Merging a small detail.
authorviric@mandarina
Fri, 07 Dec 2007 00:01:11 +0100
changeset 232 721e24aa7c18
parent 231 aeff9e4a7f0e (current diff)
parent 230 67b1f6da1d87 (diff)
child 233 f212a710a9e2
Merging a small detail.
--- a/doc/Makefile	Sat Nov 17 20:54:13 2007 +0100
+++ b/doc/Makefile	Fri Dec 07 00:01:11 2007 +0100
@@ -8,6 +8,9 @@
 	bwlines.pdf \
 	escalat422fancy.pdf \
 	escalat420fancy.pdf \
+	lumscaler422.pdf \
+	lumscaler420.pdf \
+	co-inter-situats422.pdf \
 	cadena.pdf
 
 memoria.pdf: memoria.tex $(INCLUDES)
--- a/doc/cadena.eps	Sat Nov 17 20:54:13 2007 +0100
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--- a/doc/cadena.svg	Sat Nov 17 20:54:13 2007 +0100
+++ b/doc/cadena.svg	Fri Dec 07 00:01:11 2007 +0100
@@ -122,7 +122,7 @@
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--- a/doc/cited.bib	Sat Nov 17 20:54:13 2007 +0100
+++ b/doc/cited.bib	Fri Dec 07 00:01:11 2007 +0100
@@ -7,6 +7,9 @@
 % misc.
 %  must: none
 %  opt: author, title, howpublished, month, year, note
+
+@preamble{ "\addcontentsline{toc}{chapter}{Bibliografia}" }
+
 @manual{jpegspec,
     title = "Information Technology -- Digital Compression and Coding of
         Continuous-Tone Still Images -- Requirements and Guidelines",
@@ -16,6 +19,14 @@
     year = "1992"
 }
 
+@book{eyeandbrain,
+    title = "Eye and Brain -- The Psychology of Seeing",
+    author = "Richard L. Gregory",
+    publisher = "Princeton University Press",
+    year = "1997",
+    edition = "5th"
+}
+
 @misc{libjpeg,
     organization = "The Independent JPEG's Group",
     author = "The Independent JPEG's Group",
@@ -364,3 +375,11 @@
     month = "September",
     year = "1992"
 }
+
+@book{kou1995,
+    title = "Digital Image Compression -- Algorithms and Standards",
+    author = "Weidong Kou",
+    edition = "1st",
+    publisher = "Kluwer Academic Publishers",
+    year = "1995"
+}
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--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc/images/lena.txt	Fri Dec 07 00:01:11 2007 +0100
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--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc/images/sailing1top-q70-froment-pla-15-1.pgm	Fri Dec 07 00:01:11 2007 +0100
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--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
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+         xml:space="preserve"><tspan
+           y="41.935341"
+           x="545.66931"
+           id="tspan11306"
+           sodipodi:role="line">Originals de lluminositat</tspan></text>
+      <text
+         id="text11308"
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+         x="545.66931"
+         style="font-size:8px;font-style:normal;font-weight:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1pt;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Lucida Sans Unicode"
+         xml:space="preserve"><tspan
+           y="63.271355"
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+           id="tspan11310"
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+           d="M 205.51181,201.96848 L 219.68504,201.96848"
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+        <path
+           style="fill:none;fill-opacity:0.75;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
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+      <g
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+         id="g11318">
+        <path
+           style="fill:none;fill-opacity:0.75;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
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+        <path
+           style="fill:none;fill-opacity:0.75;fill-rule:evenodd;stroke:#000000;stroke-width:0.99999952px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
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+         xml:space="preserve"><tspan
+           y="155.39734"
+           x="544.97729"
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+           sodipodi:role="line">Pla de color delmat</tspan></text>
+      <text
+         id="text11328"
+         y="168.84792"
+         x="544.47223"
+         style="font-size:8px;font-style:normal;font-weight:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1pt;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Lucida Sans Unicode"
+         xml:space="preserve"><tspan
+           y="168.84792"
+           x="544.47223"
+           id="tspan11330"
+           sodipodi:role="line">Pla de color reescalat (ponderat)</tspan></text>
+      <rect
+         ry="0"
+         y="138.18893"
+         x="517.32281"
+         height="42.519707"
+         width="155.90553"
+         id="rect12311"
+         style="opacity:1;fill:none;fill-opacity:1;fill-rule:nonzero;stroke:#000000;stroke-width:0.99921256;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1" />
+    </g>
+  </g>
+</svg>
--- a/doc/memoria.tex	Sat Nov 17 20:54:13 2007 +0100
+++ b/doc/memoria.tex	Fri Dec 07 00:01:11 2007 +0100
@@ -1,4 +1,4 @@
-\documentclass[catalan,a4paper,titlepage,12pt]{report}
+\documentclass[catalan,a4paper,titlepage,12pt]{book}
 \usepackage{graphicx}
 \usepackage{ucs}
 \usepackage[utf8x]{inputenc}
@@ -8,17 +8,23 @@
 \usepackage{amssymb}
 \usepackage{subfigure}
 \usepackage{icomma}
+\usepackage{paralist} % compact enumerations
 \usepackage{url}
-\usepackage{makeidx}%,showidx} % This latter shows the indexed in the margin
+%\usepackage{makeidx}%,showidx} % This latter shows the indexed in the margin
 \usepackage[margin=1cm,font=footnotesize]{caption}
 %\usepackage{a4wide}
-%\usepackage{natbib}
-\bibliographystyle{plain}
+%\usepackage{natbib} No volem referències que no siguin números.
+\bibliographystyle{caplain}
 
 \title{Restauració d'imatges JPEG}
 \author{Lluís Batlle}
 \date{\today}
 
+\setcounter{tocdepth}{1}
+
+% For in-paragraph lists of a), b), c).
+\newcommand{\subcite}[1]{\emph{#1)}}
+
 \newcommand{\decimal}[1]{\sepnum{,\!}{.}{}{#1}}
 
 \newcommand{\revisar}[1]{\marginpar{\scriptsize #1}}
@@ -37,30 +43,72 @@
     \caption{#3}
     \end{figure}
     }
-
+\newcommand{\reg}{$^\text{\textregistered}$}
 \makeindex
 
 \begin{document}
 
 \maketitle
 
-\chapter{Planetjament del problema}
+\tableofcontents
+% TODO: Remove subsections
+
+\chapter*{Introducció}
+
+La codificació d'imatges JPEG ofereix unes imatges bastant fidels amb molt pocs
+bits per pícsel, reduint en bona mesura la mida dels fitxers d'imatge i els
+seus temps de transferència per Internet. Per això ha estat molt popular en la publicació d'imatges
+per Internet, i ho continua sent en l'ambient domèstic degut a les càmeres
+digitals. Per
+tant, qualsevol esforç a millorar la descodificació d'aquestes imatges té molt
+valor.
+
+Ens hem decidit a analitzar i recopilar les nombroses tècniques
+de restauració de JPEG aparegudes a diferents publicacions científiques. Alhora,
+hem creat una aplicació informàtica lliure que implementa alguns dels algorismes
+més interessants de la nostra cerca, a més d'establir una base per al
+desenvolupament d'aquest tipus de tècniques.
+
+Mitjançant aquest informe i el programa informàtic desenvolupat en paral·lel
+introduïm al lector al món de la pèrdua d'informació JPEG, donem a conèixer
+l'estat de l'art de la seva restauració, i proveïm als
+manipuladors d'imatges (de centres fotogràfics, d'impressió, etc.) amb
+les tècniques més prometedores d'aquest camp. Portem les idees de les
+publicacions científiques de laboratori als
+ordinadors personals de cases i centres de fotografia.
+
+Al llarg del document trobarem vàries imatges exemplificant soroll i
+restauracions, algunes d'elles difícils d'apreciar amb una impressora làser
+convencional. Per això us remetem a la versió electrònica d'aquest document per
+a poder apreciar les diferències entre imatges en detall.
+
+\chapter{Plantejament del problema}
 
 \section{La codificació JPEG}
 
-\revisar{Això ve de la Wikipedia. Cal citar-la? Ja la referencio un altre cop
-més tard, pels dibuixos}
+\revisar{Això ve de la Viquipèdia. Cal trobar una font millor.}
 El 1986 es va crear un comitè per establir un estàndard de codificació
 d'imatges de to-continu (continous tone), per exemple, fotografies. Aquest
-comitè tenia el nom de Joint Photographic Experts Group, i el 1992 va aprovar
-l'estàndard conegut arreu amb el nom de JPEG. Va decidir tant la
-codificació de la imatge, efectivament amb les sigles del grup, i també un
-format fitxer que contindria la informació codificada
-(JIF - JPEG Interchange Format).
-Aquest primer JIF tenia algunes mancances que al llarg de la història s'han
-suplit amb nous formats, com JFIF i EXIF. La majoria de gent sol anomenar-los
-indistintament "Fitxers JPEG", o bé amb la reducció a tres lletres obligada sota
-sistemes MS-DOS, "Fitxers JPG".
+comitè tenia el nom de Joint Photographic Experts Group, i el 1992 la ISO i el
+CCITT van aprovar
+l'estàndard conegut arreu amb el nom de JPEG (ISO/IEC IS 10918-1 - ITU-T
+Recommendation T.81)\footnote{\url{http://www.jpeg.org/jpeg/index.html}}.
+Van decidir tant la
+codificació de la imatge, efectivament amb les sigles del grup, i en paral·lel
+també es va decidir un
+format de fitxer que contindria la informació codificada
+(JFIF - JPEG File Interchange Format), ja que en informàtica sovint convé
+conservar informació en fitxers. Aquest format de fitxer permetia especificar-hi
+diferents espais de
+color al JPEG, i el delmat d'alguns dels plans.
+Amb les càmeres digitals que directament
+conservaven les fotografies en JPEG i en fitxers, es va creure que la informació
+que permetia conservar el JFIF no era suficient. Va aparèixer l'EXIF, que
+permet conservar molta més informació de les circumstàncies de la fotografia
+(estat de la càmera, sensors, etc.). La majoria de gent sol anomenar els
+contenidors JFIF i EXIF indistintament ``Fitxers JPEG'',
+o bé amb la reducció a tres lletres obligada sota sistemes MS-DOS,
+``Fitxers JPG''.
 
 Dins l'estàndard s'hi inclouen dues formes diferents de codificar la imatge: amb
 pèrdues (\emph{lossy}) o sense (\emph{lossless}). Aquesta última no ha
@@ -73,7 +121,7 @@
 bits. Això significa que tenim la imatge dividida en espai, amb una malla
 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,
+quadricula de punts suficientment densa, a ull nu no podrem distingir-la.
 i el conjunt li semblarà una imatge de color contínua o de to continu --
 sobretot
 en el cas de que la imatge sigui d'orígen fotogràfic, en comptes de generada
@@ -86,7 +134,8 @@
 d'informació de 24 bits per punt (bpp). Amb la compressió JPEG podem arribar a
 relacions de 1,5 bpp sense escatimar en qualitat d'imatge. En el cas de tenir el
 mapa de bits a partir d'una fotografia, almenys a una densitat de pícsels de 72
-ppi (72 pícsels per polzada) en horitzontal i vertical, l'ull humà mitjà en prou
+ppi (72 pícsels per polzada) en horitzontal i vertical, el nostre sistema
+de visió en prou
 feines pot distingir la pèrdua de definició del JPEG a 1,5 bpp.
 
