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NAME
     Quantize - ImageMagick's color reduction algorithm.

SYNOPSIS
     #include <magick.h>

DESCRIPTION
     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 terminology.

     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 vector 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 intensity which varies linearly from 0 to a
     maximum value, cmax, which corresponds to full saturation of
     that color.  Color allocation is defined over a domain
     consisting of the cube in RGB space with opposite vertices
     at (0,0,0) and (cmax,cmax,cmax).  ImageMagick requires cmax
     = 255.

     The algorithm maps this domain onto a tree in which each
     node represents a cube within that domain.  In the following
     discussion, these cubes are defined by the coordinate of two
     opposite vertices: The vertex nearest the origin in RGB
     space and the vertex farthest from the origin.

     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 subdividing 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, Reduction, and Assignment.  Classification
     builds a color description tree for the image.  Reduction
     collapses the tree until the number it represents, 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 sufficient depth to represent each possible input
     color in a leaf.  However, 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

             _ k
                i=1 8k
     A complete tree would require 19,173,961 nodes for k = 8,
     cmax = 255.  Therefore, to avoid building a fully populated
     tree, ImageMagick: (1) Initializes data structures for nodes
     only as they are needed; (2) Chooses a maximum depth for the
     tree as a function of the desired number of colors in the
     output image (currently log4(colormap size)+2).  A tree of
     this depth generally allows the best representation of the
     source image with the fastest computational speed and the
     least amount of memory.  However, the default depth is
     inappropriate for some images.  Therefore, the caller can
     request a specific tree depth.

     For each pixel in the input image, classification scans
     downward from the root of the color description tree.  At
     each level of the tree, it identifies the single node which
     represents a cube in RGB space containing the pixel's color.
     It updates the following data for each such node:

     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 quantization 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 colors
           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 pixels whose colors should be defined
     by nodes at a lower level in the tree.

     Assignment generates the output image from the pruned tree.
     The output image consists of two parts:  (1)  A color map,
     which is an array of color descriptions (RGB triples) for
     each color present in the output image; (2)  A pixel array,
     which represents each pixel as an index into the color map
     array.

     First, the assignment phase makes one pass over the pruned
     color description tree to establish the image's color map.
     For each node with n2 > 0, it divides Sr, Sg, and Sb by n2.
     This produces the mean color of all pixels that classify no
     lower than this node.  Each of these colors becomes an entry
     in the color map.

     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 convenience, the color components are normalized to the
     range 0 to a maximum value, cmax.  The color reduction can
     then proceed as described.

MEASURING COLOR REDUCTION ERROR
     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 contour 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 perceptually-based, since the
     human eye is the final judge of quantization quality.

     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 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.

SEE ALSO
     display(1), animate(1), mogrify(1), import(1), miff(5)

COPYRIGHT
     Copyright (C) 2000 ImageMagick Studio, a non-profit
     organization dedicated to making software imaging solutions
     freely available.

     Permission is hereby granted, free of charge, to any person
     obtaining a copy of this software and associated
     documentation files ("ImageMagick"), to deal in ImageMagick
     without restriction, including without 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 conditions:

     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 merchantability, 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.

ACKNOWLEDGEMENTS
     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.

AUTHORS
     John Cristy, ImageMagick Studio

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