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| 11 | <meta name="Description" content="Libcaca study - 3. Error diffusion" /> |
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| 12 | <meta name="Keywords" content="libcaca, ASCII, ASCII ART, console, text mode, ncurses, slang, AAlib, dithering, thresholding" /> |
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| 13 | <title>Libcaca study - 3. Error diffusion</title> |
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| 21 | <?php include($_SERVER["DOCUMENT_ROOT"]."/header.inc"); ?> |
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| 22 | |
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| 23 | <p> <span style="color: #aa0000; font-weight: bold;">Warning</span>: this |
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| 24 | document is still work in progress. Feel free to send comments but do not |
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| 25 | consider it final material. </p> |
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| 26 | |
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| 27 | <div style="float: left;"> |
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| 28 | <a href="part2.html">Halftoning <<<</a> |
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| 29 | </div> |
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| 30 | <div style="float: right;"> |
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| 31 | <a href="part4.html">>>> Model-based dithering</a> |
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| 32 | </div> |
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| 33 | <div style="text-align: center;"> |
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| 34 | <a href="index.html">^^^ Index</a> |
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| 35 | </div> |
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| 36 | |
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| 37 | <h2> 3. Error diffusion </h2> |
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| 38 | |
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| 39 | <p> The idea behind error diffusion is to compute the error caused by |
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| 40 | thresholding a given pixel and propagate it to neighbour pixels, in order to |
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| 41 | compensate for the average intensity loss or gain. It is based upon the |
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| 42 | assumption that a slightly out-of-place pixel causes little visual harm. |
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| 43 | </p> |
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| 44 | |
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| 45 | <p> The error is computed by simply substracting the source value and the |
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| 46 | destination value. Destination value can be chosen by many means but does |
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| 47 | not impact the image a lot with most methods in comparison to the crucial |
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| 48 | choice of error distribution coefficients. </p> |
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| 49 | |
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| 50 | <p> This is the simplest error diffusion method. It thresholds the image |
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| 51 | to 0.5 and propagates 100% of the error to the next (right) pixel. It is |
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| 52 | quite impressive given its simplicity but causes important visual artifacts: |
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| 53 | </p> |
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| 54 | |
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| 55 | <p style="text-align: center;"> |
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| 56 | <img src="out/lena3-0-1.png" width="256" height="256" |
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| 57 | class="inline" alt="Simple error diffusion" /> |
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| 58 | <img src="out/grad3-0-1.png" width="32" height="256" |
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| 59 | class="inline" alt="Simple error diffusion gradient" /> |
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| 60 | </p> |
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| 61 | |
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| 62 | <h3> 3.1. Floyd-Steinberg and JaJuNi error diffusion </h3> |
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| 63 | |
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| 64 | <p> The most famous error diffusion method is the <b>Floyd-Steinberg</b> |
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| 65 | algorithm [5]. It propagates the error to more than one adjacent pixels using |
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| 66 | the following coefficients: </p> |
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| 67 | |
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| 68 | <p style="text-align: center;"> |
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| 69 | <img src="out/fig3-1-1.png" width="121" height="81" alt="Floyd-Steinberg" /> |
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| 70 | </p> |
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| 71 | |
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| 72 | <p> The result of this algorithm is rather impressive even compared to the |
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| 73 | best ordered dither results we could achieve: </p> |
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| 74 | |
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| 75 | <p style="text-align: center;"> |
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| 76 | <img src="out/lena3-1-1.png" width="256" height="256" |
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| 77 | class="inline" alt="Floyd-Steinberg error diffusion" /> |
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| 78 | <img src="out/grad3-1-1.png" width="32" height="256" |
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| 79 | class="inline" alt="Floyd-Steinberg error diffusion gradient" /> |
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| 80 | </p> |
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| 81 | |
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| 82 | <p> <b>Jarvis, Judice and Ninke dithering</b> [7] (sometimes nicknamed |
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| 83 | <b>JaJuNi</b>) was published almost at the same time as Floyd-Steinberg. It |
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| 84 | uses a much more complex error diffusion matrix: </p> |
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| 85 | |
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| 86 | <p style="text-align: center;"> |
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| 87 | <img src="out/fig3-1-3.png" width="201" height="121" |
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| 88 | class="matrix" alt="Jarvis, Judice and Ninke" /> |
