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gabarro.github.io/ccn/colors_of_noise.tex
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\title{Colors of image noise} | |
\section{White Gaussian noise} | |
Everybody knows about white Gaussian noise | |
\includegraphics{white.png} | |
%SCRIPT plambda zero:512x512 randg | qauto -p 0.1 - white.png & | |
White Gaussian noise is famous because it has very nice properties: | |
\begin{enumerate} | |
\item It is easy to generate using pseudorandom numbers | |
\item Each pixel is an independent, identically distributed Normal | |
variable | |
\item The discrete Fourier, Hartley and Cosine transforms are also | |
white Gaussian noise (except for the obvious symmetries) | |
\item In particular, the power spectrum is mostly flat | |
\item Applying a linear filter renders the pixel values | |
non-independent, but they are still Normal and identically | |
distributed. | |
\end{enumerate} | |
Some properties of dubious convenience: | |
\begin{enumerate} | |
\item The mean is zero, thus it cannot be directly represented as | |
a positive-valued image | |
\item Worse, the pixel values are not bounded, thus it has a-priori | |
infinite dynamic range. | |
\item When you see it from far away (zooming-out), it disappears. | |
\end{enumerate} | |
Statistics of white gaussian noise and its DFT: | |
\begin{tabular}{cccc} | |
\includegraphics{w256.png} & | |
\includegraphics{w256_h.png} & | |
\includegraphics{w256_f.png} & | |
\includegraphics{w256_p.png} \\ | |
$u(x)$ & | |
histogram of~$u$ & | |
$\log|\hat u(\xi)|$ & | |
average spectral profile | |
\end{tabular} | |
%SCRIPT plambda zero:256x256 randg -o w256.tiff | |
%SCRIPT qauto -p 0.1 w256.tiff w256.png & | |
%SCRIPT plambda w256.tiff 'x%a x%i - 300 / >1 x <1 / round <1 *' | ghisto -p | \ | |
%SCRIPT sed 's/cairo/cairo enhanced font "Verdana,8" size 256,256/' |\ | |
%SCRIPT gnuplot > w256_h.png & | |
%SCRIPT fft<w256.tiff|fftshift|plambda 'vnorm 1 + log'|qauto -p .1 - w256_f.png& | |
%SCRIPT (printf 'set term pngcairo size 256,256\nset format y ""\nset xlabel "±|ξ|"\nset ylabel "avg |û(ξ)|"\n'; fft <w256.tiff|fftshift|plambda 'vnorm'|cline nan|sed 's/data//g')|gnuplot > w256_p.png & | |
\section{Blurred white Gaussian noise} | |
White gaussian noise blurred by a small gaussian kernel: | |
\begin{tabular}{cccc} | |
\includegraphics{b256.png} & | |
\includegraphics{b256_h.png} & | |
\includegraphics{b256_f.png} & | |
\includegraphics{b256_p.png} \\ | |
$u(x)$ & | |
histogram of~$u$ & | |
$\log|\hat u(\xi)|$ & | |
average spectral profile | |
\end{tabular} | |
%SCRIPT plambda zero:256x256 randg |blur g 1 - b256.tiff | |
%SCRIPT qauto -p 0.1 b256.tiff b256.png & | |
%SCRIPT plambda b256.tiff 'x%a x%i - 300 / >1 x <1 / round <1 *' | ghisto -p | \ | |
