" Our images look noisy and grainy near the boundary of the Mandelbrot set. The escape time bands get closer and closer, while the pixel spacing is fixed. The pixel grid samples isolated points of a mathematically abstract image defined on the continuous plane. The Nyquist-Shannon sampling theorem shows that sampling isolated points from a continuum is a valid approximation only so long as the values don’t change too quickly between the points. Aliasing occurs when the values do change too quickly compared to the sampling rate, with the grainy noisy visual effects as we have been. Because the escape time bands increase in number without bound as we approach the boundary of the Mandelbrot set, no sampling rate can be high enough." Claude Heiland-Allen[6]
binary decomposition of level sets of exterior of Mandelbrot set
"To detect noise you first need to know the properties of your useful data. So if you have no prior knowledge of the input images then you Can not reliably detect noise." Spektre[7]
"an objective measurement for image quality based on" the structural similarity[8]
noise rate = number of pixels that were recognized as noise. "To differentiate normal pixels from noise, I just calculated the medium value of its neighbor pixels and if its value was bigger than some critical value, we say that this one is noise."[9]
" Instead of classifying a pixel as noise if it exceeds such a threshold, you could measure the "error" and compute the variance or standard deviation for all of the pixels in the image. This would help you distinguish between having n pixels just above the threshold and n pixels way out whack. It also avoids the need to select a threshold." Adrian McCarthy
estimate the noise variance[10] If you get sigma > 10.0, then you have a noisy image ( only for grayscale)
// https://stackoverflow.com/questions/8960462/how-can-i-measure-image-noise// noise rate by Vitalii Boiarskyiif(ABS(1-(currentPixel.R+currentPixel.G+currentPixel.B)/(neigborsMediumValues.R+neigboursMediumValues.G+neigboursMediumValues.B)))>criticalValue)then{currentPixelIsNoise=TRUE;}
# https://stackoverflow.com/questions/2440504/noise-estimation-noise-measurement-in-image# J. Immerkær, “Fast Noise Variance Estimation”, Computer Vision and Image Understanding, Vol. 64, No. 2, pp. 300-302, Sep. 1996importmath,importnumpyasnpfromscipy.signalimportconvolve2ddefestimate_noise(I):H,W=I.shape# Tuple of image array dimensions.M=[[1,-2,1],[-2,4,-2],[1,-2,1]]sigma=np.sum(np.sum(np.absolute(convolve2d(I,M))))sigma=sigma*math.sqrt(0.5*math.pi)/(6*(W-2)*(H-2))returnsigma
Enlargement of a 6×6 pixel raster graphic to 11×11 pixels (the low image resolutions in this example were chosen for better clarity; the principle is the same for higher resolutions).
1: input image; the pixels are shown here as circles.
2: The pixel grid of the output image, shown here as yellow crosses, is superimposed on the input image.
3: The color values of the output image are calculated from the nearby pixels of the input image.
4: Output image.
Image resampling
Image resampling reconstruction filter. Scaling using a radially symmetric reconstruction filter (carrier shown in green). A color value of the output image is calculated as a sum of color values of the input image weighted by the reconstruction filter.When scaling using bilinear interpolation, a color value of the output image is calculated from the four closest color values of the input image.
Dither is an intentionally applied form of noise used to randomize quantization error, preventing large-scale patterns such as color banding in images.[19][20]
dither to more accurately display graphics containing a greater range of colors than the display hardware is capable of showing:
The limited precision of 8bit (s)RGB means visible banding can occur, particularly in darker areas with smooth gradients. Adding dither trades off spatial resolution vs colour resolution, spreading the quantisation errors around so the end result is more perceptually uniform.[21]
The iterations can be approximated by bivariate linear function( A linear function with with two variables is called a bivariate linear function):
For Mandelbrot set
For Mandelbrot set this is valid when: "non-linear part of the full perturbation iterations is so small that omitting it would cause fewer problems than the rounding error of the low precision data type"[23]