 En el procés de codificació (o compressió) JPEG, es llença part de la informació
@@ -96,16 +145,18 @@
 les implementacions populars de descodificadors JPEG que hem vist
 \cite{libjpeg}, la invenció d'informació és molt simple, i no sempre encertada,
 considerant que cal obtenir una imatge el més semblant possible a l'original
-d'acord amb els criteris de l'ull humà. En aquest treball analitzarem
+d'acord amb els criteris del nostre sistema visual.
+En aquest treball analitzarem
 tècniques que pretenen aprofitar millor la informació del JPEG per a reconstruir
 una versió més agradable de la imatge original codificada.
 
 \section{Popularitat}
 
+\revisar{Cal fer que es vegi més això de la Popularitat}
 Des de l'establiment de l'estàndard JPEG com a norma ISO (10918-1) el 1994, i de
 la donació al domini públic de la implementació coneguda amb el nom de
 \texttt{libjpeg} \cite{libjpeg}, en versió 5 el mateix any, aquesta codificació
-ha esdevingut molt popular. Als inicis d'internet, JPEG oferia la major densitat
+ha esdevingut molt popular. Als inicis d'Internet, JPEG oferia la major densitat
 d'informació per a transmetre fotografies. La publicació de l'estàndard va
 alliberar l'ús de qualsevol restricció per patents, i la velocitat de procés de
 la \texttt{libjpeg} va permetre introduir suport pel format en moltíssimes
@@ -113,7 +164,7 @@
 personals.
 
 L'alta densitat de la informació JPEG, junt amb les restriccions d'ample de
-banda dels primers anys de l'internet popular, van fer que la gent publiqués les
+banda dels primers anys de l'Internet popular, van fer que la gent publiqués les
 seves fotografies en aquest format. Els límits en espai d'emmagatzematge també
 van causar que la gent conservés les fotografies només en aquest format. Aquesta
 corrent continua fins ara, tot i que els amples de banda i l'espai per
@@ -131,13 +182,14 @@
 millor densitat d'informació que el JPEG, però malgrat que han passat set anys
 des de la seva definició, no ha guanyat popularitat. Parlant exclusivament de la
 codificació d'imatges de to continu, el comitè garanteix que no està subjecte a
-patents. Però la seva implementació lliure més popular, \texttt{jasper} 
-\revisar{Referència al jasper?}
+patents. Però la seva implementació lliure més popular,
+\texttt{jasper}\footnote{\url{http://www.ece.uvic.ca/~mdadams/jasper/}}
 conté molt de càlcul i ús de memòria, fins i tot pels ordinadors
 personals actuals. Tampoc hi ha implementacions en maquinari per part dels
 fabricants de càmeres digitals. Tanmateix, per posar un exemple popular, el
 % ref http://lists.brouhaha.com/pipermail/tumble-devel/2004q4/000007.html
-programari Adobe Acrobat des de la versió 6 suporta la codificació JPEG2000 dins
+programari Adobe Acrobat\reg des de la versió 6 suporta la
+codificació JPEG2000 dins
 el seu format PDF 1.5. La majoria d'aplicacions d'Adobe, en la seva versió
 actual, suporten el format. D'acord amb experts en el camp de fotografia, per
 això, per a densitats d'informació similars, opinen que JPEG encara els dóna
@@ -164,16 +216,14 @@
 \section{Les pèrdues al JPEG}
 \label{sec:perdues}
 
-% TODO: mal sangrat
-\revisar{mal sagnat}
 El process de codificació segueix els següents passos:
-\revisar{Cal una enumeració compacta}
-\begin{enumerate}
+\begin{compactenum}
 \item Dividir la imatge en plans de color.
 \item Delmar alguns plans, si d'ells acceptem menys detall. Aquí tenim la
 primera perdua d'informació important.
 \item Dividir cada pla resultant en blocs de 8x8 pícsels.
-\item Fem la transformada DCT de dues dimensions a cada bloc, i així n'obtenim
+\item Fem la transformada DCT (Apèndix \ref{sec:dct}) de
+dues dimensions a cada bloc, i així n'obtenim
 la seva respresentació freqüencial en 64 coeficients.
 \item Quantitzem els coeficients. Aquí hi ha la segona pèrdua d'informació
 important.
@@ -183,10 +233,10 @@
 \item Col·loquem el paquet de dades en un fitxer a part (mitjançant JIF, JFIF,
 EXIF, etc.), o bé dins el contenidor que creiem convenient (Postscript, PDF,
 DJVU, etc.).
-\end{enumerate}
+\end{compactenum}
 
 \noindent La descodificació, anàlogament:
-\begin{enumerate}
+\begin{compactenum}
 \item S'obté el paquet de dades JPEG del seu contenidor.
 \item Recuperem els valors dels coeficients quantitzats del paquet (que havíem
 guardat sense pèrdues).
@@ -195,12 +245,12 @@
 decisions, inventar, el millor possible)
 \item Fem la transformada inversa DCT dels coeficients obtinguts, per
 arribar a la representació en blocs de mapa de bits de 8x8 pícsels.
-\item Els blocs constitueixen la informació espacial dels plans, i cal reescalar
-
-els plans delmats a la seva dimensió original. (Aquí el descodificador també ha
+\item Els blocs constitueixen la informació espacial dels plans, i cal
+reescalar els plans delmats a la seva dimensió original.
+(Aquí el descodificador també ha
 d'inventar els nous punts al escalar la imatge)
 \item Ja tenim una versió dels tres plans que determinaven la imatge.
-\end{enumerate}
+\end{compactenum}
 
 
 \subsection{Plans de color i delmat}
@@ -212,11 +262,12 @@
 cons distingeixen millor. Les freqüències de llum concretes s'avenen amb les de
 màxima sensibilitat pels nostres tres cons, de manera que amb la pantalla es
 produeix, pràcticament, l'excitació independent dels nostres cons, donant la
-sensació d'un espai de color continu. Un color groc, per exemple, degut a les
-responstes freqüencials dels cons, excita parcialment el con blau i el con verd.
-La pantalla enganya els cons, excitant concretament una mica el blau, i també
-una mica el verd.
-\revisar{No puc citar res per això?}
+sensació d'un espai de color continu (\cite{eyeandbrain}).
+Un color groc, per exemple, degut a les
+responstes freqüencials dels cons, excita parcialment el con vermell
+i el con verd.
+La pantalla enganya els cons, excitant concretament una mica els vermell,
+i també una mica els verd.
 
 El nostre ull distingeix variacions d'intensitat de llum millor que variacions
 de color.
@@ -231,14 +282,19 @@
 \figura{rapids-nens-rgb2ycc.pdf}{
 \label{fig:cb-rgb2ycc} Plans RGB, i els anàlogs en YCbCr}
 
-L'ull humà és menys sensible a defectes als plans Cb i Cr (ref ?), per tant, en
-la majoria de codificadors JPEG veurem que aquests plans es delmen. La majoria
+Quan parlem d'imatges amb formes reconeixedores, aquestes formes són molt
+importants a l'hora d'evaluar la percepció de la imatge. En bona part les formes
+vindran determinades per la lluminositat general, i per això l'ull humà el
+considerem
+menys sensible a defectes als plans Cb i Cr (\cite{eyeandbrain}). Per tant, en
+la majoria de codificadors JPEG veurem que aquests plans es delmen i
+es quantitzen a menys qualitat. La majoria
 de càmeres digitals delmen per maquinari a 2:1 només en horitzontal, mentre que
 molts codificadors per programari delmen a 2:1 tant en horitzontal com en
 vertical.
 
-\figura{rapids-nens-cb-rescale.pdf}{
-\label{fig:cb-rescale} Delmat de l'original de 2:1, i posterior
+\figura{rapids-nens-cb-rescale.pdf}{\label{fig:cb-rescale}Delmat
+de l'original de 2:1, i posterior
 reescalat de la versió delmada.}
 
 A la descodificació, cal reescalar els plans delmats a la seva mida original,
@@ -291,11 +347,11 @@
 relació al bloc i això vol dir que no es té en compte la conservació de
 propietats dels entorns locals dels punts. Degut a això, sovint podem
 distingir molesties visuals precisament a les vores entre blocs,
-on la informació pertany a dos blocs considerats completmaent independents pel
+on la informació pertany a dos blocs considerats completament independents pel
 codificador.
 
 Dels 64 punts d'intensitat de color que determinen cada bloc de 8x8 pícsels
-obtenim 64 coeficients mitjançant la transformada DCT 2D.
+obtenim 64 coeficients mitjançant la transformada DCT 2D (\ref{sec:dct}).
 Aquests es solen ordenar en
 freqüències de menor a major, utilitzant l'ordre de Zig Zag
 (Figura \ref{fig:jpeg-coeficients}) definit a l'estàndard de JPEG.
@@ -315,12 +371,13 @@
 l'estàndard (\cite{jpegspec}). Podem veure un exemple de quantitzadors
 a la Taula \ref{tab:example-qtable}.
 
-\figuraw{10cm}{jpeg-coeficients.pdf}{
+\figuraw{13cm}{jpeg-coeficients.pdf}{
 \label{fig:jpeg-coeficients} A l'esquerra, funció base de cada
-coeficient de la DCT 2D. A la dreta, la serialització dels coeficients en Zig
+coeficient de la DCT 2D. Al mig, contribució parcial dels coeficients de la DCT;
+cada bloc s'ha reconstruit considerant nuls els coeficients de més altes
+freqüències. A la dreta, la serialització dels coeficients en Zig
 Zag (\cite{jpegspec}, 4.3). Imatges obtingudes de la wikipedia \cite{wiki:jpeg}}
 
-
 % Taula de quantització
 \begin{table}[!htp]
 \centering
@@ -341,9 +398,12 @@
 \end{table}
 
 La quantització dels coeficients consisteix en conservar només el valor enter
-resultat d'arrodonir la divisió pel quantitzador. Per cada un dels 64
-coeficients reals provinents de la transformada DCT 2D obtenim un enter:
-\[ \hat y_i = \left[ \frac{y_i}{Q_i}\right] , \, i \in 1..64 \] 
+resultat d'arrodonir la divisió pel quantitzador. El primer coeficient (la
+component DC) no es quantitza independentment com els altres, sino que es
+quantitza la diferència amb el primer coeficient del bloc anterior. Per cada un dels 63
+coeficients reals provinents de la transformada DCT 2D ($i = 2,3,\ldots,64$) i
+per la diferència amb la component DC anterior ($i = 1$) obtenim un enter:
+\[ \hat y_i = \left[ \frac{y_i}{Q_i}\right] , \, i = 1,2,\ldots,64 \] 
 
 La recuperació convencional del coeficient, especificada a l'estàndard
 \cite{jpegspec}, consisteix en multiplicar pel
@@ -366,20 +426,21 @@
 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
+A la pèrdua d'informació deguda a la quantització dels coeficients, un cop se'ls
+ha aplicat la transformada DCT 2D inversa, se l'anomena
 \emph{soroll de quantització}. Té diverses conseqüències visibles a la imatge
-descodificada de manera convencional, i en podem veure una classificació a
+descodificada de manera convencional i en podem veure una classificació a
 la Figura \ref{fig:quant-problems}.
-\revisar{Cal comentar l'efecte de \emph{suavitzat} del senyal? (menyspreu
-d'altes freqüències)}
 
 \figura{lena-blocking.pdf}{
 \label{fig:quant-problems} A l'esquerra, pla de lluminositat de la Lena de 256x256 píxels, a
-qualitat 40 segons la \texttt{libjpeg}. A l'esquerra, tall on podem veure els efectes a) d'ones
+qualitat 40 segons la \texttt{libjpeg}. A l'esquerra, tall on podem veure els
+efectes
+\subcite{a}d'ones
 (\index{ringing}\emph{ringing})
-als blocs que tenen un canvi brusc d'intensitat pel mig, b) 
-d'escala (\index{staircase}\emph{staircase})
-i c) de blocs \index{blocking}\emph{blocking}, entre blocs a les
+als blocs que tenen un canvi brusc d'intensitat pel mig,
+\subcite{b}d'escala (\index{staircase}\emph{staircase}) i
+\subcite{c}de blocs (\index{blocking}\emph{blocking}), entre blocs a les
 superfícies de lluminositat similar. Al mig, el mateix tall de la imatge
 original, per comparar.}
 
@@ -387,7 +448,7 @@
 
 En aquest treball ens proposem millorar la descompressió d'imatges JPEG. Per
 tant, cal que definim la qualitat de la imatge resultant d'una manera
-quantitzatble, mesurable.
+quantitzable, mesurable.
 
