* Final version of the paper. Uploaded at the last minute, of course.

git-svn-id: file:///srv/caca.zoy.org/var/lib/svn/research@2293 92316355-f0b4-4df1-b90c-862c8a59935f

2008-displacement/README

@@ -66,11 +66,26 @@ set xrange [-1:1]
 set yrange [1:-1]
 set mxtics 2
 set mytics 2
+set xlabel "$dx$"
+set ylabel "$dy$"
+unset mztics
+unset ztics
 set size square
 set xzeroaxis
 set yzeroaxis
-set xlabel "dx" font "Italic,32"
-splot 'part0/lena-values.txt' with lines
+set lmargin screen 0.0 #.05
+set bmargin screen 0.2 # .15
+set tmargin screen 0.9 #.95
+set rmargin screen 0.9 #.95
+set terminal epslatex color size 5,2 blacktext
+unset colorbox
+#set lmargin 0.05
+#set rmargin 0.95
+#splot 'part0/lena-values.txt' with lines
+set output "lena-values.tex"
+splot '../part0/lena-values.txt' with lines lw 3 notitle
+unset output
+set term pop
 
 
 #set border 0
@@ -127,19 +142,21 @@ sed -ne 's/.* for //p' fs-*txt | ./xy2d >| histo-fs.txt
 # Et ensuite dans gnuplot:
 set xrange [-.1:.4]
 set yrange [.4:-.1]
+set xlabel "$dx$"
+set ylabel "$dy$"
 set grid
 set lmargin screen 0.05
-set bmargin screen 0.15
+set bmargin screen 0.25
 set tmargin screen 0.95
 set rmargin screen 0.95
 unset colorbox
 set pm3d explicit map interpolate 1.8,1.8
 set palette model RGB defined(0 "white", 1 "blue", 2 "red", 3 "yellow")
 unset key
-#splot "histo-fs.txt" notitle
-set terminal epslatex color size 1.8,1.8 blacktext
+#splot "../../subblock/histo-fs.txt" notitle
+set terminal epslatex color size 1.8,2.0 blacktext
 set output "fs-histo.tex"
-splot "histo-fs.txt" notitle
+splot "../../subblock/histo-fs.txt" notitle
 unset output
 set term pop
 
@@ -155,19 +172,21 @@ sed -ne 's/.* for //p' jajuni*txt | ./xy2d-jajuni >| histo-jajuni.txt
 # Et ensuite dans gnuplot:
 set xrange [-.1:.9]
 set yrange [.9:-.1]
+set xlabel "$dx$"
+set ylabel "$dy$"
 set grid
 set lmargin screen 0.05
-set bmargin screen 0.15
+set bmargin screen 0.25
 set tmargin screen 0.95
 set rmargin screen 0.95
 unset colorbox
 set pm3d explicit map interpolate 1.8,1.8
 set palette model RGB defined(0 "white", 1 "blue", 2 "red", 3 "yellow")
 unset key
-#splot "histo-jajuni.txt" notitle
-set terminal epslatex color size 1.8,1.8 blacktext
+#splot "../../subblock/histo-jajuni.txt" notitle
+set terminal epslatex color size 1.8,2.0 blacktext
 set output "jajuni-histo.tex"
-splot "histo-jajuni.txt" notitle
+splot "../../subblock/histo-jajuni.txt" notitle
 unset output
 set term pop
 
@@ -190,19 +209,22 @@ sed -ne 's/.* for //p' ostro*txt | ./xy2d-ostro >| histo-ostro.txt
 # Et ensuite dans gnuplot:
 set xrange [-.2:.2]
 set yrange [.3:-.1]
+set xlabel "$dx$"
+set ylabel "$dy$"
+set xtics -.2,0.1
 set grid
 set lmargin screen 0.05
-set bmargin screen 0.15
+set bmargin screen 0.25
 set tmargin screen 0.95
 set rmargin screen 0.95
 unset colorbox
 set pm3d explicit map interpolate 1.8,1.8
 set palette model RGB defined(0 "white", 1 "blue", 2 "red", 3 "yellow")
 unset key
-splot "histo-ostro.txt" notitle
-set terminal epslatex color size 1.8,1.8 blacktext
+splot "../../subblock/histo-ostro.txt" notitle
+set terminal epslatex color size 1.8,2.0 blacktext
 set output "ostro-histo.tex"
-splot "histo-ostro.txt" notitle
+splot "../../subblock/histo-ostro.txt" notitle
 unset output
 set term pop
 
