quantize(5) File Formats and Configurations quantize(5)

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

<Sigma> 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

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