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| 89 | <img src="out/lena3-1-3.png" width="256" height="256" |
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| 90 | class="inline" alt="Jarvis, Judice and Ninke error diffusion" /> |
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| 91 | <img src="out/grad3-1-3.png" width="32" height="256" |
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| 92 | class="inline" alt="Jarvis, Judice and Ninke error diffusion gradient" /> |
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| 93 | </p> |
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| 94 | |
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| 95 | <h3> 3.2. Floyd-Steinberg derivatives </h3> |
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| 96 | |
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| 97 | <p> Zhigang Fan came up with several Floyd-Steinberg derivatives. <b>Fan |
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| 98 | dithering</b> [8] just moves one coefficient around: </p> |
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| 99 | |
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| 100 | <p style="text-align: center;"> |
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| 101 | <img src="out/fig3-2-1.png" width="161" height="81" |
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| 102 | class="matrix" alt="Fan" /> |
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| 103 | <img src="out/lena3-2-1.png" width="256" height="256" |
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| 104 | class="inline" alt="Fan error diffusion" /> |
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| 105 | <img src="out/grad3-2-1.png" width="32" height="256" |
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| 106 | class="inline" alt="Fan error diffusion gradient" /> |
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| 107 | </p> |
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| 108 | |
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| 109 | <p> <b>Shiau-Fan dithering</b> use a family of matrices supposed to reduce |
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| 110 | the apparition of artifacts usually seen with Floyd-Steinberg: </p> |
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| 111 | |
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| 112 | <p style="text-align: center;"> |
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| 113 | <img src="out/fig3-2-1b.png" width="161" height="81" |
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| 114 | class="matrix" alt="Shiau-Fan" /> |
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| 115 | <img src="out/lena3-2-1b.png" width="256" height="256" |
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| 116 | class="inline" alt="Shiau-Fan error diffusion" /> |
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| 117 | <img src="out/grad3-2-1b.png" width="32" height="256" |
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| 118 | class="inline" alt="Shiau-Fan error diffusion gradient" /> |
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| 119 | </p> |
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| 120 | |
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| 121 | <p style="text-align: center;"> |
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| 122 | <img src="out/fig3-2-1c.png" width="201" height="81" |
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| 123 | class="matrix" alt="Shiau-Fan 2" /> |
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| 124 | <img src="out/lena3-2-1c.png" width="256" height="256" |
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| 125 | class="inline" alt="Shiau-Fan 2 error diffusion" /> |
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| 126 | <img src="out/grad3-2-1c.png" width="32" height="256" |
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| 127 | class="inline" alt="Shiau-Fan 2 error diffusion gradient" /> |
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| 128 | </p> |
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| 129 | |
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| 130 | <p> By the way, these matrices are covered by Shiau’s and Fan’s |
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| 131 | <a href="http://www.freepatentsonline.com/5353127.html">U.S. patent |
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| 132 | 5353127</a>. </p> |
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| 133 | |
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| 134 | <p> <b>Stucki dithering</b> [6] is a slight variation of Jarvis-Judice-Ninke |
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| 135 | dithering: </p> |
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| 136 | |
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| 137 | <p style="text-align: center;"> |
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| 138 | <img src="out/fig3-2-3.png" width="201" height="121" |
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| 139 | class="matrix" alt="Stucki" /> |
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| 140 | <img src="out/lena3-2-3.png" width="256" height="256" |
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| 141 | class="inline" alt="Stucki error diffusion" /> |
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| 142 | <img src="out/grad3-2-3.png" width="32" height="256" |
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| 143 | class="inline" alt="Stucki error diffusion gradient" /> |
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| 144 | </p> |
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| 145 | |
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| 146 | <p> <b>Burkes dithering</b> is yet another variation [10] which improves on |
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| 147 | Stucki dithering by removing a line and making the error coefficients fractions |
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| 148 | of powers of two: </p> |
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| 149 | |
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| 150 | <p style="text-align: center;"> |
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| 151 | <img src="out/fig3-2-4.png" width="201" height="81" |
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| 152 | class="matrix" alt="Burkes" /> |
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| 153 | <img src="out/lena3-2-4.png" width="256" height="256" |
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| 154 | class="inline" alt="Burkes error diffusion" /> |
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| 155 | <img src="out/grad3-2-4.png" width="32" height="256" |
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| 156 | class="inline" alt="Burkes error diffusion gradient" /> |
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| 157 | </p> |
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| 158 | |
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| 159 | <p> Frankie Sierra [11] came up with a few error diffusion matrices: <b>Sierra |
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| 160 | dithering</b> is a variation of Jarvis that is slightly faster because it |