@@ -219,19 +241,21 @@ sed -ne 's/.* for //p' optimum*txt | ./xy2d-optimum >| histo-optimum.txt
 # Et ensuite dans gnuplot:
 set xrange [-.5:.2]
 set yrange [.6:-.1]
+set xlabel "$dx$"
+set ylabel "$dy$"
 set grid
 set lmargin screen 0.05
-set bmargin screen 0.15
+set bmargin screen 0.25
 set tmargin screen 0.95
 set rmargin screen 0.95
 unset colorbox
 set pm3d explicit map interpolate 1.8,1.8
 set palette model RGB defined(0 "white", 1 "blue", 2 "red", 3 "yellow")
 unset key
-splot "histo-optimum.txt" notitle
-set terminal epslatex color size 1.8,1.8 blacktext
+splot "../../subblock/histo-optimum.txt" notitle
+set terminal epslatex color size 1.8,2.0 blacktext
 set output "optimum-histo.tex"
-splot "histo-optimum.txt" notitle
+splot "../../subblock/histo-optimum.txt" notitle
 unset output
 set term pop
 
@@ -248,19 +272,21 @@ sed -ne 's/.* for //p' serpopt*txt | ./xy2d-serpopt >| histo-serpopt.txt
 # Et ensuite dans gnuplot:
 set xrange [-.3:.3]
 set yrange [.6:-.1]
+set xlabel "$dx$"
+set ylabel "$dy$"
 set grid
 set lmargin screen 0.05
-set bmargin screen 0.15
+set bmargin screen 0.25
 set tmargin screen 0.95
 set rmargin screen 0.95
 unset colorbox
 set pm3d explicit map interpolate 1.8,1.8
 set palette model RGB defined(0 "white", 1 "blue", 2 "red", 3 "yellow")
 unset key
-splot "histo-serpopt.txt" notitle
-set terminal epslatex color size 1.8,1.8 blacktext
+splot "../../subblock/histo-serpopt.txt" notitle
+set terminal epslatex color size 1.8,2.0 blacktext
 set output "serpopt-histo.tex"
-splot "histo-serpopt.txt" notitle
+splot "../../subblock/histo-serpopt.txt" notitle
 unset output
 set term pop
 
@@ -286,4 +312,7 @@ for x in out-*.txt; do sort -k7 $x | head -20 ; done | cut -f1 -d: | sort | uniq
 ### Conversion du vieux format de résultats (deprecated)
 cat -n $x | sed 's/^  *[0-9]*\([0-9]\)[[:space:]]*/[\1] /; s/....###/###/; s/\[[27]/[1/; s/\[[38]/[2/; s/\[[49]/[3/; s/\[[50]/[4/'
 
-# 
+# Convert lena results to cropped detail
+convert output-7-3-5-1-serp.png -crop 160x160+32+160 crop-7-3-5-1-serp.png
+convert output-7-4-5-0-serp.png -crop 160x160+32+160 crop-7-4-5-0-serp.png
+

2008-displacement/paper/Makefile

@@ -1,10 +1,14 @@
 all: paper.pdf
 
-paper.pdf: paper.tex fs-histo.tex
+IMAGES = direction.tex fs-histo.tex jajuni-histo.tex lena-values.tex optimum-histo.tex ostro-histo.tex serpopt-histo.tex
+
+paper.pdf: paper.tex $(IMAGES)
 	latex paper.tex
 	latex paper.tex
-	dvipdf paper.dvi
+	dvips paper.dvi
+	sed -i 's,/Title.*,/Title (Reinstating Floyd-Steinberg: Improved Metrics for Quality Assessment of Error Diffusion Algorithms),' paper.ps
+	ps2pdf paper.ps
 
 clean:
-	rm -f paper.log paper.dvi paper.aux paper.pdf
+	rm -f paper.log paper.dvi paper.aux paper.pdf paper.ps
 

