Table of contents [expand all] [collapse all]

- Abstract
- Algorithms
- State of art available in current software (comparison)
- Animated comparison
- From there to here
- Appendix 1: Gamma correction
- Appendix 2: Threshold matrix
- Appendix 3: Color comparisons
- Appendix 4: PNG compression
- Authors
- Literature

This paper introduces a patent-free**¹** positional (ordered) dithering algorithm
that is applicable for arbitrary palettes. Such dithering algorithm
can be used to change truecolor animations into paletted ones, while
maximally avoiding unintended jitter arising from dithering.

For most of the article, we will use this example truecolor picture and palette. The scene is from a PSX game called Chrono Cross, and the palette has been manually selected for this particular task.

You may immediately notice that the palette is not regular; although there
are clearly *some* gradients, the gradients are not regularly spaced.

Undithered rendering is, in pseudo code:
`√(ΔRed² + ΔGreen² + ΔBlue²)`.
We will also begin from there.

For each pixel, Input, in the original picture: Color = FindClosestColorFrom(Palette, Input) Draw pixel using that color.It will produce a picture like that on the left. The exact appearance depends on the particular "closest color" function. Most software uses a simple euclidean RGB distance to determine how well colors match, i.e.

Error diffusion dithers work by distributing an error to neighboring
pixels in hope that one error won't show when the *whole* picture
is equally off. Although it works great for static pictures, it won't
work for animation.

On the right is an animation with the single static screenshot shown above.
A *single yellow pixel* was added to the image and moved around.
The animation has been quantized to 16 colors and dithered
using Floyd-Steinberg dithering. An entire cone of jittering
artifacts gets spawned from that single point downwards and to the right.

There exist different error diffusion dithers, but they all suffer from the same problem. Aside from Riemersma (which walks through the pixels in a non-linear order) and Scolorq (which treats an entire image at once), they all use the same algorithm, only differing on the diffusion map that they use.

Standard ordered dithering, which uses the Bayer threshold matrix, is:

There are many immediate problems one may notice in this picture, the most important being that it simply*looks bad*.
The reason why this is an inadequate algorithm *as is* is because the algorithm
assumes that the palette contains equally spaced elements on each of the R,G,B axis.
An example of such palette is the *web-safe palette* (shown on the right),
which contains colors for each combination of six-bit red, green and blue.

However, in practical applications, this is rarely the case. An example would be developing a game for a handheld device that can only display 16 simultaneous colors from a larger palette. The 16 colors would have to be an optimal representation of the colors present in original graphics.

Although the algorithm can be slightly improved by*measuring* rather than
estimating the maximum distance between successive values on each channel
in the palette (below), such improvements only rarely give a
satisfying outcome. They also tend to reduce the dithering
benefits (compare the hanging gray curtain before and after).

Threshold = COLOR(256/4, 256/4, 256/4); /* Estimated precision of the palette */ For each pixel, Input, in the original picture: Factor = ThresholdMatrix[xcoordinate % X][ycoordinate % Y]; Attempt = Input + Factor * Threshold Color = FindClosestColorFrom(Palette, Attempt) Draw pixel using ColorIf we translate this into PHP, a whole test program becomes (using the image and palette that is described after the program):

<?php /* Create a 8x8 threshold map */ $map = array_map(function($p) { $q = $p ^ ($p >> 3); return ((($p & 4) >> 2) | (($q & 4) >> 1) | (($p & 2) << 1) | (($q & 2) << 2) | (($p & 1) << 4) | (($q & 1) << 5)) / 64.0; }, range(0,63)); /* Define palette */ $pal = Array(0x080000,0x201A0B,0x432817,0x492910, 0x234309,0x5D4F1E,0x9C6B20,0xA9220F, 0x2B347C,0x2B7409,0xD0CA40,0xE8A077, 0x6A94AB,0xD5C4B3,0xFCE76E,0xFCFAE2); /* Read input image */ $srcim = ImageCreateFromPng('scene.png'); $w = ImageSx($srcim); $h = ImageSy($srcim); /* Create paletted image */ $im = ImageCreate($w,$h); foreach($pal as $c) ImageColorAllocate($im, $c>>16, ($c>>8)&0xFF, $c&0xFF); $thresholds = Array(256/4, 256/4, 256/4); /* Render the paletted image by converting each input pixel using the threshold map. */ for($y=0; $y<$h; ++$y) for($x=0; $x<$w; ++$x) { $map_value = $map[($x & 7) + (($y & 7) << 3)]; $color = ImageColorsForIndex($srcim, ImageColorAt($srcim, $x,$y)); $r = (int)($color['red'] + $map_value * $thresholds[0]); $g = (int)($color['green'] + $map_value * $thresholds[1]); $b = (int)($color['blue'] + $map_value * $thresholds[2]); /* Plot using the palette index with color that is closest to this value */ ImageSetPixel($im, $x,$y, ImageColorClosest($im, $r,$g,$b)); } ImagePng($im, 'scenebayer0.png');Here is what this program produces:

There are many immediate problems one may notice in this picture, the most important being that it simply

However, in practical applications, this is rarely the case. An example would be developing a game for a handheld device that can only display 16 simultaneous colors from a larger palette. The 16 colors would have to be an optimal representation of the colors present in original graphics.

Although the algorithm can be slightly improved by

/* Find the maximum distance between successive values on each channel in the palette */ $thresholds = Array(0, 0, 0); foreach($thresholds as $channel => &$t) { $values = array_map(function($val) use($channel) { return ($val >> ($channel*8)) & 0xFF; }, $pal); sort($values); /* Sort the color values of the palette in ascending order */ array_reduce($values, function($p,$val) use(&$t) { $t = max($t, $val-$p); return $val; }, $values[0]); }We present here several algorithms that have the "goods" from Bayer's ordered dithering algorithm (namely, the color of a pixel depends on that pixel alone, making it suitable for animations), but is applicable to arbitrary palettes.

We begin by making the observation that the ordered dithering algorithm
always mixes *two* colors together in a variable proportion.

#### Psychovisual model

#### Refinements

#### Tri-tone dithering

Using the same principle, we begin by envisioning a method of optimizing that pair of colors:

For each pixel, Input, in the original picture: Factor = ThresholdMatrix[xcoordinate % X][ycoordinate % Y]; Make a Plan, based on Input and the Palette. If Factor < Plan.ratio, Draw pixel using Plan.color2 else, Draw pixel using Plan.color1The Planning procedure can be implemented as follows:

SmallestPenalty = 10^99 /* Impossibly large number */ For each unique combination of two colors from the palette, Color1 and Color2: For each possible Ratio, 0 .. (X*Y-1): /* Determine what mixing the two colors in this proportion will produce */ Mixed = Color1 + Ratio * (Color2 - Color1) / (X*Y) /* Rate how well it matches what we want to accomplish */ Penalty = Evaluate the difference of Input and Mixed. /* Keep the result that has the smallest error */ If Penalty < SmallestPenalty, SmallestPenalty = Penalty Plan = { Color1, Color2, Ratio / (X*Y) }.This function runs for M × N × (N −1) ÷ 2 + N iterations for a palette of size N and a dithering pattern of size M = X×Y, complexity being O(N²×M), and it depends on an evaluation function.

The evaluation function might be defined using an euclidean distance
between the two colors, considered as three-dimensional vectors
formed by the Red, Green and Blue color components, i.e.
`√(ΔRed² + ΔGreen² + ΔBlue²)` as discussed earlier.

The whole program becomes in C++:

#include <gd.h> #include <stdio.h> /* 8x8 threshold map */ #define d(x) x/64.0 static const double map[8*8] = { d( 0), d(48), d(12), d(60), d( 3), d(51), d(15), d(63), d(32), d(16), d(44), d(28), d(35), d(19), d(47), d(31), d( 8), d(56), d( 4), d(52), d(11), d(59), d( 7), d(55), d(40), d(24), d(36), d(20), d(43), d(27), d(39), d(23), d( 2), d(50), d(14), d(62), d( 1), d(49), d(13), d(61), d(34), d(18), d(46), d(30), d(33), d(17), d(45), d(29), d(10), d(58), d( 6), d(54), d( 9), d(57), d( 5), d(53), d(42), d(26), d(38), d(22), d(41), d(25), d(37), d(21) }; #undef d /* Palette */ static const unsigned pal[16] = {0x080000,0x201A0B,0x432817,0x492910, 0x234309,0x5D4F1E,0x9C6B20,0xA9220F, 0x2B347C,0x2B7409,0xD0CA40,0xE8A077, 0x6A94AB,0xD5C4B3,0xFCE76E,0xFCFAE2 }; // Compare the difference of two RGB values double ColorCompare(int r1,int g1,int b1, int r2,int g2,int b2) { double diffR = (r1-r2)/255.0, diffG = (g1-g2)/255.0, diffB = (b1-b2)/255.0; return diffR*diffR + diffG*diffG + diffB*diffB; } double EvaluateMixingError(int r,int g,int b, // Desired color int r0,int g0,int b0, // Mathematical mix product int r1,int g1,int b1, // Mix component 1 int r2,int g2,int b2, // Mix component 2 double ratio) // Mixing ratio { return ColorCompare(r,g,b, r0,g0,b0); } struct MixingPlan { unsigned colors[2]; double ratio; /* 0 = always index1, 1 = always index2, 0.5 = 50% of both */ }; MixingPlan DeviseBestMixingPlan(unsigned color) { const unsigned r = color>>16, g = (color>>8)&0xFF, b = color&0xFF; MixingPlan result = { {0,0}, 0.5 }; double least_penalty = 1e99; // Loop through every unique combination of two colors from the palette, // and through each possible way to mix those two colors. They can be // mixed in exactly 64 ways, when the threshold matrix is 8x8. for(unsigned index1 = 0; index1 < 16; ++index1) for(unsigned index2 = index1; index2 < 16; ++index2) for(unsigned ratio=0; ratio<64; ++ratio) { if(index1 == index2 && ratio != 0) break; // Determine the two component colors unsigned color1 = pal[index1], color2 = pal[index2]; unsigned r1 = color1>>16, g1 = (color1>>8)&0xFF, b1 = color1&0xFF; unsigned r2 = color2>>16, g2 = (color2>>8)&0xFF, b2 = color2&0xFF; // Determine what mixing them in this proportion will produce unsigned r0 = r1 + ratio * int(r2-r1) / 64; unsigned g0 = g1 + ratio * int(g2-g1) / 64; unsigned b0 = b1 + ratio * int(b2-b1) / 64; // Determine how well that matches what we want to accomplish double penalty = EvaluateMixingError(r,g,b, r0,g0,b0, r1,g1,b1, r2,g2,b2, ratio/64.0); if(penalty < least_penalty) { // Keep the result that has the smallest error least_penalty = penalty; result.colors[0] = index1; result.colors[1] = index2; result.ratio = ratio / 64.0; } } return result; } int main() { FILE* fp = fopen("scene.png", "rb"); gdImagePtr srcim = gdImageCreateFromPng(fp); fclose(fp); unsigned w = gdImageSX(srcim), h = gdImageSY(srcim); gdImagePtr im = gdImageCreate(w, h); for(unsigned c=0; c<16; ++c) gdImageColorAllocate(im, pal[c]>>16, (pal[c]>>8)&0xFF, pal[c]&0xFF); #pragma omp parallel for for(unsigned y=0; y<h; ++y) for(unsigned x=0; x<w; ++x) { double map_value = map[(x & 7) + ((y & 7) << 3)]; unsigned color = gdImageGetTrueColorPixel(srcim, x, y); MixingPlan plan = DeviseBestMixingPlan(color); gdImageSetPixel(im, x,y, plan.colors[ map_value < plan.ratio ? 1 : 0 ] ); } fp = fopen("scenedither1.png", "wb"); gdImagePng(im, fp); fclose(fp); gdImageDestroy(im); gdImageDestroy(srcim); }The result of this program is shown below (on the right-hand side, the standard ordered-dithered version):

There are two problems with this trivial implementation:

- It is very slow.
- There is a lot of visual noise.