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| 161 | propagates to fewer pixels, <b>Two-row Sierra</b> is a simplified version |
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| 162 | thereof, and <b>Filter Lite</b> is one of the simplest Floyd-Steinberg |
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| 163 | derivatives: </p> |
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| 164 | |
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| 165 | <p style="text-align: center;"> |
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| 166 | <img src="out/fig3-2-5.png" width="201" height="121" |
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| 167 | class="matrix" alt="Sierra" /> |
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| 168 | <img src="out/lena3-2-5.png" width="256" height="256" |
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| 169 | class="inline" alt="Sierra error diffusion" /> |
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| 170 | <img src="out/grad3-2-5.png" width="32" height="256" |
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| 171 | class="inline" alt="Sierra error diffusion gradient" /> |
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| 172 | </p> |
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| 173 | |
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| 174 | <p style="text-align: center;"> |
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| 175 | <img src="out/fig3-2-6.png" width="201" height="81" |
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| 176 | class="matrix" alt="Sierra" /> |
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| 177 | <img src="out/lena3-2-6.png" width="256" height="256" |
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| 178 | class="inline" alt="Sierra error diffusion" /> |
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| 179 | <img src="out/grad3-2-6.png" width="32" height="256" |
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| 180 | class="inline" alt="Sierra error diffusion gradient" /> |
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| 181 | </p> |
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| 182 | |
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| 183 | <p style="text-align: center;"> |
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| 184 | <img src="out/fig3-2-7.png" width="121" height="81" |
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| 185 | class="matrix" alt="Sierra" /> |
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| 186 | <img src="out/lena3-2-7.png" width="256" height="256" |
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| 187 | class="inline" alt="Sierra error diffusion" /> |
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| 188 | <img src="out/grad3-2-7.png" width="32" height="256" |
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| 189 | class="inline" alt="Sierra error diffusion gradient" /> |
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| 190 | </p> |
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| 191 | |
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| 192 | <p> <b>Atkinson dithering</b> [12] only propagates 75% of the error, leading |
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| 193 | to a loss of contrast around very dark and very light areas (also called |
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| 194 | <b>highlights and shadows</b>), but better contrast in the midtones. The |
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| 195 | original Macintosh software <i>HyperScan</i> used this dithering algorithm, |
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| 196 | still considered superior to other Floyd-Steinberg derivatives by many Mac |
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| 197 | zealots: </p> |
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| 198 | |
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| 199 | <p style="text-align: center;"> |
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| 200 | <img src="out/fig3-2-8.png" width="161" height="121" |
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| 201 | class="matrix" alt="Atkinson" /> |
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| 202 | <img src="out/lena3-2-8.png" width="256" height="256" |
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| 203 | class="inline" alt="Atkinson error diffusion" /> |
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| 204 | <img src="out/grad3-2-8.png" width="32" height="256" |
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| 205 | class="inline" alt="Atkinson error diffusion gradient" /> |
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| 206 | </p> |
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| 207 | |
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| 208 | <!-- XXX: Stevenson-Arce is for hexagonal cells! |
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| 209 | <p> <b>Stevenson-Arce dithering</b>: </p> |
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| 210 | |
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| 211 | <p style="text-align: center;"> |
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| 212 | <img src="fig3-2-9.png" width="280" height="160" |
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| 213 | class="matrix" alt="Stevenson-Arce" /> |
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| 214 | <img src="out/lena3-2-9.png" width="256" height="256" |
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| 215 | class="inline" alt="Stevenson-Arce error diffusion" /> |
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| 216 | <img src="out/grad3-2-9.png" width="32" height="256" |
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| 217 | class="inline" alt="Stevenson-Arce error diffusion gradient" /> |
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| 218 | </p> |
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| 219 | --> |
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| 220 | |
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| 221 | <h3> 3.3. Changing image parsing direction </h3> |
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| 222 | |
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| 223 | <p> While image parsing order does not matter with ordered dithering, it can |
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| 224 | actually be crucial with error diffusion. The reason is that once a pixel has |
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| 225 | been processed, standard error diffusion methods do not go back. </p> |
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| 226 | |
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| 227 | <p> The usual way to parse an image is one pixel after the other, following |
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| 228 | their order in memory. When reaching the end of a line, we automatically jump |
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| 229 | to the beginning of the next line. Error diffusion methods using this |
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| 230 | parsing order are called <b>raster error diffusion</b>: </p> |
