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quantize(5) [bsd man page]

```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 knowl-
edge 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 sub-
dividing 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 mini-
mize 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.  How-
ever, 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
_
> 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 quantiza-
tion 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 maxi-
mum 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 con-
tour 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.

display(1), animate(1), mogrify(1), import(1), miff(5)

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 mer-
chantability, 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)```