2008-displacement/paper/README

@@ -0,0 +1,2 @@
+Author: Sam Hocevar <sam@zoy.org>
+

2008-displacement/paper/crop-7-3-5-1-serp.eps

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+%%CreationDate: (Tue Apr 15 23:54:59 2008)
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+%%HiResBoundingBox: 32 160 192 320
+%%DocumentData: Clean7Bit
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+%%Pages: 1
+%%EndComments
+
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+  currentfile color_packet readhexstring pop pop
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+%%EOF

2008-displacement/paper/crop-7-4-5-0-serp.eps

@@ -0,0 +1,369 @@
+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: (ImageMagick)
+%%Title: (crop-7-4-5-0-serp.eps)
+%%CreationDate: (Tue Apr 15 23:55:04 2008)
+%%BoundingBox: 32 160 192 320
+%%HiResBoundingBox: 32 160 192 320
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 1
+%%Pages: 1
+%%EndComments
+
+%%BeginDefaults
+%%EndDefaults
+
+%%BeginProlog
+%
+% Display a color image.  The image is displayed in color on
+% Postscript viewers or printers that support color, otherwise
+% it is displayed as grayscale.
+%
+/DirectClassPacket
+{
+  %
+  % Get a DirectClass packet.
+  %
+  % Parameters:
+  %   red.
+  %   green.
+  %   blue.
+  %   length: number of pixels minus one of this color (optional).
+  %
+  currentfile color_packet readhexstring pop pop
+  compression 0 eq
+  {
+    /number_pixels 3 def
+  }
+  {
+    currentfile byte readhexstring pop 0 get
+    /number_pixels exch 1 add 3 mul def
+  } ifelse
+  0 3 number_pixels 1 sub
+  {
+    pixels exch color_packet putinterval
+  } for
+  pixels 0 number_pixels getinterval
+} bind def
+
+/DirectClassImage
+{
+  %
+  % Display a DirectClass image.
+  %
+  systemdict /colorimage known
+  {
+    columns rows 8
+    [
+      columns 0 0
+      rows neg 0 rows
+    ]
+    { DirectClassPacket } false 3 colorimage
+  }
+  {
+    %
+    % No colorimage operator;  convert to grayscale.
+    %
+    columns rows 8
+    [
+      columns 0 0
+      rows neg 0 rows
+    ]
+    { GrayDirectClassPacket } image
+  } ifelse
+} bind def
+
+/GrayDirectClassPacket
+{
+  %
+  % Get a DirectClass packet;  convert to grayscale.
+  %
+  % Parameters:
+  %   red
+  %   green
+  %   blue
+  %   length: number of pixels minus one of this color (optional).
+  %
+  currentfile color_packet readhexstring pop pop
+  color_packet 0 get 0.299 mul
+  color_packet 1 get 0.587 mul add
+  color_packet 2 get 0.114 mul add
+  cvi
+  /gray_packet exch def
+  compression 0 eq
+  {
+    /number_pixels 1 def
+  }
+  {
+    currentfile byte readhexstring pop 0 get
+    /number_pixels exch 1 add def
+  } ifelse
+  0 1 number_pixels 1 sub
+  {
+    pixels exch gray_packet put
+  } for
+  pixels 0 number_pixels getinterval
+} bind def
+
+/GrayPseudoClassPacket
+{
+  %
+  % Get a PseudoClass packet;  convert to grayscale.
+  %
+  % Parameters:
+  %   index: index into the colormap.
+  %   length: number of pixels minus one of this color (optional).
+  %
+  currentfile byte readhexstring pop 0 get
+  /offset exch 3 mul def
+  /color_packet colormap offset 3 getinterval def
+  color_packet 0 get 0.299 mul
+  color_packet 1 get 0.587 mul add
+  color_packet 2 get 0.114 mul add
+  cvi
+  /gray_packet exch def
+  compression 0 eq
+  {
+    /number_pixels 1 def
