\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_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_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_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_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_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_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_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_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_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_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_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_p.png & %SCRIPT wait % vim:set tw=77 filetype=tex spell spelllang=en: