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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
_ k
i=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) 2000 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
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