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| 231 | |
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| 232 | <p style="text-align: center;"> |
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| 233 | <img src="fig3-3-1.png" width="260" height="110" |
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| 234 | class="matrix" alt="Regular parsing" /> |
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| 235 | </p> |
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| 236 | |
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| 237 | <p> Changing the parsing order can help prevent the apparition of artifacts in |
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| 238 | error diffusion algorithms. This is <b>serpentine parsing</b>, where every odd |
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| 239 | line is parsed in reverse order (right to left): </p> |
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| 240 | |
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| 241 | <p style="text-align: center;"> |
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| 242 | <img src="fig3-3-2.png" width="260" height="110" |
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| 243 | class="matrix" alt="Serpentine parsing" /> |
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| 244 | </p> |
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| 245 | |
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| 246 | <p> The major problem with Floyd-Steinberg is the <b>worm artifacts</b> it |
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| 247 | creates. Here is an example of an image made of grey 0.9 dithered with standard |
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| 248 | Floyd-Steinberg and with <b>serpentine Floyd-Steinberg</b> [13 pp.266—267]. |
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| 249 | Most of the worm artifacts have disappeared or were highly reduced: </p> |
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| 250 | |
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| 251 | <p style="text-align: center;"> |
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| 252 | <img src="out/lena3-3-1.png" width="256" height="256" |
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| 253 | class="inline" alt="Floyd-Steinberg on grey 90%" /> |
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| 254 | <img src="out/lena3-3-2.png" width="256" height="256" |
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| 255 | class="inline" alt="serpentine Floyd-Steinberg on grey 90%" /> |
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| 256 | </p> |
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| 257 | |
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| 258 | <p> And here are the results of serpentine Floyd-Steinberg on Lena. Only a |
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| 259 | very close look will show the differences with standard Floyd-Steinberg, but |
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| 260 | a few of the artifacts did disappear: </p> |
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| 261 | |
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| 262 | <p style="text-align: center;"> |
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| 263 | <img src="out/lena3-1-2.png" width="256" height="256" |
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| 264 | class="inline" alt="serpentine Floyd-Steinberg" /> |
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| 265 | <img src="out/grad3-1-2.png" width="32" height="256" |
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| 266 | class="inline" alt="serpentine Floyd-Steinberg gradient" /> |
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| 267 | </p> |
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| 268 | |
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| 269 | <p> <b>Riemersma dithering</b> [26] parses the image following a plane-filling |
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| 270 | <b>Hilbert curve</b> and only propagates the error of the last <i>q</i> pixels, |
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| 271 | weighting it with an exponential rule. The method is interesting and inventive, |
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| 272 | unfortunately the results are disappointing: structural artifacts are worse |
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| 273 | than with other error diffusion methods (shown here with <i>q = 16</i> and <i>r |
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| 274 | = 16</i>): </p> |
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| 275 | |
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| 276 | <p style="text-align: center;"> |
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| 277 | <img src="fig3-3-3.png" width="250" height="250" |
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| 278 | class="matrix" alt="Hilbert curve parsing" /> |
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| 279 | <img src="out/lena3-3-3.png" width="256" height="256" |
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| 280 | class="inline" alt="Riemersma dither on Hilbert curve" /> |
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| 281 | <img src="out/grad3-3-3.png" width="32" height="256" |
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| 282 | class="inline" alt="Riemersma dither on Hilbert curve gradient" /> |
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| 283 | </p> |
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| 284 | |
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| 285 | <p> A variation of Riemersma dithering uses a <b>Hilbert 2 curve</b>, giving |
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| 286 | slightly better results but still causing random artifacts here and there: |
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| 287 | </p> |
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| 288 | |
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| 289 | <p style="text-align: center;"> |
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| 290 | <img src="fig3-3-4.png" width="233" height="233" |
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| 291 | class="matrix" alt="Hilbert 2 curve parsing" /> |
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| 292 | <img src="out/lena3-3-4.png" width="256" height="256" |
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| 293 | class="inline" alt="Riemersma dither on Hilbert 2 curve" /> |
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| 294 | <img src="out/grad3-3-4.png" width="32" height="256" |
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| 295 | class="inline" alt="Riemersma dither on Hilbert 2 curve gradient" /> |
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| 296 | </p> |
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| 297 | |
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| 298 | <p> An inherent problem with plane-filling curves is that distances on the |
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| 299 | curve do not mean anything in image space. Riemersma dithering distributes |
