[0163]first step434 in which thedecoder110 expands theAC terms200, theDC terms201, theURCA data segment246, and theXRCA data segment248 into theYm miniature436, theUm miniature438, andXm miniature440. Thefirst step434 includes aninverse Huffman encoder458, aninverse DPCM476, adequantizer450, acombiner452, aninverse DCT476, ademultiplexer454, and anadder456.
• The[0164]decoder110 then separates theDC terms201 and theAC terms200 from theURCA data segment246 and theXRCA data segment248. Theinverse Huffman encoder458 decompresses the first layer of thedata stream118 which includes theAC terms200, theURCA data segment246, and theXRCA data segment248. Theinverse DPCM476 further expands theDC terms201 tooutput DC terms201′. Thedequantizer450 further expands theAC terms200 tooutput AC terms200′ by multiplying theoutput AC terms200′ with the quantization factors478 in thequantization table Q202 tooutput 8×8 DCT coefficient blocks482. Thequantization table Q202 is stored in the CS data segment204 (not shown).
• The[0165]combiner452 combines theoutput DC terms201′ with the 8×8 DCT coefficient blocks482. Thedecoder110 sets theinverse DCT factor480, and theinverse DCT476 outputs the DCT coefficient blocks482 that correspond to the Ym miniature436 that is 1/256th the size of the original image.
• The[0166]demultiplexer454 separates the inverse Huffman encodedURCA data segment246 from theXRCA data segment248. Theinverse DPCM476 then expands theURCA data segment246 and theXRCA data segment248 to generate the blocks that correspond to theUm miniature438 and theXm miniature440.
• The[0167]adder456 translates the blocks corresponding to theUm miniature438 and the Xm miniature440 into blocks that correspond to aXm miniature460.
• FIG. 26 illustrates the[0168]second step442 in which thedecoder110 expands theYm miniature436, theUm miniature438, and the Xm miniature460 that thedecoder110 further includes theinterpolator462 that operates on theUm miniature436, theUm miniature438 and theXm miniature460. Theinterpolator462 is controlled by aYm interpolation factor484, aUm interpolation factor486, and aXm interpolation factor496. Ascaler466 is controlled by aYm scale factor490, aUm scale factor492, aXm scale factor494. Thedecoder110 further includes thereplicator464 and the inverse color converter. Theinterpolator462 uses a linear interpolation process to enlarge theYm miniature436, theUm miniature438, and the Xm miniature460 by one, two or four times in both the horizontal and vertical directions.
• The[0169]Ym interpolation factor484, theUm interpolation factor486, and the Xm interpolation factor488 control the amount of interpolation. The size of thesource image100 in thecompressed file104 is fixed, thus thedecoder110 may need to enlarge or reduce the expanded image before display. Thedecoder110 sets theYm interpolation factor484 to a power of 2 (i.e., 1, 2, 4, etc.) in order to optimize the decoding process. However, in order to display an expanded image at the proper size, thescaler466 scales the interpolated image to accommodate different display formats.
• The[0170]interpolator462 also expands theUm miniature438 and theXm miniature440. Like theYm interpolation factor484, thedecoder110 sets theUm interpolation factor486 and theXm interpolation factor496 to a power of two. Thedecoder110 sets theYm interpolation factor484, and theUm interpolation factor486 so that theUm miniature438 and Xm miniature460 approximate the size of the interpolated and scaledYm miniature436.
• After interpolation, the[0171]scaler466 enlarges or reduces the interpolated Ym miniature based on theYm scale factor490. In the preferred embodiment, thedecoder110 sets theYm interpolation factor484 so that the interpolatedYm miniature436 is nearly twice the size of thethumbnail miniature120. Thedecoder110 then sets theYm scale factor490 to reduce the interpolated Ym miniature436 to the display size of thethumbnail miniature120. Thescaler466 interpolates theUm miniature458 and the Xm miniature460 with theUm scale factor492, and theXm scale factor494. Thedecoder110 sets theXm scale factor494, theUm scale factor492, as necessary to scale the image to the display size.
• The[0172]inverse color converter468 transforms the interpolated and scaled miniatures into a red, green, and blue pixel array or a palletized image as required by thedisplay112. When converting to a palletized image, theinverse color converter468 also dithers the converted image. Thedecoder110 displays the interpolated, scaled and color converted miniatures as thethumbnail miniature120.
• In order to create the[0173]splash image122, thedecoder110 expands the interpolated Ym miniature436, the interpolatedUm miniature438 and the interpolated Xm miniature440 with a second interpolation process that uses a Ymsplash interpolation factor498, a Umsplash interpolation factor500, and an Xmsplash interpolation factor502. Like thethumbnail miniature120, thedecoder110 also sets the splash interpolation factors to a power of two.
• The interpolated data are then expanded with the[0174]replicator464. Thereplicator464 enlarges the interpolated data one or two times by replicating the pixel information. Thereplicator464 enlarges the interpolated data based on aYm replication factor504, aUm replication factor506, and anXm replication factor508. Thedecoder110 sets theYm replication factor504, theUm replication factor506, and theXm replication factor508 so that the replicated image is one-fourth of the display size.
• The[0175]inverse color converter468 transforms the replicated image data into red, green and blue image data. Thereplicator464 then again replicates the red, green, and blue image data to match the display size. Thedecoder110 displays the resultingsplash image122 on thedisplay112.
• FIG. 27 illustrates the[0176]third step3 in which thedecoder110 generates the higher detail panels to expand thethumbnail miniature120 into astandard image124. FIG. 27 also illustrates thefourth step446 in which thedecoder110 generates generate higher detail panels to enhance the detail of the standard image in order to create an enhanced image105 that corresponds to thesource image100.
• The decoding of the[0177]standard image124 and the enhanced image105 requires theinverse Huffman encoder458, thecombiner452, thedequantizer450, theinverse DCT476, apattern matcher524, theadder456, theinterpolator462, and anedge overlay builder516. Thedecoder110 adds additional detail to the displayed image as thedecoder110 receives new layers of compressed data. The additional layers include new panels of the DCT data segment208 (containing the remainingAC terms200′), theVQ1 data segment224, theVQ2 data segment258, the enhancement location data segment510, theVQ3 data segment242, and theVQ4 data segment244.
• The[0178]decoder110 builds upon theYm miniature436, theUm miniature438 and the Xm miniature440 calculated for thethumbnail miniature120 by expanding the next layer of image detail. The next layer contains a portion of theDCT data segment208, theVQ1 data segment224, theVQ2 data segment258, the enhancement location data segment510, theVQ3 data segment242, and theVQ4 data segment244 that correspond to the standard image.
• The[0179]inverse Huffman encoder458 decompresses theDCT data segment208 and the VQ1 data segment224 (the DCT residual). Thecombiner452 combines the DCT information from the inverse Huffman encoder458 with theAC terms200 and the DC terms201. Thedequantizer450 reverses the quantization process by multiplying the DCT quantized values206 with the quantization factors478. The dequantizer obtains the correct quantization factors478 from thequantization table Q202. The dequantizer outputs 8×8 DCT coefficient blocks482 to theinverse DCT476. Theinverse DCT476 in turn, outputs the 8×8 DCT coefficient blocks482 that correspond to a Y image509 that is ¼th the size of the original image.
• The[0180]pattern matcher524 replaces the DCT residual blocks512 by finding an index to a matching pattern block in thecodebook214. Theadder456 adds the DCT residual blocks512 to the DCT coefficient blocks482 on a pixel by pixel basis. Theinterpolator462 interpolates the output of theadder456 by a factor of four to create a fullsize Y image520. Theinterpolator462 performs bilinear interpolation to enlarge theY image520 horizontally and vertically.
• The[0181]inverse Huffman encoder458 decompresses the VQ2 data segment258 (the high resolution residual) and the enhancement location data segment510. The pattern matcher524 uses the codebook indexes to retrieve the matching pattern blocks stored in thecodebook214 to expand theVQ2 data segment258 to create 16×16 high resolutionresidual blocks514. Anenhancement overlay builder516 inserts the 16×16 high resolution residual blocks into aY image overlay518 specified by the edge location data segment510. TheY image overlay518 is the size of the original image. Theadder456 adds theY image overlay518 to the fullsized Y image520.
• To calculate the full[0182]sized U image522, theinverse Huffman encoder458 expands theVQ3 data segment242. The pattern matcher524 uses the codebook indexes to retrieve the matching pattern blocks stored in thecodebook214 to expand theVQ3 data segment242 into 4×4 rU_tau4residual blocks526. Theinterpolator462 interpolates theUm miniature438 by a factor of four and theadder456 adds the 4×4 rU_tau4residual blocks526 to the interpolated Um miniature438 in order to create a Um+r miniature528. Theinterpolator462 interpolates the Um+r miniature528 by a factor of four to create the fullsized U image522.
• To calculate the full[0183]sized X image530, theinverse Huffman encoder458 expands theVQ4 data segment244. The pattern matcher524 uses the codebook indexes to retrieve the matching pattern blocks stored in thecodebook214 to expand theVQ4 data segment244 into 4×4 rX_tau4 residual blocks. Thedecoder110 then translates the 4×4 rX_tau4 residual blocks532 into 4×4 rV_tau4 residual blocks534. Theinterpolator462 interpolates the Xm miniature460 by a factor of four, and theadder456 adds the 4×4 rV_tau4 residual blocks534 to the interpolated Xm miniature460 in order to create a Xm+r miniature536. Theinterpolator462 interpolates the Xm+r miniature536 by a factor of four to create the fullsized X image530.
• The decoder stores the full[0184]sized Y image520, the fullsized U image522, and the fullsized X image530 in local memory. Theinverse color converter468 then converts the fullsized Y image520, the fullsized U image522, and the fullsized X image530 into a full sized red, green, and blue image. The panel is then added to the displayed image. This process is completed for each panel until theentire source image100 is expanded.
• In the forth step the[0185]decoder110 receives the third image layer and builds upon the fullsized Y image520, the fullsized U image522, and the fullsized X image530 stored in local memory to generate the enhanced image105. The third image data layer contains the remaining portion of theDCT data segment208, theVQ1 data segment224, theVQ2 data segment258, the enhancement location data segment510, theVQ3 data segment242, and theVQ4 data segment244 that correspond to the enhanced image105.
• The[0186]decoder110 repeats the process illustrated in FIG. 27 to generate a new fullsized Y image520, a new fullsized U image522, and a new fullsized X image530. The new fullsized Y image520 is added to the full sized Y image generated in thethird step444. The new fullsized U image522 is added to the fullsized U image522 generated in thethird step444. The new fullsized X image530 is added to the full sized X image generated in thethird step444.
• The[0187]inverse color converter468 converts the fullsized Y image520, the fullsized U image522, and the fullsized X image530 into a full sized red, green, and blue image. The panel is then added to the displayed image. This process is completed for each panel until the entire enhanced image105 is expanded.
• The[0188]inverse DCT476 of the preferred embodiment is a mathematical transformation for mapping data in the time (or spatial) domain to the frequency domain, based on the “cosine” kernel. The two dimensional version operates on a block of 8×8 elements.
• Referring to FIG. 9, the[0189]compressed DCT coefficients198 are stored asDC terms201 andAC terms200. In the preferred embodiment, theinverse DCT476 as shown in FIGS. 25 and 27 combines the process of transformation and decimation in the frequency and spatial domains (frequency and then spatial) into a single operation in the frequency domain. Theinverse DCT476 of the present invention provides at least a factor of 2 in implementation efficiency and is utilized by thedecoder110 to expand thethumbnail miniature120 andsplash image122.
• The[0190]inverse DCT476 receives a sequence ofDC terms201 andAC terms200 which are frequency coefficients. The high frequency terms are arbitrarily discarded at a predefined frequency to prevent aliasing. The discarding of the high frequency terms is equivalent to a low pass filter which passes everything below a predefine frequency while attenuating all the high frequencies to zero.
• The equation for an inverse DCT is:[0191]$f_{y,x}:=\frac{1}{4}\sum _u\sum _vC_u·C_v·F_{v,u}·\cos \left(\frac{2·x+1}{16}\right)·u·\pi ·\cos \left(\frac{2·y+1}{16}·v·\pi \right)$

  
• where[0192]
• u:=0 . . . 7 v:=0 . . . 7
• x:=0 . . . 7 y:=0 . . . 7
• [0193]$C_u:=\frac{1}{\sqrt{2}}·\left(u=0\right)+\left(u\ne 0\right)$