- Overall, there is a lot more color [than in the standard version], and the scene does not look that washed out anymore.

This is the mathematically correct result, assuming gamma of 1.0. If one substitutes temporal dithering for the spatial dithering, it is easy to see that the wild dithering patterns do indeed produce, by average, colors very close to the originals. However, the human brain just sees a lot of bright and dim pixels where there should be none. Temporal dithering will be covered later in this article.

Therefore, *psychovisual* concerns must also be accounted for when
implementing this algorithm.

Consider this example palette: #000000, #FFFFFFF, #7E8582, #8A7A76. For producing a color #808080, one might combine the two extremes, black and white. However, this produces a very nasty visual effect. It is better to combine the two slightly tinted values near the intended result, even though it produces #847F7C, a noticeably red-tinted gray value, rather than the mathematically accurate #808080 that would be acquired from combining the two other values.

The psychovisual model that we introduce, consists of three parts:

- Algorithm for comparing the similarity of two color values.
- Criteria for deciding which pixels can be paired together.
- Gamma correction (technically not
*psycho*visual, because it is just physics).

The simplest way to adjust the psychovisual model is to add some code
that considers the difference between the two pixel values that are being
mixed in the dithering process, and penalizes combinations that differ
too much.

double EvaluateMixingError(int r,int g,int b, // Desired color int r0,int g0,int b0, // Mathematical mix product int r1,int g1,int b1, // Mix component 1 int r2,int g2,int b2, // Mix component 2 double ratio) // Mixing ratio { return ColorCompare(r,g,b, r0,g0,b0) + ColorCompare(r1,g1,b1, r2,g2,b2)*0.1; }The result is shown below:

Though the result looks very nice now, there are still many ways the
algorithm can still be improved. For instance, the color comparison
function could be improved by a great deal.
Wikipedia has an entire article
about the topic of comparing two color values.
Most of the improved color comparison functions are based on the CIE
colorspace, but simple improvements can be done in the RGB space too.
Such a simple improvement is shown below. We might call this RGB^{L},
for luminance-weighted RGB.

The EvaluateMixingError function was also changed to weigh the component difference only in inverse proportion to the mixing evenness.

// Compare the difference of two RGB values, weigh by CCIR 601 luminosity: double ColorCompare(int r1,int g1,int b1, int r2,int g2,int b2) { double luma1 = (r1*299 + g1*587 + b1*114) / (255.0*1000); double luma2 = (r2*299 + g2*587 + b2*114) / (255.0*1000); double lumadiff = luma1-luma2; double diffR = (r1-r2)/255.0, diffG = (g1-g2)/255.0, diffB = (b1-b2)/255.0; return (diffR*diffR*0.299 + diffG*diffG*0.587 + diffB*diffB*0.114)*0.75 + lumadiff*lumadiff; } double EvaluateMixingError(int r,int g,int b, int r0,int g0,int b0, int r1,int g1,int b1, int r2,int g2,int b2, double ratio) { return ColorCompare(r,g,b, r0,g0,b0) + ColorCompare(r1,g1,b1, r2,g2,b2) * 0.1 * (fabs(ratio-0.5)+0.5); }

The result is shown below. Improvements can be seen in the rightside
window and in the girl's skirt, among other places.

Left: After tweaking the color comparison function.

Right: Before tweaking the color comparison function.

This version of DeviseBestMixingPlan calculates the mixing ratio mathematically
rather than by iterating. It ends up being about 64 times faster than the
iterating version, and differs only neglibly.

The function now runs for N²÷2 iterations for a palette of size N.

Left: Faster planner

Right: Slower and more thorough planner

The quality did suffer slightly, but the faster rendering might still be worth it.

The function now runs for N²÷2 iterations for a palette of size N.

MixingPlan DeviseBestMixingPlan(unsigned color) { const unsigned r = color>>16, g = (color>>8)&0xFF, b = color&0xFF; MixingPlan result = { {0,0}, 0.5 }; double least_penalty = 1e99; for(unsigned index1 = 0; index1 < 16; ++index1) for(unsigned index2 = index1; index2 < 16; ++index2) { // Determine the two component colors unsigned color1 = pal[index1], color2 = pal[index2]; unsigned r1 = color1>>16, g1 = (color1>>8)&0xFF, b1 = color1&0xFF; unsigned r2 = color2>>16, g2 = (color2>>8)&0xFF, b2 = color2&0xFF; int ratio = 32; if(color1 != color2) { // Determine the ratio of mixing for each channel. // solve(r1 + ratio*(r2-r1)/64 = r, ratio) // Take a weighed average of these three ratios according to the // perceived luminosity of each channel (according to CCIR 601). ratio = ((r2 != r1 ? 299*64 * int(r - r1) / int(r2-r1) : 0) + (g2 != g1 ? 587*64 * int(g - g1) / int(g2-g1) : 0) + (b1 != b2 ? 114*64 * int(b - b1) / int(b2-b1) : 0)) / ((r2 != r1 ? 299 : 0) + (g2 != g1 ? 587 : 0) + (b2 != b1 ? 114 : 0)); if(ratio < 0) ratio = 0; else if(ratio > 63) ratio = 63; } // Determine what mixing them in this proportion will produce unsigned r0 = r1 + ratio * int(r2-r1) / 64; unsigned g0 = g1 + ratio * int(g2-g1) / 64; unsigned b0 = b1 + ratio * int(b2-b1) / 64; double penalty = EvaluateMixingError( r,g,b, r0,g0,b0, r1,g1,b1, r2,g2,b2, ratio / double(64)); if(penalty < least_penalty) { least_penalty = penalty; result.colors[0] = index1; result.colors[1] = index2; result.ratio = ratio / double(64); } } return result; }With these changes, the rendering result becomes:

Left: Faster planner

Right: Slower and more thorough planner

The quality did suffer slightly, but the faster rendering might still be worth it.

When non-realtime rendering is not desired, such as when pre-rendering static pictures or animations for later presentation, one might want to strive for better quality and continue using the slower, looping method. The remainder of this article's pictures will continue using the loop.

The final improvement for this algorithm for now
that is covered in this article is *tri-tone dithering*.
It is a three-color dithering algorithm with a fixed 2x2 matrix, where one of
the colors occurs at 50% proportion and the others occur at 25% proportion.
An example of using this approach is shown on the right.

The complete source code is shown below.
The DeviseBestMixingPlan function now runs for
N² × (N − 1) ÷ 2
iterations for a palette of size N, for
a complexity of O(N^{3}).

