1 | % This is LLNCS.DEM the demonstration file of |
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2 | % the LaTeX macro package from Springer-Verlag |
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3 | % for Lecture Notes in Computer Science, |
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4 | % version 2.2 for LaTeX2e |
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5 | % |
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6 | \documentclass{llncs} |
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7 | % |
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8 | \usepackage{makeidx} % allows for indexgeneration |
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9 | \usepackage{graphicx} % for gnuplot epslatex stuff |
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10 | \usepackage{color} % ditto |
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11 | \usepackage{pstricks} % for inkscape TeX output |
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12 | % |
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13 | \begin{document} |
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14 | % |
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15 | \mainmatter % start of the contributions |
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16 | % |
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17 | \title{Reinstating Floyd-Steinberg: Improved Metrics for Quality Assessment |
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18 | of Error Diffusion Algorithms} |
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19 | % |
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20 | \titlerunning{Adapting Qualitative Metrics to Common Error Diffusion Algorithms} % abbreviated title (for running head) |
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21 | % also used for the TOC unless |
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22 | % \toctitle is used |
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23 | % |
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24 | \author{Sam Hocevar\inst{1} \and Gary Niger\inst{2}} |
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25 | % |
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26 | \authorrunning{Sam Hocevar et al.} % abbreviated author list (for running head) |
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27 | % |
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28 | %%%% modified list of authors for the TOC (add the affiliations) |
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29 | \tocauthor{Sam Hocevar, Gary Niger (Laboratoire d'Imagerie Bureautique et de |
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30 | Conception Artistique)} |
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31 | % |
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32 | \institute{Laboratoire d'Imagerie Bureautique et de Conception Artistique\\ |
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33 | 14 rue de Plaisance, Paris, France |
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34 | \and |
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35 | 143 Rolloffle Avenue, Tarzana, California 91356\\ |
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36 | \email{sam@hocevar.net}, \email{gary\_niger@gnaa.us}} |
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37 | |
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38 | \maketitle % typeset the title of the contribution |
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39 | |
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40 | \begin{abstract} |
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41 | In this contribution we introduce a little-known property of error diffusion |
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42 | halftoning algorithms which we call {\it error diffusion displacement}. |
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43 | By accounting for the inherent sub-pixel displacement caused by the error |
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44 | propagation, we correct an important flaw in most metrics used to assess the |
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45 | quality of resulting halftones. We find these metrics to usually highly |
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46 | underestimate the quality of error diffusion in comparison to more modern |
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47 | algorithms such as direct binary search. |
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48 | Using empirical observation, we give a method for creating computationally |
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49 | efficient, image-independent, model-based metrics for this quality assessment. |
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50 | Finally, we use the properties of error diffusion displacement to justify |
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51 | Floyd and Steinberg's well-known choice of algorithm coefficients. |
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52 | |
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53 | {\bf Keywords}: halftoning, error diffusion, image quality, human visual |
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54 | system, color quantization |
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55 | \end{abstract} |
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56 | % |
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57 | \section{Introduction} |
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58 | |
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59 | Image dithering is the process of reducing continuous-tone images to images |
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60 | with a limited number of available colours. Applications vary tremendously, |
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61 | from laser and ink-jet printing to display on small devices such as cellphones, |
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62 | or even the design of banknotes. |
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63 | |
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64 | Countless methods have been published for the last 40 years that try to best |
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65 | address the problem of colour reduction. Comparing two algorithms in terms of |
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66 | speed or memory usage is often straightforward, but how exactly a halftoning |
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67 | algorithm performs quality-wise is a far more complex issue, as it highly |
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68 | depends on the display device and the inner workings of the human eye. |
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69 | |
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70 | Though this document focuses on the particular case of bilevel halftoning, |
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71 | most of our results can be directly adapted to the more generic problem of |
