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quantize(9)					   quantize(9)

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) repre-
       sents an intensity which varies linearly from 0 to a maxi-
       mum 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 follow-
       ing 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:  Classifica-
       tion, 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 origi-
       nal 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

ImageMagick		 1 May 1994				1

quantize(9)					   quantize(9)

       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
	       _
	       > ki=1 8k

       A complete tree would require 19,173,961 nodes for k = 8,
       cmax = 255.  Therefore, to avoid building a fully popu-
       lated 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 compu-
       tational 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 ulti-
	      mately characterize the mean color of a set of pix-
	      els represented by this node.

       E:     The distance squared in RGB space between each
	      pixel contained within a node and the nodes' cen-
	      ter.  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,

ImageMagick		 1 May 1994				2

quantize(9)					   quantize(9)

       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 proce-
       dure 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 correspond-
       ing 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 repre-
       sents 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 clas-
       sify 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 differ-
       ences more closely than do distances in RGB space.  These
       color spaces may give better results when color reducing

ImageMagick		 1 May 1994				3

quantize(9)					   quantize(9)

       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.

MEASURING COLOR REDUCTION ERROR
       Depending on the image, the color reduction error may be
       obvious or invisible.  Images with high spatial frequen-
       cies (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 nor-
       malized 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)

ImageMagick		 1 May 1994				4

quantize(9)					   quantize(9)

COPYRIGHT
       Copyright 1997 E. I. du Pont de Nemours and Company

       Permission to use, copy, modify, distribute, and sell this
       software and its documentation for any purpose is hereby
       granted without fee, provided that the above copyright
       notice appear in all copies and that both that copyright
       notice and this permission notice appear in supporting
       documentation, and that the name of E. I. du Pont de
       Nemours and Company not be used in advertising or public-
       ity pertaining to distribution of the software without
       specific, written prior permission.  E. I. du Pont de
       Nemours and Company makes no representations about the
       suitability of this software for any purpose.  It is pro-
       vided "as is" without express or implied warranty.

       E. I. du Pont de Nemours and Company disclaims all war-
       ranties with regard to this software, including all
       implied warranties of merchantability and fitness, in no
       event shall E. I. du Pont de Nemours and Company be liable
       for any special, indirect or consequential damages or any
       damages whatsoever resulting from loss of use, data or
       profits, whether in an action of contract, negligence or
       other tortious action, arising out of or in connection
       with the use or performance of this software.

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 docu-
       ment is an adaptation from a document he wrote.

AUTHORS
       John Cristy, E.I. du Pont de Nemours and Company Incorpo-
       rated

ImageMagick		 1 May 1994				5

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