+  }
+  {
+    currentfile byte readhexstring pop 0 get
+    /number_pixels exch 1 add def
+  } ifelse
+  0 1 number_pixels 1 sub
+  {
+    pixels exch gray_packet put
+  } for
+  pixels 0 number_pixels getinterval
+} bind def
+
+/PseudoClassPacket
+{
+  %
+  % Get a PseudoClass packet.
+  %
+  % Parameters:
+  %   index: index into the colormap.
+  %   length: number of pixels minus one of this color (optional).
+  %
+  currentfile byte readhexstring pop 0 get
+  /offset exch 3 mul def
+  /color_packet colormap offset 3 getinterval def
+  compression 0 eq
+  {
+    /number_pixels 3 def
+  }
+  {
+    currentfile byte readhexstring pop 0 get
+    /number_pixels exch 1 add 3 mul def
+  } ifelse
+  0 3 number_pixels 1 sub
+  {
+    pixels exch color_packet putinterval
+  } for
+  pixels 0 number_pixels getinterval
+} bind def
+
+/PseudoClassImage
+{
+  %
+  % Display a PseudoClass image.
+  %
+  % Parameters:
+  %   class: 0-PseudoClass or 1-Grayscale.
+  %
+  currentfile buffer readline pop
+  token pop /class exch def pop
+  class 0 gt
+  {
+    currentfile buffer readline pop
+    token pop /depth exch def pop
+    /grays columns 8 add depth sub depth mul 8 idiv string def
+    columns rows depth
+    [
+      columns 0 0
+      rows neg 0 rows
+    ]
+    { currentfile grays readhexstring pop } image
+  }
+  {
+    %
+    % Parameters:
+    %   colors: number of colors in the colormap.
+    %   colormap: red, green, blue color packets.
+    %
+    currentfile buffer readline pop
+    token pop /colors exch def pop
+    /colors colors 3 mul def
+    /colormap colors string def
+    currentfile colormap readhexstring pop pop
+    systemdict /colorimage known
+    {
+      columns rows 8
+      [
+        columns 0 0
+        rows neg 0 rows
+      ]
+      { PseudoClassPacket } false 3 colorimage
+    }
+    {
+      %
+      % No colorimage operator;  convert to grayscale.
+      %
+      columns rows 8
+      [
+        columns 0 0
+        rows neg 0 rows
+      ]
+      { GrayPseudoClassPacket } image
+    } ifelse
+  } ifelse
+} bind def
+
+/DisplayImage
+{
+  %
+  % Display a DirectClass or PseudoClass image.
+  %
+  % Parameters:
+  %   x & y translation.
+  %   x & y scale.
+  %   label pointsize.
+  %   image label.
+  %   image columns & rows.
+  %   class: 0-DirectClass or 1-PseudoClass.
+  %   compression: 0-none or 1-RunlengthEncoded.
+  %   hex color packets.
+  %
+  gsave
+  /buffer 512 string def
+  /byte 1 string def
+  /color_packet 3 string def
+  /pixels 768 string def
+
+  currentfile buffer readline pop
+  token pop /x exch def
+  token pop /y exch def pop
+  x y translate
+  currentfile buffer readline pop
+  token pop /x exch def
+  token pop /y exch def pop
+  currentfile buffer readline pop
+  token pop /pointsize exch def pop
+  /Times-Roman findfont pointsize scalefont setfont
+  x y scale
+  currentfile buffer readline pop
+  token pop /columns exch def
+  token pop /rows exch def pop
+  currentfile buffer readline pop
+  token pop /class exch def pop
+  currentfile buffer readline pop
+  token pop /compression exch def pop
+  class 0 gt { PseudoClassImage } { DirectClassImage } ifelse
+  grestore
+} bind def
+%%EndProlog
+%%Page:  1 1
+%%PageBoundingBox: 32 160 192 320
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+%%PageTrailer
+%%Trailer
+%%EOF