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72 | colour reduction. |
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73 | |
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74 | \section{Halftoning algorithms} |
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75 | |
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76 | The most ancient halftoning method is probably classical screening. This highly |
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77 | parallelisible algorithm consists in tiling a dither matrix over the image |
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78 | and using its elements as threshold values. Classical screening is known for |
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79 | its structural artifacts such as the cross-hatch patterns caused by Bayer |
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80 | ordered dither matrices \cite{bayer}. However, modern techniques such as the |
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81 | void-and-cluster method \cite{void1}, \cite{void2} allow to generate screens |
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82 | yielding visually pleasing results. |
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83 | |
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84 | \medskip Error diffusion dithering, introduced in 1976 by Floyd and Steinberg |
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85 | \cite{fstein}, tries to compensate for the thresholding error through the use |
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86 | of feedback. Typically applied in raster scan order, it uses an error diffusion |
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87 | matrix such as the following one, where $x$ denotes the pixel being processed: |
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88 | |
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89 | \[ \frac{1}{16} \left| \begin{array}{ccc} |
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90 | - & x & 7 \\ |
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91 | 3 & 5 & 1 \end{array} \right| \] |
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92 | |
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93 | Though efforts have been made to make error diffusion parallelisable |
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94 | \cite{parfstein}, it is generally considered more computationally expensive |
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95 | than screening, but carefully chosen coefficients yield good visual results |
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96 | \cite{kite}. |
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97 | |
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98 | \medskip Model-based halftoning is the third important algorithm category. It |
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99 | relies on a model of the human visual system (HVS) and attempts to minimise |
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100 | an error value based on that model. One such algorithm is direct binary seach |
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101 | (DBS) \cite{allebach}, also referred to as least-squares model-based halftoning |
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102 | (LSMB) \cite{lsmb}. |
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103 | |
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104 | HVS models are usually low-pass filters. Nasanen \cite{nasanen}, Analoui |
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105 | and Allebach \cite{allebach} found that using Gaussian models gave visually |
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106 | pleasing results, an observation confirmed by independent visual perception |
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107 | studies \cite{mcnamara}. |
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108 | |
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109 | DBS yields halftones of impressive quality. However, despite efforts to make |
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110 | it more efficient \cite{bhatt}, it suffers from its large computational |
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111 | requirements and error diffusion remains a more widely used technique. |
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112 | |
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113 | \section{Error diffusion displacement} |
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114 | |
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115 | Most error diffusion implementations parse the image in raster scan order. |
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116 | Boustrophedonic (serpentine) scanning has been shown to cause fewer visual |
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117 | artifacts \cite{halftoning}, but other, more complex processing paths such as |
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118 | Hilbert curves \cite{spacefilling}, \cite{peano} are seldom used as they do not |
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119 | improve the image quality significantly. |
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120 | |
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121 | Intuitively, as the error is always propagated to the bottom-left or |
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122 | bottom-right of each pixel (Fig. \ref{fig:direction}), one may expect the |
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123 | resulting image to be slightly translated. This expectation is confirmed |
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124 | visually when rapidly switching between an error diffused image and the |
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125 | corresponding DBS halftone. |
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126 | |
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127 | \begin{figure} |
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128 | \begin{center} |
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129 | \input{direction} |
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130 | \caption{Floyd-Steinberg error diffusion direction in raster scan (left) |
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131 | and serpentine scan (right).}\label{fig:direction} |
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132 | \end{center} |
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133 | \end{figure} |
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134 | |
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135 | This small translation is visually innocuous but we found that it means a lot |
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136 | in terms of error computation. A common way to compute the error between an |
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137 | image $h_{i,j}$ and the corresponding binary halftone $b_{i,j}$ is to compute |
