More on quantization.
authorviric@llimona
Thu, 25 Oct 2007 12:45:59 +0200
changeset 204 3245ee20e27a
parent 203 ca3267f08a9d
child 205 819afafb5de7
More on quantization.
doc/Makefile
doc/cited.bib
doc/house-bleeding.eps
doc/house-bleeding.svg
doc/images/Dctjpeg-wikipedia.png
doc/images/JPEG_ZigZag-wikipedia.svg
doc/images/house-bleeding-color40-111.ppm
doc/images/house-bleeding-color40.ppm
doc/images/house-bleeding-original.ppm
doc/images/lenacutre40g.jpg
doc/images/lenag-40-effects.pgm
doc/images/rapids-nens-b.pgm
doc/images/rapids-nens-cb-211-rescaled.pgm
doc/images/rapids-nens-cb-211.pgm
doc/images/rapids-nens-cb.pgm
doc/images/rapids-nens-cr.pgm
doc/images/rapids-nens-g.pgm
doc/images/rapids-nens-r.pgm
doc/images/rapids-nens-y.pgm
doc/images/rapids-nens.pnm
doc/lena-blocking.eps
doc/lena-blocking.svg
doc/memoria.tex
--- a/doc/Makefile	Wed Oct 24 21:47:36 2007 +0200
+++ b/doc/Makefile	Thu Oct 25 12:45:59 2007 +0200
@@ -2,7 +2,9 @@
 INCLUDES=cited.bib block16x16simple.pdf \
 	jpeg-coeficients.pdf \
 	rapids-nens-cb-rescale.pdf \
-	rapids-nens-rgb2ycc.pdf
+	rapids-nens-rgb2ycc.pdf \
+	house-bleeding.pdf \
+	lena-blocking.pdf
 
 memoria.pdf: memoria.tex $(INCLUDES)
 	pdflatex $<
--- a/doc/cited.bib	Wed Oct 24 21:47:36 2007 +0200
+++ b/doc/cited.bib	Thu Oct 25 12:45:59 2007 +0200
@@ -17,3 +17,11 @@
     howpublished = "Software"
 }
 
+@article{robertson2004,
+    author = "Mark A. Robertson and Robert L. Stevenson",
+    title = "DCT Quantization Noise in Compressed Images",
+    year = "2004",
+    month = "February",
+    journal = "Draft"
+}
+
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
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--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc/images/rapids-nens-cb.pgm	Thu Oct 25 12:45:59 2007 +0200
@@ -0,0 +1,5 @@
+P5
+# CREATOR: The GIMP's PNM Filter Version 1.0
+128 128
+255
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+++ b/doc/images/rapids-nens-y.pgm	Thu Oct 25 12:45:59 2007 +0200
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+{currentfile /ASCII85Decode filter /RunLengthDecode filter gstr readstring pop}
+{currentfile /ASCII85Decode filter /RunLengthDecode filter bstr readstring pop}
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--- a/doc/memoria.tex	Wed Oct 24 21:47:36 2007 +0200
+++ b/doc/memoria.tex	Thu Oct 25 12:45:59 2007 +0200
@@ -6,7 +6,7 @@
 \usepackage{units}
 \usepackage{amsmath}
 \usepackage{subfigure}
-\usepackage{sepnum}
+\usepackage{icomma}
 \usepackage[margin=1cm,font=footnotesize]{caption}
 %\usepackage{a4wide}
 %\usepackage{natbib}
@@ -53,14 +53,15 @@
 sistemes MS-DOS, "Fitxers JPG".
 
 Dins l'estàndard s'hi inclouen dues formes diferents de codificar la imatge: amb
-pèrdues ({\em lossy}) o sense ({\em lossless}). Aquesta última no ha esdevingut
+pèrdues (\emph{lossy}) o sense (\emph{lossless}). Aquesta última no ha
+esdevingut
 popular, i segurament qualsevol imatge JPEG que trobem haurà estat codificada
 amb pèrdues. D'ara en endavant parlarem exclusivament de la compressió amb
 pèrdues.
 
 La imatge original, prèvia a codificar-la, la trobem definida amb un mapa de
 bits. Això significa que tenim la imatge dividida en espai, amb una malla
-quadriculada de punts de color. Cada punt (pícsel, píxel, pel, de {\em picture
+quadriculada de punts de color. Cada punt (pícsel, píxel, pel, de \emph{picture
 element}) ens defineix el color en aquell lloc de la quadrícula. Si tenim una
 quadricula de punts suficientment densa, el nostre ull no la distingirà,
 i el conjunt li semblarà una imatge de color contínua o de to continu. Sobretot
@@ -231,21 +232,39 @@
 suplir la imatge final a la pantalla, cal invertir el canvi de plans de YCbCr
 a RGB.
 