  
• The[0194]inverse DCT476 generates an 8×8 output matrix that is decimated to a 4×4 matrix then to a 2×2 matrix. Theinverse DCT476 then decimates the output matrix by subsampling with a filter. After subsampling, an averaging filter smooths the output. Smoothing is accomplished by using a running average of the adjacent elements to form the output.
• For example, for a 4×4 output matrix the 8×8 matrix from the[0195]inverse DCT476 is sub-divided into sixteen 2×2 regions, and adjacent elements within each 2×2 region is averaged to form the output. Thus the sixteen regions form a 4×4 matrix output.
• For a 2×2 output matrix, the 8×8 matrix from the[0196]inverse DCT476 is sub-divided into four 4×4 regions. The adjacent elements within each 4×4 matrix region are averaged to form the output. Thus, the four regions form a 2×2 matrix output.
• In addition, since most of the AC coefficients are zero, the[0197]inverse DCT476 is simplified by combining the inverse DCT equations with the averaging and the decimation equations. Thus, the creation of a 2×2 output matrix where a given X is an 8×8 input matrix that consists ofDC terms201 andAC terms200 is stated formally as:$X:=\left[\begin{array}{cccccccc}{\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">X</figure-callout>}}_{0,0}& {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}"><figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">X</figure-callout></figure-callout>}}_{0,1}& 0& 0& 0& 0& 0& 0\\ {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">X</figure-callout>}}_{1,0}& {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}"><figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">X</figure-callout></figure-callout>}}_{1,1}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$

  
• All elements with i or j greater than 1 are set to zero. The setting of the high frequency index to zero is equivalent to filtering out the high frequency coef ficients from the signal.[0198]
• Assigning Y as the, 2×2 output matrix, the decimated output is thus equal to:[0199]
• Y_0,0:=X_0,0+(k₁·(X_0,1))+(k₁·(X_1,0))+(k₂·(X_1,1))
• Y_0,1:=X_0,0−(k₁·(X_0,1))+(k₁·(X_1,0))−(k₂·(X_1,1))
• Y_1,0:=X_0,0+(k₁·(X_0,1))−(k₁·(X_1,0))−(k₂·(X_1,1))
• Y_1,1:=X_0,0−(k₁·(X_0,1))−(k₁·(X_1,0))+(k₂·(X_1,1))
• where[0200]$k_1:=\frac{1}{\sqrt{8}}·\left(c\left(1\right)+c\left(3\right)+c\left(5\right)+c\left(7\right)\right)c\left(k\right)=\cos \left(\pi ·\frac{k}{16}\right)$

  
• k₂:=(k₁)²
• The creation of a 4×4 output matrix where a given X is an 8×8 input matrix that consists of[0201]DC terms201 andAC terms200 is stated formally as:
• All elements with i or j greater than 3 are set to zero.[0202]
• It is possible to implement the calculations in the 2×2 case where the two dimensional equation is decomposed downward; however, performing the one dimensional approach twice reduces complexity and decreases the calculation time. In the preferred embodiment, the[0203]inverse DCT476 computes an additional one-dimensional row inverse DCT, and then a one-dimensional$X:=\left[\begin{array}{cccccccc}{\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">X</figure-callout>}}_{0,0}& {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">X</figure-callout>}}_{0,1}& {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">X</figure-callout>}}_{0,2}& {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}"><figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">X</figure-callout></figure-callout>}}_{0,3}& 0& 0& 0& 0\\ {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">X</figure-callout>}}_{1,0}& {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">X</figure-callout>}}_{1,1}& {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">X</figure-callout>}}_{1,2}& {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}"><figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">X</figure-callout></figure-callout>}}_{1,3}& 0& 0& 0& 0\\ {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">X</figure-callout>}}_{2,0}& {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">X</figure-callout>}}_{2,1}& {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">X</figure-callout>}}_{2,2}& {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}"><figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">X</figure-callout></figure-callout>}}_{2,3}& 0& 0& 0& 0\\ {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">X</figure-callout>}}_{3,0}& {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">X</figure-callout>}}_{3,1}& {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">X</figure-callout>}}_{3,2}& {\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}"><figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">X</figure-callout></figure-callout>}}_{3,3}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$

  
• column inverse DCT.[0204]
• The equation for a one dimensional case is as follows: (1dout[0205]_xare the elements of the one dimensional case)
• 1dout₀:=in₀+(k₁·in₁)+(k₂·in₂)+(k₃·in₃)
• 1dout₁:=in₀+(k₄·in₁)−(k₂·in₂)−(k₅·in₃)
• 1dout₂:=in₀−(k₄·in₁)−(k₂·in₂)+(k₅·in₃)
• 1dout₃:=in₀−(k₁·in₁)+(k₂·in₂)−(k₃·in₃)
• [0206]$\begin{array}{ccc}{k}_1:=\frac{c\left(1\right)+c\left(3\right)}{\sqrt{2}}& k_2:=\frac{c\left(2\right)+c\left(6\right)}{\sqrt{2}}& k_3:=\frac{c\left(3\right)-c\left(7\right)}{\sqrt{2}}\\ k_4:=\frac{c\left(5\right)+c\left(7\right)}{\sqrt{2}}& k_5:=\frac{c\left(5\right)+c\left(2\right)}{\sqrt{2}}& \end{array}$

  
• where c(k) is defined as in the 2×2 output matrix.[0207]
• The[0208]scaler466 of the preferred embodiment is also shown in FIG. 27. More specifically, thescaler466 utilizes a generalized routine that scales the image up or down while reducing aliasing and reconstruction noise. Scaling can be described as a combination of decimation and interpolation. The decimation step consists of downsampling and using an anti-aliasing filter; the interpolation step consists of pixel filling using a reconstruction filter for any scale factor that can be represented by a rational number P/Q, where P and Q are integers associated with the interpolation and decimation ratios.
• The[0209]scaler466 decimates the input data by dividing the source image into the desired number of output pixels and then radiometrically weights the input data to form the necessary output. FIG. 28 illustrates thescaler466 with an input to output ratio of five-to-three in the one dimensional case.Input pixel P₁538,pixel P₂540,pixel P₃542,pixel P₄544, andpixel P₂546 contain different data values. Theoutput pixel X₁548,pixel X₂550, andpixel X₃552 are computed as follows:
• X₁=P₁+(P₂) (0.67)
• X₂=(P₂) (0.33)+P₃+(P₄) (0.33)
• X₃=(P₄) (0.66)+P₅
• The decimated data is then filtered with a reconstruction filter and an area average filter. The reconstruction filter interpolates the input data by replicating the pixel data. The area average filter then area averages by integrating the area covered by the output pixel.[0210]
• If the output ratio is less than 1 (i.e, interpolation is necessary), the[0211]interpolator462 utilizes bilinear interpolation. FIG. 29 illustrates the operation of the bilinear interpolation.Input pixel A554,input pixel B556,input pixel C558, andinput pixel D560, andreference point X562 are interpolated to createoutput564. For this examplereference point X562 is α to the right ofpixel A554 and 1-α to the right ofpixel C558, andreference point X562 is βdown frompixel A554 and 1-β up frompixel B556.Reference point X562 is stated formally as:
• X=(1−α)*((1−β)*A+62 *B)+α*((1−β)*C+β*D).
• The Image Classifier[0212]
• The preferred embodiment of the[0213]image classifier152 is illustrated in FIG. 8. More specifically, theimage classifier152 uses fuzzy logic techniques to determine which compression methods will optimize the compression of various regions of thesource image100. Theimage classifier152 adds intelligence to theencoder102 by providing the means to decide, based on statistical characteristics of the image, what “tools” (combinations of compression methods) will best compress the image.
• The[0214]source image100 may include a combination of different image types. For example, a photograph could show a person framed in a graphical border, wherein the person is wearing a shirt that contains printed text. In order to optimize the compression ratio for the regions of the image that contain different image types, theimage classifier152 subdivides thesource image100 and then outputs thecontrol script196 that specifies the correct compression methods for each region. Thus, theimage classifier152 provides a customized, “most-efficient” compression ratio for multiple image types.
• The[0215]image classifier152 uses fuzzy logic to infer the correct compression steps from the image content. Image content is inherently “fuzzy” and is not amenable to simple discrete classification. Images will thus tend to belong to several “classes.” For example, a classification scheme might include one class for textual images and a second class for photographic images. Since an image may comprise a photograph of a person wearing a shirt containing printed text, the image will belong to both classes to varying degrees. Likewise, the same image may be high contrast, “grainy,” black and white and/or high activity.
• Fuzzy logic is a set-theoretic approach to classification of objects that assigns degrees of membership in a particular class. In classical set theory, an object either belongs to a set or it does not; membership is either 100% or 0%. In fuzzy set theory, an object can be partly in one set and partly in another. The fuzziness is of greater significance when the content must be categorized for the purpose of applying appropriate compression techniques. Relevant categories in image compression include photographic, graphical, noisy, and high-energy. Clearly the boundaries of these sets are not sharp. A scheme that matches appropriate compression tools to image content must reliably distinguish between content types that require different compression techniques, and must also be able to judge how to blend tools when types requiring different tools overlap.[0216]
• FIG. 30 illustrates the optimization of the compression process. The optimization process analyzes the[0217]input image600 at different levels. In thetop level analysis602 theimage classifier152 decomposes the image into a plurality of subimages604 (regions) of relatively homogeneous content as defined by aclassification map606. Theimage classifier152 then outputs thecontrol script196 that specifies which compression methods or “tools” to employ in compressing each region. The compression methods are further optimized in thesecond level analysis608 by theenhancement analyzer144 which determines which areas of an image are the most visually important (for example, text and strong luminance edges). The compression methods are then further optimized in thethird level analysis610 with the optimizedDCT156,AVQ134, and adaptive methods in thechannel encoder168. Thesecond level analysis608 and thethird level analysis610 determine how to adapt parameters and tables to a particular image.
• The fuzzy[0218]logic image classifier152 provides adaptive “intelligent” branching to appropriate compression methods with a high degree of computational simplicity. It is not feasible to provide theencoder102 with an exhaustive mapping of all possible combinations of inherently non-linear, discontinuous, multidimensional inputs (image measurements) onto desiredcontrol scripts196. The fuzzylogic image classifier152 reduces such an analysis.
• Furthermore, the fuzzy[0219]logic image classifier152 ensures that theencoder102 makes a smooth transition from one compression method (as defined by the control script196) to another compression method. As image content becomes “more like” one class than another, the fuzzy controller avoids the discrete switching from one compression method to another compression method.
• The fuzzy[0220]logic image classifier152 receives the image data and determines a set of image measurements which are mapped onto one or more input sets. Theimage classifier152 in turn maps the input sets to corresponding output sets that identify which compression methods to apply. The output sets are then blended (“defuzzified”) to generate acontrol script196. The process of mapping the input image to aparticular control script196 thus requires three sets of rules: 1) rules for mapping input measurements onto input sets (e.g., degree of membership with the “high activity” input set=F[average of AC coefficients 56-63]); 2) rules for mapping input sets onto output sets (e.g., if graphical image, use DCT quantization table 5 and 3) rules for defuzzification that mediate between membership of several output sets, i.e., how the membership of more than one output sets, should be blended to generate asingle control script196 that controls the compression process.
• Still further, the fuzzy logic rule base is easily maintained. The rules are modular. Thus, the rules can be understood, researched, and modified independently of one another. In addition, the rule bases are easily modified allowing new rules to make the[0221]image classifier152 more sensitive to different types of image content. Furthermore, the fuzzy logic rule base is extendable to include additional image types specified by the user or learned using neural network or genetic programming methods.
• FIG. 31 illustrates a block diagram of the[0222]image classifier152. Inblock612 theimage classifier152 determines a set ofinput measurements614 that correspond to thesource image100. In order to determine theinput measurements614, theimage classifier152 sub-divides thesource image100 into a plurality of blocks. To conserve computations, the user can enable theimage classifier152 to select a random sample of the plurality of blocks to use as the basis of theinput measurements614.
• The[0223]image classifier152 determines the set ofinput measurements614 from the plurality of blocks using a variety of methods. Theimage classifier152 calculates the mean, the variance, and a histogram of all three color components. Theimage classifier152 performs a discrete cosine transform of the image blocks to derive a set of DCT components wherein each DCT coefficient is histogrammed to provide a frequency domain profile of the imputed image. Theimage classifier152 performs special convolutions to gather information about edge content, texture content, and the efficacy of the Reed Spline Filter. Theimage classifier152 derives spatial domain blocks and matches the spatial domain blocks with a special VQ-like pattern list to provide information about the types of activity contained in the picture. Finally, the image classifier scans the image for common and possibly localized features that bear on the compressibility of the image (such as typed text or scanning artifacts).
• In block[0224]616 theimage classifier152 analyzes theinput measurements614 generated inblock612 to determine the extent to which thesource image100 belongs to one of the fuzzy input sets618 within the input rule base620. The input rule base620 identifies the list of image types. In the preferred embodiment, theimage classifier152 contains input sets618 for the following image types: scale, text, graphics, photographic, color depth, degree of activity, and special features.
• Membership in the activity input set and the scale image input set are determined by the[0225]input measurements614 for the DCT coefficient histogram, the spatial statistics, and the convolutions. Membership in the text image input set and the graphic input set correspond to theinput measurements614 for a linear combination of high frequency DCT coefficients and gaps in the luminance histogram. The photographic input set is the complement of the graphic input set.
• The color depth input set includes four classifications: gray scale images, 4-bit images, 8-bit images and 24-bit images. The color depth input corresponds to the[0226]input measurements614 for the Y, U and X color components. A small dynamic range in the U and X color components indicates that the picture is likely to be a gray scale image, while gaps in the Y component histogram reveals whether the image was once a palettized 4-bit or 8-bit image.
• The special feature input set corresponds to the[0227]input measurements614 for the common or localized features that bear on the compressibility of the image. Thus the special feature input set identifies such artifacts as black borders caused by inaccurate scanning and graphical titling on a photographic image.
• In block[0228]622 theimage classifier152 maps the input sets618 onto output sets624 according to theoutput rule base626. Theimage classifier152 applies theoutput rule base626 to map each input set618 onto membership of eachfuzzy output set624. The output sets624 determine, for example, how many CS terms are stored in theCS data segment204 and the optimization of theVQ1 data segment224, theVQ2 data segment258, theVQ3 data segment242, theVQ4 data segment244, and the number of VQ patterns to use. The output sets also determine whether theencoder102 performs an optimizedDCT136 and whichquantization tables Q202 to apply.
• For the second Reed Spline Filter[0229]225 and the thirdReed Spline Filter227, the output sets624 adjust the decimation factor tau and the orientation of the kernal function. Finally, the output sets determine whether thechannel encoder168 utilizes a fixed Huffman encoder, and adaptive Huffman encoder or an LZ1. FIG. 33 illustrates several examples of mapping frominput measurements614 to inputsets618 to output sets624.
• Referring to FIG. 31, in[0230]block626 the image classifier constructs aclassification map628 based upon membership within the output sets. Theclassification map628 identifies independent regions in thesource image100 that are independently compressed. Thus theimage classifier152 identifies the regions of the image that belong to compatible output sets624. These are regions that contain relatively homogenous image contrast and call for one method or set or complementary methods to be applied to the entire region.
• In[0231]block630 theimage classifier152 converts (defuzzifies), based on the defuzzification rule base632, the membership of the fuzzy output sets624 of each independent region in order to generate thecontrol script196. Thecontrol script196 contains instructions for which compression methods to perform and what parameters, tables, and optimization levels to employ for a particular region of thesource image100.
• The Enhancement Analyzer[0232]
• The preferred embodiment of the[0233]enhancement analyzer144 is illustrated in FIGS. 4, 15 and30. More specifically, theenhancement analyzer144 examines theY_tau2 miniature190, theU_tau2 miniature192, and the X_tau4 miniature228 to determine the enhancement priority of image blocks that correspond to 16×16 blocks in theoriginal source image100. Theenhancement analyzer144 prioritizes the image blocks by 1) calculating the mean of theY_tau2 miniature190, theU_tau2 miniature192, and theX_tau4 miniature228, and 2) testing every color block against a normalizedthreshold value E252 for theY_tau2 miniature190, theU_tau2 miniature192, and theX_tau4 miniature228. A list of blocks that exceed thethreshold value E252 are added to theenhancement list250.
• The[0234]enhancement analyzer144 determines a threshold value E_Yfor theY_tau2 miniature190, a threshold value E_Ufor theU_tau2 miniature192, and a threshold value E_Xfor theX_tau4 miniature228. Once theenhancement analyzer144 computes the threshold value E_Y, the threshold value F_Uand the threshold value E_X, theenhancement analyzer144 tests each 8×8 Y_tau2 block, each 4×4 U_tau4 block and each 4×4 X_tau4 block (each block corresponds to a 16×16 block in the source image100) as follows:
• Every pixel in the test block is convolved with the following filter masks:[0235]
• M₁={−1,−2,−1,0,0,0,1,2,1}
• M₂={1,0,−1,2,0,−2,1,0,−1}
• to compute two statistics S[0236]₁and S₂.
• Masks M[0237]₁and M₂are convolved with a three by three block of pixels centered on the pixel being tested. The three by three block of pixels is represented as:
• x₁₁x₁₂x₁₃
• x₂₁x₂₂x₂₃
• x₃₁x₃₂x₃₃
• where the pixel x[0238]₂₂is the pixel being tested. Thus the statistics are calculated with the following equations:
• S₁=(−1·x₁₁)−(2·x₁₂)−(1·x₁₃)+(1·x₃₁)+(2·x₃₂)+(1·x₃₃)
• S[0239]₂=(1·x₁₁)−(1·x₁₃)+(2·x₂₁)−(1·x₂₃)+(1·x₃₁)−(1·x₃₃)
• If S[0240]₁plus S₂is greater than the threshold value E_Yfor a particular 8×8 Y_tau2 block, theenhancement analyzer144 adds the 8×8 Y_tau2 block to theenhancement list250. If S₁plus S₂is greater than the threshold value E_Ufor a particular 4×4 U_tau4 block, theenhancement analyzer144 adds the 4×4 U_tau4 block to theenhancement list250. If S₁plus S₂is greater than the threshold value E_Xfor a particular 4×4 X_tau4 block theenhancement analyzer144 adds the 4×4 X_tau4 block to theenhancement list250.
• In addition to the[0241]enhancement list250, theenhancement analyzer144 also uses theDCT coefficients198 to identify visually unimportant “texture” regions where the compression ratio can be increased without significant loss to the image quality.
• Optimized DCT[0242]
• The preferred embodiment of the optimized[0243]DCT136 is illustrated in FIG. 9. More specifically, the optimizedDCT136 uses thequantization table Q202 to assign the DCT coefficients (DC terms200 and AC terms201) quantization step values. In addition, the quantization step values in thequantization table Q202 vary depending on the optimizedDCT136 operation mode. The optimizedDCT136 operates in four DCT modes as follows: 1) switched fixed uniform DCT quantization tables that correspond to image classification, 2) optimal reconstruction values, 3) adaptive uniform DCT quantization tables, and 4) adaptive non-uniform DCT quantization tables.
• The fixed DCT quantization tables are tuned to different image types, including eight standard tables corresponding to images differing along three dimensions: photographic versus graphic, small-scale versus large-scale, and high-activity versus low-activity. In the preferred embodiment, additional tables can be added to the resource file[0244]160 (not shown).
• The[0245]control script196 defines which standard table the optimizedDCT136 uses in the fixed-table DCT mode. In the fixed-table mode, quantized step values for each DCT coefficient is obtained by linearly quantizing each x_iDCT coefficient with the quantization value q_iin quantization table Q. The mathematical relationship for the quantization procedure is:
• for i=0, 1, . . . , 63[0246]
• if x[0247]_i>=0,$c_i=\frac{\left[x_i+\frac{q_i}{2}\right]}{q_i}$