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138 | the mean square error between modified versions of the images, in the form: |
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139 | |
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140 | \begin{equation} |
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141 | E(h,b) = \frac{(||v * h_{i,j} - v * b_{i,j}||_2)^2}{wh} |
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142 | \end{equation} |
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143 | |
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144 | \noindent where $w$ and $h$ are the image dimensions, $*$ denotes the |
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145 | convolution and $v$ is a model for the human visual system. |
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146 | |
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147 | To compensate for the slight translation observed in the halftone, we use the |
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148 | following error metric instead: |
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149 | |
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150 | \begin{equation} |
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151 | E_{dx,dy}(h,b) = \frac{(||v * h_{i,j} - v * t_{dx,dy} * b_{i,j}||_2)^2}{wh} |
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152 | \end{equation} |
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153 | |
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154 | \noindent where $t_{dx,dy}$ is an operator which translates the image along the |
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155 | $(dx,dy)$ vector. By design, $E_{0,0} = E$. |
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156 | |
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157 | A simple example can be given using a Gaussian HVS model: |
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158 | |
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159 | \begin{equation} |
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160 | v(x,y) = e^{\frac{x^2+y^2}{2\sigma^2}} |
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161 | \end{equation} |
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162 | |
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163 | Finding the second filter is then straightforward: |
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164 | |
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165 | \begin{equation} |
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166 | (v * t_{dx,dy})(x,y) = e^{\frac{(x-dx)^2+(y-dy)^2}{2\sigma^2}} |
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167 | \end{equation} |
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168 | |
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169 | Experiments show that for a given image and a given corresponding halftone, |
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170 | $E_{dx,dy}$ has a local minimum almost always away from $(dx,dy) = (0,0)$ (Fig. |
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171 | \ref{fig:lena-min}). Let $E$ be an error metric where this remains true. We |
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172 | call the local minimum $E_{min}$: |
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173 | |
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174 | \begin{equation} |
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175 | E_{min}(h,b) = \min_{dx,dy}E_{dx,dy}(h,b) |
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176 | \end{equation} |
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177 | |
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178 | \begin{figure} |
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179 | \begin{center} |
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180 | \input{lena-min} |
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181 | \caption{Mean square error for the \textit{Lena} image. $v$ is a simple |
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182 | $11\times11$ Gaussian convolution kernel with $\sigma = 1.2$ and |
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183 | $(dx,dy)$ vary in $[-1,1]\times[-1,1]$.} |
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184 | \label{fig:lena-min} |
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185 | \end{center} |
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186 | \end{figure} |
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187 | |
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188 | For instance, a Floyd-Steinberg dither of \textit{Lena} with $\sigma = 1.2$ |
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189 | yields a per-pixel mean square error of $3.67\times10^{-4}$. However, when |
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190 | taking the displacement into account, the error becomes $3.06\times10^{-4}$ for |
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191 | $(dx,dy) = (0.165,0.293)$. The new, corrected error is significantly smaller, |
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192 | with the exact same input and output images. |
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193 | |
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194 | Experiments show that the corrected error is always noticeably smaller except |
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195 | in the case of images that are already mostly pure black and white. The |
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196 | experiment was performed on a database of 10,000 images from common computer |
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197 | vision sets and from the image board \textit{4chan}, providing a representative |
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198 | sampling of the photographs, digital art and business graphics widely exchanged |
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199 | on the Internet nowadays \cite{4chan}. |
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200 | |
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201 | In addition to the classical Floyd-Steinberg and Jarvis-Judice-Ninke kernels, |
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202 | we tested two serpentine error diffusion algorithms: Ostromoukhov's simple |
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203 | error diffusion \cite{ostromoukhov}, which uses a variable coefficient kernel, |
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204 | and Wong and Allebach's optimum error diffusion kernel \cite{wong}. |
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205 | |
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206 | \begin{center} |
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207 | \begin{tabular}{|l|c|c|} |
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208 | \hline |
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209 | &~ $E\times10^4$ ~&~ $E_{min}\times10^4$ ~\\ \hline |
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210 | ~raster Floyd-Steinberg ~&~ 3.7902 ~&~ 3.1914 ~\\ \hline |