%SCRIPT sed 's/cairo/cairo enhanced font "Verdana,8" size 256,256/' | \ | |
%SCRIPT gnuplot > b256_h.png & | |
%SCRIPT fft<b256.tiff|fftshift|plambda 'vnorm 1 + log'|qauto -p .1 - b256_f.png& | |
%SCRIPT (printf 'set term pngcairo size 256,256\nset format y ""\nset xlabel "±|ξ|"\nset ylabel "avg |û(ξ)|"\n'; fft <b256.tiff|fftshift|plambda 'vnorm'|cline nan|sed 's/data//g')|gnuplot > b256_p.png & | |
White gaussian noise blurred by a larger gaussian kernel: | |
\begin{tabular}{cccc} | |
\includegraphics{B256.png} & | |
\includegraphics{B256_h.png} & | |
\includegraphics{B256_f.png} & | |
\includegraphics{B256_p.png} \\ | |
$u(x)$ & | |
histogram of~$u$ & | |
$\log|\hat u(\xi)|$ & | |
average spectral profile | |
\end{tabular} | |
%SCRIPT plambda zero:256x256 randg |blur g 5 - B256.tiff | |
%SCRIPT qauto -p 0.1 B256.tiff B256.png & | |
%SCRIPT plambda B256.tiff 'x%a x%i - 300 / >1 x <1 / round <1 *' | ghisto -p | \ | |
%SCRIPT sed 's/cairo/cairo enhanced font "Verdana,8" size 256,256/' | \ | |
%SCRIPT gnuplot > B256_h.png & | |
%SCRIPT fft<B256.tiff|fftshift|plambda 'vnorm 1 + log'|qauto -p .1 - B256_f.png& | |
%SCRIPT (printf 'set term pngcairo size 256,256\nset format y ""\nset xlabel "±|ξ|"\nset ylabel "avg |û(ξ)|"\n'; fft <B256.tiff|fftshift|plambda 'vnorm'|cline nan|sed 's/data//g')|gnuplot > B256_p.png & | |
White gaussian noise blurred by a Cauchy kernel: | |
\begin{tabular}{cccc} | |
\includegraphics{c256.png} & | |
\includegraphics{c256_h.png} & | |
\includegraphics{c256_f.png} & | |
\includegraphics{c256_p.png} \\ | |
$u(x)$ & | |
histogram of~$u$ & | |
$\log|\hat u(\xi)|$ & | |
average spectral profile | |
\end{tabular} | |
%SCRIPT plambda zero:256x256 randg |blur c 1 - c256.tiff | |
%SCRIPT qauto -p 0.1 c256.tiff c256.png & | |
%SCRIPT plambda c256.tiff 'x%a x%i - 300 / >1 x <1 / round <1 *' | ghisto -p | \ | |
%SCRIPT sed 's/cairo/cairo enhanced font "Verdana,8" size 256,256/' |\ | |
%SCRIPT gnuplot > c256_h.png & | |
%SCRIPT fft<c256.tiff|fftshift|plambda 'vnorm 1 + log'|qauto -p .1 - c256_f.png& | |
%SCRIPT (printf 'set term pngcairo size 256,256\nset format y ""\nset xlabel "±|ξ|"\nset ylabel "avg |û(ξ)|"\n'; fft <c256.tiff|fftshift|plambda 'vnorm'|cline nan|sed 's/data//g')|gnuplot > c256_p.png & | |
White gaussian noise blurred by a Laplace kernel: | |
\begin{tabular}{cccc} | |
\includegraphics{l256.png} & | |
\includegraphics{l256_h.png} & | |
\includegraphics{l256_f.png} & | |
\includegraphics{l256_p.png} \\ | |
$u(x)$ & | |
histogram of~$u$ & | |
$\log|\hat u(\xi)|$ & | |
average spectral profile | |
\end{tabular} | |
%SCRIPT plambda zero:256x256 randg |blur l 1 - l256.tiff | |
%SCRIPT qauto -p 0.1 l256.tiff l256.png & | |
%SCRIPT plambda l256.tiff 'x%a x%i - 300 / >1 x <1 / round <1 *' | ghisto -p | \ | |