  
• if x[0248]_i<0,$c_i=\frac{\left[x_i+\frac{q_i}{2}\right]}{q_i}$

  
• Reconstruction is also linear unless reconstruction values have been computed and stored in the[0249]CS data segment204. Letting r denote the dequantized DCT coefficients, the linear dequantization formula is:
• for i=0, 1, . . . , 63[0250]
• r_i=c_i·q_i
• In the fixed-table DCT mode, the optimized[0251]DCT136 can also compute the optimal reconstruction values stored in theCS data segment204. While theDC term201 is always calculated linearly, the CS reconstruction values represent the conditional expected value of each quantized level of eachAC term200. The CS reconstruction values are calculated for eachAC term200 by first calculating an absolute value frequency histogram, H_ifor the ith coefficient (for i=1, 2, . . . , 63) over all DCT blocks in the source image, N, as follows:
• for j=0, 1, . . . , N[0252]
• H[0253]_i(k)=frequency (abs(x_ij)=k)
• where x[0254]_ij=the value of the ith coefficient in the jth DCT block.
• Second, the centroid of coefficient values is calculated between each quantization step. The formula for the centroid of the ith coefficient in the kth quantization interval is:[0255]${\mathrm{CS}}_i\left(k\right)=\sum _{j=\mathrm{kq}-\frac{\mathrm{<figure-callout\; label="q"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">q</figure-callout>}}{2}}^{\mathrm{kq}+\frac{\mathrm{<figure-callout\; label="q"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">q</figure-callout>}}{2}}\left[\frac{H_i\left(j\right)}{T_i\left(k\right)}\right]$

  
• where[0256]$T_i\left(k\right)=\sum _{j=\mathrm{kq}-\frac{\mathrm{<figure-callout\; label="q"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">q</figure-callout>}}{2}}^{\mathrm{kq}+\frac{\mathrm{<figure-callout\; label="q"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">q</figure-callout>}}{2}}H\left(j\right)$