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211 | ~raster Ja-Ju-Ni ~&~ 9.7013 ~&~ 6.6349 ~\\ \hline |
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212 | ~Ostromoukhov ~&~ 4.6892 ~&~ 4.4783 ~\\ \hline |
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213 | ~optimum kernel ~&~ 7.5209 ~&~ 6.5772 ~\\ |
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214 | \hline |
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215 | \end{tabular} |
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216 | \end{center} |
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217 | |
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218 | We clearly see that usual metrics underestimate the quality of error-diffused |
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219 | halftones, especially in raster scan. Algorithms such as direct binary search, |
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220 | on the other hand, do not suffer from this bias since they are designed to |
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221 | minimise the very error induced by the HVS model. |
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222 | |
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223 | \section{An image-independent corrected quality metric for error-diffused |
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224 | halftones} |
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225 | |
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226 | We have seen that for a given image, $E_{min}(h,b)$ is a better and fairer |
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227 | visual error measurement than $E(h,b)$. However, its major drawback is that it |
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228 | is highly computationally expensive: for each image, the new $(dx,dy)$ values |
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229 | need to be calculated to minimise the error value. |
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230 | |
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231 | Fortunately, we found that for a given raster or serpentine scan |
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232 | error diffusion algorithm, there was often very little variation in |
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233 | the optimal $(dx,dy)$ values (Fig. \ref{fig:table-historaster} and |
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234 | \ref{fig:table-histoserp}). |
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235 | |
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236 | \begin{figure} |
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237 | \begin{center} |
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238 | \begin{minipage}[c]{0.50\textwidth} |
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239 | \input{fs-histo} |
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240 | \end{minipage} |
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241 | \begin{minipage}[c]{0.40\textwidth} |
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242 | \input{jajuni-histo} |
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243 | \end{minipage} |
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244 | \caption{error diffusion displacement histograms for the raster |
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245 | Floyd-Steinberg (left) and raster Jarvis, Judis and Ninke (right) |
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246 | algorithms applied to a corpus of 10,000 images} |
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247 | \label{fig:table-historaster} |
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248 | \end{center} |
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249 | \end{figure} |
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250 | |
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251 | \begin{figure} |
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252 | \begin{center} |
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253 | \begin{minipage}[c]{0.50\textwidth} |
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254 | \input{ostro-histo} |
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255 | \end{minipage} |
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256 | \begin{minipage}[c]{0.40\textwidth} |
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257 | \input{serpopt-histo} |
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258 | \end{minipage} |
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259 | \caption{error diffusion displacement histograms for the Ostromoukhov (left) and optimum kernel (right) algorithms applied to a corpus |
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260 | of 10,000 images} |
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261 | \label{fig:table-histoserp} |
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262 | \end{center} |
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263 | \end{figure} |
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264 | |
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265 | For each algorithm, we choose the $(dx,dy)$ values at the histogram peak and |
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266 | we refer to them as the \textit{algorithm's displacement}, as opposed to the |
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267 | \textit{image's displacement} for a given algorithm. We call $E_{fast}(h,b)$ |
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268 | the error computed at $(dx,dy)$. As $E_{fast}$ does not depend on the image, it |
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269 | is a lot faster to compute than $E_{min}$, and as it is statistically closer to |
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270 | $E_{min}$, we can expect it to be a better error estimation than $E$. |
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271 | |
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272 | \begin{center} |
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273 | \begin{tabular}{|l|c|c|c|c|} |
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274 | \hline |
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275 | &~ $E\times10^4$ ~&~ $dx$ ~&~ $dy$ ~&~ $E_{fast}\times10^4$ ~\\ \hline |
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276 | ~raster Floyd-Steinberg ~&~ 3.7902 ~&~ 0.16 ~&~ 0.28 ~&~ 3.3447 ~\\ \hline |
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277 | ~raster Ja-Ju-Ni ~&~ 9.7013 ~&~ 0.26 ~&~ 0.76 ~&~ 7.5891 ~\\ \hline |
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278 | ~Ostromoukhov ~&~ 4.6892 ~&~ 0.00 ~&~ 0.19 ~&~ 4.6117 ~\\ \hline |
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279 | ~optimum kernel ~&~ 7.5209 ~&~ 0.00 ~&~ 0.34 ~&~ 6.8233 ~\\ |
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280 | \hline |
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281 | \end{tabular} |
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282 | \end{center} |
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283 | |