%SCRIPT sed 's/cairo/cairo enhanced font "Verdana,8" size 256,256/' |\ | |
%SCRIPT gnuplot > l256_h.png & | |
%SCRIPT fft<l256.tiff|fftshift|plambda 'vnorm 1 + log'|qauto -p .1 - l256_f.png& | |
%SCRIPT (printf 'set term pngcairo size 256,256\nset format y ""\nset xlabel "±|ξ|"\nset ylabel "avg |û(ξ)|"\n'; fft <l256.tiff|fftshift|plambda 'vnorm'|cline nan|sed 's/data//g')|gnuplot > l256_p.png & | |
White gaussian noise blurred by a Disk kernel: | |
\begin{tabular}{cccc} | |
\includegraphics{d256.png} & | |
\includegraphics{d256_h.png} & | |
\includegraphics{d256_f.png} & | |
\includegraphics{d256_p.png} \\ | |
$u(x)$ & | |
histogram of~$u$ & | |
$\log|\hat u(\xi)|$ & | |
average spectral profile | |
\end{tabular} | |
%SCRIPT plambda zero:256x256 randg |blur disk 4.5 - d256.tiff | |
%SCRIPT qauto -p 0.1 d256.tiff d256.png & | |
%SCRIPT plambda d256.tiff 'x%a x%i - 300 / >1 x <1 / round <1 *' | ghisto -p | \ | |
%SCRIPT sed 's/cairo/cairo enhanced font "Verdana,8" size 256,256/' |\ | |
%SCRIPT gnuplot > d256_h.png & | |
%SCRIPT fft<d256.tiff|fftshift|plambda 'vnorm 1 + log'|qauto -p .1 - d256_f.png& | |
%SCRIPT (printf 'set term pngcairo size 256,256\nset format y ""\nset xlabel "±|ξ|"\nset ylabel "avg |û(ξ)|"\n'; fft <d256.tiff|fftshift|plambda 'vnorm'|cline nan|sed 's/data//g')|gnuplot > d256_p.png & | |
White gaussian noise blurred by a Square kernel: | |
\begin{tabular}{cccc} | |
\includegraphics{s256.png} & | |
\includegraphics{s256_h.png} & | |
\includegraphics{s256_f.png} & | |
\includegraphics{s256_p.png} \\ | |
$u(x)$ & | |
histogram of~$u$ & | |
$\log|\hat u(\xi)|$ & | |
average spectral profile | |
\end{tabular} | |
%SCRIPT plambda zero:256x256 randg |blur square 14 - s256.tiff | |
%SCRIPT qauto -p 0.1 s256.tiff s256.png | |
%SCRIPT plambda s256.tiff 'x%a x%i - 300 / >1 x <1 / round <1 *' | ghisto -p | \ | |
%SCRIPT sed 's/cairo/cairo enhanced font "Verdana,8" size 256,256/' |\ | |
%SCRIPT gnuplot > s256_h.png & | |
%SCRIPT fft<s256.tiff|fftshift|plambda 'vnorm 1 + log'|qauto -p .1 - s256_f.png& | |
%SCRIPT (printf 'set term pngcairo size 256,256\nset format y ""\nset xlabel "±|ξ|"\nset ylabel "avg |û(ξ)|"\n'; fft <s256.tiff|fftshift|plambda 'vnorm'|cline nan|sed 's/data//g')|gnuplot > s256_p.png & | |
\section{Colored gaussian noise} | |
When the spectrum of noise decays as a power-law, we say that it is | |
``colored'' noise. The exponent~$\alpha$ of the power law determines its | |
color. The particular case of~$\alpha=0$ corresponds to white noise (a flat | |
spectrum). | |
\begin{tabular}{lll} | |
\includegraphics{purple.png} & | |
\includegraphics{blue.png} & | |
\includegraphics{white2.png} \\ | |
$\alpha=2$ purple & | |
$\alpha=1$ blue & | |
$\alpha=0$ white \\ | |