#include <gd.h> #include <stdio.h> #include <math.h> /* 8x8 threshold map */ #define d(x) x/64.0 static const double map[8*8] = { d( 0), d(48), d(12), d(60), d( 3), d(51), d(15), d(63), d(32), d(16), d(44), d(28), d(35), d(19), d(47), d(31), d( 8), d(56), d( 4), d(52), d(11), d(59), d( 7), d(55), d(40), d(24), d(36), d(20), d(43), d(27), d(39), d(23), d( 2), d(50), d(14), d(62), d( 1), d(49), d(13), d(61), d(34), d(18), d(46), d(30), d(33), d(17), d(45), d(29), d(10), d(58), d( 6), d(54), d( 9), d(57), d( 5), d(53), d(42), d(26), d(38), d(22), d(41), d(25), d(37), d(21) }; #undef d /* Palette */ static const unsigned pal[16] = {0x080000,0x201A0B,0x432817,0x492910, 0x234309,0x5D4F1E,0x9C6B20,0xA9220F, 0x2B347C,0x2B7409,0xD0CA40,0xE8A077, 0x6A94AB,0xD5C4B3,0xFCE76E,0xFCFAE2 }; double ColorCompare(int r1,int g1,int b1, int r2,int g2,int b2) { double luma1 = (r1*299 + g1*587 + b1*114) / (255.0*1000); double luma2 = (r2*299 + g2*587 + b2*114) / (255.0*1000); double lumadiff = luma1-luma2; double diffR = (r1-r2)/255.0, diffG = (g1-g2)/255.0, diffB = (b1-b2)/255.0; return (diffR*diffR*0.299 + diffG*diffG*0.587 + diffB*diffB*0.114)*0.75 + lumadiff*lumadiff; } double EvaluateMixingError(int r,int g,int b, int r0,int g0,int b0, int r1,int g1,int b1, int r2,int g2,int b2, double ratio) { return ColorCompare(r,g,b, r0,g0,b0) + ColorCompare(r1,g1,b1, r2,g2,b2)*0.1*(fabs(ratio-0.5)+0.5); } struct MixingPlan { unsigned colors[4]; double ratio; /* 0 = always index1, 1 = always index2, 0.5 = 50% of both */ /* 4 = special three or four-color dither */ }; MixingPlan DeviseBestMixingPlan(unsigned color) { const unsigned r = color>>16, g = (color>>8)&0xFF, b = color&0xFF; MixingPlan result = { {0,0}, 0.5 }; double least_penalty = 1e99; for(unsigned index1 = 0; index1 < 16; ++index1) for(unsigned index2 = index1; index2 < 16; ++index2) //for(int ratio=0; ratio<64; ++ratio) { // Determine the two component colors unsigned color1 = pal[index1], color2 = pal[index2]; unsigned r1 = color1>>16, g1 = (color1>>8)&0xFF, b1 = color1&0xFF; unsigned r2 = color2>>16, g2 = (color2>>8)&0xFF, b2 = color2&0xFF; int ratio = 32; if(color1 != color2) { // Determine the ratio of mixing for each channel. // solve(r1 + ratio*(r2-r1)/64 = r, ratio) // Take a weighed average of these three ratios according to the // perceived luminosity of each channel (according to CCIR 601). ratio = ((r2 != r1 ? 299*64 * int(r - r1) / int(r2 - r1) : 0) + (g2 != g1 ? 587*64 * int(g - g1) / int(g2 - g1) : 0) + (b1 != b2 ? 114*64 * int(b - b1) / int(b2 - b1) : 0)) / ((r2 != r1 ? 299 : 0) + (g2 != g1 ? 587 : 0) + (b2 != b1 ? 114 : 0)); if(ratio < 0) ratio = 0; else if(ratio > 63) ratio = 63; } // Determine what mixing them in this proportion will produce unsigned r0 = r1 + ratio * int(r2-r1) / 64; unsigned g0 = g1 + ratio * int(g2-g1) / 64; unsigned b0 = b1 + ratio * int(b2-b1) / 64; double penalty = EvaluateMixingError( r,g,b, r0,g0,b0, r1,g1,b1, r2,g2,b2, ratio / double(64)); if(penalty < least_penalty) { least_penalty = penalty; result.colors[0] = index1; result.colors[1] = index2; result.ratio = ratio / double(64); } if(index1 != index2) for(unsigned index3 = 0; index3 < 16; ++index3) { if(index3 == index2 || index3 == index1) continue; // 50% index3, 25% index2, 25% index1 unsigned color3 = pal[index3]; unsigned r3 = color3>>16, g3 = (color3>>8)&0xFF, b3 = color3&0xFF; r0 = (r1 + r2 + r3*2) / 4; g0 = (g1 + g2 + g3*2) / 4; b0 = (b1 + b2 + b3*2) / 4; penalty = ColorCompare(r,g,b, r0,g0,b0) + ColorCompare(r1,g1,b1, r2,g2,b2)*0.025 + ColorCompare((r1+g1)/2,(g1+g2)/2,(b1+b2)/2, r3,g3,b3)*0.025; if(penalty < least_penalty) { least_penalty = penalty; result.colors[0] = index3; // (0,0) index3 occurs twice result.colors[1] = index1; // (0,1) result.colors[2] = index2; // (1,0) result.colors[3] = index3; // (1,1) result.ratio = 4.0; } } } return result; } int main(int argc, char**argv) { FILE* fp = fopen(argv[1], "rb"); gdImagePtr srcim = gdImageCreateFromPng(fp); fclose(fp); unsigned w = gdImageSX(srcim), h = gdImageSY(srcim); gdImagePtr im = gdImageCreate(w, h); for(unsigned c=0; c<16; ++c) gdImageColorAllocate(im, pal[c]>>16, (pal[c]>>8)&0xFF, pal[c]&0xFF); #pragma omp parallel for for(unsigned y=0; y<h; ++y) for(unsigned x=0; x<w; ++x) { unsigned color = gdImageGetTrueColorPixel(srcim, x, y); MixingPlan plan = DeviseBestMixingPlan(color); if(plan.ratio == 4.0) // Tri-tone or quad-tone dithering { gdImageSetPixel(im, x,y, plan.colors[ ((y&1)*2 + (x&1)) ]); } else { double map_value = map[(x & 7) + ((y & 7) << 3)]; gdImageSetPixel(im, x,y, plan.colors[ map_value < plan.ratio ? 1 : 0 ]); } } fp = fopen(argv[2], "wb"); gdImagePng(im, fp); fclose(fp); gdImageDestroy(im); gdImageDestroy(srcim); }It is also possible to implement quad-tone dithering, but it is too slow to calculate (O(N^4) runtime) using this algorithm. We'll return to that topic later.

An altogetherly different dithering algorithm can be devised by discarding
the initial assumption that the dithering mixes *two* colortones together,
and instead, assuming that each matrix value corresponds to a particular
color tone. A 8x8 matrix has 64 color tones, a 2x2 matrix has 4 color tones,
and so on.

Right: Image produced by the tri-tone dither of the previous chapter.

Left: Image produced with the C++ program above. One may immediately observe that it is better in almost all aspects. For example, the colors in the skirt, and the smooth gradients in the window and in the hanging curtain, look much better now. There are a few more scattered red pixels in this image that look like noise, but arguably, those are*exactly what there should be*
(the original always has red details at those locations).
One thing that still theoretically improves the result is *gamma correction*,
which is a core concept in high quality dithering.

#### Gamma correction

An algorithm for populating such a color array will need to find the N-term expression of color values that, when combined, will produce a the closest approximation of the input color.

One such algorithm is to start with a guess (the closest color), and then find out how much it went wrong, and then find out by experimentation which terms are needed to improve the result.

To solve the issue about pixel orientations changing, the colors will be sorted by luma. They will still change relative orientation, but such action is relatively minor.

In pseudo code, the process of converting the input bitmap into a target bitmap goes like this:

For each pixel, Input, in the original picture: Achieved = 0 // Total color sum achieved so far CandidateList.clear() LoopWhile CandidateList.Size < (X * Y) Count = 1 Candidate = 0 // Candidate color from palette Comparison.reset() Max = CandidateList.Size, Or 1 if empty For each Color in palette: AddingCount = 1 LoopWhile AddingCount <= Max Sum = Achieved + Color * AddingCount Divide = CandidateList.Size + AddingCount Test = Sum / Divide Compare Test to Input using CIEDE2000 or RGB; If it was the best match since Comparison was reset: Candidate := Color Count := AddingCount EndCompare AddingCount = AddingCount * 2 // Faster version // AddingCount = AddingCount + 1 // Slower version EndWhile CandidateList.Add(Candidate, Count times) Achieved = Achieved + Candidate * Count LoopEnd CandidateList.Sort( by: luminance ) Index = ThresholdMatrix[xcoordinate % X][ycoordinate % Y] Draw pixel using CandidateList[Index * CandidateList.Size() / (X*Y)] EndForThe color matching function runs for N × (log2(M) + 1) iterations at minimum and for N × M × log2(M) iterations at maximum.

In C++, it can be written as follows:

#include <gd.h> #include <stdio.h> #include <math.h> #include <algorithm> /* For std::sort() */ #include <map> /* For associative container, std::map<> */ /* 8x8 threshold map */ static const unsigned char map[8*8] = { 0,48,12,60, 3,51,15,63, 32,16,44,28,35,19,47,31, 8,56, 4,52,11,59, 7,55, 40,24,36,20,43,27,39,23, 2,50,14,62, 1,49,13,61, 34,18,46,30,33,17,45,29, 10,58, 6,54, 9,57, 5,53, 42,26,38,22,41,25,37,21 }; /* Palette */ static const unsigned pal[16] = { 0x080000,0x201A0B,0x432817,0x492910, 0x234309,0x5D4F1E,0x9C6B20,0xA9220F, 0x2B347C,0x2B7409,0xD0CA40,0xE8A077, 0x6A94AB,0xD5C4B3,0xFCE76E,0xFCFAE2 }; /* Luminance for each palette entry, to be initialized as soon as the program begins */ static unsigned luma[16]; bool PaletteCompareLuma(unsigned index1, unsigned index2) { return luma[index1] < luma[index2]; } double ColorCompare(int r1,int g1,int b1, int r2,int g2,int b2) { double luma1 = (r1*299 + g1*587 + b1*114) / (255.0*1000); double luma2 = (r2*299 + g2*587 + b2*114) / (255.0*1000); double lumadiff = luma1-luma2; double diffR = (r1-r2)/255.0, diffG = (g1-g2)/255.0, diffB = (b1-b2)/255.0; return (diffR*diffR*0.299 + diffG*diffG*0.587 + diffB*diffB*0.114)*0.75 + lumadiff*lumadiff; } struct MixingPlan { const unsigned n_colors = 16; unsigned colors[n_colors]; }; MixingPlan DeviseBestMixingPlan(unsigned color) { MixingPlan result = { {0} }; const unsigned src[3] = { color>>16, (color>>8)&0xFF, color&0xFF }; unsigned proportion_total = 0; unsigned so_far[3] = {0,0,0}; while(proportion_total < MixingPlan::n_colors) { unsigned chosen_amount = 1; unsigned chosen = 0; const unsigned max_test_count = std::max(1u, proportion_total); double least_penalty = -1; for(unsigned index=0; index<16; ++index) { const unsigned color = pal[index]; unsigned sum[3] = { so_far[0], so_far[1], so_far[2] }; unsigned add[3] = { color>>16, (color>>8)&0xFF, color&0xFF }; for(unsigned p=1; p<=max_test_count; p*=2) { for(unsigned c=0; c<3; ++c) sum[c] += add[c]; for(unsigned c=0; c<3; ++c) add[c] += add[c]; unsigned t = proportion_total + p; unsigned test[3] = { sum[0] / t, sum[1] / t, sum[2] / t }; double penalty = ColorCompare(src[0],src[1],src[2], test[0],test[1],test[2]); if(penalty < least_penalty || least_penalty < 0) { least_penalty = penalty; chosen = index; chosen_amount = p; } } } for(unsigned p=0; p<chosen_amount; ++p) { if(proportion_total >= MixingPlan::n_colors) break; result.colors[proportion_total++] = chosen; } const unsigned color = pal[chosen]; unsigned palcolor[3] = { color>>16, (color>>8)&0xFF, color&0xFF }; for(unsigned c=0; c<3; ++c) so_far[c] += palcolor[c] * chosen_amount; } // Sort the colors according to luminance std::sort(result.colors, result.colors+MixingPlan::n_colors, PaletteCompareLuma); return result; } int main(int argc, char**argv) { FILE* fp = fopen(argv[1], "rb"); gdImagePtr srcim = gdImageCreateFromPng(fp); fclose(fp); unsigned w = gdImageSX(srcim), h = gdImageSY(srcim); gdImagePtr im = gdImageCreate(w, h); for(unsigned c=0; c<16; ++c) { unsigned r = pal[c]>>16, g = (pal[c]>>8) & 0xFF, b = pal[c] & 0xFF; gdImageColorAllocate(im, r,g,b); luma[c] = r*299 + g*587 + b*114; } #pragma omp parallel for for(unsigned y=0; y<h; ++y) for(unsigned x=0; x<w; ++x) { unsigned color = gdImageGetTrueColorPixel(srcim, x, y); unsigned map_value = map[(x & 7) + ((y & 7) << 3)]; MixingPlan plan = DeviseBestMixingPlan(color); map_value = map_value * MixingPlan::n_colors / 64; gdImageSetPixel(im, x,y, plan.colors[ map_value ]); } fp = fopen(argv[2], "wb"); gdImagePng(im, fp); fclose(fp); gdImageDestroy(im); gdImageDestroy(srcim); }Here is what this program produces.

Right: Image produced by the tri-tone dither of the previous chapter.

Left: Image produced with the C++ program above. One may immediately observe that it is better in almost all aspects. For example, the colors in the skirt, and the smooth gradients in the window and in the hanging curtain, look much better now. There are a few more scattered red pixels in this image that look like noise, but arguably, those are

The principle and rationale for gamma correction is explained in a later chapter.
`COMPARE_RGB` hack disabled.