+El delmat dels plans de color provoca una incomoditat visual sovint detectable a
+les imatges JPEG decodificades: el \emph{vessament de color} (\emph{color
+bleeding}). A la Figura \ref{fig:house-bleeding} podem veure'l.
+
+\figura{house-bleeding.pdf}{
+\label{fig:house-bleeding} A l'esquerra, l'original. Al mig, JPEG amb màxima
+qualitat al pla de lluminositat, i qualitat 40 de \texttt{libjpeg} als plans de
+color, amb delmat 2x2:1x1:1x1 (als dos plans de color la meitat de punts en
+horitzontal i vertical que al pla de lluminositat). Es pot apreciar el vessament
+de color sobre les finestres. A la dreta, JPEG amb la mateixa codificació per
+plans, però sense delmar els plans de color. Ja no s'hi aprecia tant el
+vessament.}
+
 \subsubsection{Quantització dels coeficients}
 
 Dels 64 punts d'intensitat de color que determinen cada bloc de 8x8 pícsels
 obtenim 64 coeficients. Aquests es solen ordenar en
 freqüències de menor a major, utilitzant l'ordre de Zig Zag
-(\ref{fig:jpeg-coeficients}) definit a l'estàndard de JPEG.
+(Figura \ref{fig:jpeg-coeficients}) definit a l'estàndard de JPEG.
 
 Les taules de quantització determinen la quantitat d'informació que es llença
 per cada un dels 64 coeficients. Es sol utilitzar una taula de quantització
 diferent per cada pla del JPEG. S'han dut a terme nombrosos experiments per a
-decidir quins coeficients són més importants, per establir un grau d'importància
-entre ells. I alhora, s'han establert diferents taules de quantització relatives
-a la densitat d'informació que desitja l'usuari. En els taules de quantització
-sí que difereixen una mica les càmeres digitals, els programes de manipulació
-fotogràfica professionals, i la \texttt{libjpeg} \cite{libjpeg}.
+decidir quins coeficients són més importants per a degradar mínimament la
+imatge, establint un grau d'importància entre ells.
+En la majoria d'aplicacions que codifiquen imatges en JPEG
+l'usuari té l'opció d'escollir la densitat de la informació codificada.
+No tots els codificadors utilitzen els mateixos patrons de taules de
+quantització. Les càmeres digitals tenen limitacions als algorismes hardware,
+els programes de manipulació fotogràfica professionals poden tenir els seus
+propis estudis de les taules de coeficients, i la \texttt{libjpeg}
+\cite{libjpeg} simplement segueix les taules recomanades a
+l'estàndard (\cite{jpegspec}).
 
 \figuraw{10cm}{jpeg-coeficients.pdf}{
 \label{fig:jpeg-coeficients} A l'esquerra, funció base de cada
@@ -274,17 +293,18 @@
 \end{table}
 
 La quantització dels coeficients consisteix en conservar només el valor enter
-resultat d'arrodonir la divisió pel quantitzador:
+resultat d'arrodonir la divisió pel quantitzador. Per cada un dels 64
+coeficients reals provinents de la transformada DCT 2D obtenim un enter:
 \[ \hat y_i = \left[ \frac{y_i}{Q_i}\right] , \, i \in 1..64 \] 
 
-La recuperació convencional del coeficient consisteix en multiplicar pel
+La recuperació convencional del coeficient, especificada a l'estàndard
+\cite{jpegspec}, consisteix en multiplicar pel
 quantitzador, $ z_i = \hat y_i Q_i $. Podem veure un exemple dels coeficients
-d'un bloc a la figura \ref{fig:quant-intervals}. En realitat, a la
-descodificació,
-del coeficient original només sabem que es troba dins
+d'un bloc a la Figura \ref{fig:quant-intervals}.De tota manera, a la
+descodificació, del coeficient original només sabem que es troba dins
 l'\emph{interval de quantització}:
-\[ y_i \in Q_i \left[ \hat y_i - \decimal{0.5} \, , \,
-\hat y_i + \decimal{0.5} \right] \]
+\[ y_i \in Q_i \left[ \hat y_i - 0,5 \, , \,
+\hat y_i + 0,5 \right] \]
 % TODO comma numbers
 
 \figura{block16x16simple.pdf}{
@@ -296,6 +316,18 @@
 marca l'ordre de Zig Zag de JPEG. Es poden apreciar els intervals de
 quantització, i la pèrdua d'informació sobretot a les freqüències més altes.}
 
+A la  pèrdua d'informació deguda a la quantització se l'anomena
+\emph{soroll de quantització}. Té diverses conseqüències visibles a la imatge
+descodificada de manera convencional, i en podem veure algunes classificades a
+la Figura \ref{fig:quant-problems}.
+
+\figura{lena-blocking.pdf}{
+\label{fig:quant-problems} Pla de lluminositat de la Lena de 256x256 píxels, a
+qualitat 40 segons la \texttt{libjpeg}. Podem veure els efectes a) d'ones
+(\emph{ringing}) als blocs que tenen un canvi brusc d'intensitat pel mig, b) de
+cantonades (\emph{corner}) i c) de blocs \emph{blocking}, entre blocs a les
+superfícies de lluminositat similar.}
+
 \bibliography{cited}
 
 \end{document}