bionic (5) quantize.5.gz

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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

               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)

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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