2008-displacement/paper/fs-histo.tex

@@ -72,7 +72,7 @@
     \fi
   \fi
   \setlength{\unitlength}{0.0500bp}%
-  \begin{picture}(2590.00,2590.00)%
+  \begin{picture}(2590.00,2880.00)%
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@@ -400,29 +400,31 @@
       \color{black}%
       \color{black}%
       \csname LTb\endcsname%
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       \csname LTb\endcsname%
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+      \put(1062,435){\makebox(0,0){\strut{} 0.1}}%
       \csname LTb\endcsname%
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       \csname LTb\endcsname%
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+      \put(1994,435){\makebox(0,0){\strut{} 0.3}}%
       \csname LTb\endcsname%
-      \put(2460,104){\makebox(0,0){\strut{} 0.4}}%
+      \put(2460,435){\makebox(0,0){\strut{} 0.4}}%
+      \put(1295,105){\makebox(0,0){\strut{}$dx$}}%
       \csname LTb\endcsname%
-      \put(-42,2460){\makebox(0,0)[r]{\strut{}-0.1}}%
+      \put(-42,2735){\makebox(0,0)[r]{\strut{}-0.1}}%
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+      \put(-42,2332){\makebox(0,0)[r]{\strut{} 0}}%
       \csname LTb\endcsname%
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+      \put(-42,1527){\makebox(0,0)[r]{\strut{} 0.2}}%
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+      \put(-42,721){\makebox(0,0)[r]{\strut{} 0.4}}%
+      \put(-636,1728){\rotatebox{90}{\makebox(0,0){\strut{}$dy$}}}%
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2008-displacement/paper/jajuni-histo.tex

@@ -72,7 +72,7 @@
     \fi
   \fi
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+  \begin{picture}(2590.00,2880.00)%
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       \color{black}%
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+      \put(2227,435){\makebox(0,0){\strut{} 0.8}}%
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2008-displacement/paper/lena-min.tex → 2008-displacement/paper/lena-values.tex

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2008-displacement/paper/optimum-histo.tex

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2008-displacement/paper/ostro-histo.tex

@@ -72,7 +72,7 @@
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2008-displacement/paper/paper.tex

@@ -101,10 +101,9 @@ an error value based on that model. One such algorithm is direct binary seach
 (DBS) \cite{allebach}, also referred to as least-squares model-based halftoning
 (LSMB) \cite{lsmb}.
 
-HVS models are usually low-pass filters. Nasanen \cite{nasanen}, Analoui
-and Allebach \cite{allebach} found that using Gaussian models gave visually
-pleasing results, an observation confirmed by independent visual perception
-studies \cite{mcnamara}.
+HVS models are usually low-pass filters. Nasanen \cite{nasanen}, Analoui and
+Allebach found that using Gaussian models gave visually pleasing results, an
+observation confirmed by independent visual perception studies \cite{mcnamara}.
 
 DBS yields halftones of impressive quality. However, despite efforts to make
 it more efficient \cite{bhatt}, it suffers from its large computational
@@ -115,8 +114,8 @@ requirements and error diffusion remains a more widely used technique.
 Most error diffusion implementations parse the image in raster scan order.
 Boustrophedonic (serpentine) scanning has been shown to cause fewer visual
 artifacts \cite{halftoning}, but other, more complex processing paths such as
-Hilbert curves \cite{spacefilling}, \cite{peano} are seldom used as they do not
-improve the image quality significantly.
+Hilbert curves \cite{spacefilling} are seldom used as they do not improve the
+image quality significantly.
 