Right: Original picture.

Left: Gamma correction added. It definitely changed the picture.
It is now somewhat greener. The previous one was maybe too blue.
Mathematically this is the better picture, and the eye seems to
somewhat agree.

C++ source code for the version with gamma correction:

#include <gd.h> #include <stdio.h> #include <math.h> #include <algorithm> /* For std::sort() */ #include <vector> #include <map> /* For associative container, std::map<> */ #define COMPARE_RGB 1 /* 8x8 threshold map */ static const unsigned char map[8*8] = { 0,48,12,60, 3,51,15,63, 32,16,44,28,35,19,47,31, 8,56, 4,52,11,59, 7,55, 40,24,36,20,43,27,39,23, 2,50,14,62, 1,49,13,61, 34,18,46,30,33,17,45,29, 10,58, 6,54, 9,57, 5,53, 42,26,38,22,41,25,37,21 }; static const double Gamma = 2.2; // Gamma correction we use. double GammaCorrect(double v) { return pow(v, Gamma); } double GammaUncorrect(double v) { return pow(v, 1.0 / Gamma); } /* CIE C illuminant */ static const double illum[3*3] = { 0.488718, 0.176204, 0.000000, 0.310680, 0.812985, 0.0102048, 0.200602, 0.0108109, 0.989795 }; struct LabItem // CIE L*a*b* color value with C and h added. { double L,a,b,C,h; LabItem() { } LabItem(double R,double G,double B) { Set(R,G,B); } void Set(double R,double G,double B) { const double* const i = illum; double X = i[0]*R + i[3]*G + i[6]*B, x = X / (i[0] + i[1] + i[2]); double Y = i[1]*R + i[4]*G + i[7]*B, y = Y / (i[3] + i[4] + i[5]); double Z = i[2]*R + i[5]*G + i[8]*B, z = Z / (i[6] + i[7] + i[8]); const double threshold1 = (6*6*6.0)/(29*29*29.0); const double threshold2 = (29*29.0)/(6*6*3.0); double x1 = (x > threshold1) ? pow(x, 1.0/3.0) : (threshold2*x)+(4/29.0); double y1 = (y > threshold1) ? pow(y, 1.0/3.0) : (threshold2*y)+(4/29.0); double z1 = (z > threshold1) ? pow(z, 1.0/3.0) : (threshold2*z)+(4/29.0); L = (29*4)*y1 - (4*4); a = (500*(x1-y1) ); b = (200*(y1-z1) ); C = sqrt(a*a + b+b); h = atan2(b, a); } LabItem(unsigned rgb) { Set(rgb); } void Set(unsigned rgb) { Set( (rgb>>16)/255.0, ((rgb>>8)&0xFF)/255.0, (rgb&0xFF)/255.0 ); } }; /* From the paper "The CIEDE2000 Color-Difference Formula: Implementation Notes, */ /* Supplementary Test Data, and Mathematical Observations", by */ /* Gaurav Sharma, Wencheng Wu and Edul N. Dalal, */ /* Color Res. Appl., vol. 30, no. 1, pp. 21-30, Feb. 2005. */ /* Return the CIEDE2000 Delta E color difference measure squared, for two Lab values */ static double ColorCompare(const LabItem& lab1, const LabItem& lab2) { #define RAD2DEG(xx) (180.0/M_PI * (xx)) #define DEG2RAD(xx) (M_PI/180.0 * (xx)) /* Compute Cromanance and Hue angles */ double C1,C2, h1,h2; { double Cab = 0.5 * (lab1.C + lab2.C); double Cab7 = pow(Cab,7.0); double G = 0.5 * (1.0 - sqrt(Cab7/(Cab7 + 6103515625.0))); double a1 = (1.0 + G) * lab1.a; double a2 = (1.0 + G) * lab2.a; C1 = sqrt(a1 * a1 + lab1.b * lab1.b); C2 = sqrt(a2 * a2 + lab2.b * lab2.b); if (C1 < 1e-9) h1 = 0.0; else { h1 = RAD2DEG(atan2(lab1.b, a1)); if (h1 < 0.0) h1 += 360.0; } if (C2 < 1e-9) h2 = 0.0; else { h2 = RAD2DEG(atan2(lab2.b, a2)); if (h2 < 0.0) h2 += 360.0; } } /* Compute delta L, C and H */ double dL = lab2.L - lab1.L, dC = C2 - C1, dH; { double dh; if (C1 < 1e-9 || C2 < 1e-9) { dh = 0.0; } else { dh = h2 - h1; /**/ if (dh > 180.0) dh -= 360.0; else if (dh < -180.0) dh += 360.0; } dH = 2.0 * sqrt(C1 * C2) * sin(DEG2RAD(0.5 * dh)); } double h; double L = 0.5 * (lab1.L + lab2.L); double C = 0.5 * (C1 + C2); if (C1 < 1e-9 || C2 < 1e-9) { h = h1 + h2; } else { h = h1 + h2; if (fabs(h1 - h2) > 180.0) { /**/ if (h < 360.0) h += 360.0; else if (h >= 360.0) h -= 360.0; } h *= 0.5; } double T = 1.0 - 0.17 * cos(DEG2RAD(h - 30.0)) + 0.24 * cos(DEG2RAD(2.0 * h)) + 0.32 * cos(DEG2RAD(3.0 * h + 6.0)) - 0.2 * cos(DEG2RAD(4.0 * h - 63.0)); double hh = (h - 275.0)/25.0; double ddeg = 30.0 * exp(-hh * hh); double C7 = pow(C,7.0); double RC = 2.0 * sqrt(C7/(C7 + 6103515625.0)); double L50sq = (L - 50.0) * (L - 50.0); double SL = 1.0 + (0.015 * L50sq) / sqrt(20.0 + L50sq); double SC = 1.0 + 0.045 * C; double SH = 1.0 + 0.015 * C * T; double RT = -sin(DEG2RAD(2 * ddeg)) * RC; double dLsq = dL/SL, dCsq = dC/SC, dHsq = dH/SH; return dLsq*dLsq + dCsq*dCsq + dHsq*dHsq + RT*dCsq*dHsq; #undef RAD2DEG #undef DEG2RAD } static double ColorCompare(int r1,int g1,int b1, int r2,int g2,int b2) { double luma1 = (r1*299 + g1*587 + b1*114) / (255.0*1000); double luma2 = (r2*299 + g2*587 + b2*114) / (255.0*1000); double lumadiff = luma1-luma2; double diffR = (r1-r2)/255.0, diffG = (g1-g2)/255.0, diffB = (b1-b2)/255.0; return (diffR*diffR*0.299 + diffG*diffG*0.587 + diffB*diffB*0.114)*0.75 + lumadiff*lumadiff; } /* Palette */ static const unsigned palettesize = 16; static const unsigned pal[palettesize] = { 0x080000,0x201A0B,0x432817,0x492910, 0x234309,0x5D4F1E,0x9C6B20,0xA9220F, 0x2B347C,0x2B7409,0xD0CA40,0xE8A077, 0x6A94AB,0xD5C4B3,0xFCE76E,0xFCFAE2 }; /* Luminance for each palette entry, to be initialized as soon as the program begins */ static unsigned luma[palettesize]; static LabItem meta[palettesize]; static double pal_g[palettesize][3]; // Gamma-corrected palette entry inline bool PaletteCompareLuma(unsigned index1, unsigned index2) { return luma[index1] < luma[index2]; } typedef std::vector<unsigned> MixingPlan; MixingPlan DeviseBestMixingPlan(unsigned color, size_t limit) { // Input color in RGB int input_rgb[3] = { ((color>>16)&0xFF), ((color>>8)&0xFF), (color&0xFF) }; // Input color in CIE L*a*b* LabItem input(color); // Tally so far (gamma-corrected) double so_far[3] = { 0,0,0 }; MixingPlan result; while(result.size() < limit) { unsigned chosen_amount = 1; unsigned chosen = 0; const unsigned max_test_count = result.empty() ? 1 : result.size(); double least_penalty = -1; for(unsigned index=0; index<palettesize; ++index) { const unsigned color = pal[index]; double sum[3] = { so_far[0], so_far[1], so_far[2] }; double add[3] = { pal_g[index][0], pal_g[index][1], pal_g[index][2] }; for(unsigned p=1; p<=max_test_count; p*=2) { for(unsigned c=0; c<3; ++c) sum[c] += add[c]; for(unsigned c=0; c<3; ++c) add[c] += add[c]; double t = result.size() + p; double test[3] = { GammaUncorrect(sum[0]/t), GammaUncorrect(sum[1]/t), GammaUncorrect(sum[2]/t) }; #if COMPARE_RGB double penalty = ColorCompare( input_rgb[0],input_rgb[1],input_rgb[2], test[0]*255, test[1]*255, test[2]*255); #else LabItem test_lab( test[0], test[1], test[2] ); double penalty = ColorCompare(test_lab, input); #endif if(penalty < least_penalty || least_penalty < 0) { least_penalty = penalty; chosen = index; chosen_amount = p; } } } // Append "chosen_amount" times "chosen" to the color list result.resize(result.size() + chosen_amount, chosen); for(unsigned c=0; c<3; ++c) so_far[c] += pal_g[chosen][c] * chosen_amount; } // Sort the colors according to luminance std::sort(result.begin(), result.end(), PaletteCompareLuma); return result; } int main(int argc, char**argv) { FILE* fp = fopen(argv[1], "rb"); gdImagePtr srcim = gdImageCreateFromPng(fp); fclose(fp); unsigned w = gdImageSX(srcim), h = gdImageSY(srcim); gdImagePtr im = gdImageCreate(w, h); for(unsigned c=0; c<palettesize; ++c) { unsigned r = pal[c]>>16, g = (pal[c]>>8) & 0xFF, b = pal[c] & 0xFF; gdImageColorAllocate(im, r,g,b); luma[c] = r*299 + g*587 + b*114; meta[c].Set(pal[c]); pal_g[c][0] = GammaCorrect(r/255.0); pal_g[c][1] = GammaCorrect(g/255.0); pal_g[c][2] = GammaCorrect(b/255.0); } #pragma omp parallel for for(unsigned y=0; y<h; ++y) for(unsigned x=0; x<w; ++x) { unsigned color = gdImageGetTrueColorPixel(srcim, x, y); unsigned map_value = map[(x & 7) + ((y & 7) << 3)]; MixingPlan plan = DeviseBestMixingPlan(color, 16); map_value = map_value * plan.size() / 64; gdImageSetPixel(im, x,y, plan[ map_value ]); } fp = fopen(argv[2], "wb"); gdImagePng(im, fp); fclose(fp); gdImageDestroy(im); gdImageDestroy(srcim); }An attentive reader may also notice that the code has CIEDE2000 comparisons written, though disabled. That's because it did not go as well as anticipated. Below is the result of the same program with the

So, yeah. CIE works better for some pictures than for others. Even a mere euclidean CIE76 ΔE brought forth the yellow scattered pixels. Disclaimer: I'm new to the CIE colorspace. I may have a fundamental misunderstanding or two somewhere.