 Una mesura molt utilitzada,
 simplement pel fet de ser purament de teoria del senyal,
@@ -404,7 +465,8 @@
 
 Als articles que fan referència a comparació d'imatges amb l'objectiu de mesurar
 la similitud encara es fa servir més una funció del MSE, la \emph{relació senyal
-soroll de pic} (PSNR \index{PNSR} -- \emph{Peak Signal-to-Noise Ratio}),
+soroll de pic} (PSNR \index{PNSR}\label{sec:psnr}
+-- \emph{Peak Signal-to-Noise Ratio}),
 que es sol expressar en decibels i es defineix així:
 \[ PSNR = 10 \log_{10} \frac{255^2}{MSE} \]
 
@@ -469,7 +531,7 @@
 qualitat, el pendent d'intensitat abans del canvi de bloc serà diferent al que
 hi ha entre blocs. Aquestes diferències es sumen, tant les mesurades en
 horitzontal com en vertical i en diagonal, per a donar un únic valor NR.
-\item \index{GBIM}\emph{GBIM -- Generalized Block-edge Impariment Metric}:
+\item \index{GBIM}\emph{GBIM -- Generalized Block-edge Impairment Metric}:
 a \cite{gbim1997} proposen mesurar les diferències d'intensitat dels
 punts que estan al límit dels blocs consecutius, i comparar-los amb les
 diferències d'intensitat de punts que no estan a límits de blocs. Comparteix
@@ -491,6 +553,12 @@
 llavors en la majoria dels casos $1 < S < 10$.
 \end{itemize}
 
+El programari vinculat a aquest informe, tot i que ofereix mesures matemàtiques,
+fonamentalment serveix per a permetre comparar imatges a ull nu. Tenim diferents
+mesures per evaluar les restauracions d'imatges JPEG, i per tant ens
+interessa proveïr a l'usuari d'un entorn on ell pugui evaluar les imatges segons
+el seu criteri, que segurament serà el més adequat a les seves circumstàncies.
+
 \chapter{Tècniques}
 
 \label{sec:tecniques}
@@ -504,6 +572,7 @@
 
 \section{Projecció sobre l'espai de quantització}
 
+\label{sec:qcsproject}
 Aquesta tècnica no millora les imatges en si, però sí que la utilitzen a molts
 mètodes. Podem diferenciar els mètodes en dos grans grups: els de \emph{dins la
 cadena de descodificació}, i els de {post-processat}. Els primers consideren
@@ -516,10 +585,33 @@
 convencional. Si al recodificar el pla amb
 les mateixes taules de quantització obtenim la mateixa versió codificada 
 que de l'original,
-vol dir que hem aplicat una descodificació \emph{no destructiva}.
-Els mètodes de \emph{post-processat} això no poden garantir-ho, ja que
-per començar desconeixen les taules de quantització.\revisar{Citar-ne alguns,
-perquè fan falta.}
+vol dir que hem aplicat una descodificació \emph{no destructiva}. Els
+coeficients que hem restaurat podrien ben ser de la imatge original, ja que són
+coherents amb la informació del JPEG. Ara bé, com podem veure a la Figura
+\ref{fig:qcs-margins} el marge pot ser bastant gran segons les taules de
+quantització.
+Els mètodes de \emph{post-processat} no poden garantir aquesta coherència amb el
+JPEG ja que desconeixen les taules de quantització. Per exemple, amb programes
+informàtics
+de retoc fotogràfic es sol utilitzar el filtrat
+Gaussià sel·lectiu per a eliminar
+soroll, entre d'altres, de la descodificació JPEG convencional.
+
+\begin{figure}[!htp]
+\centering
+\begin{tabular}{ccc}
+\includegraphics[width=4cm]{images/lena-40-jpg.png} &
+\includegraphics[width=4cm]{images/lenag-40-lowfreq.png} &
+\includegraphics[width=4cm]{images/lenag-40-highfreq.png} \\
+a) & b) & c)
+\end{tabular}
+\caption{\label{fig:qcs-margins}\subcite{a}Imatge Lena
+codificada a qualitat 40 amb
+la \texttt{libqjpeg}, \subcite{b}Descodificació segons els valors absoluts més
+baixos dins dels intervals de quantització, \subcite{c}Descodificació segons els
+valors absoluts més alts.
+}
+\end{figure}
 
 Alguns mètodes de descodificació basen la seva estimació d'imatge en
 filtres espacials, a vegades fins i tot en
@@ -534,9 +626,9 @@
 l'interval.
 \begin{equation}
 z'_i = \begin{cases}
-    (\hat y_i - 0.5) Q_i & z_i < (\hat y_i - 0.5) Q_i \\
-    (\hat y_i + 0.5) Q_i & z_i > (\hat y_i + 0.5) Q_i \\
-    z_i & \textrm{altrament}
+    (\hat y_i - 0.5) Q_i & \text{si} \, z_i < (\hat y_i - 0.5) Q_i \\
+    (\hat y_i + 0.5) Q_i & \text{si} \, z_i > (\hat y_i + 0.5) Q_i \\
+    z_i & \text{altrament}
     \end{cases}
 \end{equation}
 
@@ -565,13 +657,30 @@
 
 Alguns articles suggereixen altres distribucions de probabilitat. Per exemple,
 a \cite{trianta2003} proposen utilitzar una v.a. exponencial, de
-mitjana a determinar segons la imatge. A \cite{robertson2004},
+mitjana a determinar segons la imatge (veieu la Figura \ref{fig:fexp}).
+A \cite{robertson2004},
 \cite{trianta2002} i a
 \cite{queiroz1998}
 mencionen utilitzar una v.a. Laplaciana. El primer, basant-la amb la imatge a
 descodificar. El segon, basant-la amb estadístiques d'altres
 imatges.\revisar{Com ho fa exactament el primer?}
 
+\begin{figure}[!htp]
+\centering
+\begin{tabular}{ccc}
+\includegraphics[width=4cm]{images/lena.png} &
+\includegraphics[width=4cm]{images/lena-40-jpg.png} &
+\includegraphics[width=4cm]{images/lenag-40-fexp-1.png} \\
+a) & b) & c)
+\end{tabular}
+\caption{\label{fig:fexp}\subcite{a}Imatge Lena
+original, \subcite{b}Descodificació convencional de qualitat 40 segons la
+\texttt{libjpeg}, considerant els
+intervals com v.a. uniformes, \subcite{c}descodificació de la mateixa imatge
+considerant v.a.  exponencials de mitjana $\mu = 1$.
+}
+\end{figure}
+
 \section{Segmentació i suavitzat}
 
 Un dels efectes que més es distingeixen a les imatges codificades amb una
@@ -593,8 +702,9 @@
 fan servir filtres 1D; i a \cite{liu2002}
 fan servir interpolació lineal sobre la component contínua.
 
-Alguns algorismes més complexos suggereixen processar les zones d'altes
-freqüències de manera especial. A \cite{chen2001} i
+Alguns algorismes més complexos suggereixen distingir i processar també
+les zones d'altes
+freqüències. A \cite{chen2001} i
 \cite{liu2002} fan servir un filtre amb molt
 poc pes dels punts del voltant; i a \cite{alfahoum2001}, \cite{pham2005} i
 \cite{trianta2003} fan servir filtres direccionals. 
@@ -602,7 +712,8 @@
 \section{\index{POCS}POCS - Projecció sobre conjunts convexos}
 
 Els mètodes basats en POCS solen donar molts bons resultats, i apareixen a la
-literatura des dels inicis de l'ús del JPEG. Malauradament l'algorisme és lent,
+literatura des dels inicis de l'ús del JPEG. Malauradament l'algorisme és
+bastant més lent que altres,
 ja que consisteix en dur a terme passos iteratius.
 
 Referint-me a l'explicació de POCS de \cite{yang1993}, assumim que
@@ -649,7 +760,7 @@
 definits pels conjunts convexes involucrats. Amb la tècnica de MAP el que es
 busca és la minimització d'una funció.
 
-En general l'objectiu és minimitzar una funció, sempre i quan mentenim els
+En general l'objectiu és minimitzar la funció sempre i quan mentenim els
 coeficients dins els intervals de quantització. Si $\hat z$ és la nostra imatge
 estimada a la descodificació, i $Z$ és el conjunt d'imatges amb coeficients dins
 el QCS,
@@ -687,17 +798,20 @@
 utilitzen amb un algorisme adaptatiu
 per corregir els vessaments de color deguts a l'escalat. A \cite{sugita2006}
 ponderen els canvis d'intensitat dels plans de color d'acord amb els canvis
-d'intensitat del pla de lluminositat.
+d'intensitat del pla de lluminositat. Aquest l'últim l'hem adaptat al delmat del
+JPEG, explicat en detall a l'Apèndix \ref{sec:colorponderat}.
 
 \chapter{Programari}
+% TODO: Add program captures: cadena de descodificació, algunes imatges
+% analitzades
 
 Com hem vist a l'apartat anterior, trobem nombrosos articles sobre la millora de
 la descompressió JPEG. Malauradament, no trobem els algorismes implementats en
-cap mena d'aplicació per a l'usuari final. GIMP i Adobe Photoshop
-\textregistered, possiblement els processadors de mapes de bits més populars,
+cap aplicació popular. GIMP i Adobe
+Photoshop\reg, possiblement els processadors de mapes de bits més populars,
 només suporten la descodificació convencional. A la última versió (6b) de la
 \texttt{libjpeg} tampoc hi trobem res d'especial, tot i que a la documentació
-fan una declaració d'intencions per incorporar mètodes de restauració a la
+hi trobem una declaració d'intencions per incorporar mètodes de restauració a la
 versió 7, però sembla que el desenvolupament d'aquesta llibreria es va aturar
 precisament a la 6b el 1998.
 
@@ -707,24 +821,25 @@
 la codificació JPEG es continua utilitzant activament, i per tant molts usuaris
 es poden beneficiar d'una iniciativa com aquesta.
 