  
• This provides a non-linear mapping of quantized coefficients onto reconstructed values as follows:[0257]
• r_i=CS_i(q_i) for i=1, 2, . . . , 63
• In the adaptive uniform DCT quantization mode, the image the[0258]classifier152 outputs thecontrol script196 that directs the optimizedDCT136 to adjust a given DCT uniformquantization table Q202 to provide more efficient compression while holding the visual quality constant. This method adjusts the DCT quantization step sizes such that the compressed bit rate (entropy) after quantizing the DCT coefficients is minimized subject to the constraint that the visually-weighted mean squared error arising from the DCT quantization is held constant with respect to the base quantization table and the user-supplied quantization parameter L.
• The optimized[0259]DCT136 uses marginal analysis to adjust the DCT quantization step sizes. A “marginal rate of transformation (MRT)” is computed for each DCT coefficient. The MRT represents the rate at which bits are “transformed” into (a reduction of) the visually weighted mean squared error (VMSE). The MRT of a coefficient is defined as the ratio of 1) the marginal change in the encoded bit rate with respect to a quantization step value q to 2) the marginal change in the visual mean square error with respect to the quantization step value q.
• MRT (bits/VMSE) ratio is calculated as follows:[0260]
• MRT (bits/VMSE)=((Δbits/Δq)/I(ΔVMSE/Δq)).[0261]
• Increasing the quantization step value q will add more bits to the representation of the corresponding DCT coefficient. However, adding more bits to the representation of a DCT coefficient will reduce the VMSE. Since the bits added to the step value q are usually transformed into VMSE reduction, the MRT is generally negative.[0262]
• The MRT is calculated for all of the DCT coefficients. The adaptive method utilized by the optimized[0263]DCT136 adjusts the quantization step values q of thequantized table Q202 by reducing the quantization step value q corresponding to the maximum MRT and increasing the quantization step value q corresponding to the minimum MRT. The optimizedDCT136 repeats the process until the MRT is equalized across all of the DCT coefficients while holding the VMSE constant.
• FIG. 32 shows a flow chart of the process of creating an adaptive uniform DCT quantization table. In a[0264]step700 the optimizedDCT136 computes the MRT values for all DCT coefficients i. Instep702 the optimizedDCT136 compares the MRT values, if the MRT values are the same, the optimizedDCT136 uses the resultingquantization table Q202. If the MRT values are not equal, the optimizedDCT136 finds the minimum MRT value and the maximum MRT value for the DCT coefficients i instep706.
• In step[0265]708, the optimizedDCT136 increases the quantization step value q_lowcorresponding to the minimum MRT value and decreases the quantization step value q_highassociated with the maximum MRT value. Increasing q_lowwhich reduces the number of bits devoted to the corresponding DCT coefficient but does not increase VMSE appreciably. Reducing the quantization step value q_highincreases the number of bits devoted to the corresponding dCT coefficient and reduces the VMSE significantly. The optimizedDCT136 offsets the adjustments for the quantization step values q_lowand q_highin order to keep the VMSE constant.
• The optimized[0266]DCT136 returns to step700, where the process is repeated until all MRT values are equal. Once all of the quantization step values q are determined the resultingquantization table Q202 is complete.
• The Reed Spline Filter[0267]
• FIGS.[0268]34-57 illustrate a preferred embodiment of theReed Spline Filter138 which is advantageously used for the first, second and thirdReed Spline Filters148,225, and227. The Reed Spline Filter described in FIG. 34-57 is in terms of a generic image format. In particular the image input data comprises Y image input which corresponds for example to the red, green and blue image data in the firstReed Spline Filter148 in the foregoing discussion. In like manner the outputs of theReed Spline Filter138 described as reconstruction values should be understood to correspond, for example, to theR_tau2 miniature180, theG_tau2 miniature182 and theB_tau2 miniature184 of the firstReed Spline Filter138.
• The Reed Spline Filter is based on the a least-mean-square error (LMS)-error spline approach, which is extendable to N dimensions. One- and two-dimensional image data compression utilizing linear and planar splines, respectively, are shown to have compact, closed-form optimal solutions for convenient, effective compression. The computational efficiency of this new method is of special interest, because the compression/reconstruction algorithms proposed herein involve only the Fast Fourier Transform (FFT) and inverse FFT types of processors or other high-speed direct convolution algorithms. Thus, the compression and reconstruction from the compressed image can be extremely fast and realized in existing hardware and software. Even with this high computational efficiency, good image quality is obtained upon reconstruction. An important and practical consequence of the disclosed method is the convenience and versatility with which it is integrated into a variety of hybrid digital data compression systems.[0269]
• I. SPLINE FILTER OVERVIEW[0270]
• The basic process of digital image coding entails transforming a source image X into a “compressed” image Y such that the signal energy of Y is concentrated into fewer elements than the signal energy of X, with some provisions regarding error. As depicted in FIG. 34, digital[0271]source image data1002 represented by an appropriate N-dimensional array X is supplied tocompression block1004, whereupon image data X is transformed to compressed data Y′ via a first generalized process represented here as G (X)=Y′. Compressed data may be stored or transmitted (process block1006) to a “remote”reconstruction block1008, whereupon a second generalized process, G′(Y′)=X′, operates to transform compressed data Y′ into a reconstructed image X′.
• G and G′ are not necessarily processes of mutual inversion, and the processes may not conserve the full information content of image data X. Consequently, X′ will, in general, differ from X, and information is lost through the coding/reconstruction process. The residual image or so-called residue is generated by supplying compressed data Y′ to a “local”[0272]reconstruction process1005 followed by adifference process1010 which computes the residue ΔX=X-X′1012. Preferably, X and X′ are sufficiently close, so that theresidue ΔX1012 is small and may be transmitted, stored along with the compressed data Y′, or discarded. Subsequent to theremote reconstruction process1008, theresidue ΔX1012 and reconstructed image X′ are supplied to addingprocess1007 to generate a restored image X′+ΔX=X″1003.
• In practice, to reduce computational overhead associated with large images during compression, a decimating or subsampling process may be performed to reduce the number of samples. Decimation is commonly characterized by a reduction factor τ (tau), which indicates a measure of image data elements to compressed data elements. However, one skilled in the art will appreciate that image data X must be filtered in conjunction with decimation to avoid aliasing. As shown in FIG. 35, a low-pass input filter may take the form of a pointwise convolution of image data X with a[0273]suitable convolution filter1014, preferably implemented using a matrix filter kernel. Adecimation process1016 then produces compressed data Y′, which is substantially free of aliasing prior to subsequent process steps. While the convolution ordecimation filter1014 attenuates aliasing effects, it does so by reducing the number of bits required to represent the signal. It is “low-pass” in nature, reducing the information content of the reconstructed image X′. Consequently, theresidue ΔX1012 will be larger, and in part, will offset the compression attained through decimation.
• The present invention disclosed herein solves this problem by providing a method of optimizing the compressed data such that the mean-square-residue <ΔX[0274]²> is minimized, where “< >” shall herein denote an averaging process. As shown in FIG. 36, compressed data Y′, generated in a manner similar to that shown in FIG. 35, is further processed by anoptimization process1018. Accordingly, theoptimization process1018 is dependent upon the properties ofconvolution filter1014 and is constrained such that the variance of the mean-square-residue is zero, δ<ΔX²>=0. The disclosed method of filter optimization “matches” the filter response to the image data, thereby minimizing the residue. Since thedecimation filter1014 is low-pass in nature, theoptimization process1018, in part, compensates by effectively acting as a “self-tuned” high-pass filter. A brief descriptive overview of the optimization procedure is provided in the following sections.
• A. Image Approximation by Spline Functions[0275]
• As will become clear in the following detailed description, the[0276]input decimation filter1014 of FIG. 36 may be regarded as a projection of an image data vector X onto a set of basis functions that constitute shifted, but overlapping, spline functions {ψ_k(x)} such that$\underset{\_}{X}\cong {\underset{\_}{X}}^{\prime }=\sum _k{\chi }_k{\Psi }_k\left(\underset{\_}{x}\right),$

  
• where X′ is the reconstructed image vector and χ[0277]_kis the decomposition weight. The image data vector X is thus approximated by an array of preferably computationally simple, continuous functions, such as lines or planes, allowing also an efficient reconstruction of the original image.
• According to the method, the basis functions need not be orthogonal and are preferably chosen to overlap in order to provide a continuous approximation to image data, thereby rendering a non-diagonal basis correlation matrix:[0278]
• A_jk=ψ_j(x)·ψ_k(x).
• This property is exploited by the method of the present invention, since it allows the user to “adapt” the response of the filter by the nature and degree of cross-correlation. Furthermore, the basis of spline functions need not be complete in the sense of spanning the space of all image data, but preferably generates a close approximation to image X. It is known that the decomposition of image vector X into components of differing spline basis functions {ψ[0279]_k(X) } is not unique. The method herein disclosed optimizes the projection by adjusting the weights χ_ksuch that the differential variations of the average residue vanishes, δ<ΔX²>=0, or equivalently <ΔX²>=min. In general, it will be expected that a more complete basis set will provide a smaller residue and better compression, which, however, requires greater computational overhead and greater compression. Accordingly, it is preferable to utilize a computationally simple basis set, which is easy to manipulate in closed form and which renders a small residual image. This residual image or residue ΔX is preferably retained for subsequent processing or reconstruction. In this respect there is a compromise between computational complexity, compression, and the magnitude of the residue.
• In a schematic view, a set of spline basis functions S′={ψ[0280]_k} may be regarded as a subset of vectors in the domain of possible image vectors S={X}, as depicted in FIG. 37. The decomposition on projection of X onto components of S′ is not unique and may be accomplished in a number of ways. A preferable criterion set forth in the present description is a least-mean-square (LMS) error, which minimizes the overall difference between the source image X and the reconstructed image X′. Geometrically, the residual image ΔX can be thought of as a minimal vector in the sense that it is the shortest possible vector connecting X to X′. That is, ΔX might, for instance, be orthogonal to the subspace S′, as shown in FIG. 37. As it will be elaborated in the next section, the projection of image vector X onto S′ is approximated by an expression of the form:$\underset{\_}{X}={\underset{\_}{X}}^{\prime }=\sum _k{\chi }_k{\Psi }_k\left(\underset{\_}{x}\right)$

  
• The “best” X′ is determined by the constraint that ΔX=X−X′ is minimized with respect to variations in the weights χ[0281]_j:$\frac{\partial }{\partial {\chi }_j}〈\Delta {\underset{\_}{x}}^2〉=\frac{\partial }{\partial {\chi }_j}〈{\left(\underset{\_}{X}-\sum _k{\chi }_k{\Psi }_k\left(\underset{\_}{x}\right)\right)}^2〉=0,$

  
• which by analogy to FIG. 37, described an orthogonal projection of X onto S′.[0282]
• Generally, the above system of equations which determines the optimal χ[0283]_kmay be regarded as a linear transformation, which maps X onto S′ optimally, represented here by:
• A (χ_k)=X*ψ_k(X)
• where A[0284]_ij=ψ_i*ψ_jis a transformation matrix having elements representing the correlation between bases vectors ψ_iand ψ_j. The optimal weights ψ_kare determined by the inverse operation A⁻¹:
• χ_k=A⁻¹(X*ψ_k(x)),
• rendering compression with the least residue. One skilled in the art of LMS criteria will know how to express the processes given here in the geometry of multiple dimensions. Hence, the processes described herein are applicable to a variety of image data types.[0285]
• The present brief and general description has direct processing counterparts depicted in FIG. 36. The operation[0286]
• X*ψ_k(x)
• represents a[0287]convolution filtering process1014, and
• A⁻¹(X*ψ_k(x))
• represents the optimizing[0288]process1018.
• In addition, as will be demonstrated in the following sections, the inverse operation A[0289]⁻¹is equivalent to a so-called inverse eigenfilter when taken over to the conjugate image domain. Specifically,$\mathrm{DFT}{\chi }_k=\frac{1}{{\lambda }_m}\mathrm{DFT}\left(\underset{\_}{X}·{\Psi }_k\left(\underset{\_}{x}\right)\right),$

  
• where DFT is the familiar discrete Fourier transform (DFT) and λ[0290]_mare the eigenvalues of A. Theequivalent optimization block1018, shown in FIG. 38, comprises three steps: (1) a discrete Fourier transformation (DFT)1020; (2)inverse eigenfiltering1022; and (3) an inverse discrete Fourier transformation (DFT⁻¹)1024. The advantages of this embodiment, in part, rely on the fast coding/reconstruction speed, since only DFT and DFT⁻¹are the primary computations, where now the optimization is a simple division. Greater elaboration into the principles of the method are provided in Section II where also the presently contemplated preferred embodiments are derived as closed form solutions for a one-dimensional linear spline basis and two-dimensional planar spline bases. Section III provides an operational description for the preferred method of compression and reconstruction utilizing the optimal procedure disclosed in Section II. Section IV discloses results of a reduction to practice of the preferred embodiments applied to one- and two-dimensional images. Finally, Section V discloses a preferred method of the filter optimizing process implemented in the image domain.
• II. IMAGE DATA COMPRESSION BY OPTICAL SPLINE INTERPOLATION[0291]
• A. One-Dimensional Data Compression by LMS-Error Linear Splines[0292]
• For one-dimensional image data, bi-linear spline functions are combined to approximate the image data with a resultant linear interpolation, as shown in FIG. 39. The resultant closed-form approximating and optimizing process has a significant advantage in computational simplicity and speed.[0293]
• Letting the decimation index τ and image sampling period t be fixed, positive integers τ, t=1,2, . . . , and letting X(t) be a periodic sequence of data of period nτ, where n is also an integer, consider a periodic,[0294]linear spline1014 of period nτ of the type,
• F(t)=F(t+nτ),   (1)
• where   (2)[0295]
• as shown by the functions ψ[0296]_k(t)1014 of FIG. 39.
• The family of shifted linear splines F(t) is defined as follows:[0297]
• ψ_k(t)=F(t-kτ) for (k=0,1,2, . . . , (n−1)).   (3)
• One object of the present embodiment is to approximate X(t) by the n-point sum:[0298]$\begin{array}{cc}S\left(t\right)=\sum _{k=0}^{n-1}X_k{\Psi }_k\left(t\right),& \left(4\right)\end{array}$

  
• in a least-mean-squares fashion where X[0299]₀, . . . , X_n−1are n reconstruction weights. Observe that the two-point sum in theinterval 0<t<τ is:$\begin{array}{cc}L\left({\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">X</figure-callout>}}_0,{\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">X</figure-callout>}}_1,\dots ,X_{n-1}\right)=\sum _{t=-\tau }^{n\tau }〈{\left[X\left(t\right)-\sum _{k=0}^{n-1}X_k{\Psi }_k\left(t\right)\right]}^2〉,& \left(6\right)\end{array}$

  
• Hence, S(t)[0300]1030 inEquation 4 represents a linear interpolation of the original waveform X(t)1002, as shown in FIG. 39.
• To find the “best” weights X[0301]₀, . . . , X_n−1, the quality L (X₀,X₁, . . . , X_n−1) is minimized:$\begin{array}{cc}\begin{array}{c}{\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">X</figure-callout>}}_0{\mathrm{<figure-callout\; label="\psi "\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">\psi </figure-callout>}}_0\left(t\right)+{\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">X</figure-callout>}}_1{\mathrm{<figure-callout\; label="\Psi "\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\Psi </figure-callout>}}_1\left(t\right)={\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">X</figure-callout>}}_0\left(1-\frac{t}{\tau }\right)+{\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">X</figure-callout>}}_1\left(1-\frac{t-\tau }{\tau }\right)\\ ={\mathrm{<figure-callout\; label="X"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">X</figure-callout>}}_0+\left(X_1-X_0\right)\frac{t}{\tau }\end{array},& \left(5\right)\end{array}$

  
• where the sum has been taken over one period plus τ of the data. X[0302]_kis minimized by differentiating as follows:$\begin{array}{cc}\begin{array}{c}\frac{\partial L}{\partial X_j}=\sum _{t=\tau }^{n\tau }2\left[X\left(t\right)-\sum _{k=0}^{n-1}X_k{\Psi }_k\left(t\right)\right]{\Psi }_j\left(t\right)\\ =〈2\left[\sum _{t=-\tau }^{n\tau }X\left(t\right){\Psi }_j\left(t\right)-\sum _{k=0}^{n-1}X_k\sum _{t=-\tau }^{n\tau }{\Psi }_k\left(t\right){\Psi }_j\left(t\right)\right]〉\equiv 0.\end{array}& \left(7\right)\end{array}$