 Intuitively, as the error is always propagated to the bottom-left or
 bottom-right of each pixel (Fig. \ref{fig:direction}), one may expect the
@@ -168,7 +167,7 @@ Finding the second filter is then straightforward:
 
 Experiments show that for a given image and a given corresponding halftone,
 $E_{dx,dy}$ has a local minimum almost always away from $(dx,dy) = (0,0)$ (Fig.
-\ref{fig:lena-min}). Let $E$ be an error metric where this remains true. We
+\ref{fig:lena-values}). Let $E$ be an error metric where this remains true. We
 call the local minimum $E_{min}$:
 
 \begin{equation}
@@ -176,12 +175,14 @@ call the local minimum $E_{min}$:
 \end{equation}
 
 \begin{figure}
+  \begin{minipage}[c]{0.8\textwidth}
+    \input{lena-values}
+  \end{minipage}
   \begin{center}
-   \input{lena-min}
-   \caption{Mean square error for the \textit{Lena} image. $v$ is a simple
-            $11\times11$ Gaussian convolution kernel with $\sigma = 1.2$ and
-            $(dx,dy)$ vary in $[-1,1]\times[-1,1]$.}
-   \label{fig:lena-min}
+    \caption{Mean square error for the \textit{Lena} image ($\times10^4$). $v$
+             is a simple $11\times11$ Gaussian convolution kernel with $\sigma
+             = 1.2$ and $(dx,dy)$ vary in $[-1,1]\times[-1,1]$.}
+    \label{fig:lena-values}
   \end{center}
 \end{figure}
 
@@ -201,7 +202,7 @@ on the Internet nowadays \cite{4chan}.
 In addition to the classical Floyd-Steinberg and Jarvis-Judice-Ninke kernels,
 we tested two serpentine error diffusion algorithms: Ostromoukhov's simple
 error diffusion \cite{ostromoukhov}, which uses a variable coefficient kernel,
-and Wong and Allebach's optimum error diffusion kernel \cite{wong}.
+and Wong and Allebach's optimum error diffusion kernel \cite{wong}:
 
 \begin{center}
   \begin{tabular}{|l|c|c|}
@@ -267,16 +268,16 @@ we refer to them as the \textit{algorithm's displacement}, as opposed to the
 \textit{image's displacement} for a given algorithm. We call $E_{fast}(h,b)$
 the error computed at $(dx,dy)$. As $E_{fast}$ does not depend on the image, it
 is a lot faster to compute than $E_{min}$, and as it is statistically closer to
-$E_{min}$, we can expect it to be a better error estimation than $E$.
+$E_{min}$, we can expect it to be a better error estimation than $E$:
 