$\phantom{a}$ &&\\ | |
\includegraphics{pink.png} & | |
\includegraphics{brown.png} & | |
\includegraphics{smooth.png} \\ | |
$\alpha=-1$ pink & | |
$\alpha=-2$ brown & | |
$\alpha=-3$ smooth \\ | |
\end{tabular} | |
%SCRIPT plambda zero:256x256 randg -o white.tiff | |
%SCRIPT cat white.tiff|qauto -p 0.1 - white2.png & | |
%SCRIPT cat white.tiff|fft|plambda ':R 2 ^ *'|ifft|qauto -p 0.1 - purple.png & | |
%SCRIPT cat white.tiff|fft|plambda ':R 1 ^ *'|ifft|qauto -p 0.1 - blue.png & | |
%SCRIPT cat white.tiff|fft|plambda ':R -1 ^ *'|ifft|qauto -p 0.1 - pink.png & | |
%SCRIPT cat white.tiff|fft|plambda ':R -2 ^ *'|ifft|qauto -p 0.1 - brown.png & | |
%SCRIPT cat white.tiff|fft|plambda ':R -3 ^ *'|ifft|qauto -p 0.1 - smooth.png & | |
Statistics of Pink noise ($\alpha=-1$): | |
\begin{tabular}{cccc} | |
\includegraphics{KK-1.png} & | |
\includegraphics{KK-1_h.png} & | |
\includegraphics{KK-1_f.png} & | |
\includegraphics{KK-1_p.png} \\ | |
$u(x)$ & | |
histogram of~$u$ & | |
$\log|\hat u(\xi)|$ & | |
average spectral profile | |
\end{tabular} | |
%SCRIPT plambda zero:256x256 randg|fft|plambda ':R -1 ^ *'|ifft >KK-1.tiff | |
%SCRIPT qauto -p 0.1 KK-1.tiff KK-1.png & | |
%SCRIPT plambda KK-1.tiff 'x%a x%i - 300 / >1 x <1 / round <1 *' | ghisto -p | \ | |
%SCRIPT sed 's/cairo/cairo enhanced font "Verdana,8" size 256,256/' |\ | |
%SCRIPT gnuplot > KK-1_h.png & | |
%SCRIPT fft<KK-1.tiff|fftshift|plambda 'vnorm 1 + log'|qauto -p .1 - KK-1_f.png& | |
%SCRIPT (printf 'set term pngcairo size 256,256\nset format y ""\nset xlabel "±|ξ|"\nset ylabel "log avg |û(ξ)|"\nset logscale y\n'; fft <KK-1.tiff|fftshift|plambda 'vnorm'|cline nan|sed 's/data//g')|gnuplot > KK-1_p.png & | |
Statistics of Brown noise ($\alpha=-2$): | |
\begin{tabular}{cccc} | |
\includegraphics{KK-2.png} & | |
\includegraphics{KK-2_h.png} & | |
\includegraphics{KK-2_f.png} & | |
\includegraphics{KK-2_p.png} \\ | |
$u(x)$ & | |
histogram of~$u$ & | |
$\log|\hat u(\xi)|$ & | |
average spectral profile | |
\end{tabular} | |
%SCRIPT plambda zero:256x256 randg|fft|plambda ':R -2 ^ *'|ifft >KK-2.tiff | |
%SCRIPT qauto -p 0.1 KK-2.tiff KK-2.png & | |
%SCRIPT plambda KK-2.tiff 'x%a x%i - 300 / >1 x <1 / round <1 *' | ghisto -p | \ | |
%SCRIPT sed 's/cairo/cairo enhanced font "Verdana,8" size 256,256/' | \ | |
%SCRIPT gnuplot > KK-2_h.png & | |
%SCRIPT fft<KK-2.tiff|fftshift|plambda 'vnorm 1 + log'|qauto -p .1 - KK-2_f.png& | |
%SCRIPT (printf 'set term pngcairo size 256,256\nset format y ""\nset xlabel "±|ξ|"\nset ylabel "avg |û(ξ)|"\nset logscale y\n'; fft <KK-2.tiff|fftshift|plambda 'vnorm'|cline nan|sed 's/data//g')|gnuplot > KK-2_p.png & | |
Statistics of Smooth noise ($\alpha=-3$): | |
\begin{tabular}{cccc} | |
\includegraphics{KK-3.png} & | |
\includegraphics{KK-3_h.png} & | |