-Quan hem buscat implementacions dels algorismes, n'hem trobat molt poques,
-i la majoria d'aquestes poques estaven escrites en Matlab. Els interprets de
+Quan hem buscat implementacions dels algorismes de restauració analitzats,
+n'hem trobat molt poques,
+i la majoria d'aquestes poques estaven escrites en Matlab. Els intèrprets de
 Matlab els trobem pràcticament només a les universitats o dins departaments
-d'investigació d'empreses, però la majoria de professionals de la imatge ni el
-posseiran ni el sabran utilitzar. Nosaltres hem desenvolupat tota l'aplicació i
+d'investigació d'empreses, però la majoria de professionals de la imatge ni els
+posseiran ni els sabran utilitzar. Nosaltres hem desenvolupat tota l'aplicació i
 els algorismes en C i C++, ja que ens semblen els llenguatges més populars i a
 l'abast de qualsevol gràcies a diversos compiladors i enllaçadors lliures
 disponibles. Aquí tenim un llistat de les principals característiques
 del programa:
 \begin{itemize}
-\item Només permet manipular un JPEG alhora, junt amb versions
+\item Només permet manipular una imatge JPEG alhora, junt amb versions
 recuperades o que teníem guardades a disc.
 \item Tot el procés i emmagatzematge intern de la informació de les imatges
-es realitza en punt flotant (\texttt{float} en C), per evitar pèrdues per
+es realitza en coma flotant (\texttt{float} en C), per evitar pèrdues per
 arrodoniment als càlculs.
 \item El programa requereix que l'usuari entengui la part de descodificació JPEG
 que hem explicat al primer apartat, ja que
-és l'usuari qui determinarà com es realitzaran diversos passos d'aquesta
+l'usuari determinarà com es realitzaran diversos passos d'aquesta
 descodificació.
 \end{itemize}
 
@@ -737,10 +852,12 @@
 \item No ens importa la complexitat de l'algorisme en temps de computació o
 memòria. No tractarem vídeo, sino imatges estàtiques. Alguns articles presenten
 novetats exclusivament en el camp de la restauració ràpida, tot i que no
-aconsegueixen recuperacions tan bones com els algorismes lents.
+aconsegueixen recuperacions tan bones com alguns algorismes més lents.
 \item Hem considerat que els exemples d'imatge presentats al mateix article
-ofereixen una bona idea del millor que pot assolir l'algorisme. Creiem que
-alguns articles presenten molt poca millora com per a tenir-los en compte en una
+ofereixen una bona idea del millor resultat que l'algorisme permet assolir.
+Creiem que
+alguns articles presenten molt poca millora (valorant subjectivament a ull nu)
+com per a tenir-los en compte en una
 implementació.
 \item L'algorisme ha de permetre millorar imatges en general. Alguns només
 milloren les zones de baixes freqüències, o fins i tot només serveixen per a
@@ -748,9 +865,9 @@
 \item L'efecte de l'algorisme s'ha de caracteritzar per pocs paràmetres, i
 aquests han de tenir una relació monòtona amb els seus efectes.
 En la majoria de casos l'usuari no
-entendrà internament algorisme, i variarà intuitivament els paràmetres. Si
+entendrà internament algorisme, i en variarà intuitivament els paràmetres. Si
 l'usuari no pot entendre l'efecte que té cada paràmetre, segurament refusarà
-manipular-lo.
+manipular-lo, o fins i tot utilitzar-lo.
 \end{itemize}
 
 Podem classificar els algorismes segons la seva funció. A la nostra aplicació
@@ -762,7 +879,7 @@
 \end{itemize}
 
 A qualsevol imatge que haguem carregat dins de l'aplicació li podem aplicar els
-algorismes de mesura, i obtindrem els resultats en un nou quadre de
+algorismes de mesura, i obtindrem els resultats en un nou quadre flotant de
 texte. Per a les mesures referencials haurem d'escollir les dues imatges a
 comparar, en comptes de només una.
 
@@ -786,16 +903,27 @@
 baules (amb el contorn més gruixut) on hi podem col·locar diversos 
 algorismes. Les línies de punts separen els diferents tipus d'algorismes
 acceptats. D'esquerra a dreta: \emph{desquantitzador i IDCT2D},
-\emph{restaurador de pla}, \emph{escalador}, \emph{canvi de plans de color}.}
+\emph{restauradors de pla}, \emph{escalador}, \emph{canvis de plans de color}.}
 
-L'aplicació posarà a la disposició de l'usuari uns quants algorismes de cada
-tipus, i aquest podrà col·locar els que cregui adients a la cadena, en forma de
-baula. Alguns dels algorismes que col·locarem a les baules són configurables.
+L'aplicació posarà a disposició de l'usuari uns quants algorismes de cada
+tipus, i aquest podrà col·locar els que cregui adients a la cadena, com si es
+tractéssin de baules.
+Alguns dels algorismes que col·locarem a les baules són configurables.
 Això vol dir que el resultat que donin dependrà tant de l'entrada de
-l'algorisme (de la baula anterior) com de la configuració dels seus
-paràmetres interns.
+l'algorisme (provinent de la baula anterior) com de la configuració dels seus
+paràmetres interns. Podem veure l'aspecte de l'aplicació mentre intentem
+restaurar un JPEG de poca qualitat de la imatge Lena a la Figura
+\ref{fig:qjpeg-screenshot}. Per entendre els noms d'algorismes que s'hi
+distingeixen, podem consultar la Taula \ref{tab:algnames}, pàg.
+\pageref{tab:algnames}.
 
-\subsection{Desquantització i IDCT 2D de cada pla}
+\figura{images/projecteui.png}{\label{fig:qjpeg-screenshot}Mostra de les
+principals finestres del programa. La principal, amb versions de la imatge a
+restaurar. A sota, la cadena de descompressió. A dalt a la dreta, de fons, la
+configuració de la baula de Froment. A l'esquerra, configuració de la baula
+Hcontrast.}
+
+\section{Desquantització i IDCT 2D de cada pla}
 
 \begin{description}
 \item[entrada] els intervals de quantització d'un pla de la imatge ocdificada
@@ -811,9 +939,9 @@
 convencional de JPEG escollint el punt mig dels intervals. I el segon considera
 la informació dels coeficients AC (tots menys la component contínua DC, el
 primer coeficient de cada bloc) com variables aleatòries exponencials d'igual
-mitjana, com a \cite{trianta2003}. L'usuari pot configurar la mitjana.
+mitjana, com a \cite{trianta2003}. L'usuari pot configurar-ne la mitjana.
 
-\subsection{Restaurador de pla}
+\section{Restaurador de pla}
 
 \begin{description}
 \item[entrada] els intervals de quantització d'un pla de la imatge ocdificada
@@ -823,13 +951,13 @@
 
 La majoria d'algorismes de restauració de JPEGs que hem trobat encaixen dins
 aquesta categoria. Partint d'un pla inicial (als articles sol ser una
-descompressió convencional) utilitzen diverses tècniques per obtenir una versió
+descodificació convencional) utilitzen diverses tècniques per obtenir una versió
 millor del mateix pla. A l'apartat \ref{sec:tecniques} ja hem parlat d'aquest
 tipus d'algorismes.
 
 Cal destacar que als articles utilitzen imatges d'escala de grisos ja que
 exclusivament intenten millorar un pla de la imatge. A la nostra aplicació podem
-utilitzar els algorismes a qualsevol dels plans del JPEG. Igualment, podem
+utilitzar aquests algorismes a qualsevol dels plans del JPEG. Igualment, podem
 encadenar tants algorismes d'aquests tipus com volguem, on cada un determinarà
 la imatge inicial del seu successor.
 
@@ -841,76 +969,82 @@
 projecció com a algorisme independent. A part de les
 propostes d'articles, inclou un
 senzill algorisme de millora de les zones amb textura desenvolupat per nosaltres
-mateixos que aprofita la naturalesa de l'aplicació: combinar i encadenar
+mateixos que aprofita la naturalesa del programa: combinar i encadenar
 tècniques. L'explicarem més endavant a l'apartat \ref{sec:hcontrast}.
 Finalment, la implementació actual també inclou uns mòduls especials
 que ens permeten alterar els
 plans mitjançant una altra aplicació informàtica. Són els següents:
 \begin{itemize}
-\item \emph{salvar pla a disc}. Si l'usuari col·loca aquesta funció dins una baula de
+\item \emph{salvar pla a disc}. Si l'usuari col·loca aquesta funció dins
+'duna baula de
 restauració de pla, al aplicar la descodificació definida per la cadena, el pla
 d'entrada de la baula s'escriurà a disc. Llavors l'usuari pot utilitzar el seu
-programa preferit de manipulació fotogràfica amb les tècniques que cregui
-convenients, i guardar-les a disc, utilitzant després la funció que ve a
-continuació.
+programa preferit de manipulació fotogràfica aplicant les tècniques que cregui
+convenients al pla, i guardar els canvis per a reincorporar-los a la cadena amb
+la funció que descrivim a continuació.
 \item \emph{carregar pla de disc}. El pla d'entrada d'aquest algorisme
 s'ignorarà completament, i el pla de sortida vindrà determinat per un fitxer
 guardat a disc.
 \end{itemize}
 Aquestes dues funcions les hem utilitzat a la pràctica per millorar els plans
 amb un filtre del GIMP\footnote{GNU Image Manipulation Program
-\url{http://www.gimp.org/}}, el suavitzat Gaussià sel·lectiu.
+\url{http://www.gimp.org/}}:, el suavitzat Gaussià sel·lectiu.
 Aquesta tècnica també la utilitzen
 combinada amb altres a \cite{queiroz1998} i \cite{kim1999}, i alguns manuals de
-GIMP la recomanen per a eliminar part del soroll d'algunes imatges JPEG.
+GIMP la recomanen per eliminar part del soroll d'algunes imatges JPEG.
 Si utilitzem les funcions de \emph{salvar el
 pla}, apliquem el filtre amb el GIMP, \emph{el carreguem a la cadena}, i després
 hi apliquem una \emph{projecció sobre l'espai de quantització}, obtenim
 resultats competitius amb altres algorismes, segons el tipus de imatge i el
 tipus de codificació JPEG.
 
-\subsection{Escalador dels plans}
+\section{Escalador dels plans}
 
 \begin{description}
 \item[entrada] $n$ plans independents, possiblement alguns delmats
-\item[sortida] $n$ plans de color
+\item[sortida] $n$ plans, tots de la mateixa mida
 \end{description}
 
-Tal com hem explicat a l'apartat \ref{sec:plans-color}, alguns dels plans de color de la imatge poden haver estat delmats abans de codificar independent els
+Tal com hem explicat a l'apartat \ref{sec:plans-color}, alguns dels plans de
+color de la imatge poden haver estat delmats abans de codificar
+independentment els
 plans (DCT 2D i quantització). La imatge final descodificada ha de tenir tots
-els plans de color de la mateixa mida. Per això, ens cal interpolar els plans
-que hem descodificat, que havien set delmats, de manera que adquireixin les
-seves dimensions originals. Cal recordar que els delmats sempre seran dividint
-les dimensions originals per un número enter.
+els plans de color de la mateixa mida. Per això al descodificar
+ens cal interpolar els plans delmats, de manera que adquireixin les
+seves dimensions originals. Cal recordar que el codificador delmarà els plans
+dividint
+les dimensions originals per un número enter, i els delmats més comuns són 4:2:2
+i 4:2:0.
 