Algorithm 3 is a variant to algorithm 2. It uses an array of color candidates
per pixel, and it has gamma based mixing rules and CIE color evalations in its core.
However, it is more thorough on its algorithm of devising the array of
color candidates.

In this algorithm, the color candidate array is first preinitialized with
the single closest-resembling palette index to the input color. Then, it
is iteratively subdivided by finding whether one of the palette indices
can be *replaced* with two other palette indices in equal proportion,
that would produce a better substitute for the original single palette index.

Here is how this algorithm can be described in pseudo code:

For each pixel, Input, in the original picture: Mapping.clear() // ^ An associative array, key:palette index, value:count Color,CurrentPenalty = FindClosestColorFrom(Palette, Input) // ^ The palette index that closest resembles Input // ^ CurrentPenalty is a quantitive difference between Input and Color. Mapping[Color] = M LoopWhile CurrentPenalty <> 0 // Loop until we've got a perfect match. BestPenalty = CurrentPenalty For each pair of SplitColor, SplitCount in Mapping: Sum = 0 For each pair of OtherColor, OtherCount in Mapping: If OtherColor <> SplitColor, Then Sum = Sum + OtherColor * OtherCount EndFor Portion1 = SplitCount / 2 // Equal portion 1 Portion2 = SplitCount - Portion1 // Equal portion 2 For each viable two-color combination, Index1,Color1 and Index,Color2, in Palette: Test = (Sum + Color1 * Portion1 + Color2 * Portion2) / (M) TestPenalty = CompareColors(Input, Test) If TestPenalty < BestPenalty, Then: BestPenalty = TestPenalty BestSplitData = { SplitColor, Color1, Color2 } EndIf EndFor EndFor If BestPenalty = CurrentPenalty, Then Exit Loop. // ^ Break loop if we cannot improve the result anymore. SplitCount = Mapping[BestSplitData.SplitColor] Portion1 = SplitCount / 2 // Equal portion 1 Portion2 = SplitCount - Portion1 // Equal portion 2 Mapping.Erase(BestSplitData.SplitColor) If Portion1 > 0, Then Mapping[BestSplitData.Color1] += Portion1 If Portion2 > 0, Then Mapping[BestSplitData.Color2] += Portion2 CurrentPenalty = BestPenalty EndLoop CandidateList.Clear() For each pair of Candidate, Count in Mapping: CandidateList.Add(Candidate, Count times) EndFor CandidateList.Sort( by: luminance ) Index = ThresholdMatrix[xcoordinate % X][ycoordinate % Y] Draw pixel using CandidateList[Index * CandidateList.Size() / (X*Y)] EndForIt is slow, but very thorough. It can be made to utilize psychovisual analysis by precalculating all two-color combinations from the palette and only saving those that don't look too odd when combined. Such as, only saving those pairs where their luma (luminance) does not differ more, than the average luma difference between two successive items in the luma-sorted array, scaled by a sufficient factor.

Using a matrix of size 8x8, M of 4, gamma of 2.2, the CIEDE2000 color comparison algorithm, and a luminance difference threshold of 500% of the average, we get the following picture:

C++ source code that implements this algorithm is listed below.
Since most of the program is the same as in algorithm 2, we will
only include the modified part, which is the `DeviseBestMixingPlan` function):

MixingPlan DeviseBestMixingPlan(unsigned color, size_t limit) { // Input color in RGB int input_rgb[3] = { ((color>>16)&0xFF), ((color>>8)&0xFF), (color&0xFF) }; // Input color in CIE L*a*b* LabItem input(color); std::map<unsigned, unsigned> Solution; // The penalty of our currently "best" solution. double current_penalty = -1; // First, find the closest color to the input color. // It is our seed. if(1) { unsigned chosen = 0; for(unsigned index=0; index<palettesize; ++index) { const unsigned color = pal[index]; #if COMPARE_RGB unsigned r = color>>16, g = (color>>8)&0xFF, b = color&0xFF; double penalty = ColorCompare( input_rgb[0],input_rgb[1],input_rgb[2], r,g,b); #else LabItem test_lab(color); double penalty = ColorCompare(input, test_lab); #endif if(penalty < current_penalty || current_penalty < 0) { current_penalty = penalty; chosen = index; } } Solution[chosen] = limit; } double dbllimit = 1.0 / limit; while(current_penalty != 0.0) { // Find out if there is a region in Solution that // can be split in two for benefit. double best_penalty = current_penalty; unsigned best_splitfrom = ~0u; unsigned best_split_to[2] = { 0,0}; for(std::map<unsigned,unsigned>::iterator i = Solution.begin(); i != Solution.end(); ++i) { //if(i->second <= 1) continue; unsigned split_color = i->first; unsigned split_count = i->second; // Tally the other colors double sum[3] = {0,0,0}; for(std::map<unsigned,unsigned>::iterator j = Solution.begin(); j != Solution.end(); ++j) { if(j->first == split_color) continue; sum[0] += pal_g[ j->first ][0] * j->second * dbllimit; sum[1] += pal_g[ j->first ][1] * j->second * dbllimit; sum[2] += pal_g[ j->first ][2] * j->second * dbllimit; } double portion1 = (split_count / 2 ) * dbllimit; double portion2 = (split_count - split_count/2) * dbllimit; for(unsigned a=0; a<palettesize; ++a) { //if(a != split_color && Solution.find(a) != Solution.end()) continue; unsigned firstb = 0; if(portion1 == portion2) firstb = a+1; for(unsigned b=firstb; b<palettesize; ++b) { if(a == b) continue; //if(b != split_color && Solution.find(b) != Solution.end()) continue; int lumadiff = int(luma[a]) - int(luma[b]); if(lumadiff < 0) lumadiff = -lumadiff; if(lumadiff > 80000) continue; double test[3] = { GammaUncorrect(sum[0] + pal_g[a][0] * portion1 + pal_g[b][0] * portion2), GammaUncorrect(sum[1] + pal_g[a][1] * portion1 + pal_g[b][1] * portion2), GammaUncorrect(sum[2] + pal_g[a][2] * portion1 + pal_g[b][2] * portion2) }; // Figure out if this split is better than what we had #if COMPARE_RGB double penalty = ColorCompare( input_rgb[0],input_rgb[1],input_rgb[2], test[0]*255, test[1]*255, test[2]*255); #else LabItem test_lab( test[0], test[1], test[2] ); double penalty = ColorCompare(input, test_lab); #endif if(penalty < best_penalty) { best_penalty = penalty; best_splitfrom = split_color; best_split_to[0] = a; best_split_to[1] = b; } if(portion2 == 0) break; } } } if(best_penalty == current_penalty) break; // No better solution was found. std::map<unsigned,unsigned>::iterator i = Solution.find(best_splitfrom); unsigned split_count = i->second, split1 = split_count/2, split2 = split_count-split1; Solution.erase(i); if(split1 > 0) Solution[best_split_to[0]] += split1; if(split2 > 0) Solution[best_split_to[1]] += split2; current_penalty = best_penalty; } // Sequence the solution. MixingPlan result; for(std::map<unsigned,unsigned>::iterator i = Solution.begin(); i != Solution.end(); ++i) { result.resize(result.size() + i->second, i->first); } // Sort the colors according to luminance std::sort(result.begin(), result.end(), PaletteCompareLuma); return result; }

After reviewing the ideas in algorithms 2 and 3, the algorithm 1 can
be improved significantly by precalculating all the combinations of
1…M colors, and simply finding the best matching color from
that list and using that mix as the array of candidates.

The precalculated array can also be gamma corrected in advance, and the component list of each combination sorted by luminance in advance.

The algorithm thus becomes:

For each pixel, Input, in the original picture: SmallestPenalty = 10^99 /* Impossibly large number */ CandidateList.clear() For each combination, Mixed, in the precalculated list of combinations: Penalty = Evaluate the difference of Input and Mixed. If Penalty < SmallestPenalty, SmallestPenalty = Penalty CandidateList = Mixed.components EndIf EndFor Index = ThresholdMatrix[xcoordinate % X][ycoordinate % Y] Draw pixel using CandidateList[Index * CandidateList.Size() / (X*Y)] EndForLeft: Running this algorithm with gamma level 2.2 and color comparison operator CIE2000, with 8x8 matrix and 8 components at maximum (all unique), we get this following picture. (7258 combinations.)

The combination list was formed from all unique combinations of 1…8 slots
from the palette where the difference of the luminance of the brightest
and dimmest elements in the mix is less than 276% of the maximum difference
between the luminance of successive items in the palette.

Better psychovisual quality might be achieved by comparing the chroma as well.

Left: The same, with all combinations that have max. 2 unique
elements and max. 4 elements in total (4796 combinations).

Creating the list of combinations is fast for small palettes (1..16), but
on larger palettes (say, 256 colors), you might want to restrict the parameters
(M, luminance threshold) lest the list become millions of combinations long.
Though if you use a simple euclidean distance in either RGB or L*a*b* colorspaces,
using a kd-tree for the search will still preserve quickness of the algorithm.

Left: Adobe Photoshop CS4's take on this same challenge.
It uses “Pattern Dithering” invented by Thomas Knoll.
It is very good, and faster than any of my algorithms
(though #2 has a better minimal time and
the improved #1 can be faster, if the combination list is short).
It does not appear to use gamma correction.
A description of how the algorithm works is included later in this article.

Right: Gimp 2.6's take on this same challenge.### Pattern dithering, the patented algorithm used in Adobe® Photoshop®

Right: Gimp 2.6's take on this same challenge.

I also tried Paintshop Pro, but I could not get it
to ordered-dither using a custom palette at all.

Left: In Imagemagick 6.6.0, ordered-dithering ignores palette completely,
for it is a thresholding filter.
You specify for it the threshold levels, and it ordered-dithers.
When applying the palette, it either uses a diffusion filter
(Floyd-Steinberg or Hilbert-Peano), or does not.
This image was created with the commandline
`convert scene.png -ordered-dither o8x8,8,16,8 +dither -map scenepal.png scene_imagick.png`
and it is the best I could get from Imagemagick.

I tried a dozen opensource image manipulation libraries, and all of those
tested that implemented an ordered dithering algorithm, produced
a) an explicitly monochrome image,
b) an image of a hardcoded, fixed palette or
c) simply thresholded the image like Imagemagick did.
From this study I have a reason to assume that Yliluoma's algorithms
described on this page are the *best* that are available as
free software at the time of this writing (early 2011).

Adobe Systems Incorporated is currently in possession of US Patent number 6606166,
applied for in 1999-30-04, granted in 2003-08-12.
It describes an algorithm called *pattern dithering* invented by Thomas Knoll.
For the sake of documentation, we will explain how that algorithm works as well.