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284 | \section{Using error diffusion displacement for optimum kernel design} |
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285 | |
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286 | We believe that our higher quality $E_{min}$ error metric may be useful in |
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287 | kernel design, because it is the very same error that admittedly superior yet |
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288 | computationally expensive algorithms such as DBS try to minimise. |
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289 | |
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290 | Our first experiment was a study of the Floyd-Steinberg-like 4-block error |
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291 | diffusion kernels. According to the original authors, the coefficients were |
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292 | found "mostly by trial and error" \cite{fstein}. With our improved metric, we |
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293 | now have the tools to confirm or infirm Floyd and Steinberg's initial choice. |
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294 | |
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295 | We chose to do an exhaustive study of every $\frac{1}{16}\{a,b,c,d\}$ integer |
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296 | combination. We deliberately chose positive integers whose sum was 16: error |
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297 | diffusion coefficients smaller than zero or adding up to more than 1 are known |
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298 | to be unstable \cite{stability}, and diffusing less than 100\% of the error |
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299 | causes important loss of detail in the shadow and highlight areas of the image. |
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300 | |
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301 | We studied all possible coefficients on a pool of 3,000 images with an error |
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302 | metric $E$ based on a standard Gaussian HVS model. $E_{min}$ is only given here |
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303 | as an indication and only $E$ was used to elect the best coefficients: |
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304 | |
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305 | \begin{center} |
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306 | \begin{tabular}{|c|c|c|c|} |
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307 | \hline |
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308 | ~ rank ~&~ coefficients ~&~ $E\times10^4$ ~&~ $E_{min}\times10^4$ ~\\ \hline |
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309 | ~ 1 ~&~ 7 3 6 0 ~&~ 4.65512 ~&~ 3.94217 ~\\ \hline |
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310 | ~ 2 ~&~ 8 3 5 0 ~&~ 4.65834 ~&~ 4.03699 ~\\ \hline |
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311 | ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\ \hline |
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312 | ~ 5 ~&~ 7 3 5 1 ~&~ 4.68588 ~&~ 3.79556 ~\\ \hline |
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313 | ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\ \hline |
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314 | ~ 18 ~&~ 6 3 5 2 ~&~ 4.91020 ~&~ 3.70465 ~\\ \hline |
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315 | ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\ |
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316 | \hline |
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317 | \end{tabular} |
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318 | \end{center} |
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319 | |
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320 | The exact same operation using $E_{min}$ as the decision variable yields very |
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321 | different results. Similarly, $E$ is only given here as an indication: |
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322 | |
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323 | \begin{center} |
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324 | \begin{tabular}{|c|c|c|c|} |
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325 | \hline |
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326 | ~ rank ~&~ coefficients ~&~ $E_{min}\times10^4$ ~&~ $E\times10^4$ ~\\ \hline |
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327 | ~ 1 ~&~ 6 3 5 2 ~&~ 3.70465 ~&~ 4.91020 ~\\ \hline |
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328 | ~ 2 ~&~ 7 3 5 1 ~&~ 3.79556 ~&~ 4.68588 ~\\ \hline |
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329 | ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\ \hline |
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330 | ~ 15 ~&~ 7 3 6 0 ~&~ 3.94217 ~&~ 4.65512 ~\\ \hline |
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331 | ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\ \hline |
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332 | ~ 22 ~&~ 8 3 5 0 ~&~ 4.03699 ~&~ 4.65834 ~\\ \hline |
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333 | ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\ |
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334 | \hline |
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335 | \end{tabular} |
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336 | \end{center} |
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337 | |
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338 | Our improved metric allowed us to confirm that the original Floyd-Steinberg |
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339 | coefficients were indeed amongst the best possible for raster scan. |
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340 | More importantly, using $E$ as the decision variable may have elected |
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341 | $\frac{1}{16}\{8,4,4,0\}$, which is in fact a poor choice. |
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342 | |
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343 | For serpentine scan, however, our experiment suggests that |
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344 | $\frac{1}{16}\{7,4,5,0\}$ is a better choice than the Floyd-Steinberg |
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345 | coefficients that have nonetheless been widely in use so far (Fig. |
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346 | \ref{fig:lena7450}). |
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347 | |
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348 | \begin{figure} |
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349 | \begin{center} |
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350 | \includegraphics[width=0.8\textwidth]{lena.eps} |
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351 | \caption{halftone of \textit{Lena} using serpentine error diffusion and |