 Al programa hem implementat tres tipus d'escaladors. El primer simplement
-replica els punts per aconseguir les dimensions desitjades. Per exemple, si
+duplica els punts per aconseguir les dimensions desitjades. Per exemple, si
 tenim els valors per punt en una fila $1,5,3,2,6$, i ens cal doblar la seva
 mida horitzontal, l'algorisme ens donarà $1,1,5,5,3,3,2,2,6,6$. La
 \texttt{libjpeg} també inclou una implementació d'aquest escalador.
 
-El segon escalador, provinent també de la \texttt{libjpeg}, interpola
-els punts del pla final a partir de la versió delmada descodificada, tot i que
-només funciona pels delmats 4:2:2 i 4:2:0. \marginpar{Cal citar on explico la
+El segon escalador, provinent també de la \texttt{libjpeg}
+(on l'anomenen \emph{fancy upsampler}), interpola
+linealment els punts del pla final a partir de la versió delmada
+descodificada, tot i que
+només funciona pels delmats 4:2:2 i 4:2:0. \revisar{Cal citar on explico la
 notació 4:x:y?}. En el cas de 4:2:2, interpola cada punt final amb els dos punts
 més propers del pla delmat.  Hem de recordar que segons l'especificació JFIF
 \cite{jfif}, els plans
-delmats a la meitat horitzontal, no tindran punts coincidents amb els del pla
+delmats a la meitat en horitzontal, no tindran punts coincidents amb els del pla
 final. Així doncs, obtenim els punts de color per les posicions dels punts de
 lluminositat de la següent manera:
 \[ \hat p_o = \frac{3}{4} p_1 + \frac{1}{4} p_2 \]
-on $\hat p_o$ és el valor d'un punt d'una línia final,
-$p_1$ és el del punt més proper a
-$\hat p_o$
-de la línia delmada, i $p_2$ el del segon punt més proper. La Figura
+on $\hat p_o$ és el valor d'un punt d'una fila final,
+$p_1$ és el del punt de la fila delmada més proper a
+$\hat p_o$, i $p_2$ el del segon punt més proper. La Figura
 \ref{fig:fancy422} representa visualment el procés.
 
-\figuraw{10cm}{escalat422fancy.pdf}{\label{fig:fancy422}Reescalat d'una línia
+\figuraw{10cm}{escalat422fancy.pdf}{\label{fig:fancy422}Reescalat d'una fila
 de la component de color de delmat 4:2:2 (a dalt)
 a la mida final determinada per la lluminositat (a baix), mitjançant
 interpolació dels dos punts més propers.}
 
-En el cas 4:2:0 l'escalador interpola cada punt final mitjançant
+En el cas 4:2:0 el segon escalador interpola cada punt final mitjançant
 els quatre punts més propers:
 \[ \hat p_o = \frac{9}{16} p_1 + \frac{3}{16} (p_2 + p_3) + \frac{1}{16} p_4 \]
 on $\hat p_o$ és el valor d'un punt del pla final, $p_2$ i $p_3$
@@ -921,9 +1055,29 @@
 \figuraw{10cm}{escalat420fancy.pdf}{\label{fig:fancy420}Reescalat d'una porció
 de la component de color de delmat 4:2:0 (en forma de creu dreta)
 a la mida final determinada per la lluminositat (en forma de cercles),
-mitjançant interpolació entre els quatre punts més propers.}
+mitjançant interpolació linial entre els quatre punts més propers.}
+
+Finalment, hem implementat un tercer escalador, basat en \cite{sugita2006},
+on utilitzem una tècnica similar a la interpolació linial,
+però a més d'obtenir el
+color de cada punt final basant-nos amb els quatre punts de color més propers,
+ho ponderem segons les variacions de llum del pla de lluminositat (quan no ha
+estat delmat). A
+l'Apèndix \ref{sec:colorponderat} expliquem l'algorisme en detall.
 
-\subsection{Canvi de plans de color}
+Tot i que fins ara hem parlat del cas general de que les imatges JPEG tenen
+unes mides múltiples de la mida de bloc (8 pícsels), en realitat aquesta
+condició no s'exigeix a les imatges a codificar. La mida original de la imatge
+ESTà escrita al contenidor del JPEG, i l'escalador del descodificador és
+l'encarregat de retallar
+els últims blocs en horitzontal i vertical per a aconseguir la imatge final de
+la mida que toca. Així, si una imatge medeix 124 pícsels d'amplada,
+s'haurà codificat amb 16 blocs de 8 pícsels, però de l'últim
+bloc de cada fila (codificada d'esquerra a dreta)
+només tindrem en compte la meitat esquerra descodificada
+de 4 pícsels d'amplada.
+
+\section{Canvi de plans de color}
 
 \begin{description}
 \item[entrada] $n>1$ plans de color
@@ -932,16 +1086,475 @@
 
 Aquests algorismes transformen els plans de color codificats del JPEG als plans
 exigits pel sistema gràfic del nostre ordinador. La nostra aplicació ara mateix
-necessita que les imatges finals tinguin a) un
-pla de lluminositat o b) tres plans vermell-ved-blau. Segurament la majoria dels
+necessita que les imatges finals tinguin \subcite{a}un
+pla de lluminositat o \subcite{b}tres plans vermell-ved-blau.
+Segurament la majoria dels
 nostres JPEG a color estaran codificats amb lluminositat-blavor-vermellor, i per
 tant, per a mostrar-los correctament a l'aplicació,
 necessitarem un algorisme conversor de YCbCr a RGB.
 