  
• This leads to the system,[0303]$\begin{array}{cc}\sum _{k=0}^{n-1}A_{\mathrm{jk}}X_k=Y_j,& \left(8\right)\end{array}$

  
• of linear equations for X[0304]_k, where$\begin{array}{cc}{A}_{\mathrm{jk}}=\sum _{t=-\tau }^{\mathrm{n\tau }}{\Psi }_j\left(t\right){\Psi }_k\left(t\right)\mathrm{for}\left(j,k=0,1,\dots ,n-1\right)& \left(9\right)\end{array}$

  
• and[0305]$\begin{array}{cc}{Y}_j=\sum _{t=-\tau }^{\mathrm{n\tau }}X\left(t\right){\Psi }_j\left(t\right)\mathrm{for}\left(j=0,1,\dots ,n-1\right)& \left(10\right)\end{array}$

  
• The term Y[0306]_jinEquation 10 is reducible as follows:$\begin{array}{cc}\begin{array}{c}{Y}_j=\sum _{t=-\tau }^{\mathrm{n\tau }}X\left(t\right)F\left(t-j\tau \right)\\ =\sum _{t=\left(j-1\right)\tau }^{\left(j+1\right)\tau }X\left(t\right)F\left(t-j\tau \right).\end{array}& \left(11\right)\end{array}$

  
• Letting (t-jτ)=m, then:[0307]$\begin{array}{cc}{Y}_j=\sum _{m=-\mathrm{<figure-callout\; label="\tau \; +"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\tau </figure-callout>}+1}^{\tau -1}X\left(m+j\tau \right)F\left(m\right)\mathrm{for}\left(j=0,1,2,\dots ,n-1\right).& \left(12\right)\end{array}$

  
• The Y[0308]_j's inEquation 12 represent the compressed data to be transmitted or stored. Note that this encoding scheme involves n correlation operations on only 2τ−1 points.
• Since F(t) is assumed to be periodic with period nτ, the matrix form of A[0309]_jkinEquation 9 can be reduced bysubstitution Equation 3 intoEquation 9 to obtain:$\begin{array}{cc}\begin{array}{c}{A}_{\mathrm{jk}}=\sum _{m=-\mathrm{<figure-callout\; label="T\; +"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">T</figure-callout>}+1}^{\tau -1}F\left(m+\left(j-k\right)\tau \right)F\left(m\right)\\ =\{\begin{array}{c}\sum _{m=-\mathrm{<figure-callout\; label="\tau \; +"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\tau </figure-callout>}+1}^{\tau -1}{\left(F\left(m\right)\right)}^2\stackrel{\Delta }{=}\alpha \mathrm{if}j-k\equiv 0\mathrm{mod}n\\ \sum _{m=-\mathrm{<figure-callout\; label="\tau \; +"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\tau </figure-callout>}+1}^{\tau -1}F\left(m\pm \tau \right)F\left(m\right)\stackrel{\Delta }{=}\beta \mathrm{if}j-k\equiv \pm 1\mathrm{mod}n\\ 0\mathrm{otherwise}\end{array}\end{array}& \left(13\right)\end{array}$

  
• By Equation 13, A[0310]_jkcan be expressed also in circulant form in the following manner:
• A_jk=a_(k-j)n,   (14)
• where (k-j)[0311]_ndenotes (k-j) mod n, and
• a₀=α, a₁=β, a₂=0, . . . , a_n−1=β  (15)
• Therefore, A[0312]_jkinEquations 14 and 15 has explicitly the following equivalent circulant matrix representations:$\begin{array}{cc}\begin{array}{c}\left[A_{\mathrm{jk}}\right]\stackrel{\Delta }{=}\left[\begin{array}{cccc}{A}_{0,0}& {\mathrm{<figure-callout\; label="A"\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}"><figure-callout\; label="A"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">A</figure-callout></figure-callout>}}_{0,1}& \cdots & A_{o,n-1}\\ {\mathrm{<figure-callout\; label="A"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">A</figure-callout>}}_{1,0}& {\mathrm{<figure-callout\; label="A"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}"><figure-callout\; label="A"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">A</figure-callout></figure-callout>}}_{1,1}& \cdots & A_{1,n-1}\\ ⋮& ⋮& ⋰& ⋮\\ A_{n-1,0}& A_{n-1,1}& \cdots & A_{n-1,n-1}\end{array}\right]\\ =\left[\left\{a_{{\left(k-j\right)}_a}\right\}\right]\\ \stackrel{\Delta }{=}\left[\begin{array}{ccccc}{a}_0& a_1& a_2& \cdots & a_{n-1}\\ a_{n-1}& a_0& a_1& \cdots & a_{n-2}\\ a_{n-2}& a_{n-1}a_0& \cdots & a_{n-3}& \\ ⋮& ⋮& ⋮& ⋰& ⋮\\ a_1& a_2& a_3& \cdots & a_0\end{array}\right]\\ =\left[\begin{array}{ccccc}\mathrm{<figure-callout\; label="\alpha \; \beta "\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">\alpha </figure-callout>}& \beta & 0& \cdots & \beta \\ \beta & \alpha & \beta & \cdots & 0\\ 0& \beta & \alpha & \cdots & 0\\ ⋮& ⋮& ⋮& ⋰& ⋮\\ \beta & 0& 0& \cdots & \alpha .\end{array}\right]\end{array}& \left(16\right)\end{array}$

  
• One skilled in the art of matrix and filter analysis will appreciate that the periodic boundary conditions imposed on the data lie outside the window of observation and may be defined in a variety of ways. Nevertheless, periodic boundary conditions serve to simplify the process implementation by insuring that the correlation matrix [A[0313]_jk] has a calculable inverse. Thus, the optimization process involves an inversion of [A_jk], of which the periodic boundary conditions and consequent circulant character play a preferred role. It is also recognized that for certain spline functions, symmetry rendered in the correlation matrix allows inversion in the absence of periodic image boundary conditions.
• B. Two-Dimensional Data Compression by Planar Splines[0314]
• For two-dimensional image data, multi-planar spline functions are combined to approximate the image data with a resultant planar interpolation. In FIG. 40, X(t[0315]₁,t₂) is a doubly periodic array of image data (e.g., still image) of periods n₁τ and n₂τ, with respect to the integer variables t₁and t₂where τ is a multiple of both t₁and t₂. Theactual image1002 to be compressed can be viewed as being repeated periodically throughout the plane as shown in the FIG. 40. Each subimage of the extended picture is separated by a border1032 (or gutter) of zero intensity of width τ. This border is one of several possible preferred “boundary conditions” to achieve a doubly-periodic image.
• Consider now a doubly periodic planar spline, F(t[0316]₁, t₂) which has the form of a six-sided pyramid or tent, centered at the origin and is repeated periodically with periods n₁τ and n₂τ with respect to integer variables t₁and t₂, respectively. A perspective view of such aplanar spline function1034 is shown in FIG. 41aand may hereinafter be referred to as “hexagonal tent.” Following the one-dimensional case by analogy, letting:
• ψ_k1k2(t₁,t₂)=F(t₁-k₁τ, t₂-k₂τ)   (17)
• for (k[0317]₁=0,1, . . . , n₁−1) and (k₂=0,1, . . . ,n₂−1), the “best” weights X_k1k2are found such that:$\begin{array}{cc}L\left({\mathrm{<figure-callout\; label="X\; k"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">X</figure-callout>}}_{k_{}}\right)=\sum _{{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=-\mathrm{<figure-callout\; label="\tau \; n"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\tau </figure-callout>}}^{n_{}}〈{\left[X\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right)-\underset{k_{}}{\overset{{}_1,{\mathrm{<figure-callout\; label="k"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">k</figure-callout>}}_2=0}{<figure-callout\; label="\sum \; k"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\sum </figure-callout>}}{\mathrm{<figure-callout\; label="X\; k"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">X</figure-callout>}}_{k_{}}{\mathrm{<figure-callout\; label="\Psi \; k"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\Psi </figure-callout>}}_{k_{}}\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right)\right]}^2〉& \left(18\right)\end{array}$

  
• is a minimum.[0318]
• A condition for L to be a minimum is[0319]$\begin{array}{cc}\begin{array}{c}\frac{\partial L}{\partial {\mathrm{<figure-callout\; label="X\; j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">X</figure-callout>}}_{j_{}}}=2\sum _{{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=-\mathrm{<figure-callout\; label="\tau \; n"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\tau </figure-callout>}}^{n_{}}〈[X\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right)-\\ \sum _{{\mathrm{<figure-callout\; label="k"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,{\mathrm{<figure-callout\; label="k"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">k</figure-callout>}}_2=0}^{n_1-1,n_2-1}{\mathrm{<figure-callout\; label="X\; k"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">X</figure-callout>}}_{k_{}}{\mathrm{<figure-callout\; label="\Psi \; k"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\Psi </figure-callout>}}_{k_{}}\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right)]{\mathrm{<figure-callout\; label="\Psi \; j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\Psi </figure-callout>}}_{j_{}}\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right)〉\\ =2〈[\sum _{{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=-\mathrm{<figure-callout\; label="\tau \; n"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\tau </figure-callout>}}^{n_{}}X\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right){\mathrm{<figure-callout\; label="\Psi \; j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\Psi </figure-callout>}}_{j_{}}\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right)-\\ \sum _{{\mathrm{<figure-callout\; label="k"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,{\mathrm{<figure-callout\; label="k"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">k</figure-callout>}}_2=0}^{n_1-1,n_2-1}{\mathrm{<figure-callout\; label="X\; k"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">X</figure-callout>}}_{k_{}}\sum _{{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=-\mathrm{<figure-callout\; label="\tau \; n"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\tau </figure-callout>}}^{n_{}}{\Psi }_{{\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">j</figure-callout>}}_1{\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">j</figure-callout>}}_2}\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right){\mathrm{<figure-callout\; label="\Psi \; k"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\Psi </figure-callout>}}_{k_{}}\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right)]〉\\ \equiv 0.\end{array}& \left(19\right)\end{array}$

  
• The best efficients X[0320]_k1k2are the solution of the 2nd-order tensor equation,
• A_j1j2k1k2X_k1k2=Y_j1j2,   (20)
• where the ummation is on k[0321]₁and k₂,$\begin{array}{cc}{A}_{{\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">j</figure-callout>}}_1{\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">j</figure-callout>}}_2{\mathrm{<figure-callout\; label="k"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">k</figure-callout>}}_1{\mathrm{<figure-callout\; label="k"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}=\sum _{{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=-\mathrm{<figure-callout\; label="\tau \; n"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\tau </figure-callout>}}^{n_{}}{\Psi }_{{\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">j</figure-callout>}}_1{\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">j</figure-callout>}}_2}\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right){\mathrm{<figure-callout\; label="\Psi \; k"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\Psi </figure-callout>}}_{k_{}}\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right)& \left(21\right)\end{array}$

  
• and[0322]$\begin{array}{cc}{\mathrm{<figure-callout\; label="Y\; j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">Y</figure-callout>}}_{j_{}}=\sum _{{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=-\mathrm{<figure-callout\; label="\tau \; n"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\tau </figure-callout>}}^{n_{}}X\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right){\mathrm{<figure-callout\; label="\Psi \; j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\Psi </figure-callout>}}_{j_{}}\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right).& \left(22\right)\end{array}$

  
• With the visual aid of FIG. 41[0323]a, the tensor Y_j1j2reduces as follows:$\begin{array}{cc}\begin{array}{c}{\mathrm{<figure-callout\; label="Y\; j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">Y</figure-callout>}}_{j_{}}=\sum _{{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=-\mathrm{<figure-callout\; label="T\; n"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">T</figure-callout>}}^{n_{}}X\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right){\mathrm{<figure-callout\; label="\Psi \; j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\Psi </figure-callout>}}_{j_{}}\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right)\\ =\sum _{{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=-\mathrm{<figure-callout\; label="T\; n"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">T</figure-callout>}}^{n_{}}X\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right)F\left(t_1-{\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">j</figure-callout>}}_1\tau ,t_2-{\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">j</figure-callout>}}_2\tau \right)\\ =\sum _{{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1=\left(j_1-1\right)\tau }^{\left({\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">j</figure-callout>}}_1+1\right)\tau }\sum _{{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=\left(j_2-1\right)\tau }^{\left({\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">j</figure-callout>}}_2+1\right)\tau }X\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right)F\left(t_1-{\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">j</figure-callout>}}_1\tau ,t_2-{\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">j</figure-callout>}}_2\tau \right).\end{array}& \left(23\right)\end{array}$

