Inmathematics,deconvolution is theinverse ofconvolution. Both operations are used insignal processing andimage processing. For example, it may be possible to recover the original signal after a filter (convolution) by using a deconvolution method with a certain degree of accuracy.^[1] Due to the measurement error of the recorded signal or image, it can be demonstrated that the worse thesignal-to-noise ratio (SNR), the worse the reversing of a filter will be; hence, inverting a filter is not always a good solution as the error amplifies. Deconvolution offers a solution to this problem.

The foundations for deconvolution andtime-series analysis were largely laid byNorbert Wiener of theMassachusetts Institute of Technology in his bookExtrapolation, Interpolation, and Smoothing of Stationary Time Series (1949).^[2] The book was based on work Wiener had done duringWorld War II but that had been classified at the time. Some of the early attempts to apply these theories were in the fields ofweather forecasting andeconomics.

In general, the objective of deconvolution is to find the solutionf of a convolution equation of the form:

${\displaystyle f\ast g^{-1}=h}$

Usually,h is some recorded signal, andf is some signal that we wish to recover, but has been convolved with a filter or distortion functiong, before we recorded it. Usually,h is a distorted version off and the shape off can't be easily recognized by the eye or simpler time-domain operations. The functiong represents theimpulse response of an instrument or a driving force that was applied to a physical system. If we knowg, or at least know the form ofg, then we can perform deterministic deconvolution. However, if we do not knowg in advance, then we need to estimate it. This can be done using methods ofstatisticalestimation or building the physical principles of the underlying system, such as the electrical circuit equations or diffusion equations.

There are several deconvolution techniques, depending on the choice of the measurement error and deconvolution parameters:

When the measurement error is very low (ideal case), deconvolution collapses into a filter reversing. This kind of deconvolution can be performed in the Laplace domain. By computing theFourier transform of the recorded signalh and the system response functiong, you getH andG, withG as thetransfer function. Using theConvolution theorem,

${\displaystyle F=H/G}$

whereF is the estimated Fourier transform off. Finally, theinverse Fourier transform of the functionF is taken to find the estimated deconvolved signalf. Note thatG is at the denominator and could amplify elements of the error model if present.

In physical measurements, the situation is usually closer to

${\displaystyle (f\ast g^{-1})+\epsilon =h}$

In this caseε isnoise that has entered our recorded signal. If a noisy signal or image is assumed to be noiseless, the statistical estimate ofg will be incorrect. In turn, the estimate ofƒ will also be incorrect. The lower thesignal-to-noise ratio, the worse the estimate of the deconvolved signal will be. That is the reason whyinverse filtering the signal (as in the "raw deconvolution" above) is usually not a good solution. However, if at least some knowledge exists of the type of noise in the data (for example,white noise), the estimate ofƒ can be improved through techniques such asWiener deconvolution.

The concept of deconvolution had an early application inreflection seismology. In 1950,Enders Robinson was a graduate student atMIT. He worked with others at MIT, such asNorbert Wiener,Norman Levinson, and economistPaul Samuelson, to develop the "convolutional model" of a reflectionseismogram. This model assumes that the recorded seismograms(t) is the convolution of an Earth-reflectivity functione(t) and aseismicwaveletw(t) from apoint source, wheret represents recording time. Thus, our convolution equation is

${\displaystyle s(t)=(e\ast w)(t).}$

The seismologist is interested ine, which contains information about the Earth's structure. By theconvolution theorem, this equation may beFourier transformed to

${\displaystyle S(\omega )=E(\omega )W(\omega )}$

in thefrequency domain, where${\displaystyle \omega }$ is the frequency variable. By assuming that the reflectivity is white, we can assume that thepower spectrum of the reflectivity is constant, and that the power spectrum of the seismogram is the spectrum of the wavelet multiplied by that constant. Thus,

${\displaystyle |S(\omega )|\approx k|W(\omega )|.}$

If we assume that the wavelet isminimum phase, we can recover it by calculating the minimum phase equivalent of the power spectrum we just found. The reflectivity may be recovered by designing and applying aWiener filter that shapes the estimated wavelet to aDirac delta function (i.e., a spike). The result may be seen as a series of scaled, shifted delta functions (although this is not mathematically rigorous):

${\displaystyle e(t)=\sum _{i=1}^N{r}_i\delta (t-{\tau }_i),}$

whereN is the number of reflection events,${\displaystyle r_i}$ are thereflection coefficients,${\displaystyle t-{\tau }_i}$ are the reflection times of each event, and${\displaystyle \delta }$ is theDirac delta function.