-Les extensions JFIF permeten codifcar la imatge en varis plans de color. Per
-defecte amb YCbCr, però també s'hi poden indicar codificacions RGB o CMYK.
+Les extensions JFIF permeten anotar en quins plans de color s'ha codificat la
+imatge. Per defecte es codifica amb YCbCr,
+però també s'hi poden indicar codificacions RGB o CMYK. Nosaltres només hem
+implementat el conversor YCbCr a RGB, ja que no ens hem trobat imatges que
+utilitzessin altres configuracions de color.
+
+\chapter{Anàlisi dels mètodes implementats}
+% TODO: one or two draws (logos), taken from webs. Real cases.
+
+Al programa hem donat uns noms als algorismes que hem implementat. A la Taula
+\ref{tab:algnames} els teniu tots llistats, i permetran seguir millor els
+comentaris dels seus resultats.
+
+\begin{table}[!htp]
+\centering
+\begin{tabular}{|ll|}
+\hline
+IDCTFloat & Descodificació convencional per a un pla \\
+IDCTFExp & Coeficients com v.a. exponencials, de \cite{trianta2003} \\
+Froment & \cite{froment2005} \\
+O'Rourke & \cite{orourke1995} \\
+Robertson & \cite{robertson2004} \\
+Trianta & \cite{trianta2003} \\
+Nosratinia & \cite{nosratinia2002} \\
+HP & \cite{hp2004} \\
+QCS Projection &  Apartat \ref{sec:qcsproject} \\
+HContrast &  Apèndix \ref{sec:hcontrast} \\
+% QCSKeepHF &  XXXXXXXXXXXXXX \\    % Val més que no fem cas d'aquest mètode
+LoadPlane &  Carregar el pla de disc \\
+SavePlane &  Salvar el pla a disc \\
+IntScaler &  Escalador que duplica el valor dels punts delmats \\
+FancyScaler &  Escalador interpolador lineal \\
+LumScaler & \cite{sugita2006}, Apèndix \ref{sec:colorponderat} \\
+\hline
+\end{tabular}
+\caption{\label{tab:algnames}Noms curts dels mètodes
+implementats al programa amb l'algorisme corresponent.}
+\end{table}
+
+En aquest anàlisi no ens hem basat gaire amb mesures matemàtiques, ja que
+precisament un dels objectius que ens havíem proposat a l'hora de fer aquest
+programari consisteix en pemetre a l'usuari comparar resultats variant
+paràmetres manualment. L'usuari dels algorismes, en aquest cas, provarà
+d'aplicar diversos mètodes i a anar variant els paràmetres, fins a obtenir
+una imatge que el contenti pel seu objectiu.
+Els articles ofereixen mesures objectives dels
+resultats de diversos algorismes, i no sempre hi ha una correspondència clara
+entre aquestes mesures i l'opinió d'individuals. A l'avaluació hi
+prenen part almenys: el contingut de la imatge (persones, dibuixos,
+paisatges), la
+resolució (de quina distància es miraran els punts de la imatge), i
+l'objectiu de la restauració (impressió, postprocessat). Les mesures objectives
+que hem analitzat no tenen en consideració aquests paràmetres, i per tant no ens
+són suficients per avaluar els resultats dels algorismes. I és amb això que
+justifiquem l'existència d'una aplicació informàtica com la que presentem. En
+aquest apartat, quan comparem algorismes, ens referirem a les opinions
+d'uns pocs individus després d'haver escollit manualment tant els algorismes com
+els paràmetres de la cadena de descodificació.
+
+\section{Plans a molt poca qualitat}
+
+Aquest tipus d'imatges (qualitat$<40$ a \texttt{libjpeg})
+es caracteritzen per efectes de bloc molt importants
+i poc detall de textures.
+Partint d'una descodificació convencional, els algorismes iteratius
+no-adaptatius de \emph{Froment}, \emph{O'Rourke},
+i \emph{Robertson} són els que produeixen millors resultats.
+El de Trianta, com que funciona de manera
+adaptativa segons informació espacial, no distingeix bé els blocs de
+cantondes reals de la imatge. Els de \emph{Nosratinia} i
+\emph{HP} suavitzen excesivament la
+imatge, comparat amb els tres primers que hem mencionat. Podeu veure un exemple
+de baixa qualitat a la Figura \ref{fig:lena40g}.
+
+\begin{figure}[!htp]
+\centering
+\begin{tabular}{cc}
+\includegraphics[width=6cm]{images/lena.png} &
+\includegraphics[width=6cm]{images/lena-40-jpg.png} \\
+a) & b) \\
+\includegraphics[width=6cm]{images/lena-40-froment-8-1-f-sharp30.png} &
+\includegraphics[width=6cm]{images/lena-40-jp2.png} \\
+c) & d)
+\end{tabular}
+\caption{\label{fig:lena40g}Codificació/descodificació de la imatge Lena de 256x256
+pícsels:
+\subcite{a}original,
+\subcite{b}JPEG qualitat 40 - PSNR 44,09 dB,
+\subcite{c}descodificació amb
+\emph{Froment} \cite{froment2005} (pesos de Froment, 8 iteracions,
+mida inicial de pas 1) i posterior esmolat amb GIMP
+(Filtre ``afila'', amb afilament 30) - PSNR 44,99 dB, i
+\subcite{d}versió JPEG2000 (amb \texttt{jasper} versió 1.701.0) amb la mateixa
+relació de compressió que la JPEG - PSNR 68,89 dB.}
+\end{figure}
+
+L'algorisme de \emph{Trianta}, però, si l'encadanem després d'algun dels altres
+algorismes que suavitzen la imatge, permet recuperar bé moltes cantonades
+accentuant-les. Cal mencionar que aquest algorisme no manté els coeficients dins
+l'espai de quantització, i per això després d'aplicar-lo, recomanem una
+\emph{projecció sobre el QCS}. Podeu veure un exemple de compressió a molt baixa
+qualitat a la Figura \ref{fig:lena10g}.
+
+\begin{figure}[!htp]
+\centering
+\begin{tabular}{cc}
+\includegraphics[width=6cm]{images/lena.png} &
+\includegraphics[width=6cm]{images/lena-10-jpg.png} \\
+a) & b) \\
+\includegraphics[width=6cm]{images/lena-10-froment-12-1-f.png} &
+\includegraphics[width=6cm]{images/lena-10-froment-12-1-f-trianta-2-2,5-1-5-500-qcs.png} \\
+c) & d)
+\end{tabular}
+\caption{\label{fig:lena10g}Codificació/descodificació de la imatge Lena de
+256x256 pícsels a molt baixa qualitat:
+\subcite{a}original,
+\subcite{b}JPEG qualitat 10 - PSNR 26,04 dB,
+\subcite{c}descodificació amb
+\emph{Froment} \cite{froment2005} (pesos de Froment, 12 iteracions,
+mida inicial de pas 1) - PSNR 28,92 dB, i
+\subcite{d}descodificació amb \emph{Froment} (com abans) seguit de
+\emph{Trianta} \cite{trianta2002}
+($\sigma_{lf}=2, \sigma_{hf}=2,5, \sigma_{\text{WSMM}}=1, \mu = 5, T=500$) i
+\emph{projecció sobre QCS} - PSNR 27,90 dB}
+\end{figure}
+
+Respecte als plans de color, que en el cas de poca qualitat segurament hauran
+estat delmats i codificats amb unes taules de quantització de valors molt alts,
+donaran grans blocs diferenciats de color a la imatge. El que ens ha semblat més
+correcte ha estat utilitzar algorismes que suavitzin molt, com els de
+\emph{Nosratinia} i \emph{HP}.
+Utilitzant l'escalador ponderat per lluminositat ens disminuirà
+els vessaments de color importants que podem trobar en un cas com aquest.
+
+Si l'ús excessiu d'algorismes suavitzants ens ha reduït el contrast de la
+imatge, podem arreglar-ho a la imatge final mitjançant un programa de retoc
+fotogràfic.
+
+\section{Efecte d'ones superior al de blocs}
+
+Quan els plans es codifiquen a la qualitat sovint oferta per defecte als
+programes de manipulació gràfica ($\approx 75$ a la \texttt{libjpeg}), els
+efectes d'ones sovint superen els de blocs. En aquests casos,
+\emph{Nosratinia} i \emph{HP} no
+els eliminen del tot, i els algorismes iteratius de \emph{Froment},
+\emph{O'Rourke} i
+\emph{Robertson} ofereixen bons resultats, tot i que amb masses iteracions o passos
+per iteració massa grans (més de
+10) suavitzarem la imatge més del compte. Per exemple, quan apliquem
+excessivament \emph{O'Rourke} o \emph{Robertson} (molt similars), els gradients d'intensitat
+de la imatge sofriran una mena de reducció de color. Les petites diferències de
+tonalitat desapareixeran, deixant només les cantonades més marcades.
+
+També hem comprovat l'efectivitat de restaurar el pla de lluminositat amb el
+\emph{filtre Gaussià selectiu} del GIMP, i involucrar el resultat al mig de la
+cadena de descodificació. Si es tracta d'una codificació d'alta qualitat (amb
+intervals de quantització prou petits), aquest algorisme no ens suavitzarà zones
+d'altes freqüències. Això sí, caldrà aplicar una projecció sobre el QCS abans de
+l'escalador per a obtenir una versió possiblement més fidel a la codificació
+original.
+
+Respecte als plans de color, l'escalador de color que pondera segons la
+lluminositat ens dóna els millors resultats dels tres que hi ha implementats en
+totes les imatges que hem provat, també en el cas d'alta qualitat si és que hi
+ha delmat de color. Podem veure un exemple a la Figura \ref{fig:nenufar75}
+
+\begin{figure}[!htp]
+\centering
+\begin{tabular}{cc}
+\includegraphics[width=6cm]{images/nenufar.png} &
+\includegraphics[width=6cm]{images/nenufar-75.png} \\
+a) & b) \\
+\includegraphics[width=6cm]{images/nenufar-75-froment-6-fancy.png} &
+\includegraphics[width=6cm]{images/nenufar-75-froment-6-lumscaler.png} \\
+c) & d)
+\end{tabular}
+\caption{\label{fig:nenufar75}Codificació/descodificació d'una fotografia d'un
+nenúfar de 256x256 pícsels, amb la qualitat per defecte de la \texttt{libjpeg}.
+\subcite{a}original,
+\subcite{b}JPEG qualitat 75, delmat 4:2:0 - PSNR R/G/B 38,47/47,33/33,87 dB,
+\subcite{c}descodificació amb
+\emph{Froment} \cite{froment2005} (pesos de Froment, 6 iteracions,
+mida inicial de pas 1) i escalador \emph{Fancy} - PSNR R/G/B 41,26/46,43/37,52 dB, i
+\subcite{d}descodificació amb \emph{Froment} (com abans) seguit de
+l'escalador \emph{LumScale} - PSNR R/G/B 42,12/47,92/39,82 dB}
+\end{figure}
+
+En el cas de que algunes textures es perdin massa pel suavitzat d'alguns
+algorismes, podem utilitzar l'algorisme \emph{HContrast} (Apèndix \ref{sec:hcontrast})
+per a intentar recuperar-les. Podem veure un exemple del seu ús a la Figura
+\ref{fig:masxa}. A la mateixa figura també hi podem distingir clarament les
+millores de color gràcies a l'algorisme \emph{O'Rourke} i el \emph{LumScaler}.
+
+\begin{figure}[!htp]
+\centering
+\begin{tabular}{cc}
+\includegraphics[width=6cm]{images/masxa-crop-jpg.png} &
+\includegraphics[width=6cm]{images/masxa-crop-froment-5.png} \\
+a) & b) \\
+\includegraphics[width=6cm]{images/masxa-crop-froment-5-orourke-lum.png} &
+\includegraphics[width=6cm]{images/masxa-crop-froment-5-orourke-lum-hcontrast-0,5.png} \\
+c) & d)
+\end{tabular}
+\caption{\label{fig:masxa}Codificació/descodificació d'una fotografia
+publicada a Internet, de 256x256 pícsels.
+\subcite{a}publicació original,
+\subcite{b}descodificació amb Froment \cite{froment2005} (pesos de Froment, 5 iteracions, mida
+inicial de pas 1),
+\subcite{c}descodificació amb Froment com abans, més O'Rourke \cite{orourke1995}
+(10 iteracions, $T=0,1$, mida inical de pas 1) als canals de color, i
+l'escalador LumScaler ($\sigma_h = \sigma_v = 0,25$,
+Apèndix \ref{sec:colorponderat}).
+\subcite{d}descodificació amb Froment i O'Rourke com abans, amb HContrast
+(Llindar 0,05 i finestra de 7x7), i LumScaler també com abans.}
+\end{figure}
+
+\section{Dibuixos}
 
