Provided by: graphicsmagick_1.4+really1.3.42-1.1build3_amd64 
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 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
ImageMagick $Date$ quantize(5)