1quantize(5) File Formats Manual quantize(5)
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6 Quantize - ImageMagick's color reduction algorithm.
7
9 #include <magick.h>
10
12 This document describes how ImageMagick performs color reduction on an
13 image. To fully understand this document, you should have a knowledge
14 of basic imaging techniques and the tree data structure and terminol‐
15 ogy.
16 For purposes of color allocation, an image is a set of n pixels, where
18 each pixel is a point in RGB space. RGB space is a 3-dimensional vec‐
19 tor space, and each pixel, pi, is defined by an ordered triple of red,
20 green, and blue coordinates, (ri, gi, bi).
21 Each primary color component (red, green, or blue) represents an inten‐
23 sity which varies linearly from 0 to a maximum value, cmax, which cor‐
24 responds to full saturation of that color. Color allocation is defined
25 over a domain consisting of the cube in RGB space with opposite ver‐
26 tices at (0,0,0) and (cmax,cmax,cmax). ImageMagick requires cmax =
27 255.
28 The algorithm maps this domain onto a tree in which each node repre‐
30 sents a cube within that domain. In the following discussion, these
31 cubes are defined by the coordinate of two opposite vertices: The ver‐
32 tex nearest the origin in RGB space and the vertex farthest from the
33 origin.
34 The tree's root node represents the the entire domain, (0,0,0) through
36 (cmax,cmax,cmax). Each lower level in the tree is generated by subdi‐
37 viding one node's cube into eight smaller cubes of equal size. This
38 corresponds to bisecting the parent cube with planes passing through
39 the midpoints of each edge.
40 The basic algorithm operates in three phases: Classification, Reduc‐
42 tion, and Assignment. Classification builds a color description tree
43 for the image. Reduction collapses the tree until the number it repre‐
44 sents, at most, is the number of colors desired in the output image.
45 Assignment defines the output image's color map and sets each pixel's
46 color by reclassification in the reduced tree. Our goal is to minimize
47 the numerical discrepancies between the original colors and quantized
48 colors. To learn more about quantization error, see MEASURING COLOR
49 REDUCTION ERROR later in this document.
50 Classification begins by initializing a color description tree of suf‐
52 ficient depth to represent each possible input color in a leaf. How‐
53 ever, it is impractical to generate a fully-formed color description
54 tree in the classification phase for realistic values of cmax. If
55 color components in the input image are quantized to k-bit precision,
56 so that cmax = 2k-1, the tree would need k levels below the root node
57 to allow representing each possible input color in a leaf. This
58 becomes prohibitive because the tree's total number of nodes is
59 Σ ki=1 8k
61 A complete tree would require 19,173,961 nodes for k = 8, cmax = 255.
63 Therefore, to avoid building a fully populated tree, ImageMagick: (1)
64 Initializes data structures for nodes only as they are needed; (2)
65 Chooses a maximum depth for the tree as a function of the desired num‐
66 ber of colors in the output image (currently log4(colormap size)+2). A
67 tree of this depth generally allows the best representation of the
68 source image with the fastest computational speed and the least amount
69 of memory. However, the default depth is inappropriate for some
70 images. Therefore, the caller can request a specific tree depth.
71 For each pixel in the input image, classification scans downward from
73 the root of the color description tree. At each level of the tree, it
74 identifies the single node which represents a cube in RGB space con‐
75 taining the pixel's color. It updates the following data for each such
76 node:
77 n1: Number of pixels whose color is contained in the RGB cube which
79 this node represents;
80 n2: Number of pixels whose color is not represented in a node at
82 lower depth in the tree; initially, n2 = 0 for all nodes
83 except leaves of the tree.
84 Sr, Sg, Sb:
86 Sums of the red, green, and blue component values for all pixels
87 not classified at a lower depth. The combination of these sums
88 and n2 will ultimately characterize the mean color of a set of
89 pixels represented by this node.
90 E: The distance squared in RGB space between each pixel contained
92 within a node and the nodes' center. This represents the quan‐
93 tization error for a node.
94 Reduction repeatedly prunes the tree until the number of nodes with n2
96 > 0 is less than or equal to the maximum number of colors allowed in
97 the output image. On any given iteration over the tree, it selects
98 those nodes whose E value is minimal for pruning and merges their color
99 statistics upward. It uses a pruning threshold, Ep, to govern node
100 selection as follows:
101 Ep = 0
103 while number of nodes with (n2 > 0) > required maximum number of col‐
104 ors
105 prune all nodes such that E <= Ep
106 Set Ep to minimum E in remaining nodes
107 This has the effect of minimizing any quantization error when merging
109 two nodes together.
110 When a node to be pruned has offspring, the pruning procedure invokes
112 itself recursively in order to prune the tree from the leaves upward.
113 The values of n2 Sr, Sg, and Sb in a node being pruned are always
114 added to the corresponding data in that node's parent. This retains
115 the pruned node's color characteristics for later averaging.