In practice, since we are dealing with noisy, finitebandwidth, finite length,discretely sampled datasets, the above procedure only yields an approximation of the filter required to deconvolve the data. However, by formulating the problem as the solution of aToeplitz matrix and usingLevinson recursion, we can relatively quickly estimate a filter with the smallestmean squared error possible. We can also do deconvolution directly in the frequency domain and get similar results. The technique is closely related tolinear prediction.

In optics and imaging, the term "deconvolution" is specifically used to refer to the process of reversing theoptical distortion that takes place in an opticalmicroscope,electron microscope,telescope, or other imaging instrument, thus creating clearer images. It is usually done in the digital domain by asoftwarealgorithm, as part of a suite ofmicroscope image processing techniques. Deconvolution is also practical to sharpen images that suffer from fast motion or jiggles during capturing. EarlyHubble Space Telescope images were distorted by aflawed mirror and were sharpened by deconvolution.

The usual method is to assume that the optical path through the instrument is optically perfect, convolved with apoint spread function (PSF), that is, amathematical function that describes the distortion in terms of the pathway a theoreticalpoint source of light (or other waves) takes through the instrument.^[3] Usually, such a point source contributes a small area of fuzziness to the final image. If this function can be determined, it is then a matter of computing itsinverse or complementary function, and convolving the acquired image with that. The result is the original, undistorted image.

In practice, finding the true PSF is impossible, and usually an approximation of it is used, theoretically calculated^[4] or based on some experimental estimation by using known probes. Real optics may also have different PSFs at different focal and spatial locations, and the PSF may be non-linear. The accuracy of the approximation of the PSF will dictate the final result. Different algorithms can be employed to give better results, at the price of being more computationally intensive. Since the original convolution discards data, some algorithms use additional data acquired at nearby focal points to make up some of the lost information.Regularization in iterative algorithms (as inexpectation-maximization algorithms) can be applied to avoid unrealistic solutions.

When the PSF is unknown, it may be possible to deduce it by systematically trying different possible PSFs and assessing whether the image has improved. This procedure is calledblind deconvolution.^[3] Blind deconvolution is a well-establishedimage restoration technique inastronomy, where the point nature of the objects photographed exposes the PSF thus making it more feasible. It is also used influorescence microscopy for image restoration, and in fluorescencespectral imaging for spectral separation of multiple unknownfluorophores. The most commoniterative algorithm for the purpose is theRichardson–Lucy deconvolution algorithm; theWiener deconvolution (and approximations) are the most common non-iterative algorithms.

For some specific imaging systems such as laser pulsed terahertz systems, PSF can be modeled mathematically.^[6] As a result, as shown in the figure, deconvolution of the modeled PSF and the terahertz image can give a higher resolution representation of the terahertz image.

When performing image synthesis in radiointerferometry, a specific kind ofradio astronomy, one step consists of deconvolving the produced image with the "dirty beam", which is a different name for thepoint spread function. A commonly used method is theCLEAN algorithm.

Typical use of deconvolution is in tracer kinetics. For example, when measuring a hormone concentration in the blood, its secretion rate can be estimated by deconvolution. Another example is the estimation of the blood glucose concentration from the measured interstitial glucose, which is a distorted version in time and amplitude of the real blood glucose.^[7]

Deconvolution has been applied extensively toabsorption spectra.^[8] TheVan Cittert algorithm (article in German) may be used.^[9]

Deconvolution maps to division in theFourier co-domain. This allows deconvolution to be easily applied with experimental data that are subject to aFourier transform. An example isNMR spectroscopy where the data are recorded in the time domain, but analyzed in the frequency domain. Division of the time-domain data by an exponential function has the effect of reducing the width of Lorentzian lines in the frequency domain.

• Convolution
• Bit plane
• Digital filter
• Filter (signal processing)
• Filter design
• Minimum phase
• Independent component analysis
• Wiener deconvolution
• Richardson–Lucy deconvolution
• Digital room correction
• Free deconvolution
• Point spread function
• Deblurring
• Unsharp masking

• Signal processing
• Image processing

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