 \begin{center}
-  \begin{tabular}{|l|c|c|c|c|}
+  \begin{tabular}{|l|c|c|c|c|c|}
   \hline
-  &~ $E\times10^4$ ~&~ $dx$ ~&~ $dy$ ~&~ $E_{fast}\times10^4$ ~\\ \hline
-  ~raster Floyd-Steinberg ~&~ 3.7902 ~&~ 0.16 ~&~ 0.28 ~&~ 3.3447 ~\\ \hline
-  ~raster Ja-Ju-Ni        ~&~ 9.7013 ~&~ 0.26 ~&~ 0.76 ~&~ 7.5891 ~\\ \hline
-  ~Ostromoukhov           ~&~ 4.6892 ~&~ 0.00 ~&~ 0.19 ~&~ 4.6117 ~\\ \hline
-  ~optimum kernel         ~&~ 7.5209 ~&~ 0.00 ~&~ 0.34 ~&~ 6.8233 ~\\
+  &~ $E\times10^4$ ~&~ $E_{min}\times10^4$ ~&~ $dx$ ~&~ $dy$ ~&~ $E_{fast}\times10^4$ ~\\ \hline
+  ~raster Floyd-Steinberg ~&~ 3.7902 ~&~ 3.1914 ~&~ 0.16 ~&~ 0.28 ~&~ 3.3447 ~\\ \hline
+  ~raster Ja-Ju-Ni        ~&~ 9.7013 ~&~ 6.6349 ~&~ 0.26 ~&~ 0.76 ~&~ 7.5891 ~\\ \hline
+  ~Ostromoukhov           ~&~ 4.6892 ~&~ 4.4783 ~&~ 0.00 ~&~ 0.19 ~&~ 4.6117 ~\\ \hline
+  ~optimum kernel         ~&~ 7.5209 ~&~ 6.5772 ~&~ 0.00 ~&~ 0.34 ~&~ 6.8233 ~\\
   \hline
   \end{tabular}
 \end{center}
@@ -308,9 +309,9 @@ as an indication and only $E$ was used to elect the best coefficients:
   ~ rank ~&~ coefficients ~&~ $E\times10^4$ ~&~ $E_{min}\times10^4$ ~\\ \hline
   ~ 1 ~&~ 7 3 6 0 ~&~ 4.65512 ~&~ 3.94217 ~\\ \hline
   ~ 2 ~&~ 8 3 5 0 ~&~ 4.65834 ~&~ 4.03699 ~\\ \hline
-  ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\ \hline
+  \hline
   ~ 5 ~&~ 7 3 5 1 ~&~ 4.68588 ~&~ 3.79556 ~\\ \hline
-  ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\ \hline
+  \hline
   ~ 18 ~&~ 6 3 5 2 ~&~ 4.91020 ~&~ 3.70465 ~\\ \hline
   ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\
   \hline
@@ -326,9 +327,9 @@ different results. Similarly, $E$ is only given here as an indication:
   ~ rank ~&~ coefficients ~&~ $E_{min}\times10^4$ ~&~ $E\times10^4$ ~\\ \hline
   ~ 1 ~&~ 6 3 5 2 ~&~ 3.70465 ~&~ 4.91020 ~\\ \hline
   ~ 2 ~&~ 7 3 5 1 ~&~ 3.79556 ~&~ 4.68588 ~\\ \hline
-  ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\ \hline
+  \hline
   ~ 15 ~&~ 7 3 6 0 ~&~ 3.94217 ~&~ 4.65512 ~\\ \hline
-  ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\ \hline
+  \hline
   ~ 22 ~&~ 8 3 5 0 ~&~ 4.03699 ~&~ 4.65834 ~\\ \hline
   ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\
   \hline
@@ -338,7 +339,8 @@ different results. Similarly, $E$ is only given here as an indication:
 Our improved metric allowed us to confirm that the original Floyd-Steinberg
 coefficients were indeed amongst the best possible for raster scan.
 More importantly, using $E$ as the decision variable may have elected
-$\frac{1}{16}\{8,4,4,0\}$, which is in fact a poor choice.
+$\frac{1}{16}\{7,3,6,0\}$ or $\frac{1}{16}\{8,3,5,0\}$, which are in fact poor
+choices.
 
 For serpentine scan, however, our experiment suggests that
 $\frac{1}{16}\{7,4,5,0\}$ is a better choice than the Floyd-Steinberg
@@ -347,11 +349,21 @@ coefficients that have nonetheless been widely in use so far (Fig.
 
 \begin{figure}
   \begin{center}
-    \includegraphics[width=0.8\textwidth]{lena.eps}
-    \caption{halftone of \textit{Lena} using serpentine error diffusion and
-             the optimum coefficients $\frac{1}{16}\{7,4,5,0\}$ that improve
-             on the standard Floyd-Steinberg coefficients in terms of visual
-             quality for the HVS model studied in section 3.}
+    \includegraphics[width=0.4\textwidth]{output-7-3-5-1-serp.eps}
+    ~
+    \includegraphics[width=0.4\textwidth]{output-7-4-5-0-serp.eps}
+  \end{center}
+  \begin{center}
+    \includegraphics[width=0.4\textwidth]{crop-7-3-5-1-serp.eps}
+    ~
+    \includegraphics[width=0.4\textwidth]{crop-7-4-5-0-serp.eps}
+    \caption{halftone of \textit{Lena} using serpentine error diffusion
+             (\textit{left}) and the optimum coefficients
+             $\frac{1}{16}\{7,4,5,0\}$ (\textit{right}) that improve on the
+             standard Floyd-Steinberg coefficients in terms of visual quality
+             for the HVS model used in section 3. The detailed area
+             (\textit{bottom}) shows fewer structure artifacts in the regions
+             with low contrast.}
     \label{fig:lena7450}
   \end{center}
 \end{figure}
@@ -418,59 +430,54 @@ L. Velho and J. Gomes,
 \textit{Digital halftoning with space-filling curves}.
 Computer Graphics (Proceedings of SIGGRAPH 91), 25(4):81--90, 1991
 
