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R2653/teaching/COM303/lectures/99 - image processing/5_quantization.tex
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\documentclass[aspectratio=169]{beamer}
\def\stylepath{../../styles}
\usepackage{\stylepath/com303}
\begin{document}
\begin{frame} \frametitle{Quantization}
\begin{itemize}
\item digital devices can only deal with integers ($R$ bits per sample)
\item we need to map the range of a signal onto a finite set of values
\item e.g. pixel sensor level is represented with 8 bits
\item irreversible loss of information $\rightarrow$ quantization noise
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Uniform scalar quantization}
\begin{figure}
\center
\begin{dspBlocks}{1.2}{.1}
$x[n]$~~ & {\BDfilter{$\mathcal{Q}\{\cdot\}$}} & ~~$\hat{x}[n]$ \\
& & {$R$ bps} \\
\ncline{->}{1,1}{1,2}\ncline{->}{1,2}{1,3}
\end{dspBlocks}
\end{figure}
\vspace{1em}
The simplest quantizer:
\begin{itemize}
\item each sample is encoded individually (hence {\em scalar}) using $R$ bits
\item each sample is quantized independently (memoryless quantization)
\item the input range is divided into $2^R$ equal-size intervals
\end{itemize}
\end{frame}
\def\btick#1{\psline[linewidth=0.6pt](#1,-.1)(#1,.1)}
\def\qtick#1#2{\btick{#1}\uput{10pt}[-90](#1,0){$i_{#2}$}}
\def\qdot#1#2{\pscircle*(#1,0){2pt} \uput[90](#1,0){$\hat{x}_{#2}$}}
\def\qbrace#1#2#3#4{%
\psbrace[braceWidth=0.5pt,braceWidthOuter=3pt,rot=90,nodesepB=10pt]%
(! #1 0.1 add -0.3)(! #2 0.1 sub -0.3) {$I_{#3}$}%
\uput[-90](! #1 #2 add 0.5 mul 0.7){$k=#4$}}
\def\dbrace#1#2#3{%
\psbrace[braceWidth=0.5pt,braceWidthOuter=3pt,rot=90,nodesepB=10pt]%
(! #1 0.1 add -0.3)(! #2 0.1 sub -0.3){#3}}
\begin{frame} \frametitle{Uniform scalar quantization}
\begin{itemize}
\item input assumed uniformly distributed over $[A, B]$
\item range is split into $2^R$ \emph{equal} intervals of width $\Delta = (B-A)2^{-R}$
\item quantized value is interval's midpoint
\end{itemize}
\begin{figure}[t]
\center
\psset{xunit=2.4cm,yunit=2cm}
\begin{pspicture}(0.5,-1)(5.5,1)
\psline[linewidth=1pt,tickwidth=2pt,](1,0)(5,0)
\uput{8pt}[90](1,0){$A$} \uput{8pt}[90](5,0){$B$}
\qtick{1}{0} \qtick{2}{1} \qdot{1.5}{0} \qbrace{1}{2}{0}{00}
\qtick{3}{2} \qdot{2.5}{1} \qbrace{2}{3}{1}{01}
\qtick{4}{3} \qdot{3.5}{2} \qbrace{3}{4}{2}{10}
\qtick{5}{4} \qdot{4.5}{3} \qbrace{4}{5}{3}{11}
\end{pspicture}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Uniform 3-Bit quantization function}
\begin{figure}
\center
\psset{xunit=3cm,yunit=3cm}
\begin{pspicture}(-1.2,-1.2)(1.2,1.2)
\tiny
\psaxes[linewidth=1pt,tickwidth=0.5pt,%
Dx=.25,Dy=.25,%
showorigin=false](0,0)(-1.1,-1)(1.1,1)
\uput[90](0,1){$\hat{x}[n]$}
\uput[90](1.1,0){$x[n]$}
\multido{\n=0.875+-.250}{8}{%
\pscircle*[linecolor=orange](0,\n){2pt}
}
\psline[linecolor=blue,linestyle=dotted](-1,-1)(1,1)
\psplot[plotpoints=800,%
linewidth=1pt,linecolor=orange]{-.99}{.99}%
{x abs 0.25 div floor 0.25 mul 0.125 add x abs x div mul}
\uput[90](! -1 0.125 add 0.25 0 mul add dup){\gray 000}
\uput[90](! -1 0.125 add 0.25 1 mul add dup){\gray 001}
\uput[90](! -1 0.125 add 0.25 2 mul add dup){\gray 010}
%\uput[90](! -1 0.125 add 0.25 3 mul add dup){\gray 011}
\uput[90](! -1 0.125 add 0.25 4 mul add dup){\gray 100}
\uput[90](! -1 0.125 add 0.25 5 mul add dup){\gray 101}
\uput[90](! -1 0.125 add 0.25 6 mul add dup){\gray 110}
\uput[90](! -1 0.125 add 0.25 7 mul add dup){\gray 111}
\end{pspicture}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Quantization Error}
\[
e[n] = \mathcal{Q}\{ x[n] \} - x[n] = \hat{x}[n] - x[n]
\]
\vspace{1em}
\begin{itemize}
\item model $x[n]$ as a stochastic process
\item model error as a white noise sequence:
\begin{itemize}
\item error samples are uncorrelated
\item all error samples have the same distribution
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Rate/Distortion Curve }
\begin{figure}
\begin{dspPlot}[width=5cm,height=5cm,sidegap=0,xticks=none,yticks=none,xlabel={rate ($R$)},ylabel={distortion ($\sigma_e^2$)}]{0,6}{0,1}
\moocStyle
\dspFunc{x -2 mul 2 exch exp }
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Dynamic range after quantization}
Convenient rule of thumb:
\vspace{1em}
\begin{itemize}
\item signal to noise ratio:
\[
\mbox{SNR} = 2^{2R}
\]
\item in dBs:
\[
\mbox{SNR}_{\mbox{dB}} = 10\log_{10} 2^{2R} \approx 6R \,\,\mbox{dB}
\]
\end{itemize}
\end{frame}
\begin{frame} \frametitle{The ``6dB/bit'' rule of thumb}
\begin{itemize}[<+->]
\item a compact disk has 16 bits/sample:
\[
\mbox{max SNR} = 96\mbox{dB}
\]
\item a DVD has 24 bits/sample:
\[
\mbox{max SNR} = 144\mbox{dB}
\]
\end{itemize}
\end{frame}
\end{document}
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