QUANTIZE(9) QUANTIZE(9)
NAME
Quantize - ImageMagick's color reduction algorithm.
SYNOPSIS
#include <image.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.
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
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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 com-
bination of these sums and n2 will ultimately charac-
terize the mean color of a set of pixels represented
by this 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 n1 count is minimal for pruning and merges their
color statistics upward. It uses a pruning threshold, np,
to govern node selection as follows:
np = 0
while number of nodes with (n2 > 0) > required maximum
number of colors
prune all nodes such that n1 <= np
Set np to minimum n1 in remaining nodes
When a node to be pruned has offspring, the pruning
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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 cor-
responding 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
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
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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)
COPYRIGHT
Copyright 1992 E. I. du Pont de Nemours & 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 & Company not be used in advertising or publicity
pertaining to distribution of the software without spe-
cific, written prior permission. E. I. du Pont de Nemours
ImageMagick 10 October 1992 4
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& Company makes no representations about the suitability
of this software for any purpose. It is provided "as is"
without express or implied warranty.
E. I. du Pont de Nemours & 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 & 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 & Company Incorpo-
rated
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