In the patented form of pattern dithering, the threshold matrix is strictly restricted to 16 values (4x4 Bayer matrix), although there is no reason why no other size of matrix can be used. In this article, we will continue to use the 8x8 size so as to not infringe on the patent¹.

In pattern dithering, the pixel mixing plan constitutes of an array of colors, just like in Yliluoma dithering #2, where the size of the array is exactly the same as the size of the threshold matrix. The threshold matrix itself consists of integers rather than floats. These integers work as indices to the color array.

In pseudo code, the process of converting the input bitmap into target bitmap goes like this:

Threshold = 0.5 // This parameter is constant and controls the dithering. // 0.0 = no dithering, 1.0 = maximal dithering. For each pixel, Input, in the original picture: Error = 0 CandidateList.clear() LoopWhile CandidateList.Size < (X * Y) Attempt = Input + Error * Threshold Candidate = FindClosestColorFrom(Palette, Attempt) CandidateList.Add(Candidate) Error = Input - Candidate // The difference between these two color values. LoopEnd CandidateList.Sort( by: luminance ) Index = ThresholdMatrix[xcoordinate % X][ycoordinate % Y] Draw pixel using CandidateList[Index] EndForFinding the closest color from the palette can be done in a number of ways, including a naive euclidean distance in RGB space, or a ΔE comparison in CIE L*a*b* color space. We will continue to use the ColorCompare function from Yliluoma dithering version 1 to further avoid infringing on the patent¹.

The complete source code is shown below.
The `DeviseBestMixingPlan` function runs for
N × M
iterations for a palette of size N and a dithering pattern of size M = X×Y,
complexity being O(N), and it depends on a
color comparison function, `ColorCompare`.

#include <gd.h> #include <stdio.h> #include <math.h> #include <algorithm> /* For std::sort() */ /* 8x8 threshold map (note: the patented pattern dithering algorithm uses 4x4) */ static const unsigned char map[8*8] = { 0,48,12,60, 3,51,15,63, 32,16,44,28,35,19,47,31, 8,56, 4,52,11,59, 7,55, 40,24,36,20,43,27,39,23, 2,50,14,62, 1,49,13,61, 34,18,46,30,33,17,45,29, 10,58, 6,54, 9,57, 5,53, 42,26,38,22,41,25,37,21 }; /* Palette */ static const unsigned pal[16] = { 0x080000,0x201A0B,0x432817,0x492910, 0x234309,0x5D4F1E,0x9C6B20,0xA9220F, 0x2B347C,0x2B7409,0xD0CA40,0xE8A077, 0x6A94AB,0xD5C4B3,0xFCE76E,0xFCFAE2 }; /* Luminance for each palette entry, to be initialized as soon as the program begins */ static unsigned luma[16]; bool PaletteCompareLuma(unsigned index1, unsigned index2) { return luma[index1] < luma[index2]; } double ColorCompare(int r1,int g1,int b1, int r2,int g2,int b2) { double luma1 = (r1*299 + g1*587 + b1*114) / (255.0*1000); double luma2 = (r2*299 + g2*587 + b2*114) / (255.0*1000); double lumadiff = luma1-luma2; double diffR = (r1-r2)/255.0, diffG = (g1-g2)/255.0, diffB = (b1-b2)/255.0; return (diffR*diffR*0.299 + diffG*diffG*0.587 + diffB*diffB*0.114)*0.75 + lumadiff*lumadiff; } struct MixingPlan { unsigned colors[64]; }; MixingPlan DeviseBestMixingPlan(unsigned color) { MixingPlan result = { {0} }; const int src[3] = { color>>16, (color>>8)&0xFF, color&0xFF }; const double X = 0.09; // Error multiplier int e[3] = { 0, 0, 0 }; // Error accumulator for(unsigned c=0; c<64; ++c) { // Current temporary value int t[3] = { src[0] + e[0] * X, src[1] + e[1] * X, src[2] + e[2] * X }; // Clamp it in the allowed RGB range if(t[0]<0) t[0]=0; else if(t[0]>255) t[0]=255; if(t[1]<0) t[1]=0; else if(t[1]>255) t[1]=255; if(t[2]<0) t[2]=0; else if(t[2]>255) t[2]=255; // Find the closest color from the palette double least_penalty = 1e99; unsigned chosen = c%16; for(unsigned index=0; index<16; ++index) { const unsigned color = pal[index]; const int pc[3] = { color>>16, (color>>8)&0xFF, color&0xFF }; double penalty = ColorCompare(pc[0],pc[1],pc[2], t[0],t[1],t[2]); if(penalty < least_penalty) { least_penalty = penalty; chosen=index; } } // Add it to candidates and update the error result.colors[c] = chosen; unsigned color = pal[chosen]; const int pc[3] = { color>>16, (color>>8)&0xFF, color&0xFF }; e[0] += src[0]-pc[0]; e[1] += src[1]-pc[1]; e[2] += src[2]-pc[2]; } // Sort the colors according to luminance std::sort(result.colors, result.colors+64, PaletteCompareLuma); return result; } int main(int argc, char**argv) { FILE* fp = fopen(argv[1], "rb"); gdImagePtr srcim = gdImageCreateFromPng(fp); fclose(fp); unsigned w = gdImageSX(srcim), h = gdImageSY(srcim); gdImagePtr im = gdImageCreate(w, h); for(unsigned c=0; c<16; ++c) { unsigned r = pal[c]>>16, g = (pal[c]>>8) & 0xFF, b = pal[c] & 0xFF; gdImageColorAllocate(im, r,g,b); luma[c] = r*299 + g*587 + b*114; } #pragma omp parallel for for(unsigned y=0; y<h; ++y) for(unsigned x=0; x<w; ++x) { unsigned color = gdImageGetTrueColorPixel(srcim, x, y); unsigned map_value = map[(x & 7) + ((y & 7) << 3)]; MixingPlan plan = DeviseBestMixingPlan(color); gdImageSetPixel(im, x,y, plan.colors[ map_value ]); } fp = fopen(argv[2], "wb"); gdImagePng(im, fp); fclose(fp); gdImageDestroy(im); gdImageDestroy(srcim); }Here is an illustration comparing the various different error multiplier levels in Thomas Knoll's algorithm. I apologize for the large inline image size.

Miscellaneous observations:

- Apparently, as the error multiplier grows, so does the number of distinct palette entries it mixes together when necessary.
- Increasing the threshold matrix size will improve the quality while decreasing the regular patterns, but its usefulness maxes out at 8x8 or 16x16.
- Although not shown here, there was a very small visible difference between 8x8 and 16x16. There is probably no practical scenario that justifies the 64x64 matrix, aside from curiosity.

- It is possible to adjust the matrix size and the candidate list size separately. For example, if you choose a 16x16 matrix but limit the number of candidates to 4, you will get a rendering result that looks very much as if it was just 2x2 matrix, but is of better quality. The bottom-middle video is an example of such set-up. The optimize for speed, you might pay attention to the fact that the speed of the algorithm is directly proportional to the size of the candidate list.
- As with all positional-dithering algorithms, it is possible to extend part of the dithering to a temporal rather than to a spatial axis. It will appear as flickering. In the industry, temporal dithering is actually performed by some low-end TFT displays that cannot achieve their advertised color depth by honest means. The bottom right corner video uses temporal dithering; however, because of how dramatically it makes GIF files larger, a combination of settings was chosen that only yields a very modest amount of temporal dithering.

Here is an animated comparison of the four palette-aware
ordered dithering algorithms covered in this article.
Apologies for the fact that it is not exactly the same scene
as in the static picture: I had no savegame near that particular
event, so I did the best I could. It is missing the gowned character.

Note: The vertical moving line is due to screen scrolling and imperfect image stitching, and the changes around the left window are due to sunshine rays that the game animates in that region. The jitter in the rightside window is also caused by screen scrolling and imperfect image stitching.

Left: Yliluoma's ordered dithering algorithm 1 with tri-tone dithering enabled.

Matrix size = 8x8, comparison = RGB^{L}.

Right: Still image from the animation, prior to quantization and dithering.

Left: Improved Yliluoma's ordered dithering algorithm 1 with gamma correction enabled (γ = 2.2).

Matrix size = 8x8, candidate list size = 8, comparison = CIEDE2000,
luminance threshold = 700% of average (45% of maximum contrast).
This set of parameters resulted in 211396 precalculated combinations.
Quad-core rendering time with cache: About 17 seconds per frame.
That is how long it takes to do CIEDE2000 comparisons against 211469
candidates for each uniquely-colored pixel (though I cheated by stripping
two lowest-order bits from each channel, making it about 4× faster than
it would have been). I call this algorithm
*"improved"* because of how it is now extensible to any number of
pixel color combinations and how the actual combining is pre-done,
making rendering faster (an advantage which is profoundly countered
with the extreme parameters used here).
It is also most accurate one, because it is
completely about "measuring" rather than "estimating". But it is
also most subject to the need of psychovisual pruning, lest the result
look too coarse (as is still the case here).

Left: Yliluoma's ordered dithering algorithm 2 with gamma correction enabled (γ = 2.2).

Matrix size = 8x8, candidate list size = 16, comparison = RGB^{L}.

Left: Yliluoma's ordered dithering algorithm 3 with gamma correction enabled (γ = 2.2).

Matrix size = 8x8, candidate list size = 64, comparison = CIEDE2000,
luminance threshold = 700% of average (45% of maximum contrast).

Left: Thomas Knoll's pattern dithering algorithm (patented by Adobe®), with
the following non-spec parameters so as to not infringe on the patent¹:

Matrix size = 8x8, candidate list size = 64, comparison = RGB^{L}.

Left: Just to illustrate where I started from, here is a screenshot
from a PC game that uses (pre-rendered) dithering very well
(Princess Maker 2).

Left: I wanted to see if my algorithm can reproduce this similar effect, so first I undithered the image using a simple recursively-2x2 undithering algorithm that I wrote. The undithered image is shown here. A gamma correction of 2.2 was applied to the dithering calculations.

Left: The undithered image, without gamma correction. The dithering algorithms profiled in this list do not do gamma correction, so a gamma correction is not warranted for in the undithered image either, if we want to recreate the original dithered image.

Left: What Thomas Knoll's dithering algorithm (pattern dithering) produces for the undithered image, when it is re-dithered using the original image's palette with X=1.0, threshold matrix size 8x8, candidate list size 64. Different settings were tested, but none were found that would make it resemble more the original picture.

Left: After tweaking the parameters outside the specifications, I was able to get this picture from Thomas Knoll's dithering algorithm. Here, X=1.5. The other settings are the same as before.

Left: What Joel Yliluoma's dithering algorithm 1 produces for the same challenge. However, to get this particular result, the tri-tone call to ColorCompare had to be explicitly modified to disregard the difference between the component pixels. The EvaluateMixingError was likewise changed in a similar manner. Note that even though did discover the tri-tone dithering patterns for the dress, it has pixel artifacts around the clouds due to shifts between different dithering algorithms. It also did not discover the four-color patterns (only tri-tone search was enabled), and chose a grayish solid brown for those regions instead.