  
• Letting t[0324]_k-j_kτ=m_kfor k=1,2, then$\begin{array}{cc}{\mathrm{<figure-callout\; label="Y\; j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">Y</figure-callout>}}_{j_{}}=\sum _{m_1,{\mathrm{<figure-callout\; label="m"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">m</figure-callout>}}_2=-\mathrm{<figure-callout\; label="\tau \; +"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\tau </figure-callout>}+1}^{\tau -1}X\left({\mathrm{<figure-callout\; label="m"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">m</figure-callout>}}_1+{\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">j</figure-callout>}}_1\tau ,{\mathrm{<figure-callout\; label="m"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">m</figure-callout>}}_2+{\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">j</figure-callout>}}_2\tau \right)F\left({\mathrm{<figure-callout\; label="m"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">m</figure-callout>}}_1,m_2\right)& \left(24\right)\end{array}$

  
• for (j[0325]₁=0,1, . . . , n₁−1) and (j₂=0,1, . . . , n₂−1), where F(m₁,m₂) is the doubly periodic, six-sided pyramidal function, shown in FIG. 41a. The tensor transform in Equation 21 is treated in a similar fashion to obtain$\begin{array}{cc}\begin{array}{c}{A}_{j_1j_2k_1k_2}=\sum _{t_1,t_2=-\tau }^{n_1\tau ,n_2\tau }{\Psi }_{j_1,j_2}\left(t_1,t_2\right){\Psi }_{k_1k_2}\left(t_1,t_2\right)\\ =\sum _{m_1,m_2=-\tau +1}^{\tau -1}F\left(m_1+\left(j_1-k_1\right)\tau ,m_2+\left(j_2-k_2\right)\tau \right)F\left(m_1,m_2\right)\\ =\{\begin{array}{c}\sum _{m_1,m_2=-\tau +1}^{\tau -1}{\left(\left[F\left(m_1,m_2\right)\right]\right)}^2\stackrel{\Delta }{=}\alpha \\ \mathrm{if}\left(j_1-k_1\right)\equiv 0\mathrm{mod}n_1\left(j_2-k_2\right)\equiv 0\mathrm{mod}n_2\\ \sum _{m_1,m_2=-\tau +1}^{\tau -1}F\left(m_1\pm \tau ,m_2\right)F\left(m_1,m_2\right)\stackrel{\Delta }{=}\beta \\ \mathrm{if}\left(j_1-k_1\right)\equiv \pm 1\mathrm{mod}n_1\left(j_2-k_2\right)\equiv 0\mathrm{mod}n_2\\ \sum _{m_1,m_2=-\tau +1}^{\tau -1}F\left(m_1,m_2\pm \tau \right)F\left(m_1,m_2\right)\stackrel{\Delta }{=}\gamma \\ \mathrm{if}\left(j_1-k_1\right)\equiv 0\mathrm{mod}n_1\left(j_2-k_2\right)\equiv \pm 1\mathrm{mod}n_2\\ \sum _{m_1,m_2=-\tau +1}^{\tau -1}F\left(m_1\pm \tau ,m_2\pm \tau \right)F\left(m_1,m_2\right)\stackrel{\Delta }{=}\xi \\ \mathrm{if}\left(j_1-k_1\right)\equiv \pm 1\mathrm{mod}n_1\left(j_2-k_2\right)\equiv \pm 1\mathrm{mod}n_2\\ \sum _{m_1,m_2=-\tau +1}^{\tau -1}F\left(m_1\mp \tau ,m_2\pm \tau \right)F\left(m_1,m_2\right)\stackrel{\Delta }{=}\eta \\ \mathrm{if}\left(j_1-k_1\right)\equiv \mp 1\mathrm{mod}n_1\left(j_2-k_2\right)\equiv \pm 1\mathrm{mod}n_2\end{array}\end{array}& \left(25\right)\end{array}$

  
• The values of α, β, γ, and ξ depend on τ, and the shape and orientation of the hexagonal tent with respect to the image domain, where for example m[0326]₁and m₂represent row and column indices. For greater flexibility in tailoring the hexagonal tent function, it is possible to utilize all parameters of the [A_j1j2k1k2]. However, to minimize calculational overhead it is preferable to employ symmetric hexagons, disposed over the image domain with a bi-directional period τ. Under these conditions, β=γ=ξ and η=0, simplifying [A_j1j2k1k2] considerably. Specifically, the hexagonal tent depicted in FIG. 41aand having an orientation depicted in FIG. 41bis described by the preferred case in which β=γ=ξ and η=0. It will be appreciated that other orientations and shapes of the hexagonal tent are possible, as depicted, for example, in FIG. 41c. Combinations of hexagonal tents are also possible and embody specific preferable attributes. For example, a superposition of the hexagonal tents shown in FIG. 41band41ceffectively “symmetrizes” the compression process.
• From[0327]Equation 25 above, A_j1j2k1k2can be expressed in circulant form by the following expression:
• A_j1j2k1k2=a_{(k1-j1)n1,(k2-j2)n2}.   (26)
• where (k[0328]_l-j_l)_nldenote (k₁-j₁) mod n_l, l=1,2, and$\begin{array}{cc}\begin{array}{c}\left[a_{{\mathrm{<figure-callout\; label="s"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">s</figure-callout>}}_1s_2}\right]=\left[\begin{array}{ccccc}{a}_{00}& a_{01}& a_{02}& \cdots & a_{a,n_2-1}\\ a_{10}& a_{11}& a_{12}& \cdots & a_{1,n_2-1}\\ a_{20}& a_{21}& a_{22}& \cdots & a_{2,n_2-1}\\ ⋮& ⋮& ⋮& ⋰& ⋮\\ a_{n_1-1,0}& a_{n_1-1,1}& a_{n_1-1,2}& \cdots & a_{n_1-1,n_2-1}\end{array}\right]\\ =\left[\begin{array}{cccccc}\mathrm{<figure-callout\; label="\alpha \; \beta "\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">\alpha </figure-callout>}& \beta & 0& \cdots & 0& \beta \\ \beta & \beta & 0& \cdots & 0& 0\\ 0& 0& 0& \cdots & 0& 0\\ ⋮& ⋮& ⋮& ⋰& ⋮& ⋮\\ 0& 0& 0& \cdots & 0& 0\\ \mathrm{<figure-callout\; label="\beta "\; filenames="US20010019630A1-drawings-page-21.png,US20010019630A1-drawings-page-22.png"\; state="\{\{state\}\}">\beta </figure-callout>}& 0& 0& \cdots & 0& \beta \end{array}\right],\end{array}& \left(27\right)\end{array}$

  
• where (s[0329]₁=0, 1, 2, . . . n₁−1) and (s₂=1, 2, 3, . . . , n₂−1). Note that when [a_s1,s2] is represented in matrix form, it is “block circulant.”
• C. Compression-Reconstruction Alqorithms[0330]
• Because the objective is to apply the above-disclosed LMS error linear spline interpolation techniques to image sequence coding, it is advantageous to utilize the tensor formalism during the course of the analysis in order to readily solve the linear systems in[0331]equations 8 and 20. Here, the tensor summation convention is used in the analysis for one and two dimensions. It will be appreciated that such convention may readily apply to the general case of N dimensions.
• 1. Linear Transformation of Tensors[0332]
• A linear transformation of a 1st-order tensor is written as[0333]
• Y_r=A_rsX_s(sum on s),   (28)
• where A[0334]_rsis a linear transformation, and Y_r,X_sare 1st-order tensors. Similarly, a linear transformation of a second order tensor is written as:
• Y_r1r2=A_r1r2s1s2X_s1s2(sum on s₁,s₂).   (29)
• The product or composition of linear transformations is defined as follows. When the above Equation 29 holds, and[0335]
• Z_1aq2=B_q1q2r1r2Y_r1r2,   (30)
• then[0336]
• Z_q1q2=B_q1q2r1r2A_r1r2s1s2.   (31)
• Hence,[0337]
• C_q1q2s1s2=B_q1q2r1r2A_r1r2s1s2  (32)
• is the composition or product of two linear transformations.[0338]
• 2. Circulant Transformation of 1st-Order Tensors[0339]
• The tensor method for solving[0340]equations 8 and 20 is illustrated for the 1-dimensional case below: Letting A_rsrepresent a circulant tensor of the form:
• A_rs=a_{(s-r)mod n}for(r,s=0, 1, 2, . . . , n−1),   (33)
• and considering the n special 1st-order tensors as[0341]
• W_S^(l)≡(ω^l)^sfor (t=0, 1, 2, . . . ,n−1),   (34)
• where ω is the n-th root of unity, then[0342]
• A_rsW_S^(l)=λ(l)W_r^(l),   (35)
• where[0343]$\begin{array}{cc}\lambda \left(l\right)=\sum _{j=0}^{n-1}{a_j\left({\omega }^l\right)}^j& \left(36\right)\end{array}$

  
• are the distinct eigenvalues of A[0344]_rs. The terms W_S^(l)are orthogonal.$\begin{array}{cc}{W}_s^{\left(l\right)}W_s^{\left(j\right)*}=\{\begin{array}{c}0\mathrm{for}l\ne j\\ n\mathrm{for}l=j.\end{array}& \left(37\right)\end{array}$

  
• At this point it is convenient to normalize these tensors as follows:[0345]$\begin{array}{cc}\varphi \begin{array}{c}\left(l\right)\\ S\end{array}\stackrel{\Delta }{=}\frac{1}{\sqrt{n}}W\begin{array}{c}\left(l\right)\\ S\end{array}\mathrm{for}\left(l=0,1,2,\dots ,n-1\right).& \left(38\right)\end{array}$

  
• Φ[0346]_s^(l)evidently also satisfies the orthonormal property, i.e.,
• Φ_S^(l)Φ_S^(j)*=δ_lj  (39)
• where δ[0347]_ljis the Kronecker delta function and * represents complex conjugation.
• A linear transformation is formed by summing the n dyads Φ[0348]_r^(l)Φ_s^(l)*for l=0,1, . . . ,n−1 under the summation sign as follows:$\begin{array}{cc}{\tilde{A}}_{\mathrm{rs}}=\sum _{l=0}^{n-1}\lambda \left(l\right)\varphi \begin{array}{c}\left(l\right)\\ r\end{array}\varphi {\begin{array}{c}\left(l\right)\\ S\end{array}}^{*}.& \left(40\right)\end{array}$

  
• Then[0349]$\begin{array}{cc}\begin{array}{c}{\tilde{A}}_{\mathrm{rs}}\varphi \begin{array}{c}\left(j\right)\\ S\end{array}=\sum _{j\ne 0}^{n-1}\lambda \left(l\right){\varphi }_r\left(l\right){\varphi }_s\left(l\right)*{\varphi }_s\left(j\right)\\ =\sum _{n-1}^{l=0}\lambda \left(l\right){\varphi }_r\left(l\right){\delta }_{\mathrm{lj}}\\ =\lambda \left(j\right){\varphi }_r\left(j\right).\end{array}& \left(41\right)\end{array}$

  
• Since Ã[0350]_rshas by a simple verification the same eigenvectors and eigenvalues as the transformation A_rshas inEquations 9 and 33, the transformation Ã_rsand A_rsare equal.
• 3. Inverse Transformation of 1st-Order Tensors.[0351]
• The inverse transformation of A[0352]_rsis shown next to be$\begin{array}{cc}{A}_{\mathrm{rs}}^{-1}=\sum _{l=0}^{n-1}\frac{1}{\lambda \left(l\right)}{\varphi }_r^l{\varphi }_s^{l^{*}}.& \left(42\right)\end{array}$

  
• This is proven easily, as shown below:[0353]$\begin{array}{cc}\begin{array}{c}{A}_{\mathrm{rs}}A_{\mathrm{st}}^{-1}=\sum _{l=0}^{n-1}\sum _{l^{\prime }=0}^{n-1}\lambda \left(l\right)\frac{1}{\lambda \left(l^{\prime }\right)}{\varphi }_r^l{\varphi }_s^{l^{*}}{\varphi }_s^{l^{\prime }}{\varphi }_t^{l^{\prime *}}\\ =\sum _{l=0}^{n-1}\sum _{l^{\prime }=0}^{n-1}\lambda \left(l\right)\frac{1}{\lambda \left(l^{\prime }\right)}{\varphi }_{\tau }^l{\delta }_{ll^{\prime }}{\varphi }_t^{l^{\prime *}}=\sum _{l=0}^{n-1}{\varphi }_r^l{\varphi }_t^{l^{*}}\\ =\sum _{l=0}^{n-1}\frac{1}{n}{\left({\omega }^l\right)}^{\mathrm{rt}}=\sum _{l=0}^{n-1}\frac{1}{n}{\left({\omega }^{\mathrm{rt}}\right)}^l={\delta }_{\mathrm{rt}}\end{array}& \left(43\right)\end{array}$

