## table of contents

quantize(5) | File Formats Manual | 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

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

$Date$ | ImageMagick |