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| 300 | error to pixels according to their distance on the curve rather than their |
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| 301 | distance in the image. </p> |
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| 302 | |
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| 303 | <p> We introduce <b>spatial Hilbert dithering</b> that addresses this issue |
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| 304 | by distributing the error according to spatial coordinates. We also get rid |
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| 305 | of the <i>r</i> parameter, choosing to distribute 100% of the error. </p> |
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| 306 | |
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| 307 | <p> This is spatial Hilbert dithering on a Hilbert curve and on a Hilbert 2 |
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| 308 | curve. The results show a clear improvement over the original Riemersma |
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| 309 | algorithm, with far less noise and smoother low-gradient areas: </p> |
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| 310 | |
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| 311 | <p style="text-align: center;"> |
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| 312 | <img src="out/lena3-3-5.png" width="256" height="256" |
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| 313 | class="inline" alt="spatial Hilbert dither on Hilbert curve" /> |
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| 314 | <img src="out/grad3-3-5.png" width="32" height="256" |
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| 315 | class="inline" alt="spatial Hilbert dither on Hilbert curve gradient" /> |
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| 316 | <img src="out/lena3-3-6.png" width="256" height="256" |
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| 317 | class="inline" alt="spatial Hilbert dither on Hilbert 2 curve" /> |
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| 318 | <img src="out/grad3-3-6.png" width="32" height="256" |
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| 319 | class="inline" alt="spatial Hilbert dither on Hilbert 2 curve gradient" /> |
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| 320 | </p> |
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| 321 | |
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| 322 | <p> <b>Dot diffusion</b> [14] is an error diffusion method by Donald E. Knuth |
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| 323 | that uses tileable matrices just like ordered dithering, except that the cell |
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| 324 | value order is taken into account for error propagation. Diagonal cells get |
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| 325 | half as much error as directly adjacent cells: </p> |
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| 326 | |
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| 327 | <p style="text-align: center;"> |
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| 328 | <img src="out/fig3-3-7b.png" width="121" height="121" |
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| 329 | class="matrix" alt="Dot diffusion" /> |
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| 330 | </p> |
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| 331 | |
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| 332 | <p> For instance, in the following example, cell 25’s error is propagated to |
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| 333 | cells 44, 36, 30, 34 and 49. Given the diagonal cells rule, cells 44, 30 and |
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| 334 | 49 each get 1/7 of the error and cells 36 and 34 each get 2/7 of the error. |
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| 335 | Similarly, cell 63 gets 100% of cell 61’s error. </p> |
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| 336 | |
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| 337 | <p style="text-align: center;"> |
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| 338 | <img src="fig3-3-7.png" width="240" height="240" |
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| 339 | class="matrix" alt="Dot diffusion matrix sample" /> |
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| 340 | <img src="out/lena3-3-7.png" width="256" height="256" |
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| 341 | class="inline" alt="Dot diffusion" /> |
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| 342 | <img src="out/grad3-3-7.png" width="32" height="256" |
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| 343 | class="inline" alt="Dot diffusion gradient" /> |
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| 344 | </p> |
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| 345 | |
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| 346 | <p> The initial result is not extraordinary. But Knuth suggests applying a |
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| 347 | sharpen filter to the original image before applying dot diffusion. He also |
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| 348 | introduces a <i>zeta</i> value to deal with the size of laser printer dots, |
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| 349 | pretty similar to what we’ll see later as <b>gamma correction</b>. The |
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| 350 | following two images had a sharpening value of 0.9 applied to them. The image |
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| 351 | on the right shows <i>zeta = 0.2</i>: </p> |
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| 352 | |
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| 353 | <p style="text-align: center;"> |
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| 354 | <img src="out/lena3-3-8.png" width="256" height="256" |
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| 355 | class="inline" alt="Dot diffusion sharpen 0.9" /> |
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| 356 | <img src="out/grad3-3-8.png" width="32" height="256" |
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| 357 | class="inline" alt="Dot diffusion sharpen 0.9 gradient" /> |
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| 358 | <img src="out/lena3-3-9.png" width="256" height="256" |
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| 359 | class="inline" alt="Dot diffusion sharpen 0.9 zeta 0.2" /> |
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| 360 | <img src="out/grad3-3-9.png" width="32" height="256" |
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| 361 | class="inline" alt="Dot diffusion sharpen 0.9 zeta 0.2 gradient" /> |
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| 362 | </p> |
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| 363 | |
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| 364 | <p> Do not get fooled by Knuth’s apparent good results. They specifically |
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| 365 | target dot printers and do not give terribly good results on a computer |