-\printindex
+Tot i que la codificació JPEG va ser pensada per a fotografies, hi ha gent que
+l'ha utilitzat per a conservar i distribuir imatges artificials com dibuixos o
+logotips. En aquests casos el soroll ocasionat per la pèrdua d'informació pot
+ser més molest que en una fotografia codificada amb els mateixos paràmetres.
+Fins el 1996 els dibuixos artificials a través d'Internet estaven dominats pel
+format GIF, ja que fins el 1995 aquest format va ser de lliure ús. Des de
+llavors, l'empresa Unisys va decidir fer complir la patent inherent en el GIF. A
+més, el format GIF limitava la imatge a 256 colors d'una paleta de 24 bits, amb
+un dels colors respresentant la transparència (útil en la composició d'imatges
+en una web). Quan Internet es va popularitzar, poc programari suportava el
+format PNG, sobretot els navegadors web\footnote{Per exemple,
+el Microsoft Internet Explorer
+suporta PNGs només des de la versió 5, i el canal alfa de transparència del PNG
+des de la versió 7.}. Llavors, bé pel límit de colors, bé per costum, molta gent
+ha conservat imatges artificials en JPEG.
+
+Els algorismes de recuperació analitzats dónen resultats especialment bons en
+aquest tipus d'imatges. Ens referim a imatges amb un contrast molt alt,
+amb pocs colors i les cantonades molt ben marcades. Precisament això fa que ens
+trobem més vessaments de color, i un acusat efecte d'ones.
+Els mètodes iteratius,
+especialment \emph{O'Rourke} i \emph{Robertson}, combinats amb l'escalador
+ponderat per lluminositat, ofereixen recuperacions
+excel·lents en aquestes condicions. En podem veure un exemple a les Figures
+\ref{fig:dibuix-logo} i \ref{fig:dibuix-stimpy}.
+
+\urldef{\logocopamon}\url{http://www.futsalcat.com/seleccioabsoluta/copamon/LOGO%20COPA%20DEL%20MON%202007.jpg}
+\urldef{\stimpy}\url{http://www.tvacres.com/images/stimpy.jpg}
+
+\begin{figure}[!htp]
+\centering
+\begin{tabular}{ccc}
+\includegraphics[width=6cm]{images/logo-copa-mon-jpg.png} &
+\includegraphics[width=6cm]{images/logo-copa-mon-3orourke-7s-lum.png} \\
+a) & b)
+\end{tabular}
+\caption{\label{fig:dibuix-logo}\subcite{a}Logotip d'un campionat del món
+codificat amb JPEG, agafat d'Internet
+(\logocopamon). \subcite{b}Recuperació segons una baula
+d'\emph{O'Rourke} (7 passos, mida de pas de 1, $T=0,1$) per canal, i
+\emph{LumScaler}.
+}
+\end{figure}
+
+\begin{figure}[!htp]
+\centering
+\begin{tabular}{ccc}
+\includegraphics[width=4cm]{images/stimpy-jpg.png} &
+\includegraphics[width=4cm]{images/stimpy-3orourke-5steps.png} \\
+a) & b)
+\end{tabular}
+\caption{\label{fig:dibuix-stimpy}\subcite{a}Personatge dibuixat
+(Stimpy, de John Kricfalusi)
+codificat amb JPEG, agafat d'Internet
+(\stimpy). \subcite{b}Recuperació segons una baula
+d'\emph{O'Rourke} (5 passos, mida de pas de 1, $T=0,1$) per canal, i
+\emph{LumScaler}.
+}
+\end{figure}
+
+\chapter{Conclusions i treball futur}
+% Parlar amb els dels articles
+% Acompliment d'objectius
+% Usuaris contents
+% Futures millores al programa
+% - Publicació, Linux, Mac, Windows
+% - Traducció
+% - Manteniment
+% - Vincles de col·laboració
+% - Atendre recomanacions d'usuaris
+
+\appendix
+
+\chapter{Transformada DCT}
+
+\label{sec:dct}
+
+Al codificar els blocs de 8x8 pícsels de cada pla de la imatge, els apliquem una
+transformada que es coneix amb el nom de FDCT (\emph{Forward Discrete Cosine
+Transform}). Donats els 64 valors d'intensitat de cada pícsel, obtenim 64 nous
+valors del domini DCT.  El primer d'aquests nous 64 valors l'anomenem el
+coeficient DC (de freqüència zero, component contínua), i els 63 restants
+coeficients AC (de freqüència diferent de zero). Mitjançant la transformada
+IDCT (\emph{Inverse DCT}) podem tornar a recuperar els valors dels pícsels,
+donats els 64 valors del domini DCT. Per a un bloc de pícsels $s_{xy}$
+($x,y=0,1,\ldots,7$), la FDCT està establerta com (\cite{kou1995}):
+\[
+S_{uv} = \frac{1}{4} C_u C_v \sum_{x=0}^7 \sum_{y=0}^7 s_{xy}
+\cos \frac{(2x+1)u\pi}{16}
+\cos \frac{(2y+1)v\pi}{16}
+\]
+donant els 64 valors del domini freqüencial $S_{uv}$
+($u,v=0,1,\ldots,7$) i la IDCT com:
+\[
+s_{xy} = \frac{1}{4} C_u C_v S_{uv} \sum_{x=0}^7 \sum_{y=0}^7
+\cos \frac{(2x+1)u\pi}{16}
+\cos \frac{(2y+1)v\pi}{16}
+\]
+on
+\[
+C_m = \begin{cases}
+   \frac{1}{\sqrt{2}} & \text{si} \, m = 0 \\
+   1 & \text{altrament}
+   \end{cases}
+\]
+
+Podem veure gràficament les bases de la transformada per coeficient, i la
+influència gradual de les freqüències més altes a la Figura
+\ref{fig:jpeg-coeficients}, pàg. \pageref{fig:jpeg-coeficients}.
+
+\chapter{Algorismes en detall}
+
+\section{Manteniment de textures}
+
+\label{sec:hcontrast} La majoria dels mètodes de millora de pla estudiats
+utilitzen alguna mena de suavitzat per eliminar, entre altres, els efectes de
+bloc. Si aquest suavitzat s'aplica a zones on hi ha textures d'alta freqüència,
+el resultat pot empitjorar respecte a una descodificació convencional. En la
+majoria de casos això es traduirà en un augment de la PSNR, però ja hem explicat
+a l'apartat \ref{sec:psnr} com la PSNR no ens serveix com a mesura de qualitat
+en zones d'altes freqüències.
+
+És per això que hem proposat un mètode adaptatiu per a no empitjorar
+aquest tipus de zones. Ens basem en que si suavitzem una
+zona d'altes freqüències, el seu contrast es veurà reduit. Per cada punt de la
+imatge, considerem que si hem
+reduit molt el contrast local, val més recuperar el punt de la descompressió
+convencional.
+Pas per pas, evaluem segons el contrast 
+local per cada punt $i,j$ de la imatge. El contrast el valorem segons
+Michelson: \revisar{Ho he tret de la wikipedia. Crec que és prou conegut com per
+no citar-ne cap font.}
+\newcommand{\Itext}[1]{I_{\text{#1}}}
+\[ C(i,j) = \frac{\Itext{max}(i,j) - \Itext{min}(i,j)}{\Itext{max}(i,j)
++ \Itext{min}(i,j)} \]
+
+A l'entorn local:
+\begin{align}
+    \Itext{max}(i,j) &= \max_{k,l \in S} y(k,l) \\
+    \Itext{min}(i,j) &= \min_{k,l \in S} y(k,l) \\
+\end{align}
+on $y(k,l)$ és el valor associat al punt $k,l$ de la imatge, i $S$ el conjunt
+de punts d'una finestra quadrada de 3x3, 5x5, 7x7, \ldots a l'entorn de $i,j$.
+
+Calculem doncs el contrast local al pla restaurat $C(i,j)$, i 
+al pla descodificat de manera convencional $C'(i,j)$ per cada $i,j$ de la
+imatge. Llavors, si $y_1(i,j)$ fa referència a un punt del pla restaurat, i
+$y_c(i,j)$ a un del pla convencional, el nou punt restaurat $y_2(i,j)$
+ve determinat per:
+\[ y_2(i,j) = \begin{cases}
+   y_1(i,j) & \text{si} \, C'(i,j) - C(i,j) > \epsilon \\
+   y_c(i,j) & \text{altrament}
+   \end{cases}
+\]
+
+L'usuari pot escollir tant el llindar $\epsilon$ com la mida de la finestra de
+$S$.
+
+\section{Escalat dels plans de color ponderats amb la lluminositat}
+
+\label{sec:colorponderat}A l'article \cite{sugita2006} proposen el ponderat
+per a imatges que tenen els
+plans de color delmats co-situats (\emph{cosited}). Això vol dir que els punts
+del pla de color delmat coincideixen amb la posició de punts del pla sense
+delmar. A les imatges JPEG, tal com diu l'especificació del JFIF \cite{jfif},
+els plans delmats tenen els
+punts inter-situats (\emph{intersited}) respecte al pla més gran (vegeu la
+Figura \ref{fig:co-inter-situats422}. Això fa una
+mica més complexes els algorismes.
+
+\figuraw{7cm}{co-inter-situats422.pdf}{\label{fig:co-inter-situats422}Diferència entre
+un pla de color delmat co-situat i un d'inter-situat pel cas 4:2:2.}
+
+\subsection{Delmat 4:2:2}
+Comencem pel cas 4:2:2. El valor de croma del pla delmat, a la
+codificació, ha estat calculat segons la mitjana aritmètica dels dos punts
+originals més propers.
+Aquest algorisme proposa fer el mateix amb el pla de
+lluminositat, com si el delméssim.
+Del pla delmat anomenem $Y_1$ al punt més proper al nostre punt d'interès,
+i $Y_2$ al segon més proper. Com que el pla de lluminància el tenim també
+sense delmar, sabem el valor del punt d'interès $Y_o$. Llavors, podem utilitzar
+la relació
+$\tfrac{Y_o-Y_1}{Y_2-Y_1}$ per a ponderar el pla de color. Podem veure-ho
+gràficament a la Figura \ref{fig:lumscaler422}
+
+\figura{lumscaler422.pdf}{\label{fig:lumscaler422}Escalat de color de 4:2:2 a
+4:4:4 amb ponderació per lluminositat. Calculem la versió delmada de la
+lluminositat, i obtenim $Y_1$ i $Y_2$, on $Y_1$ és el punt delmat més proper a
+$Y_o$, i $Y_2$ el segon més proper. Veiem que la interpolació linial entre $Y_1$
+i $Y_2$ ens donaria un punt inferior a $Y_o$ - i aquesta és la correcció que es
+fa palesa al pla de color. Les creus dretes ens mostren la versió delmada, i
+ponderant el pla de color amb $\tfrac{Y_o-Y_1}{Y_2-Y_1}$ aconseguim també
+l'anàlog a $Y_o$ més alt que si utilitzéssim interpolació linial.}
+
+Ara podem reescriure l'algorisme en funció de cada punt de la imatge (delmat i
+sense delmar), i limitar la ponderació segons un paràmetre $\epsilon_h$.
+En general, per cada fila tenim $1..N$ columnes de pla de color
+i $1..2N$ columnes de pla
+de lluminositat. Llavors, per cada punt $i$ del pla de color obtenim:
+\begin{align}
+Y(2i) &= C(i) + (C(i+1) - C(i)) \, w(Y(2i), Y_m(2i-1), Y_m(2i+1) \\
+Y(2i+1) &= C(i+1) + (C(i) - C(i+1)) \, w(Y(2i+1), Y_m(2i+1), Y_m(2i-1))
+\end{align}
+on el pes $w(p,n,f)$ i el delmat de lluminositat $Y_m(j)$ els definim així:
+\begin{align}
+w(p,n,f) &= \begin{cases}
+    0,25 + \epsilon_h & \text{si} \, w'(p,n,f) > 0,25 + \epsilon_h \\
+    0,25 - \epsilon_h & \text{si} \, w'(p,n,f) < 0,25 - \epsilon_h \\
+    w'(p,n,f) & \text{altrament}
+    \end{cases} \\
+w'(p,n,f) &= \frac{p - n}{f-n} \\
+Y_m(j) &= \frac{Y(j) + Y(j+1)}{2}
+\end{align}
+on l'usuari pot escollir $\epsilon_h$.
+
+\subsection{Delmat 4:2:0}
+
+L'algorisme de 4:2:2 involucra dos punts $Y_1$ i $Y_2$ per cada punt de la
+imatge final. En canvi, en el cas 4:2:0 tenim quatre punts involucrats, i ens
+cal adaptar l'algorisme a aquests quatre punts. Com bé sabem, tenim tant delmat
+horitzontal com vertical. Així, la nostra adaptació consisteix en primer
+considerar el cas horitzontal i després el vertical, per cada punt de la imatge.
+
+\figuraw{8cm}{lumscaler420.pdf}{\label{fig:lumscaler420}Escalat de color 4:2:0 a
+4:4:4 ponderant segons la lluminositat.}
+
+Seguint l'esquema de la Figura \ref{fig:lumscaler420}, i amb l'objectiu de
+determinar el color al punt $Y_o$, considerem els punts
+$Y_{1\ldots4}$ com els punts ordenats segons el més proper,
+el segon més proper en
+horitzontal, el tercer més proper en vertical, i el quart més proper. Utilitzant
+la tècnica del cas de 4:2:2, considerem el punt temporal $Y_{o1}$
+i en calculem la
+croma ponderada segons $Y_1$ i $Y_2$ (amb els càlculs pertinents de la
+lluminositat) i $\epsilon_h$.
+Fem el mateix amb el punt $Y_{o2}$, $Y_3$, $Y_4$, i la mateixa $\epsilon_h$.
+Finalment, a partir d'aquestes noves cromes dels punts $Y_o1$ i $Y_o2$, i d'unes
+versions de la lluminostat de $Y_{o1}$
+i $Y_{o2}$ (segons la mitjana dels seus punts
+de lluminostiat més propers), fem la ponderació per aconseguir $Y_o$, aquest copsegons un
+paràmetre $\epsilon_v$ també configurable per l'usuari.
+
+%\printindex
 
 \bibliography{cited}
 
--- a/qjpeg/plugins/QCS.cpp	Sat Nov 17 20:54:13 2007 +0100
+++ b/qjpeg/plugins/QCS.cpp	Fri Dec 07 00:01:11 2007 +0100
@@ -141,10 +141,38 @@
             std::fclose(f);
         }
     }
-    if (DCTimage.getPtr()[j] > DCTmax.getPtr()[j])
-      DCTimage.getPtr()[j] = DCTmax.getPtr()[j];
-    else if (DCTimage.getPtr()[j] < DCTmin.getPtr()[j])
-      DCTimage.getPtr()[j] = DCTmin.getPtr()[j];
+
+    if (1) /* normal flow */
+    {
+        if (DCTimage.getPtr()[j] > DCTmax.getPtr()[j])
+          DCTimage.getPtr()[j] = DCTmax.getPtr()[j];
+        else if (DCTimage.getPtr()[j] < DCTmin.getPtr()[j])
+          DCTimage.getPtr()[j] = DCTmin.getPtr()[j];
+    } else if (0) /* Minimum coefs */
+    {
+        float &min = DCTmin.getPtr()[j];
+        float &max = DCTmax.getPtr()[j];
+        float &final = DCTimage.getPtr()[j];
+
+        if (min > 0 && max > 0)
+            final = min;
+        else if (min < 0 && max > 0)
+            final = 0;
+        else
+            final = max;
+    } else if (0) /* Maximum coefs */
+    {
+        float &min = DCTmin.getPtr()[j];
+        float &max = DCTmax.getPtr()[j];
+        float &final = DCTimage.getPtr()[j];
+
+        if (min > 0 && max > 0)
+            final = max;
+        else if (min < 0 && max > 0)
+            final = max;
+        else
+            final = min;
+    }
   }
 
   fdct_inverse_image(DCTimage.getPtr(), out.getPtr(), width, height);
--- a/src/Makefile	Sat Nov 17 20:54:13 2007 +0100
+++ b/src/Makefile	Fri Dec 07 00:01:11 2007 +0100
@@ -6,7 +6,8 @@
 LDFLAGS_JPEG=
 