\includegraphics{KK-3_f.png} & | |
\includegraphics{KK-3_p.png} \\ | |
$u(x)$ & | |
histogram of~$u$ & | |
$\log|\hat u(\xi)|$ & | |
average spectral profile | |
\end{tabular} | |
%SCRIPT plambda zero:256x256 randg|fft|plambda ':R -3 ^ *'|ifft >KK-3.tiff | |
%SCRIPT qauto -p 0.1 KK-3.tiff KK-3.png & | |
%SCRIPT plambda KK-3.tiff 'x%a x%i - 300 / >1 x <1 / round <1 *' | ghisto -p | \ | |
%SCRIPT sed 's/cairo/cairo enhanced font "Verdana,8" size 256,256/' | \ | |
%SCRIPT gnuplot > KK-3_h.png & | |
%SCRIPT fft<KK-3.tiff|fftshift|plambda 'vnorm 1 + log'|qauto -p .1 - KK-3_f.png& | |
%SCRIPT (printf 'set term pngcairo size 256,256\nset format y ""\nset xlabel "±|ξ|"\nset ylabel "avg |û(ξ)|"\nset logscale y\n'; fft <KK-3.tiff|fftshift|plambda 'vnorm'|cline nan|sed 's/data//g')|gnuplot > KK-3_p.png & | |
Statistics of Blue noise ($\alpha=1$): | |
\begin{tabular}{cccc} | |
\includegraphics{KK+1.png} & | |
\includegraphics{KK+1_h.png} & | |
\includegraphics{KK+1_f.png} & | |
\includegraphics{KK+1_p.png} \\ | |
$u(x)$ & | |
histogram of~$u$ & | |
$\log|\hat u(\xi)|$ & | |
average spectral profile | |
\end{tabular} | |
%SCRIPT plambda zero:256x256 randg|fft|plambda ':R 1 ^ *'|ifft >KK+1.tiff | |
%SCRIPT qauto -p 0.1 KK+1.tiff KK+1.png & | |
%SCRIPT plambda KK+1.tiff 'x%a x%i - 300 / >1 x <1 / round <1 *' | ghisto -p | \ | |
%SCRIPT sed 's/cairo/cairo enhanced font "Verdana,8" size 256,256/' | \ | |
%SCRIPT gnuplot > KK+1_h.png & | |
%SCRIPT fft<KK+1.tiff|fftshift|plambda 'vnorm 1 + log'|qauto -p .1 - KK+1_f.png& | |
%SCRIPT (printf 'set term pngcairo size 256,256\nset format y ""\nset xlabel "±|ξ|"\nset ylabel "avg |û(ξ)|"\nset logscale y\n'; fft <KK+1.tiff|fftshift|plambda 'vnorm'|cline nan|sed 's/data//g')|gnuplot > KK+1_p.png & | |
Statistics of Purple noise ($\alpha=2$): | |
\begin{tabular}{cccc} | |
\includegraphics{KK+2.png} & | |
\includegraphics{KK+2_h.png} & | |
\includegraphics{KK+2_f.png} & | |
\includegraphics{KK+2_p.png} \\ | |
$u(x)$ & | |
histogram of~$u$ & | |
$\log|\hat u(\xi)|$ & | |
average spectral profile | |
\end{tabular} | |
%SCRIPT plambda zero:256x256 randg|fft|plambda ':R 2 ^ *'|ifft >KK+2.tiff | |
%SCRIPT qauto -p 0.1 KK+2.tiff KK+2.png & | |
%SCRIPT plambda KK+2.tiff 'x%a x%i - 300 / >1 x <1 / round <1 *' | ghisto -p | \ | |
%SCRIPT sed 's/cairo/cairo enhanced font "Verdana,8" size 256,256/' | \ | |
%SCRIPT gnuplot > KK+2_h.png & | |
%SCRIPT fft<KK+2.tiff|fftshift|plambda 'vnorm 1 + log'|qauto -p .1 - KK+2_f.png& | |
%SCRIPT (printf 'set term pngcairo size 256,256\nset format y ""\nset xlabel "±|ξ|"\nset ylabel "avg |û(ξ)|"\nset logscale y\n'; fft <KK+2.tiff|fftshift|plambda 'vnorm'|cline nan|sed 's/data//g')|gnuplot > KK+2_p.png & | |
%SCRIPT wait | |
% vim:set tw=77 filetype=tex spell spelllang=en: |