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           #include <magick.h>


           This document describes how ImageMagick performs color reduction on an
           image.  To fully understand this document, you should have a knowledge
           of basic imaging techniques and the tree data structure and terminol-
           For purposes of color allocation, an image is a set of n pixels, where
           each pixel is a point in RGB space.  RGB space is a 3-dimensional vec-
           tor space, and each pixel, pi,  is defined by an ordered triple of red,
           green, and blue coordinates, (ri, gi, bi).
           Each primary color component (red, green, or blue) represents an inten-
           sity which varies linearly from 0 to a maximum value, cmax, which cor-
           responds to full saturation of that color.  Color allocation is defined
           over a domain consisting of the cube in RGB space with opposite ver-
           tices at (0,0,0) and (cmax,cmax,cmax).  ImageMagick requires cmax =
           The algorithm maps this domain onto a tree in which each node repre-
           sents a cube within that domain.  In the following discussion, these
           cubes are defined by the coordinate of two opposite vertices: The ver-
           tex nearest the origin in RGB space and the vertex farthest from the
           The tree's root node represents the the entire domain, (0,0,0) through
           (cmax,cmax,cmax).  Each lower level in the tree is generated by subdi-
           viding one node's cube into eight smaller cubes of equal size.  This
           corresponds to bisecting the parent cube with planes passing through
           the midpoints of each edge.
           The basic algorithm operates in three phases:  Classification, Reduc-
           tion, and Assignment.  Classification builds a color description tree
           for the image.  Reduction collapses the tree until the number it repre-
           sents, at most, is the number of colors desired in the output image.
           Assignment defines the output image's color map and sets each pixel's
           color by reclassification in the reduced tree. Our goal is to minimize
           the numerical discrepancies between the original colors and quantized
           colors.  To learn more about quantization error, see MEASURING COLOR
           REDUCTION ERROR later in this document.
           Classification begins by initializing a color description tree of suf-
           ficient depth to represent each possible input color in a leaf.  How-
           ever, it is impractical to generate a fully-formed color description
           tree in the classification phase for realistic values of cmax.  If
           color components in the input image are quantized to k-bit precision,
           so that cmax = 2k-1, the tree would need k levels below the root node
           to allow representing each possible input color in a leaf.  This
           becomes prohibitive because the tree's total number of nodes is
           identifies the single node which represents a cube in RGB space con-
           taining the pixel's color.  It updates the following data for each such
           n1:    Number of pixels whose color is contained in the RGB cube which
                  this node represents;
           n2:    Number of pixels whose color is not represented in a node at
                  lower depth in the tree;  initially,  n2 = 0 for all nodes
                  except leaves of the tree.
           Sr, Sg, Sb:
                  Sums of the red, green, and blue component values for all pixels
                  not classified at a lower depth.  The combination of these sums
                  and n2 will ultimately characterize the mean color of a set of
                  pixels represented by this node.
           E:     The distance squared in RGB space between each pixel contained
                  within a node and the nodes' center.  This represents the quan-
                  tization error for a node.
           Reduction repeatedly prunes the tree until the number of nodes with n2
           > 0 is less than or equal to the maximum number of colors allowed in
           the output image.  On any given iteration over the tree, it selects
           those nodes whose E value is minimal for pruning and merges their color
           statistics upward.  It uses a pruning threshold, Ep, to govern node
           selection as follows:
             Ep = 0
             while number of nodes with (n2 > 0) > required maximum number of col-
                 prune all nodes such that E <= Ep
                 Set Ep  to minimum E in remaining nodes
           This has the effect of minimizing any quantization error when merging
           two nodes together.
           When a node to be pruned has offspring, the pruning procedure invokes
           itself recursively in order to prune the tree from the leaves upward.
           The values of n2  Sr, Sg,  and Sb in a node being pruned are always
           added to the corresponding data in that node's parent.  This retains
           the pruned node's color characteristics for later averaging.
           For each node,  n2 pixels exist for which that node represents the
           smallest volume in RGB space containing those pixel's colors.  When n2
           > 0 the node will uniquely define a color in the output image.  At the
           beginning of reduction, n2 = 0  for all nodes except the leaves of the
           tree which represent colors present in the input image.
           The other pixel count, n1,  indicates the total number of colors within
           the cubic volume which the node represents.  This includes n1 - n2 pix-
           els whose colors should be defined by nodes at a lower level in the
           Finally, the assignment phase reclassifies each pixel in the pruned
           tree to identify the deepest node containing the pixel's color.  The
           pixel's value in the pixel array becomes the index of this node's mean
           color in the color map.
           Empirical evidence suggests that distances in color spaces such as YUV,
           or YIQ correspond to perceptual color differences more closely than do
           distances in RGB space.  These color spaces may give better results
           when color reducing an image.  Here the algorithm is as described
           except each pixel is a point in the alternate color space.  For conve-
           nience, the color components are normalized to the range 0 to a maximum
           value, cmax.  The color reduction can then proceed as described.


           Depending on the image, the color reduction error may be obvious or
           invisible.  Images with high spatial frequencies (such as hair or
           grass) will show error much less than pictures with large smoothly
           shaded areas (such as faces).  This is because the high-frequency con-
           tour edges introduced by the color reduction process are masked by the
           high frequencies in the image.
           To measure the difference between the original and color reduced images
           (the total color reduction error), ImageMagick sums over all pixels in
           an image the distance squared in RGB space between each original pixel
           value and its color reduced value. ImageMagick prints several error
           measurements including the mean error per pixel, the normalized mean
           error, and the normalized maximum error.
           The normalized error measurement can be used to compare images.  In
           general, the closer the mean error is to zero the more the quantized
           image resembles the source image.  Ideally, the error should be percep-
           tually-based, since the human eye is the final judge of quantization
           These errors are measured and printed when -verbose and -colors are
           specified on the command line:
           mean error per pixel:
                  is the mean error for any single pixel in the image.
           normalized mean square error:
                  is the normalized mean square quantization error for any single
                  pixel in the image.
                  This distance measure is normalized to a range between 0 and 1.
                  It is independent of the range of red, green, and blue values in
                  the image.
           normalized maximum square error:
                  is the largest normalized square quantization error for any sin-
                  gle pixel in the image.
           limitation the rights to use, copy, modify, merge, publish, distribute,
           sublicense, and/or sell copies of ImageMagick, and to permit persons to
           whom the ImageMagick is furnished to do so, subject to the following
           The above copyright notice and this permission notice shall be included
           in all copies or substantial portions of ImageMagick.
           The software is provided "as is", without warranty of any kind, express
           or implied, including but not limited to the warranties of mer-
           chantability, fitness for a particular purpose and noninfringement.  In
           no event shall ImageMagick Studio be liable for any claim, damages or
           other liability, whether in an action of contract, tort or otherwise,
           arising from, out of or in connection with ImageMagick or the use or
           other dealings in ImageMagick.
           Except as contained in this notice, the name of the ImageMagick Studio
           shall not be used in advertising or otherwise to promote the sale, use
           or other dealings in ImageMagick without prior written authorization
           from the ImageMagick Studio.


           Paul Raveling, USC Information Sciences Institute, for the original
           idea of using space subdivision for the color reduction algorithm.
           With Paul's permission, this document is an adaptation from a document
           he wrote.


           John Cristy, ImageMagick Studio

    ImageMagick $Date: 2001/12/07 18:43:53 $ quantize(5)


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