Left: Knoll-Yliluoma dithering algorithm. In this improvised algorithm, the temporary colors are also tested against all two-color mixes from the palette (N²), and if such a mix was chosen, then*both* colors are
added to the candidate list at the same time (the psychovisual model
discussed earlier is being ignored for now).
In this picture, the matrix size is 64x64 and the number of candidates
generated is 1024.

Left: Yliluoma's dithering algorithm #2. A gamma correction level of 2.2 was used (and the gamma-corrected undithered source image). In this picture, the matrix size is 4x4 and the number of candidates generated is 16. The dithering patterns for the sky and the clouds are somewhat noticeably uneven. It turned out to be quite difficult to fix.

Left: I wanted to see if my algorithm can reproduce this similar effect, so first I undithered the image using a simple recursively-2x2 undithering algorithm that I wrote. The undithered image is shown here. A gamma correction of 2.2 was applied to the dithering calculations.

Left: The undithered image, without gamma correction. The dithering algorithms profiled in this list do not do gamma correction, so a gamma correction is not warranted for in the undithered image either, if we want to recreate the original dithered image.

Left: What Thomas Knoll's dithering algorithm (pattern dithering) produces for the undithered image, when it is re-dithered using the original image's palette with X=1.0, threshold matrix size 8x8, candidate list size 64. Different settings were tested, but none were found that would make it resemble more the original picture.

Left: After tweaking the parameters outside the specifications, I was able to get this picture from Thomas Knoll's dithering algorithm. Here, X=1.5. The other settings are the same as before.

Left: What Joel Yliluoma's dithering algorithm 1 produces for the same challenge. However, to get this particular result, the tri-tone call to ColorCompare had to be explicitly modified to disregard the difference between the component pixels. The EvaluateMixingError was likewise changed in a similar manner. Note that even though did discover the tri-tone dithering patterns for the dress, it has pixel artifacts around the clouds due to shifts between different dithering algorithms. It also did not discover the four-color patterns (only tri-tone search was enabled), and chose a grayish solid brown for those regions instead.

Left: Knoll-Yliluoma dithering algorithm. In this improvised algorithm, the temporary colors are also tested against all two-color mixes from the palette (N²), and if such a mix was chosen, then

Left: Yliluoma's dithering algorithm #2. A gamma correction level of 2.2 was used (and the gamma-corrected undithered source image). In this picture, the matrix size is 4x4 and the number of candidates generated is 16. The dithering patterns for the sky and the clouds are somewhat noticeably uneven. It turned out to be quite difficult to fix.

Dithering mixes colours together. Although it is surprising at first,
for *proper* mixing one needs to pay attention to *gamma correction*
in order to do proper mixing. Otherwise, one ends up creating mixes that
do not represent the original color. Illustrated here.

First, without gamma correction:

Gamma=1.0

One can immediately observe that these two gradients don't line up.
The bottom, dithered one is much brighter than the top, original one.

Clearly, the 50 % white in the upper bar is not really a 50 % white;
it is much darker than that. To produce that color from a mix of black
and white, you have to put a lot more black into the mix than white.

Then, at varying levels of gamma correction:

Gamma=0.1

Gamma=0.5

Gamma=1.5

Gamma=2.0

Gamma=2.2

Gamma=2.5

Gamma=10.0

Assuming your monitor has a gamma of 2.2, according to this graph,
the true 50 % mix of 0 % white and 100 % white is generated by
the monitor only when a *73 % white* is signalled.

To achieve a match for what the monitor claims is 50 % white,
we can solve the equation
((0%^{γ}×X + 100%^{γ}×(1−X)))^{(1÷γ)} = 50%,
or rather, the general case of
((a^{γ} × X + b^{γ} × (1−X)))^{(1÷γ)} = c),
and we get the following equation:
X = (b^{γ} − c^{γ}) ÷ (b^{γ} − a^{γ}).

Substituting a = 0 % (black), b = 100 % (white), c = 50 %,
we get that X = 78 %, so we must mix 78 % of black and 22 % of
white to match the appearance of monitor's 50 % white.

Computer displays usually have a gamma level of around 1.8 or 2.2, though it is convenient if the user has the choice of selecting their own gamma value.

The process of gamma-aware and gamma-unaware
mixing of colors `a` and `b`, ranging from 0..1,
according to mixing ratio `ratio`, from 0..1:

Gamma-unaware mixing | Gamma-aware mixing |
---|---|

result = a + (b−a)*ratio | a‘ = a^{γ}b‘ = b ^{γ}result‘ = a‘ + (b‘−a‘)*ratio result = result‘ ^{1÷γ} |

Or mixing of three colors in an equal proportion:

Gamma-unaware mixing | Gamma-aware mixing |
---|---|

result = (a+b+c) ÷ 3 | a‘ = a^{γ}b‘ = b ^{γ}c‘ = c ^{γ}result‘ = (a‘ + b‘ + c‘) ÷ 3 result = result‘ ^{1÷γ} |

Or in a more general fashion, you might always opt to work with linear colors (un-gamma the original pictures and the palette), and then gamma-correct the final result just before writing it into the image file or sending it to the screen.

The algorithms outlined on this page can easily be adapted to work with gamma correction. However, there is still one unsolved "gotcha". Both Yliluoma dithering #2 and our implementation of Knoll's pattern dithering suffer from it.

Below, we use the EGA palette (four levels of grayscale: 0%, 33%, 66% and 100%):

Gamma=0.1

Gamma=0.5

Gamma=1.0

Gamma=2.2

Gamma=10.0

The algorithm hits the solid-color spots (0%, 33%, 66% and 100%)
just fine, but on any other gamma level than 1.0,
the *between*s of the solid spots are wrong.
Somehow, it starts "borrowing" colors
from the outside of the two anchor spots rather than logarithmically
mixing the colors of the two anchor spots.
It happens all according to the algorithm.
It is currently unknown how to fix it.

We introduce a threshold matrix that is defined as follows.
### Algorithm for generating a rectangle-shaped matrix

### Comparison of different matrix sizes

- Let the matrix size be X horizontally and Y vertically. Let N be the number of nodes, i.e.
`X × Y`. - The matrix contains all the successive fractions from
`0 ÷ N`to`(N − 1) ÷ N`, at step`1 ÷ N`. - The fractions are ordered geometrically in the matrix so that the pseudo-toroidal distance of any two numbers in the matrix correlates inversely to their difference.
- Pseudo-toroidal distance between two points
`(x1, y1)`and`(x2, y2)`is defined as:`min(abs(x1−x2),X−abs(x1−x2)) + min(abs(y1−y2),Y−abs(y1−y2))`.

This is the same or almost the same as is commonly known as Bayer's threshold matrix.

We define a simple algorithm for assigning the values to a square-shaped
threshold matrix. The value for slot `(x, y)` is calculated as follows:

- Take two values, the y coordinate and the XOR of x and y coordinates
- Interleave the bits of these two values in reverse order.
- Floating-point divide the result by N. (Optional, required by some algorithms but not all.)

Example implementation in C++ language:

for(unsigned M=0; M<=3; ++M) { const unsigned dim = 1 << M; printf(" X=%u, Y=%u:\n", dim,dim); for(unsigned y=0; y<dim; ++y) { printf(" "); for(unsigned x=0; x<dim; ++x) { unsigned v = 0, mask = M-1, xc = x ^ y, yc = y; for(unsigned bit=0; bit < 2*M; --mask) { v |= ((yc >> mask)&1) << bit++; v |= ((xc >> mask)&1) << bit++; } printf("%4d", v); } printf(" |"); if(y == 0) printf(" 1/%u", dim * dim); printf("\n"); } }The algorithm can easily be extended for other rectangular matrices, as long as the lengths of both sides are powers of two. This is useful when you are rendering for an output device that uses a significantly non-square pixel aspect ratio (for example, 640×200 on CGA). Example:

for(unsigned M=0; M<=3; ++M) for(unsigned L=0; L<=3; ++L) { const unsigned xdim = 1 << M; const unsigned ydim = 1 << L; printf(" X=%u, Y=%u:\n", xdim,ydim); for(unsigned y=0; y<ydim; ++y) { printf(" "); for(unsigned x=0; x<xdim; ++x) { unsigned v = 0, offset=0, xmask = M, ymask = L; if(M==0 || (M > L && L != 0)) { unsigned xc = x ^ ((y << M) >> L), yc = y; for(unsigned bit=0; bit < M+L; ) { v |= ((yc >> --ymask)&1) << bit++; for(offset += M; offset >= L; offset -= L) v |= ((xc >> --xmask)&1) << bit++; } } else { unsigned xc = x, yc = y ^ ((x << L) >> M); for(unsigned bit=0; bit < M+L; ) { v |= ((xc >> --xmask)&1) << bit++; for(offset += L; offset >= M; offset -= M) v |= ((yc >> --ymask)&1) << bit++; } } printf("%4d", v); } printf(" |"); if(y == 0) printf(" 1/%u", xdim * ydim); printf("\n"); } }It is not perfect (for example, in 4x4, the #7 and #8 are right next to each others, and #3—#4 and #11—#12 are just one diagonal across), but it is good enough in practice.

Example outputs:

X=4, Y=2: X=4, Y=8: 0 4 2 6 | 1/8 0 12 3 15 | 1/32 3 7 1 5 | 16 28 19 31 | 8 4 11 7 | X=4, Y=4: 24 20 27 23 | 0 12 3 15 | 1/16 2 14 1 13 | 8 4 11 7 | 18 30 17 29 | 2 14 1 13 | 10 6 9 5 | 10 6 9 5 | 26 22 25 21 |

X=8, Y=2: X=2, Y=2: 0 8 4 12 2 10 6 14 | 1/16 0 3 | 1/4 3 11 7 15 1 9 5 13 | 2 1 |

X=8, Y=4: X=2, Y=4: 0 16 8 24 2 18 10 26 | 1/32 0 3 | 1/8 12 28 4 20 14 30 6 22 | 4 7 | 3 19 11 27 1 17 9 25 | 2 1 | 15 31 7 23 13 29 5 21 | 6 5 |

X=8, Y=8: X=2, Y=8: 0 48 12 60 3 51 15 63 | 1/64 0 3 | 1/16 32 16 44 28 35 19 47 31 | 8 11 | 8 56 4 52 11 59 7 55 | 4 7 | 40 24 36 20 43 27 39 23 | 12 15 | 2 50 14 62 1 49 13 61 | 2 1 | 34 18 46 30 33 17 45 29 | 10 9 | 10 58 6 54 9 57 5 53 | 6 5 | 42 26 38 22 41 25 37 21 | 14 13 |Such matrices where the sides are not powers of two can be designed by hand by mimicking the same principles. However, they can have a visibly uneven look, and thus are rarely worth using. Examples:

X=5, Y=3: X=3, Y=3:

0 12 7 3 9| 1/15 0 5 2 | 1/9 14 8 1 5 11| 3 8 7 | 6 4 10 13 2| 6 1 4 |

In the following illustration, the above picture (animated a bit) is rendered using various different dithering methods. The image is magnified by a factor of 2 to make the pixels visible:

This palette was used (regular palette, red,green,blue incrementing at even intervals 255/3, 255/3 and 255/2 respectively):

Pattern dithering refers to the patented algorithm used by Adobe® Photoshop® by Adobe Systems Incorporated. (Hyenas note, I am not claiming ownership of their trademark.) It uses a fixed 4x4 threshold matrix and a configurable error multiplier; here 0.25 was used. The algorithm was explained in detail earlier in this article.