116 For each node, n2 pixels exist for which that node represents the
118 smallest volume in RGB space containing those pixel's colors. When n2
119 > 0 the node will uniquely define a color in the output image. At the
120 beginning of reduction, n2 = 0 for all nodes except the leaves of the
121 tree which represent colors present in the input image.
122 The other pixel count, n1, indicates the total number of colors within
124 the cubic volume which the node represents. This includes n1 - n2 pix‐
125 els whose colors should be defined by nodes at a lower level in the
126 tree.
127 Assignment generates the output image from the pruned tree. The output
129 image consists of two parts: (1) A color map, which is an array of
130 color descriptions (RGB triples) for each color present in the output
131 image; (2) A pixel array, which represents each pixel as an index into
132 the color map array.
133 First, the assignment phase makes one pass over the pruned color
135 description tree to establish the image's color map. For each node
136 with n2 > 0, it divides Sr, Sg, and Sb by n2. This produces the mean
137 color of all pixels that classify no lower than this node. Each of
138 these colors becomes an entry in the color map.
139 Finally, the assignment phase reclassifies each pixel in the pruned
141 tree to identify the deepest node containing the pixel's color. The
142 pixel's value in the pixel array becomes the index of this node's mean
143 color in the color map.
144 Empirical evidence suggests that distances in color spaces such as YUV,
146 or YIQ correspond to perceptual color differences more closely than do
147 distances in RGB space. These color spaces may give better results
148 when color reducing an image. Here the algorithm is as described
149 except each pixel is a point in the alternate color space. For conve‐
150 nience, the color components are normalized to the range 0 to a maximum
151 value, cmax. The color reduction can then proceed as described.
152
154 Depending on the image, the color reduction error may be obvious or
155 invisible. Images with high spatial frequencies (such as hair or
156 grass) will show error much less than pictures with large smoothly
157 shaded areas (such as faces). This is because the high-frequency con‐
158 tour edges introduced by the color reduction process are masked by the
159 high frequencies in the image.
160 To measure the difference between the original and color reduced images
162 (the total color reduction error), ImageMagick sums over all pixels in
163 an image the distance squared in RGB space between each original pixel
164 value and its color reduced value. ImageMagick prints several error
165 measurements including the mean error per pixel, the normalized mean
166 error, and the normalized maximum error.
167 The normalized error measurement can be used to compare images. In
169 general, the closer the mean error is to zero the more the quantized
170 image resembles the source image. Ideally, the error should be percep‐
171 tually-based, since the human eye is the final judge of quantization
172 quality.
173 These errors are measured and printed when -verbose and -colors are
175 specified on the command line:
176 mean error per pixel:
178 is the mean error for any single pixel in the image.
179 normalized mean square error:
181 is the normalized mean square quantization error for any single
182 pixel in the image.
183 This distance measure is normalized to a range between 0 and 1.
185 It is independent of the range of red, green, and blue values in
186 the image.
187 normalized maximum square error:
189 is the largest normalized square quantization error for any sin‐
190 gle pixel in the image.
191 This distance measure is normalized to a range between 0 and 1.
193 It is independent of the range of red, green, and blue values in
194 the image.
195
197 display(1), animate(1), mogrify(1), import(1), miff(5)
198
200 Copyright (C) 2002 ImageMagick Studio, a non-profit organization dedi‐
201 cated to making software imaging solutions freely available.
202 Permission is hereby granted, free of charge, to any person obtaining a
204 copy of this software and associated documentation files ("ImageMag‐
205 ick"), to deal in ImageMagick without restriction, including without
206 limitation the rights to use, copy, modify, merge, publish, distribute,
207 sublicense, and/or sell copies of ImageMagick, and to permit persons to
208 whom the ImageMagick is furnished to do so, subject to the following
209 conditions:
210 The above copyright notice and this permission notice shall be included
212 in all copies or substantial portions of ImageMagick.
213 The software is provided "as is", without warranty of any kind, express
215 or implied, including but not limited to the warranties of mer‐
216 chantability, fitness for a particular purpose and noninfringement. In
217 no event shall ImageMagick Studio be liable for any claim, damages or
218 other liability, whether in an action of contract, tort or otherwise,
219 arising from, out of or in connection with ImageMagick or the use or
220 other dealings in ImageMagick.
221 Except as contained in this notice, the name of the ImageMagick Studio
223 shall not be used in advertising or otherwise to promote the sale, use
224 or other dealings in ImageMagick without prior written authorization
225 from the ImageMagick Studio.
226
228 Paul Raveling, USC Information Sciences Institute, for the original
229 idea of using space subdivision for the color reduction algorithm.
230 With Paul's permission, this document is an adaptation from a document
231 he wrote.
232
234 John Cristy, ImageMagick Studio
235ImageMagick $Date$ quantize(5)