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| 366 | screen. Actually, a sharpening filter makes just any dithering method look |
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| 367 | better, even basic Floyd-Steinberg dithering (shown here with a sharpening |
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| 368 | value of 0.9, too): </p> |
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| 369 | |
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| 370 | <p style="text-align: center;"> |
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| 371 | <img src="out/lena3-3-10.png" width="256" height="256" |
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| 372 | class="inline" alt="FS with sharpening" /> |
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| 373 | <img src="out/grad3-3-10.png" width="32" height="256" |
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| 374 | class="inline" alt="FS with sharpening gradient" /> |
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| 375 | </p> |
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| 376 | |
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| 377 | <p> Dot diffusion was reinvented 14 years later by Arney, Anderson and Ganawan |
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| 378 | without even citing Knuth. They call their method <b>omni-directional error |
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| 379 | diffusion</b>. Instead of using a clustered dot matrix like Knuth recommends |
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| 380 | for dot diffusion, they use a dispersed dot matrix, which gives far better |
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| 381 | results on a computer display. This is a 16×12 portion of that matrix: </p> |
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| 382 | |
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| 383 | <p style="text-align: center;"> |
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| 384 | <img src="out/fig3-3-11b.png" width="320" height="240" |
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| 385 | class="matrix" alt="omni-directional ED matrix sample" /> |
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| 386 | </p> |
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| 387 | |
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| 388 | <p> The preferred implementation of omni-directional error diffusion uses |
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| 389 | a slightly different propagation matrix, where top and bottom neighbours get |
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| 390 | more error than the others: </p> |
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| 391 | |
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| 392 | <p style="text-align: center;"> |
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| 393 | <img src="out/fig3-3-11.png" width="121" height="121" |
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| 394 | class="matrix" alt="omni-directional ED" /> |
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| 395 | <img src="out/lena3-3-11.png" width="256" height="256" |
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| 396 | class="inline" alt="omni-directional ED" /> |
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| 397 | <img src="out/grad3-3-11.png" width="32" height="256" |
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| 398 | class="inline" alt="omni-directional ED gradient" /> |
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| 399 | </p> |
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| 400 | |
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| 401 | <h3> 3.4. Variable coefficients error diffusion </h3> |
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| 402 | |
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| 403 | <p> Small error diffusion matrices usually cause artifacts to appear because |
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| 404 | the error is not propagated in enough directions. At the same time, such |
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| 405 | matrices also reduce the sharpened aspect common in error diffusion |
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| 406 | techniques. </p> |
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| 407 | |
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| 408 | <p> Ostromoukhov suggests error diffusion values that vary according to the |
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| 409 | input value. The list of 256 discrete value triplets for <i>d1</i>, <i>d2</i> |
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| 410 | and <i>d3</i> he provides [1] give pretty good results with serpentine parsing: |
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| 411 | </p> |
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| 412 | |
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| 413 | <p style="text-align: center;"> |
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| 414 | <img src="out/fig3-4-1.png" width="121" height="81" |
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| 415 | class="matrix" alt="Ostromoukhov ED matrix" /> |
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| 416 | <img src="out/lena3-4-1.png" width="256" height="256" |
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| 417 | class="inline" alt="Ostromoukhov ED" /> |
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| 418 | <img src="out/grad3-4-1.png" width="32" height="256" |
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| 419 | class="inline" alt="Ostromoukhov ED gradient" /> |
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| 420 | </p> |
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| 421 | |
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| 422 | <h3> 3.5. Block error diffusion </h3> |
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| 423 | |
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| 424 | <p> Sometimes, due to physical restrictions of the target media, output |
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| 425 | is limited to some combinations of pixel blocks, such as the ones shown |
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| 426 | below: </p> |
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| 427 | |
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| 428 | <p style="text-align: center;"> |
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| 429 | <img src="fig3-5-1.png" width="613" height="80" |
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| 430 | class="matrix" alt="list of 2×2 pixel blocks" /> |
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| 431 | </p> |
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| 432 | |
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| 433 | <p> It is still possible to dither the image, by doing it 4 pixels at a |
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| 434 | time and simply choosing the block from the list that minimises the global |
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| 435 | error within the 2×2 block: </p> |
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| 436 | |