Most of the example pictures of this article used the 8x8 matrix.

There exist a number of algorithms for comparing the similarity
between two colors.
### Using kd-tree to optimize palette searches

For purposes of illustration, I use this graphical image and the result of quantizing it using the websafe palette, without dithering. I also provided a dithered version (improved Yliluoma dithering 1 with matrix size 2x2, 2 candidates, selected from all 2-colors combinations (23436 of them)).

Note that it is often sufficient to compare the *squared*
delta-E rather than the delta-E itself.
The results are the same, but are achieved faster.

A few standard algorithms are listed below. I took the liberty
of implementing some optimizations to the more complex formulae.

Each of these algorithms have strengths and weaknesses, and the algorithm chosen for a task depends on which features one wants to emphasize. The appearance of CIE L*a*b* based methods also depends on the whitepoint and matrix used for converting the RGB value to XYZ.

When using a color comparison method based on an euclidean search,
the search can be made significantly faster by inserting each palette
color into a kd-tree,
from which nearest-neighbor searches can be done very quickly:
Instead of an `O(N)` complexity, a `O(log(N))` complexity is acquired.

An example C++ kd-tree implementation (simple, unbalanced one) is included below, along with example code for utilizing it in palette searches, is shown below:

#include <utility> // for std::pair #include "alloc/FSBAllocator.hh" /* kd-tree implementation translated to C++ * from java implementation in VideoMosaic * at http://www.intelegance.net/video/videomosaic.shtml. */ template<typename V, unsigned K = 3> class KDTree { public: struct KDPoint { double coord[K]; KDPoint() { } KDPoint(double a,double b,double c) { coord[0] = a; coord[1] = b; coord[2] = c; } KDPoint(double v[K]) { for(unsigned n=0; n<K; ++n) coord[n] = v[n]; } bool operator==(const KDPoint& b) const { for(unsigned n=0; n<K; ++n) if(coord[n] != b.coord[n]) return false; return true; } double sqrdist(const KDPoint& b) const { double result = 0; for(unsigned n=0; n<K; ++n) { double diff = coord[n] - b.coord[n]; result += diff*diff; } return result; } }; private: struct KDRect { KDPoint min, max; KDPoint bound(const KDPoint& t) const { KDPoint p; for(unsigned i=0; i<K; ++i) if(t.coord[i] <= min.coord[i]) p.coord[i] = min.coord[i]; else if(t.coord[i] >= max.coord[i]) p.coord[i] = max.coord[i]; else p.coord[i] = t.coord[i]; return p; } void MakeInfinite() { for(unsigned i=0; i<K; ++i) { min.coord[i] = -1e99; max.coord[i] = 1e99; } } }; struct KDNode { KDPoint k; V v; KDNode *left, *right; public: KDNode() : k(),v(),left(0),right(0) { } KDNode(const KDPoint& kk, const V& vv) : k(kk), v(vv), left(0), right(0) { } virtual ~KDNode() { delete (left); delete (right); } static KDNode* ins( const KDPoint& key, const V& val, KDNode*& t, int lev) { if(!t) return (t = new KDNode(key, val)); else if(key == t->k) return 0; /* key duplicate */ else if(key.coord[lev] > t->k.coord[lev]) return ins(key, val, t->right, (lev+1)%K); else return ins(key, val, t->left, (lev+1)%K); } struct Nearest { const KDNode* kd; double dist_sqd; }; // Method Nearest Neighbor from Andrew Moore's thesis. Numbered // comments are direct quotes from there. Step "SDL" is added to // make the algorithm work correctly. static void nnbr(const KDNode* kd, const KDPoint& target, KDRect& hr, // in-param and temporary; not an out-param. int lev, Nearest& nearest) { // 1. if kd is empty then set dist-sqd to infinity and exit. if (!kd) return; // 2. s := split field of kd int s = lev % K; // 3. pivot := dom-elt field of kd const KDPoint& pivot = kd->k; double pivot_to_target = pivot.sqrdist(target); // 4. Cut hr into to sub-hyperrectangles left-hr and right-hr. // The cut plane is through pivot and perpendicular to the s // dimension. KDRect& left_hr = hr; // optimize by not cloning KDRect right_hr = hr; left_hr.max.coord[s] = pivot.coord[s]; right_hr.min.coord[s] = pivot.coord[s]; // 5. target-in-left := target_s <= pivot_s bool target_in_left = target.coord[s] < pivot.coord[s]; const KDNode* nearer_kd; const KDNode* further_kd; KDRect nearer_hr; KDRect further_hr; // 6. if target-in-left then nearer is left, further is right if (target_in_left) { nearer_kd = kd->left; nearer_hr = left_hr; further_kd = kd->right; further_hr = right_hr; } // 7. if not target-in-left then nearer is right, further is left else { nearer_kd = kd->right; nearer_hr = right_hr; further_kd = kd->left; further_hr = left_hr; } // 8. Recursively call Nearest Neighbor with parameters // (nearer-kd, target, nearer-hr, max-dist-sqd), storing the // results in nearest and dist-sqd nnbr(nearer_kd, target, nearer_hr, lev + 1, nearest); // 10. A nearer point could only lie in further-kd if there were some // part of further-hr within distance sqrt(max-dist-sqd) of // target. If this is the case then const KDPoint closest = further_hr.bound(target); if (closest.sqrdist(target) < nearest.dist_sqd) { // 10.1 if (pivot-target)^2 < dist-sqd then if (pivot_to_target < nearest.dist_sqd) { // 10.1.1 nearest := (pivot, range-elt field of kd) nearest.kd = kd; // 10.1.2 dist-sqd = (pivot-target)^2 nearest.dist_sqd = pivot_to_target; } // 10.2 Recursively call Nearest Neighbor with parameters // (further-kd, target, further-hr, max-dist_sqd) nnbr(further_kd, target, further_hr, lev + 1, nearest); } // SDL: otherwise, current point is nearest else if (pivot_to_target < nearest.dist_sqd) { nearest.kd = kd; nearest.dist_sqd = pivot_to_target; } } private: void operator=(const KDNode&); public: KDNode(const KDNode& b) : k(b.k), v(b.v), left( b.left ? new KDNode(*b.left) : 0), right( b.right ? new KDNode(*b.right) : 0 ) { } }; private: KDNode* m_root; public: KDTree() : m_root(0) { } virtual ~KDTree() { delete (m_root); } bool insert(const KDPoint& key, const V& val) { return KDNode::ins(key, val, m_root, 0); } const std::pair<V,double> nearest(const KDPoint& key) const { KDRect hr; hr.MakeInfinite(); typename KDNode::Nearest nn; nn.kd = 0; nn.dist_sqd = 1e99; KDNode::nnbr(m_root, key, hr, 0, nn); if(!nn.kd) return std::pair<V,double> ( V(), 1e99 ); return std::pair<V,double> ( nn.kd->v, nn.dist_sqd); } public: KDTree& operator=(const KDTree&b) { if(this != &b) { if(m_root) delete (m_root); m_root = b.m_root ? new KDNode(*b.m_root) : 0; m_count = b.m_count; } return *this; } KDTree(const KDTree& b) : m_root( b.m_root ? new KDNode(*b.m_root) : 0 ), m_count( b.m_count ) { } };It can be initialized and used like below (in this example, used with CIE76, i.e. euclidean distance in CIE L*a*b* space):

KDTree<unsigned> tree; // Initialize tree: for(unsigned index=0; index<PaletteSize; ++index) { LabItem lab = palette[index]; tree.ins( KDTree<unsigned>::KDPoint( lab.L, lab.a, lab.b), index ); } ... // Example for searching the tree for nearest match for particular input: std::pair<unsigned,double> result = tree.nearest( KDPoint( input.L, input.a, input.b ) ); unsigned nearest_paletteindex = result.first; double match_penalty = result.second; // euclidean distance squared.This code puts only palette values into the tree, but you can also put there pre-mixed color combinations, etc.

It does not make much difference to the speed whether you search 16 palette items linearly or with a kd-tree, but it does make a fantastic difference whether you search 2882880 combinations linearly or by using a kd-tree.

I have used this script to compress all PNG images on this webpage.
It utilizes AdvPNG,
OptiPNG,
DeflOpt (run under Wine)
and PNGOUT.

in="$1" tmp="compress.sh-tmp-"$$".png" fin="_$1" rm -f "$fin" bsize="`stat -c %s $in`" sizes="-n1 -n2 -n3 -n4 -n5 -n6 -n7 -n8 -n9 -n10 -n11 -n12 -n13" filters="0 1 2 3 4 5" advpng -z -4 "$in" optipng -o7 "$in" wine /usr/local/bin/DeflOpt.exe "$in" for filter in $filters;do for bufsize in $sizes;do rm -f "$tmp" while [ $(jobs -p|wc -l) -ge 4 ]; do sleep 0.2; done if true; then f="$tmp"."$BASHPID".png pngout -v -f$filter $bufsize "$in" "$f" flock -s 333 size="`stat -c %s "$f"`" if [ $bsize -gt $size ]; then wine /usr/local/bin/DeflOpt.exe "$f" mv -f "$f" "$fin" bsize="$size" else rm -f "$f" fi fi & done done 333< "$in" wait mv -f "$fin" "$in"

- Joel Yliluoma 2011-01-24, http://iki.fi/bisqwit/

- "Ordered Dithering, Stephen Hawley, Graphics Gems, Academic Press, 1990"
- Alejo Hausner, "Hierarchical Graph Color Dither", Durham, NH, USA, 2008
- Stephen J. Sangwine & Robin E. N. Horne, "The colour image processing handbook", ISBN 0412806207
- Argyll CMS
- Imagemagick
- Wikipedia article Color difference
- Wikipedia article Dithering
- Wikipedia article kd-tree
- Pattern Dithering, Thomas Knoll, Ann Arbor, MI (US), US patent n.o. 6606166

¹) Do not take this statement as legal guidance.
All code on this page is provided as-is, without warranty.
`#include <GPL/warranty.txt>`.

Last edited: 2014-07-02 13:20:03