# Fractals/Image noise

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 Brady
Noise can be in different situations. Here only noise in the rendered images is incuded. So there is no original image, no photo, no signal, ...

# key words

* pixel spacing

# types

* in mathworks
* ﻿ 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."

## aliasing

* binary decomposition of level sets of exterior of Mandelbrot set
* Mandelbrot set using level set method

# Commonly Used Image Quality Metrics

* PSNR (Peak Signal-to-Noise Ratio)
* SSIM (Structural similarity)
* NIQE (Naturalness Image Quality Evaluator)

# How to
## measure noise

* "an objective measurement for image quality based on" the structural similarity
* 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."
* " 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 If you get sigma > 10.0, then you have a noisy image ( only for grayscale)

## remove noise

names:
* denoising
* remove noise
* noise reduction
software
* open image denoise
* nvidia optix-denoiser
Methods
* High precision
* perturbation technique
* rebasing : rebase to new reference and carry on
* 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 )
* jittering the sampling points
* Poisson Disk Sampling
* 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.

### postprocessing

* with Image Magic

### 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 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.
It is simply and ease of implementation

### 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"
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 renders using perturbation techique
* pixel which dynamics differ significantly from the dynamics of the reference pixel
* pixel which is unevaluated ( unknown pixel) or glitched or otherwise bad
Types of glitches:
* 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)
** variation of a periodic item
* 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 graphic
* band limiting is a good solution for noisy functions ( Inigo Quilez)

# References

* ↑ wikipedia: Image noise
* ↑ digital-photography-school : how-to-avoid-and-reduce-noise-in-your-images
* ↑ scikit-image.org docs: denoise
* ↑ mathworks: imnoise
* ↑ fractalforums.org : jitter-for-moire-reduction
* ↑ mandelbrot-book: noise
* ↑ stackoverflow question: javascript-image-noise-detection
* ↑ Image Quality Assessment: From Error Visibility to Structural Similarity by Zhou Wang, at al.
* ↑ stackoverflow question: how-can-i-measure-image-noise
* ↑ stackoverflow question : noise-estimation-noise-measurement-in-image
* ↑ there-is-detail-in-noise by Mark Scott Abeln
* ↑ wenbihan: reproducible-image-denoising-state-of-the-art
* ↑ fractalforums.org: Another solution to perturbation glitches
* ↑ fractalforums.org : restrict-recomputed-areas-with-iteration-increase
* ↑ fractalforums.org: jitter-for-moire-reduction
* ↑ fractalforums.org sampling
* ↑ poisson_disk_sampling by Cem Yuksel
* ↑ fractalforums.org: analytic-logde-post-processed-with-imagemagick
* ↑ Dither in wikipedia
* ↑ Marek Fiser: Is-16-million-colors-enough
* ↑ fractalforums: dithering-for-better-image-quality
* ↑ dither by Øyvind Kolås, pippin@gimp.org , august 2013
* ↑ fraktaler by Claude Heiland-Allen
* ↑ dinkydauset at deviantar :Perturbation-for-the-Mandelbrot-set-450766847
* ↑ math.stackexchange question: selecting-reference-orbit-for-fractal-rendering-with-perturbation-theory
* ↑ fractalforums.org : how-to-get-second-reference-when-using-perturbation-theory
* ↑ fractalforums.org: are-the-mandelbrot-sets-generated-different-in-appearance-to-the-actual-set
* ↑ Jitter in wikipedia
* ↑ matteo-basei : noise