-\bibitem[9]{peano}
-I.~H. Witten and R.~M. Neal,
-\textit{Using peano curves for bilevel display of continuous-tone images}.
-IEEE Computer Graphics \& Appl., 2:47--52, 1982
-
-\bibitem[10]{nasanen}
+\bibitem[9]{nasanen}
 R. Nasanen,
 \textit{Visibility of halftone dot textures}.
 IEEE Trans. Syst. Man. Cyb., vol. 14, no. 6, pp. 920--924, 1984
 
-\bibitem[11]{allebach}
+\bibitem[10]{allebach}
 M. Analoui and J.~P. Allebach,
 \textit{Model-based halftoning using direct binary search}.
 Proc. of SPIE/IS\&T Symp. on Electronic Imaging Science and Tech.,
 February 1992, San Jose, CA, pp. 96--108
 
-\bibitem[12]{mcnamara}
+\bibitem[11]{mcnamara}
 Ann McNamara,
 \textit{Visual Perception in Realistic Image Synthesis}.
 Computer Graphics Forum, vol. 20, no. 4, pp. 211--224, 2001
 
-\bibitem[13]{bhatt}
+\bibitem[12]{bhatt}
 Bhatt \textit{et al.},
 \textit{Direct Binary Search with Adaptive Search and Swap}.
 \url{http://www.ima.umn.edu/2004-2005/MM8.1-10.05/activities/Wu-Chai/halftone.pdf}
 
-\bibitem[14]{4chan}
+\bibitem[13]{4chan}
 moot,
 \url{http://www.4chan.org/}
 
-\bibitem[15]{wong}
+\bibitem[14]{wong}
 P.~W. Wong and J.~P. Allebach,
 \textit{Optimum error-diffusion kernel design}.
 Proc. SPIE Vol. 3018, p. 236--242, 1997
 
-\bibitem[16]{ostromoukhov}
+\bibitem[15]{ostromoukhov}
 Victor Ostromoukhov,
 \textit{A Simple and Efficient Error-Diffusion Algorithm}.
 in Proceedings of SIGGRAPH 2001, in ACM Computer Graphics,  Annual Conference
 Series, pp. 567--572, 2001
 
-\bibitem[17]{lsmb}
+\bibitem[16]{lsmb}
 T.~N. Pappas and D.~L. Neuhoff,
 \textit{Least-squares model-based halftoning}.
 in Proc. SPIE, Human Vision, Visual Proc., and Digital Display III, San Jose,
 CA, Feb. 1992, vol. 1666, pp. 165--176
 
-\bibitem[18]{stability}
+\bibitem[17]{stability}
 R. Eschbach, Z. Fan, K.~T. Knox and G. Marcu,
 \textit{Threshold Modulation and Stability in Error Diffusion}.
 in Signal Processing Magazine, IEEE, July 2003, vol. 20, issue 4, pp. 39--50
 
-\bibitem[19]{sullivan}
+\bibitem[18]{sullivan}
 J. Sullivan, R. Miller and G. Pios,
 \textit{Image halftoning using a visual model in error diffusion}.
 J. Opt. Soc. Am. A, vol. 10, pp. 1714--1724, Aug. 1993
@@ -478,4 +485,3 @@ J. Opt. Soc. Am. A, vol. 10, pp. 1714--1724, Aug. 1993
 \end{thebibliography}
 
 \end{document}
-

2008-displacement/paper/serpopt-histo.tex

@@ -72,7 +72,7 @@
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