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| 437 | <p style="text-align: center;"> |
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| 438 | <img src="out/lena3-5-1.png" width="256" height="256" |
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| 439 | class="inline" alt="2×2 pixel block quantisation" /> |
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| 440 | <img src="out/grad3-5-1.png" width="32" height="256" |
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| 441 | class="inline" alt="2×2 pixel block quantisation gradient" /> |
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| 442 | </p> |
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| 443 | |
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| 444 | <p> Damera-Venkata and Evans introduce <b>block error diffusion</b> [23], which |
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| 445 | reuses traditional error diffusion methods such as Floyd-Steinberg but applies |
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| 446 | the same error value to all pixels of a given block. Only one error value is |
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| 447 | propagated, <i>a+b+c+d</i>, which is the global error within the block: </p> |
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| 448 | |
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| 449 | <p style="text-align: center; font-size: 2em;"> |
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| 450 | <img src="out/fig3-1-1.png" width="121" height="81" |
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| 451 | class="math" alt="Floyd-Steinberg" /> |
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| 452 | ⊗ |
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| 453 | <img src="out/fig3-5-2b.png" width="81" height="81" |
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| 454 | class="math" alt="2×2 balanced matrix" /> |
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| 455 | = |
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| 456 | <img src="out/fig3-5-2.png" width="241" height="161" |
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| 457 | class="math" alt="2×2-expanded Floyd-Steinberg" /> |
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| 458 | </p> |
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| 459 | |
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| 460 | <p> Here are the results using the previous pixel blocks: </p> |
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| 461 | |
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| 462 | <p style="text-align: center;"> |
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| 463 | <img src="out/lena3-5-2.png" width="256" height="256" |
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| 464 | class="inline" alt="2×2 block Floyd-Steinberg" /> |
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| 465 | <img src="out/grad3-5-2.png" width="32" height="256" |
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| 466 | class="inline" alt="2×2 block Floyd-Steinberg gradient" /> |
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| 467 | </p> |
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| 468 | |
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| 469 | <p> Carefully chosen blocks create constraints on the final picture that may |
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| 470 | be of artistic interest: </p> |
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| 471 | |
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| 472 | <p style="text-align: center;"> |
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| 473 | <img src="fig3-5-3.png" width="354" height="207" |
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| 474 | class="matrix" alt="artistic 3×3 blocks" /> |
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| 475 | <img src="out/lena3-5-3.png" width="256" height="256" |
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| 476 | class="inline" alt="3×3 block Floyd-Steinberg" /> |
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| 477 | <img src="out/grad3-5-3.png" width="32" height="256" |
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| 478 | class="inline" alt="3×3 block Floyd-Steinberg gradient" /> |
|---|
| 479 | </p> |
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| 480 | |
|---|
| 481 | <p> Using all possible pixel blocks is not equivalent to dithering the image |
|---|
| 482 | pixel by pixel. This is due to both the block-choosing method, which only |
|---|
| 483 | minimises the difference of mean values within blocks intead of the sum of |
|---|
| 484 | local distances, and to the inefficient matrix coefficients, which propagate |
|---|
| 485 | the error beyond immediate neighbours, causing the image to look sharpened. |
|---|
| 486 | </p> |
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| 487 | |
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| 488 | <p> This example shows standard block Floyd-Steinberg using all possible 2×2 |
|---|
| 489 | blocks: </p> |
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| 490 | |
|---|
| 491 | <p style="text-align: center;"> |
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| 492 | <img src="fig3-5-4.png" width="200" height="200" |
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| 493 | class="matrix" alt="all possible 2×2 blocks" /> |
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| 494 | <img src="out/lena3-5-4.png" width="256" height="256" |
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| 495 | class="inline" alt="full 2×2 block Floyd-Steinberg" /> |
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| 496 | <img src="out/grad3-5-4.png" width="32" height="256" |
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| 497 | class="inline" alt="full 2×2 block Floyd-Steinberg gradient" /> |
|---|
| 498 | </p> |
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| 499 | |
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| 500 | <p> The results on the vertical gradient indicate poor block-choosing. In |
|---|
| 501 | order to improve it, we introduce a modified, weighted intra-block error |
|---|
| 502 | distribution matrix, still based on the original Floyd-Steinberg matrix: </p> |
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| 503 | |
|---|
| 504 | <p style="text-align: center; font-size: 2em;"> |
|---|
| 505 | <img src="out/fig3-1-1.png" width="121" height="81" |
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| 506 | class="math" alt="Floyd-Steinberg" /> |
|---|
| 507 | ⊗ |
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| 508 | <img src="out/fig3-5-5b.png" width="81" height="81" |
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| 509 | class="math" alt="weighted 2×2 matrix" /> |
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| 510 | = |
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| 511 | <img src="out/fig3-5-5.png" width="241" height="161" |
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| 512 | class="math" alt="weighted 2×2 propagation matrix" /> |
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| 513 | </p> |
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| 514 | |
