Quantize - ImageMagick's color reduction algorithm.
#include <magick.h>
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.
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.
display(1),
animate(1),
mogrify(1),
import(1),
miff(5)
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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.
John Cristy, ImageMagick Studio