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352 | the optimum coefficients $\frac{1}{16}\{7,4,5,0\}$ that improve |
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353 | on the standard Floyd-Steinberg coefficients in terms of visual |
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354 | quality for the HVS model studied in section 3.} |
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355 | \label{fig:lena7450} |
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356 | \end{center} |
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357 | \end{figure} |
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358 | |
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359 | \section{Conclusion} |
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360 | |
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361 | We have disclosed an interesting property of error diffusion algorithms |
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362 | allowing to more precisely measure the quality of such halftoning methods. |
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363 | Having showed that such quality is often underestimated by usual metrics, |
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364 | we hope to see even more development in simple error diffusion methods. |
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365 | |
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366 | Confirming Floyd and Steinberg's 30-year old "trial-and-error" result with our |
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367 | work is only the beginning: future work may cover more complex HVS models, |
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368 | for instance by taking into account the angular dependance of the human eye |
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369 | \cite{sullivan}. We plan to use our new metric to improve all error diffusion |
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370 | methods that may require fine-tuning of their propagation coefficients. |
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371 | |
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372 | % |
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373 | % ---- Bibliography ---- |
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374 | % |
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375 | \begin{thebibliography}{} |
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376 | % |
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377 | \bibitem[1]{bayer} |
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378 | B. Bayer, |
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379 | \textit{Color imaging array}. |
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380 | U.S. patent 3,971,065 (1976) |
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381 | |
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382 | \bibitem[2]{void1} |
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383 | R.A. Ulichney (Digital Equipment Corporation), |
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384 | \textit{Void and cluster apparatus and method for generating dither templates}. |
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385 | U.S. patent 5,535,020 (1992) |
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386 | |
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387 | \bibitem[3]{void2} |
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388 | H. Ancin, A. Bhattacharjya and J. Shu (Seiko Epson Corporation), |
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389 | \textit{Void-and-cluster dither-matrix generation for better half-tone |
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390 | uniformity}. |
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391 | U.S. patent 6,088,512 (1997) |
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392 | |
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393 | \bibitem[4]{fstein} |
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394 | R.W. Floyd, L. Steinberg, |
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395 | \textit{An adaptive algorithm for spatial grey scale}. |
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396 | Proceedings of the Society of Information Display 17, (1976) 75--77 |
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397 | |
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398 | \bibitem[5]{parfstein} |
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399 | P. Metaxas, |
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400 | \textit{Optimal Parallel Error-Diffusion Dithering}. |
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401 | Color Imaging: Device-Indep. Color, Color Hardcopy, and Graphic Arts IV, Proc. |
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402 | SPIE 3648, 485--494 (1999) |
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403 | |
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404 | \bibitem[6]{kite} |
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405 | T. D. Kite, |
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406 | \textit{Design and Quality Assessment of Forward and Inverse Error-Diffusion |
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407 | Halftoning Algorithms}. |
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408 | PhD thesis, Dept. of ECE, The University of Texas at Austin, Austin, TX, Aug. |
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409 | 1998 |
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410 | |
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411 | \bibitem[7]{halftoning} |
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412 | R. Ulichney, |
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413 | \textit{Digital Halftoning}. |
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414 | MIT Press, 1987 |
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415 | |
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416 | \bibitem[8]{spacefilling} |
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417 | L. Velho and J. Gomes, |
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418 | \textit{Digital halftoning with space-filling curves}. |
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419 | Computer Graphics (Proceedings of SIGGRAPH 91), 25(4):81--90, 1991 |
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420 | |
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421 | \bibitem[9]{peano} |
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422 | I.~H. Witten and R.~M. Neal, |
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423 | \textit{Using peano curves for bilevel display of continuous-tone images}. |
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424 | IEEE Computer Graphics \& Appl., 2:47--52, 1982 |
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425 | |
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480 | \end{document} |
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481 | |
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