  
• 4. Solving 1st-Order Tensor Equations[0354]
• The solution of a[0355]1st-order tensor equation Y_r=A_rsX is given by
• A_qr⁻¹Y_r=A_gr⁻¹A_rsX_s=δ_qsX_s=X_q,   (44)
• so that[0356]$\begin{array}{cc}\begin{array}{c}{X}_r=A_{\mathrm{rs}}^{-1}Y_s\\ =\sum _{l=0}^{n-1}\frac{1}{\lambda \left(l\right)}{\varphi }_r^l{\varphi }_s^{l^{*}}Y_s\\ =\sum _{l=0}^{n-1}\left[\frac{{\varphi }_s^{l^{*}}Y_s}{\lambda \left(l\right)}\right]{\varphi }_r^l=\sum _{l=0}^{n-1}\left[\frac{1}{\lambda \left(l\right)}\left[\frac{1}{n}\sum _{k=0}^{n-1}Y_k{\omega }^{-\mathrm{lk}}\right]\right]{\omega }^{\mathrm{lr}}\\ =\mathrm{DFT}\left[\frac{1}{\lambda \left(l\right)}{\mathrm{DFT}}^{-1}\left(Y_k\right)\right].\end{array}& \left(45\right)\end{array}$

  
• where DFT denotes the discrete Fourier Transform and DFT[0357]⁻¹denotes its inverse discrete Fourier Transform.
• An alternative view of the above solution method is derived below for one dimension using standard matrix methods. A linear transformation of a 1st-order tensor can be represented by a matrix. For example, let A denote A[0358]_rsin matrix form. If A_rsis a circulant transformation, then A is also a circulant matrix. From matrix theory it is known that every circulant matrix is “similar” to a DFT matrix. If Q denotes the DFT matrix of dimension (n×n), and Q^tthe complex conjugate of the DFT matrix, and Λ is defined to be the eigenmatrix of A, then:
• A=QΛQ^t.   (46)
• The solution to y=Ax is then[0359]
• x=A⁻¹y=QΛ⁻¹(Q^ty).
• For the one-dimensional process described above, the eigenvalues of the transformation operators are:[0360]$\begin{array}{cc}\begin{array}{c}\lambda \left(l\right)=\sum _{j=0}^{n-1}{a_j\left(w^l\right)}^j\\ =\mathrm{DFT}\left(a_j\right).\end{array}& \left(47\right)\end{array}$

  
• where a[0361]₀=α, a₁=β, . . . , a_n−2=0, a_n−1=β, and ωⁿ=1. Hence:
• λ(l)=α+βω^l+βω^(n−1)l=α+β(ω^l+ω^−l).   (48)
• A direct extension of the 1st-order tensor concept to the 2nd-order tensor will be apparent to those skilled in the art. By solving the 2nd-order tensor equations, the results are extended to compress a 2-D image. FIG. 42 depicts three possible hexagonal tent functions for 2-dimensioned image compression indices τ=2,3,4. The following table exemplifies the relevant parameters for implementing the hexagonal tent functions:[0362]

  Decimation Index	τ = 2	τ = 3	τ = 4
  (τ)
  Compression Ratio	4	9	16
  (τ2)
  α	a2+ 6b2	a2+ 6b2+ 12c2	a2+ 6b2+ 12c2+ 18d2
  β	b2	2(c2+ bc)	2d2+ 2db + 4dc + c2
  gain	a + 6b	a + 6b + 12c	a + 6b + 12c + 18d

• The algorithms for compressing and reconstructing a still image are explained in the succeeding sections.[0363]
• III. OVERVIEW OF CODING-RECONSTRUCTION SCHEME[0364]
• A block diagram of the compression/reconstruction scheme is shown in FIG. 43. The[0365]signal source1002, which can have dimension up to N, is first passed through a low-pass filter (LPF). This low-pass filter is implemented by convolving (in a process block1014) a chosen spline filter1013 with theinput source1002. For example, the normalizedfrequency response1046 of a one-dimensional linear spline is shown in FIG. 44. Referring again to FIG. 43, it can be seen that immediately following the LPF, a subsampling procedure is used to reduce thesignal size1016 by a factor τ. The information contained in the subsampled source is not optimized in the least-mean-square sense. Thus, an optimization procedure is needed to obtain the best reconstruction weights. The optimization process can be divided into three consecutive parts. ADFT1020 maps the non-optimized weights into the image conjugate domain. Thereafter, aninverse eigenfilter process1022 optimizes the compressed data. The frequency response plots for some typical eigenfilters and inverse eigenfilters are shown in FIG. 45 and46. After theinverse eigenfilter1022, a DFT⁻¹process block1024 maps its input back to the original image domain. When the optimized weights are derived, reconstruction can proceed. The reconstruction can be viewed as oversampling followed by a reconstruction low-pass filter.
• The embodiment of the optimized spline filter described above may employ a DFT and DFT[0366]⁻¹type transform processes. However, those skilled in the art of digital image processing will appreciate that it is preferable to employ a Fast Fourier Transform (FFT) and FFT⁻¹processes, which substantially reduce computation overhead associated with conjugate transform operations. Typically, such an improvement is given by the ratio of computation steps required to transform a set of N elements:$\frac{\mathrm{FFT}}{\mathrm{DFT}}=\frac{\frac{\mathrm{<figure-callout\; label="N"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">N</figure-callout>}}{2}{\mathrm{<figure-callout\; label="log"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">log</figure-callout>}}_2\left(N\right)}{{\mathrm{<figure-callout\; label="N"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">N</figure-callout>}}^2}=\frac{1}{2N}{\mathrm{<figure-callout\; label="log"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">log</figure-callout>}}_2\left(N\right),$

  
• which improves with the size of the image.[0367]
• A. The Compression Method[0368]
• The coding method is specified in the following steps:[0369]
• 1. A suitable value of τ (an integer) is chosen. The compression ratio is τ[0370]²for two-dimensional images.
• 2. Equation 23 is applied to find Y[0371]_j1,j2, which is the compressed data to be transmitted or stored:$\begin{array}{c}{\mathrm{<figure-callout\; label="Y\; j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">Y</figure-callout>}}_{j_{}}=\sum _{{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=-\mathrm{<figure-callout\; label="T\; n"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">T</figure-callout>}}^{n_{}}X\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right){\mathrm{<figure-callout\; label="\Psi \; j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\Psi </figure-callout>}}_{j_{}}\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right)\\ =\sum _{{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=-\mathrm{<figure-callout\; label="T\; n"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">T</figure-callout>}}^{n_{}}X\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right)F\left(t_1-{\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">j</figure-callout>}}_1\tau ,t_2-{\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">j</figure-callout>}}_2\tau \right)\\ =\sum _{{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1=\left(j_1-1\right)\tau }^{\left({\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">j</figure-callout>}}_1+1\right)\tau }\sum _{{\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=\left(j_2-1\right)\tau }^{\left({\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">j</figure-callout>}}_2+1\right)\tau }X\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right)F\left(t_1-{\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">j</figure-callout>}}_1\tau ,t_2-{\mathrm{<figure-callout\; label="j"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">j</figure-callout>}}_2\tau \right)\end{array}$

  
• B. The Reconstruction Method[0372]
• The reconstruction method is shown below in the following steps:[0373]
• 1. Find the FFT[0374]⁻¹of Y_j1,j2(the compressed data).
• 2. The results of[0375]step 1 are divided by the eigenvalues λ(l, m) set forth below. The eigenvalues λ(l,m) are found by extending Equation 48 to the two-dimensional case to obtain:
• λ(l,m)=α+β(ω₁^l+ω₁^−l+ω₂^m+ω₂^−m+ω₁^lω₂^−m+ω₁^−lω₂^m),   (49)
• where ω[0376]₁is the n₁-th root of unity and ω₂is the n₂-th root of unity.
• 3. The FFT of the results from[0377]step 2 is then taken. After computing the FFT, X_k1k2(the optimized weights) are obtained.
• 4. The recovered or reconstructed image is:[0378]$\begin{array}{cc}S\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right)=\sum _{{\mathrm{<figure-callout\; label="k"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,{\mathrm{<figure-callout\; label="k"\; filenames="US20010019630A1-drawings-page-13.png,US20010019630A1-drawings-page-21.png"\; state="\{\{state\}\}">k</figure-callout>}}_2=0}^{n_1-1,n_2-1}{\mathrm{<figure-callout\; label="X\; k"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">X</figure-callout>}}_{k_{}}{\mathrm{<figure-callout\; label="\Psi \; k"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">\Psi </figure-callout>}}_{k_{}}\left({\mathrm{<figure-callout\; label="t"\; filenames="US20010019630A1-drawings-page-10.png,US20010019630A1-drawings-page-11.png"\; state="\{\{state\}\}">t</figure-callout>}}_1,t_2\right).& \left(50\right)\end{array}$

  
• 5. Preferably, the residue is computed and retained with the optimized weights:[0379]
• ΔX(t₁, t₂)=X(t₁,t₂)−S(t₁, t₂).
• Although the optimizing procedure outlined above appears to be associated with an image reconstruction process, it may be implemented at any stage between the aforementioned compression and reconstruction. It is preferable to implement the optimizing process immediately after the initial compression so as to minimize the residual image. The preferred order has an advantage with regard to storage, transmission and the incorporation of subsequent image processes.[0380]
• C. Response Considerations[0381]
• The inverse eigenfilter in the conjugate domain is described as follows:[0382]$\begin{array}{cc}H\left(i,j\right)=\frac{1}{\lambda \left(i,j\right)}.& \left(51\right)\end{array}$

  
• where λ(i,j) can be considered as an estimation of the frequency response of the combined decimation and interpolation filters. The optimization process H(i,j) attempts to “undo” what is done in the combined decimation/interpolation process. Thus, H(i,j) tends to restore the original signal bandwidth. For example, for τ=2, the decimation/interpolation combination is described as having an impulse response resembling that of the following 3×3 kernel:[0383]$\begin{array}{cc}R=\left(\begin{array}{ccc}0& \beta & \beta \\ \beta & \alpha & \beta \\ \beta & \beta & 0\end{array}\right).& \left(52\right)\end{array}$

  
• Then, its conjugate domain counterpart, λ(i,j)|[0384]_{α,β, N}, will be$\begin{array}{cc}{(\lambda \left(i,j\right))}_{\alpha ,\beta ,N}\equiv \alpha +2\beta \left[\cos \left(\frac{2\pi i}{N}\right)+\cos \left(\frac{2\pi j}{N}\right)+\cos \left[2\pi \left(\frac{i}{N}-\frac{j}{N}\right)\right]\right],& \left(53\right)\end{array}$

  
• where i,j are frequency indexes and N represents the number of frequency terms. Hence, the implementation accomplished in the image conjugate domain is the conjugate equivalent of the inverse of the above 3×3 kernel. This relationship will be utilized more explicitly for the embodiment disclosed in Section V.[0385]
• IV. NUMERICAL SIMULATIONS[0386]
• A. One-Dimensional Case[0387]
• For a one-dimensional implementation, two types of signals are demonstrated. A first test is a cosine signal which is useful for observing the relationship between the standard error, the size of τ and the signal frequency. The standard error is defined herein to be the square root of the average error:[0388]${\left[\frac{1}{N}\sum _t{\left(\Delta X\left(t\right)\right)}^2\right]}^{1/2}.$

