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

               Σ ki=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) 2002 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
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       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