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| 515 | <p> The result still looks sharpened, but shows considerably less noise: </p> |
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| 516 | |
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| 517 | <p style="text-align: center;"> |
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| 518 | <img src="out/lena3-5-5.png" width="256" height="256" |
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| 519 | class="inline" alt="weighted full 2×2 block Floyd-Steinberg" /> |
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| 520 | <img src="out/grad3-5-5.png" width="32" height="256" |
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| 521 | class="inline" alt="weighted full 2×2 block Floyd-Steinberg gradient" /> |
|---|
| 522 | </p> |
|---|
| 523 | |
|---|
| 524 | <h3> 3.6. Sub-block error diffusion </h3> |
|---|
| 525 | |
|---|
| 526 | <p> We introduce <b>sub-block error diffusion</b>, a novel technique improving |
|---|
| 527 | on block error diffusion. It addresses the following observations: </p> |
|---|
| 528 | |
|---|
| 529 | <ul> |
|---|
| 530 | <li> it is not a requirement to propagate the error beyond the immediate |
|---|
| 531 | neighbours; since it causes a sharpen effect, we decide not to do it. |
|---|
| 532 | </li> |
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| 533 | <li> the individual subpixels’ error should be propagated, not the |
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| 534 | global block error. </li> |
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| 535 | <li> subpixel <b>a</b>’s error is harder to compensate than subpixel |
|---|
| 536 | <b>d</b>’s because its immediate neighbours are already in the block |
|---|
| 537 | being processed, so we weight the sub-block matching in order to |
|---|
| 538 | prioritise pixel <b>a</b>’s matching. </li> |
|---|
| 539 | </ul> |
|---|
| 540 | |
|---|
| 541 | <p> We use <i>m⋅n</i> error diffusion matrices, one for each of the current |
|---|
| 542 | block’s pixels. Here are four error diffusion matrices for 2×2 blocks, |
|---|
| 543 | generated from the standard Floyd-Steinberg matrix: </p> |
|---|
| 544 | |
|---|
| 545 | <p style="text-align: center;"> |
|---|
| 546 | <img src="out/fig3-6-1a.png" width="161" height="121" |
|---|
| 547 | class="math" alt="sub-block 0,0 Floyd-Steinberg" /> |
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| 548 | <img src="out/fig3-6-1b.png" width="161" height="121" |
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| 549 | class="math" alt="sub-block 1,0 Floyd-Steinberg" /> |
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| 550 | </p> |
|---|
| 551 | |
|---|
| 552 | <p style="text-align: center;"> |
|---|
| 553 | <img src="out/fig3-6-1c.png" width="161" height="121" |
|---|
| 554 | class="math" alt="sub-block 0,1 Floyd-Steinberg" /> |
|---|
| 555 | <img src="out/fig3-6-1d.png" width="161" height="121" |
|---|
| 556 | class="math" alt="sub-block 1,1 Floyd-Steinberg" /> |
|---|
| 557 | </p> |
|---|
| 558 | |
|---|
| 559 | <p> The results are far better than with the original block error diffusion |
|---|
| 560 | method. On the left, sub-block error diffusion with all possible 2×2 blocks. |
|---|
| 561 | On the right, sub-block error diffusion restricted to the tiles seen in |
|---|
| 562 | 3.5: </p> |
|---|
| 563 | |
|---|
| 564 | <p style="text-align: center;"> |
|---|
| 565 | <img src="out/lena3-6-1.png" width="256" height="256" |
|---|
| 566 | class="inline" alt="full 2×2 sub-block Floyd-Steinberg" /> |
|---|
| 567 | <img src="out/grad3-6-1.png" width="32" height="256" |
|---|
| 568 | class="inline" alt="full 2×2 sub-block Floyd-Steinberg gradient" /> |
|---|
| 569 | <img src="out/lena3-6-2.png" width="256" height="256" |
|---|
| 570 | class="inline" alt="2×2 lines sub-block Floyd-Steinberg" /> |
|---|
| 571 | <img src="out/grad3-6-2.png" width="32" height="256" |
|---|
| 572 | class="inline" alt="2×2 lines sub-block Floyd-Steinberg gradient" /> |
|---|
| 573 | </p> |
|---|
| 574 | |
|---|
| 575 | <p> Similar error diffusion matrices can be generated for 3×3 blocks: </p> |
|---|
| 576 | |
|---|
| 577 | <p style="text-align: center;"> |
|---|
| 578 | <img src="out/fig3-6-3a.png" width="150" height="120" |
|---|
| 579 | class="math" alt="sub-block 0,0/3×3 Floyd-Steinberg" /> |
|---|
| 580 | <img src="out/fig3-6-3b.png" width="150" height="120" |
|---|
| 581 | class="math" alt="sub-block 1,0/3×3 Floyd-Steinberg" /> |
|---|
| 582 | <img src="out/fig3-6-3c.png" width="150" height="120" |
|---|
| 583 | class="math" alt="sub-block 2,0/3×3 Floyd-Steinberg" /> |
|---|
| 584 | </p> |
|---|
| 585 | |
|---|
| 586 | <p style="text-align: center;"> |
|---|
| 587 | <img src="out/fig3-6-3d.png" width="150" height="120" |
|---|
| 588 | class="math" alt="sub-block 0,1/3×3 Floyd-Steinberg" /> |
|---|
| 589 | <img src="out/fig3-6-3e.png" width="150" height="120" |
|---|
| 590 | class="math" alt="sub-block 1,1/3×3 Floyd-Steinberg" /> |
|---|
| 591 | <img src="out/fig3-6-3f.png" width="150" height="120" |
|---|
| 592 | class="math" alt="sub-block 2,1/3×3 Floyd-Steinberg" /> |
|---|
| 593 | </p> |
|---|
| 594 | |
|---|
| 595 | <p style="text-align: center;"> |
|---|
| 596 | <img src="out/fig3-6-3g.png" width="150" height="120" |
|---|
| 597 | class="math" alt="sub-block 0,2/3×3 Floyd-Steinberg" /> |
|---|
| 598 | <img src="out/fig3-6-3h.png" width="150" height="120" |
|---|
| 599 | class="math" alt="sub-block 1,2/3×3 Floyd-Steinberg" /> |
|---|
| 600 | <img src="out/fig3-6-3i.png" width="150" height="120" |
|---|
| 601 | class="math" alt="sub-block 2,2/3×3 Floyd-Steinberg" /> |
|---|
| 602 | </p> |
|---|
| 603 | |
|---|
| 604 | <p> Here are the results with all the possible 3×3 blocks, and with the |
|---|
| 605 | artistic 3×3 blocks seen in 3.5: </p> |
|---|
| 606 | |
|---|
| 607 | <p style="text-align: center;"> |
|---|
| 608 | <img src="out/lena3-6-3.png" width="256" height="256" |
|---|
| 609 | class="inline" alt="3×3 sub-block Floyd-Steinberg" /> |
|---|
| 610 | <img src="out/grad3-6-3.png" width="32" height="256" |
|---|
| 611 | class="inline" alt="3×3 sub-block Floyd-Steinberg gradient" /> |
|---|
| 612 | <img src="out/lena3-6-4.png" width="256" height="256" |
|---|
| 613 | class="inline" alt="3×3 artistic sub-block Floyd-Steinberg" /> |
|---|
| 614 | <img src="out/grad3-6-4.png" width="32" height="256" |
|---|
| 615 | class="inline" alt="3×3 artistic sub-block Floyd-Steinberg gradient" /> |
|---|
| 616 | </p> |
|---|
| 617 | |
|---|
| 618 | <div style="float: left;"> |
|---|
| 619 | <a href="part2.html">Halftoning <<<</a> |
|---|
| 620 | </div> |
|---|
| 621 | <div style="float: right;"> |
|---|
| 622 | <a href="part4.html">>>> Model-based dithering</a> |
|---|
| 623 | </div> |
|---|
| 624 | <div style="text-align: center;"> |
|---|
| 625 | <a href="index.html">^^^ Index</a> |
|---|
| 626 | </div> |
|---|
| 627 | |
|---|
| 628 | <?php $rev = '$Id$'; |
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| 629 | include($_SERVER['DOCUMENT_ROOT'].'/footer.inc'); ?> |
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| 630 | |
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| 631 | </body> |
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| 632 | </html> |
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