  
• A second one-dimensional signal is taken from one line of a grey-scale still image, which is considered to be realistic data for practical image compression.[0389]
• FIG. 47 shows the plots of standard error versus frequency of the cosine signal for different degrees of decimation τ[0390]1056. The general trend is that as the input signal frequency becomes higher, the standard error increases. In the low frequency range, smaller values of τ yield a better performance. One abnormal phenomenon exists for the τ=2 case and a normalized input frequency of 0.25. For this particular situation, the linear spline and the cosine signal at discrete grid points can match perfectly so that the standard error is substantially equal to 0.
• Another test example comes from one line of realistic still image data. FIG. 48[0391]aand48bshow thereconstructed signal waveform1060 for τ=2 and τ=4, respectively, superimposed on theoriginal image data1058. FIG. 48ashows a good quality of reconstruction for τ=2. For τ=4, in FIG. 48b, some of the high frequency components are lost due to the combined decimation/interpolation procedure. FIG. 48cpresents theerror plot1062 for this particular test example. It will be appreciated that the non-linear error accumulation versus decimation parameter τ may be exploited to minimize the combination of optimized weights and image residue.
• B. Two-Dimensional Case[0392]
• For the two-dimensional case, realistic still image data are used as the test. FIG. 49 and[0393]50 show the original and reconstructed images for τT=2 and τ=4. For τ=2, thereconstructed image1066,1072 is substantially similar to the original. However, for τ=4, there are zig-zag patterns along specific edges in images. This is due to the fact that the interpolation less accurately tracks the high frequency components. As described earlier, substantially complete reconstruction is achieved by retaining the minimized residue ΔX and adding it back to the approximated image. In the next section, several methods are proposed for implementing this process. FIG. 51 shows the error plots as functions of τ for both images.
• An additional aspect of interest is to look at the optimized weights directly. When these optimal weights are viewed in picture form, high-[0394]quality miniatures1080,1082 of the original image are obtained, as shown in FIG. 52. Hence, the present embodiment is a very powerful and accurate method for creating a “thumbnail” reproduction of the original image.
• V. ALTERNATIVE EMBODIMENTS[0395]
• Video compression is a major component of high-definition television (HDTV) According to the present invention, video compression is formulated as an equivalent three-dimensional approximation problem, and is amenable to the technique of optimum linear or more generally by hyperplanar spline interpolation. The main advantages of this approach are seen in its fast speed in coding/reconstruction, its suitability in a VLSI hardware implementation, and a variable compression ratio. A principal advantage of the present invention is the versatility with which it is incorporated into other compression systems. The invention can serve as a “front-end” compression platform from which other signal processes are applied. Moreover, the invention can be applied iteratively, in multiple dimensions and in either the image or image conjugate domain. The optimizing method can for example apply to a compressed image and further applied to a corresponding compressed residual image. Due to the inherent low-pass filtering nature of the interpolation process, some edges and other high-frequency features may not be preserved in the reconstructed images, but which are retained through the residue. To address this problem, the following procedures are set forth:[0396]
• Procedure (a)[0397]
• Since the theoretical formulation, derivation, and implementation of the disclosed compression method do not depend strongly on the choice of the interpolation kernel function, other kernel functions can be applied and their performances compared. So far, due to its simplicity and excellent performance, only the linear spline function has been applied. Higher-order splines, such as the quadratic spline, cubic spline could also be employed. Aside from the polynomial spline functions, other more complicated function forms can be used.[0398]
• Procedure (b)[0399]
• Another way to improve the compression method is to apply certain adaptive techniques. FIG. 53 illustrates such an adaptive scheme. For a 2-[0400]D image1002, the whole image can be divided into subimages ofsmaller size1084. Since different subimages have different local features and statistics, different compression schemes can be applied to these different subimages. An error criterion is evaluated in aprocess step1086. If the error is below a certain threshold determined in aprocess step1088, a higher compression ratio is chosen for that subimage. If the error goes above this threshold, then a lower compression ratio is chosen in astep1092 for that subimage. Bothmulti-kernel functions1090 and multi-local-compression ratios provide good adaptive modification.
• Procedure (c)[0401]
• Subband coding techniques have been widely used in digital speech coding. Recently, subband coding is also applied to digital image data compression. The basic approach of subband coding is to split the signal into a set of frequency bands, and then to compress each subband with an efficient compression algorithm which matches the statistics of that band. The subband coding techniques divide the whole frequency band into smaller frequency subbands. Then, when these subbands are demodulated into the baseband, the resulting equivalent bandwidths are greatly reduced. Since the subbands have only low frequency components, one can use the above described, linear or planar spline, data compression technique for coding these data. A 16-band filter compression system is shown in FIG. 54, and the corresponding reconstruction system in FIG. 55. There are, of course, many ways to implement this filter bank, as will be appreciated by those skilled in the art. For example, a common method is to exploit the Quadrature Mirror Filter structure.[0402]
• V. IMAGE DOMAIN IMPLEMENTATION[0403]
• The embodiments described earlier utilize a spline filter optimization process in the image conjugate domain using an FFT processor or equivalent thereof. The present invention also provides an equivalent image domain implementation of a spline filter optimization process which presents distinct advantages with regard to speed, memory and process application.[0404]
• Referring back to[0405]Equation 45, it will be appreciated that the transform processes DFT and DFT⁻¹may be subsummed into an equivalent conjugate domain convolution, shown here briefly:$\begin{array}{cc}\begin{array}{c}{X}_j=\mathrm{DFT}\left[\frac{1}{{\lambda }_m}{\mathrm{DFT}}^{-1}\left(Y_k\right)\right]\\ =\mathrm{DFT}\left[{\mathrm{DFT}}^{-1}\left[\mathrm{DFT}\left(\frac{1}{{\lambda }_m}\right)\right]{\mathrm{DFT}}^{-1}\left(Y_k\right)\right]\end{array}& \left(54\right)\end{array}$

  
• If Ω=DFT (1/λ[0406]_m), then:
• X_j=DFT[DFT⁻¹(Ω) DFT⁻¹(Y_k)]=Ω*Y_k.
• Furthermore, with λ[0407]_m=DFT(a_j), the optimization process may be completely carried over to an image domain implementation knowing only the form of the input spline filter function. The transform processes can be performed in advance to generate the image domain equivalent of the inverse eigenfilter. As shown in FIG. 57, the image domain spline optimizer Ω operates on compressed image data Y′ generated by afirst convolution process1014 followed by adecimation process1016, as previously described. Off-line or perhaps adaptively, the tensor transformation A (as shown for example inEquation 25 above) is supplied to anFFT type processor1032, which computes the transformation eigenvalues λ. The tensor of eigenvalues is then inverted atprocess block1034, followed by FFT⁻¹process block1036, generating the image domain tensor Ω. The tensor Ω is supplied to asecond convolution process1038, whereupon Ω is convolved with the non-optimized compressed image data Y′ to yield optimized compressed image data Y″.
• In practice, there is a compromise between accuracy and economy with regard to the specific form of Ω. The optimizer tensor Ω should be of sufficient size for adequate approximation of:[0408]${\mathrm{DFT}}^{-1}\left(\frac{1}{\mathrm{DFT}\left(A\right)}\right).$

  
• On the other hand, the term Ω should be small enough to be computationally tractable for the[0409]online convolution process1038. It has been found that two-dimensional image compression using the preferred hexagonal tent spline is adequately optimized by a 5×5 matrix, and preferably a 7×7 matrix, for example, with the following form:$\Omega =\left\{\begin{array}{ccccccc}0& h& -g& g& e& e& g\\ h& f& e& d& c& d& e\\ -g& e& c& b& b& c& e\\ g& d& b& a& b& d& g\\ e& c& b& b& c& e& -g\\ e& d& c& d& e& f& h\\ g& e& e& g& -g& h& 0\end{array}\right\}.$

  
• Additionally, to reduce computational overhead, the smallest elements (i.e., the elements near the perimeter) such as f, g, and h may be set to zero with little noticeable effect in the reconstruction.[0410]
• The principal advantages of the present preferred embodiment are in computational saving above and beyond that of the previously described conjugate domain inverse eigenfilter process (FIG. 38, 1018). For example, a two-dimensional FFT process may typically require about N[0411]²log₂N complex operations or equivalently 6N²log₂N multiplications. The total number of image conjugate filter operations is of order 10N²log₂N. On the other hand, the presently described (7×7) kernel with 5 distinct operations per image element will require only 5N²operations, lower by an important factor of log₂N. Hence, even for reasonably small images, there is significant improvement in computation time.
• Additionally, there is substantial reduction in buffer demands because the[0412]image domain process1038 requires only a 7×7 image block at a given time, in contrast to the conjugate process which requires a full-frame buffer before processing. In addition to the lower demands on computation with theimage domain process1038, there is virtually no latency in transmission as the process is done in pipeline. Finally, “power of 2” constraints desirable for efficient FFT processing is eliminated, allowing convenient application to a wider range of image dimensions.
• The above detailed description is intended to be exemplary and not limiting. From this detailed description, taken in conjunction with the appended drawings, the advantages of the present invention will be readily understood by one who is skilled in the relevant technology. The present apparatus and method provides a unique encoder, compressed file format and decoder which compresses images and decodes compressed images. The unique compression system increases the compression ratios for comparable image quality while achieving relatively quick encoding and decoding times, optimizes the encoding process to accommodate different image types, selectively applies particular encoding methods for a particular image type, layers the image quality components in the compressed image, and generates a file format that allows the addition of other compressed data information.[0413]
• While the above detailed description has shown, described and pointed out the fundamental novel features of the invention as applied to various embodiments, it will be understood that various omissions and substitutions and changes in the form and details of the illustrated device may be made by those skilled in the art, without departing from the spirit of the invention.[0414]

Claims (17)

What is claimed is:

1. A method of transferring a progressively-rendered, compressed image, over a finite bandwidth channel, comprising:

producing a coarse quality compressed image at a source and transmitting said coarse quality compressed image over a channel as a first part of a transmission to a destination end;

receiving the coarse quality compressed image at a receiver at the destination end at a first time and displaying an image based on said coarse quality compressed image on a display system of the receiver when received at said first time;

creating additional information about the image, at the source end, from which a standard quality image can be displayed, said standard quality image being of a higher quality than said coarse quality image, and sending compressed information over said channel indicative of information for said standard quality image, said sending said standard quality image information occurring subsequent in time to said sending of all of said information for said coarse quality image;

receiving said standard quality information at the receives at a second time, subsequent to the first time, and decompressing said standard quality image information, to improve the quality of the image displayed on said display system, and to display said standard quality image;

obtaining further information about the image beyond the information in said standard quality image, to provide an enhanced quality image, and compressing said information for said enhanced quality image, said enhanced quality image having more image details than said standard quality image;

transmitting said information for said enhanced quality image, at a time subsequent to transmitting said information for said coarse quality image and said standard quality image; and

receiving said enhanced quality image information at said receiver, at a third time subsequent to said first and second times, and updating a display on said display system to display the additional enhanced quality image.

2. A method as in

claim 1

, wherein said producing the coarse quality image uses a different compression technique than said creating additional information indicative of the standard quality image.

3. A method as in

claim 1

, wherein said coarse quality image includes information indicative of a miniature version of an original image, and said displaying the coarse quality image comprises interpolating said miniature to a size of the original image and displaying said image.

4. A method as in

claim 2

, wherein said creating additional information comprises determining a characteristic of the image, determining which of a plurality of different as compression technique will best compress the characteristic determined; and compressing said image using the determined technique.

5. A method as in

claim 4

, further comprising determining a plurality of areas in said image, and determining, for each area, which of the plurality of different compression techniques will optimize the compression ratio.

6. A method as in

claim 5

, further comprising interleaving and channel encoding different portions of the compressed image.

7. A method as in

claim 5

, wherein said compression technique include vector quantization and discrete cosine transform.

8. A method as in

claim 3

, wherein said obtaining a miniature comprises decimating along vertical and horizontal axes.

9. A method of transmitting and displaying a compressed image comprising:

first obtaining and sending a first layer of information indicative of a compressed miniature image at a first time;

first receiving said first layer at said decoder end and decompressing and displaying a first coarse image indicative thereof;

second obtaining and sending information indicative of a compressed improved resolution image having more details than said first coarse image, and transmitting said information at a second time subsequent to said first time; and

second receiving and decompressing said improved resolution image information to provide an updated display which improves the resolution of said first coarse image.

10. A method as in

claim 9

, wherein said obtaining coarse information comprises:

transmitting information indicative of a compressed miniature of the image;

receiving the compressed miniature of the image;

interpolating the compressed miniature of the image into a full sized image; and

displaying the full sized image.

11. A method as in

claim 10

, wherein the first coarse image is compressed using a first compression technique and the second image is compressed using a second compression technique which is different from the first compression technique.

12. A method as in

claim 11

, further comprising determining which of a plurality of different image compression technique will most efficiently code information indicative of said image.

13. A method as in

claim 12

, wherein said determining uses fuzzy logic technique.

14. A method as in

claim 11

, wherein said first obtaining comprises decimating data on the image to form a reduced quality image, fitting the decimated data to a first model which partially restores source image detail lost by decimation, and calculating reconstruction values from the fitting.

15. A method as in

claim 14

, further comprising using said reconstruction weights to interpolate the decimated data into a full sized image while minimizing a mean squared error between original image components and interpolated image components.

16. A method as in

claim 11

, wherein said first step comprises forming miniature versions of the original source image for each of a plurality of primary colors.

17. A method as in

claim 9

, wherein said first obtaining comprises obtaining a miniature image, and further comprising analyzing the miniature image to classify the image into one of a plurality of classes indicative of which of a plurality of compression techniques will best compress said image.

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