Fractals/Image noise

Image noise wikipedia: Image noise set of all pixels which are not rendered properly "aberrant pixels. That means pixels that are not representing the colour ... correctly. " Barry J Bradydigital-photography-school : how-to-avoid-and-reduce-noise-in-your-images Noise can be in different situations. Here only noise in the rendered images is incuded. So there is no original imagescikit-image.org docs: denoise, no photo, no signal, ... key words pixel spacing types in mathworks mathworks: imnoise  snibgo's ImageMagick pages: Noise Moire patterns "A uniform grid is known to produce Moire patterns from the interaction of thin near-parallel lines with the regularly spaced sampling points."fractalforums.org : jitter-for-moire-reduction aliasing " 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-Allenmandelbrot-book: noise Commonly Used Image Quality Metrics PSNR (Peak Signal-to-Noise Ratio) SSIM (Structural similarity) NIQE (Naturalness Image Quality Evaluator) How to detect noise "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." Spektrestackoverflow question: javascript-image-noise-detection measure noise "an objective measurement for image quality based on" the structural similarity Image Quality Assessment: From Error Visibility to Structural Similarity by Zhou Wang, at al. 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."stackoverflow question: how-can-i-measure-image-noise " 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 variancestackoverflow question : noise-estimation-noise-measurement-in-image 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 Boiarskyi if (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. 1996 import math, import numpy as np from scipy.signal import convolve2d def estimate_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)) return sigma remove noise "Noise reduction therefore quickly becomes an ‘AI-complete’ problem " Mark Scott Abeln there-is-detail-in-noise by Mark Scott Abeln names: denoisingwenbihan: reproducible-image-denoising-state-of-the-art remove noise noise reduction software open image denoise nvidia optix-denoiser Methods High precision perturbation technique rebasing : rebase to new reference and carry onfractalforums.org: Another solution to perturbation glitches Bivariate Linear Approximation ( BLA) Distance Estimation Interior Detection: Keep track of derivatives at the critical point. When the absolute value of the derivative drops below a threshold such as 0.001 , classify it as interior and stop iterating. precision increasing precision sampling increasing sampling (subpixel accuracy )fractalforums.org : restrict-recomputed-areas-with-iteration-increase jittering the sampling pointsfractalforums.org: jitter-for-moire-reductionfractalforums.org sampling Poisson Disk Samplingpoisson_disk_sampling by Cem Yuksel in matlab 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. center|Image resampling thumb|center|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. thumb|center|When scaling using bilinear interpolation, a color value of the output image is calculated from the four closest color values of the input image. postprocessing with Image Magicfractalforums.org: analytic-logde-post-processed-with-imagemagick mask mask the grainy noise from aliasing ( fading-out) dither Dither is an intentionally applied form of noise used to randomize quantization error, preventing large-scale patterns such as color banding in images. Dither in wikipediaMarek Fiser: Is-16-million-colors-enough 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.fractalforums: dithering-for-better-image-quality It is simply and ease of implementationdither by Øyvind Kolås, pippin@gimp.org , august 2013 BLA 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"fraktaler by Claude Heiland-Allen BLA can be used for acceleration avoiding glitches Variants: steps Single Step BLA Multiple step BLA Non-Conformal BLA ABS Variation BLA Hybrid BLA Multiple Critical Points Examples glitches glitches Incorrect parts of rendersdinkydauset at deviantar :Perturbation-for-the-Mandelbrot-set-450766847 using perturbation technique pixel which dynamics differ significantly from the dynamics of the reference pixelmath.stackexchange question: selecting-reference-orbit-for-fractal-rendering-with-perturbation-theory pixel which is unevaluated ( unknown pixel) or glitched or otherwise bad Types of glitches:fractalforums.org : how-to-get-second-reference-when-using-perturbation-theory noisy = isolated pixels solid = The largest connected component of the set of such pixels See also: Glitches in perturbation method Code Efficient computational noise in GLSL by Ian McEwan, David Sheets, Stefan Gustavson, Mark Richardson aizvorski: video-quality See also image quality jagged and pixelated edges, colloquially known as "jaggies" ( see antialiasing) Jitter is an Jittered grid in supresampling = "random displacements of the center of the pixel withing the square of the actual pixel" ( Gerrit)fractalforums.org: are-the-mandelbrot-sets-generated-different-in-appearance-to-the-actual-set variation of a periodic item Jitter in wikipedia thumb|right|Supersampling - Jittering commons categorise Noise Image noise procedural noise State of the Art in Procedural Noise Functions by A. Lagae, S. Lefebvre, R. Cook, T. DeRose, G. Drettakis, D.S. Ebert, J.P. Lewis K. Perlin7 M. Zwicker8 pseudo-random noise in image graphicmatteo-basei : noise band limiting is a good solution for noisy functions ( Inigo Quilez) References