 # Programs
-PROGRAMS=pgmtosli pgm2errormat mat2pgm imgerror jpegdescribe test_openjpeg
+PROGRAMS=pgmtosli pgm2errormat mat2pgm imgerror jpegdescribe test_openjpeg \
+		dctexample
 LIBRARIES=libjpegrest.a
 
 # Liblibjpegrest
@@ -29,6 +30,7 @@
 imgerror.o: imgerror.c libjpegrest.h
 test_openjpeg.o: test_openjpeg.c libjpegrest.h
 jpegdescribe.o: jpegdescribe.c
+dctexample.o: dctexample.c libjpegrest.h
 
 # Objects of libjpegrest
 error_psnr.o: error_psnr.c libjpegrest.h
@@ -53,6 +55,11 @@
 imgerror: imgerror.o libjpegrest.a jpeg-7a/libjpeg.a
 	$(CC) $(LDFLAGS_NETPBM) $(LDFLAGS_GSL) \
 		-o $@ $< libjpegrest.a jpeg-7a/libjpeg.a
+
+dctexample: dctexample.o libjpegrest.a jpeg-7a/libjpeg.a
+	$(CC) $(LDFLAGS_NETPBM) $(LDFLAGS_GSL) \
+		-o $@ $< libjpegrest.a jpeg-7a/libjpeg.a
+
 test_openjpeg: test_openjpeg.o libjpegrest.a jpeg-7a/libjpeg.a
 	$(CC) $(LDFLAGS_NETPBM) $(LDFLAGS_GSL) \
 		-o $@ $< libjpegrest.a jpeg-7a/libjpeg.a
--- a/src/bitmap.c	Sat Nov 17 20:54:13 2007 +0100
+++ b/src/bitmap.c	Fri Dec 07 00:01:11 2007 +0100
@@ -19,6 +19,9 @@
 		assert(bmap.m[plane] != NULL);
 	}
 
+    bmap.width = width;
+    bmap.height = height;
+
 	bmap.is_ready = 1;
 
 	return bmap;
--- a/src/bitmap_pam.c	Sat Nov 17 20:54:13 2007 +0100
+++ b/src/bitmap_pam.c	Fri Dec 07 00:01:11 2007 +0100
@@ -1,5 +1,7 @@
 #include <stdio.h>
 #include <pam.h>
+#include <assert.h>
+#include <string.h>
 #include "libjpegrest.h"
 
 void bitmap_pam_init(int *argn, char **argv)
@@ -36,7 +38,6 @@
 	tuple **tuples;
 	int row, column, plane;
 
-
 	/* Reading the input files PAM header */
 	tuples = pnm_readpam(input, &inpam, PAM_STRUCT_SIZE(tuple_type));
 	if (tuples == NULL)
@@ -61,3 +62,80 @@
 	pnm_freepamarray(tuples, &inpam);
 	return 0;
 }
+
+int
+bitmap_filename_pam_put(const char *filename, struct bitmap *bmap)
+{
+	FILE *output;
+	int res;
+
+	/* Opening the FILE */
+	output = fopen(filename,"w");
+	if (output == NULL)
+	{
+		fprintf(stderr,"The file \"%s\" cannot be opened.\n",
+			filename);
+		res = -1;
+	} else
+	{
+		res = bitmap_stdio_pam_put(output, bmap);
+		fclose(output);
+	}
+	
+	return res;
+}
+
+static unsigned char roundsample(float val)
+{
+    const float MAXVALUE = 255.;
+    const float MINVALUE = 0.;
+
+    val += 0.5;
+    if (val > MAXVALUE)
+        val = MAXVALUE;
+    else if (val < MINVALUE)
+        val = MINVALUE;
+
+    return (unsigned char) val;
+}
+
+
+int
+bitmap_stdio_pam_put(FILE *output, struct bitmap *bmap)
+{
+	struct pam outpam;
+	tuple *tuplerow;
+	int row, column, plane;
+
+    assert(bmap->nplanes == 1); /* TODO: only implemented for PGM. */
+
+    outpam.size = sizeof(outpam);
+    outpam.len = outpam.size;
+    outpam.file = output;
+    outpam.format = PGM_FORMAT;
+    outpam.plainformat = 0; /* false */
+    outpam.width = bmap->width;
+    outpam.height = bmap->height;
+    outpam.depth = 1; /* Grayscale - 1 sample per tuple */
+    outpam.allocation_depth = 0; /* Same as depth. */
+    outpam.maxval = 255;
+    strcpy(outpam.tuple_type, "GRAYSCALE"); /* PGM non transparent */
+
+	pnm_writepaminit(&outpam);
+
+    tuplerow = pnm_allocpamrow(&outpam);
+
+	for (row = 0; row < bmap->height; row++) {
+		for (column = 0; column < bmap->width; ++column) {
+			for (plane = 0; plane < bmap->nplanes; plane++)
+                tuplerow[column][plane] = roundsample(
+                        bmap->m[plane]->data[row *
+                            bmap->m[plane]->tda + column]);
+		}
+        pnm_writepamrow(&outpam, tuplerow);
+	}
+
+	/* Free the pnm memory */
+	pnm_freepamrow(tuplerow);
+	return 0;
+}
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/dctexample.c	Fri Dec 07 00:01:11 2007 +0100
@@ -0,0 +1,172 @@
+#include <stdio.h>
+#include <math.h>
+#include "libjpegrest.h"
+
+#define PI 3.141592657
+
+/* This opens a 8x8 PGM file from argv[1], and stores
+ * a 64+7x64+7 PGM file with each DCT-IDCT result, considering
+ * only the coefficients from top-left until the concerning
+ * DCT-IDCT result. */
+
+static void mydct(float *in, float *out)
+{
+    int x,y,u,v;
+
+    float Cu, Cv;
+
+    for (u=0; u < 8; ++u)
+        for (v=0; v < 8; ++v)
+        {
+            float *optr;
+            float sum;
+            optr = out+v*8+u;
+
+            sum = 0.;
+            for (x=0; x < 8; ++x)
+                for (y=0; y < 8; ++y) {
+                    float *iptr;
+                    iptr = in+y*8+x;
+
+                    sum += *iptr
+                        * cos(((float) ((2*x+1)*u) )*PI/16.)
+                        * cos(((float) ((2*y+1)*v) )*PI/16.);
+                }
+
+            if (u == 0) Cu = 1. / sqrt(2); else Cu = 1.;
+            if (v == 0) Cv = 1. / sqrt(2); else Cv = 1.;
+            *optr = 0.25 * Cu * Cv * sum;
+        }
+}
+
+static void myidct(float *in, float *out)
+{
+    int x,y,u,v;
+
+    float Cu, Cv;
+
+    for (x=0; x < 8; ++x)
+        for (y=0; y < 8; ++y)
+        {
+            float *optr;
+            float sum;
+            optr = out+y*8+x;
+
+            sum = 0.;
+            for (v=0; v < 8; ++v)
+                for (u=0; u < 8; ++u)
+                {
+                    float *iptr;
+                    iptr = in+v*8+u;
+
+                    if (u == 0) Cu = 1. / sqrt(2); else Cu = 1.;
+                    if (v == 0) Cv = 1. / sqrt(2); else Cv = 1.;
+
+                    sum += Cu * Cv * *iptr
+                        * cos(((float) ((2*x+1)*u) )*PI/16.)
+                        * cos(((float) ((2*y+1)*v) )*PI/16.);
+                }
+
+            *optr = 0.25 * sum;
+        }
+}
+
+void zero_block(float *block)
+{
+    int i;
+    for (i=0; i < 64; ++i)
+        block[i] = 0.;
+}
+
+void prepare_subblock(float *block, int x, int y, const float *input)
+{
+    /* Fill the subsquare from 0..x and 0..y */
+    int i,j;
+
+    zero_block(block);
+
+    for (j=0; j <= y; ++j)
+        for (i=0; i <= x; ++i)
+        {
+            block[j*8+i] = input[j*8+i];
+        }
+}
+
+void write_block_to_big(struct bitmap *bdest, const float *sub, int x, int y)
+{
+    int i,j;
+
+    for (j=0; j < 8; ++j)
+        for (i=0; i < 8; ++i)
+        {
+            gsl_matrix *dest = bdest->m[0];
+            dest->data[(j+y*9)*dest->tda+x*9+i] = sub[j*8+i];
+        }
+
+    if (x < 7)
+        for (j=0; j < 8; ++j)
+        {
+            gsl_matrix *dest = bdest->m[0];
+            dest->data[(j+y*9)*dest->tda+x*9+8] = 255;
+        }
+    if (y < 7)
+        for (i=0; i < 8; ++i)
+        {
+            gsl_matrix *dest = bdest->m[0];
+            dest->data[(8+y*9)*dest->tda+x*9+i] = 255;
+        }
+
+    if (y < 7 && x < 7)
+    {
+        gsl_matrix *dest = bdest->m[0];
+        dest->data[(8+y*9)*dest->tda+x*9+8] = 255;
+    }
+}
+
+int main(int argn, char **argv)
+{
+    struct bitmap bmap;
+    float input[64];
+    float input_test[64];
+    float input_dct[64];
+    float output_dct[64];
+    float output[64];
+    struct bitmap obmap;
+    int i,j,k;
+
+    bitmap_pam_init(&argn, argv);
+
+    bitmap_filename_pam_get(argv[1], &bmap);
+
+    if (bmap.nplanes != 1 || bmap.width != 8 || bmap.height != 8)
+    {
+        fprintf(stderr, "Error: The file should be 8x8 PGM\n");
+        return -1;
+    }
+
+    k=0;
+    for(j=0;j < 8; ++j)
+        for(i=0;i < 8; ++i)
+        {
+            gsl_matrix *mat = bmap.m[0];
+            input[k++] = mat->data[j*mat->tda + i];
+        }
+
+    mydct(input, input_dct);
+    myidct(input_dct, input_test);
+
+    obmap = bitmap_alloc(1, 64+7, 64+7);
+
+    for(j=0;j < 8; ++j)
+        for(i=0;i < 8; ++i)
+        {
+            prepare_subblock(output_dct,i,j,input_dct);
+            myidct(output_dct, output);
+            write_block_to_big(&obmap, output, i, j);
+        }
+
+    bitmap_filename_pam_put(argv[2], &obmap);
+
+    bitmap_free(obmap);
+    return 0;
+}
--- a/src/libjpegrest.h	Sat Nov 17 20:54:13 2007 +0100
+++ b/src/libjpegrest.h	Fri Dec 07 00:01:11 2007 +0100
@@ -50,7 +50,9 @@
 /* bitmap_pam.c */
 void bitmap_pam_init(int *argn, char **argv);
 int bitmap_stdio_pam_get(FILE *input, struct bitmap *bmap);
+int bitmap_stdio_pam_put(FILE *output, struct bitmap *bmap);
 int bitmap_filename_pam_get(const char *filename, struct bitmap *bmap);
+int bitmap_filename_pam_put(const char *filename, struct bitmap *bmap);
 
 /* bitmap_jpeg.c */
 enum jpeg_conditions