An efficient color quantization algorithm
/*
* @(#)Quantize.java 0.90 9/19/00 Adam Doppelt
*/
/**
* An efficient color quantization algorithm, adapted from the C++
* implementation quantize.c in <a
* href="http://www.imagemagick.org/">ImageMagick</a>. The pixels for
* an image are placed into an oct tree. The oct tree is reduced in
* size, and the pixels from the original image are reassigned to the
* nodes in the reduced tree.<p>
*
* Here is the copyright notice from ImageMagick:
*
* <pre>
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% 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 %
% E. I. du Pont de Nemours and Company 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 E. I. du Pont de %
% Nemours and Company shall not be used in advertising or otherwise to %
% promote the sale, use or other dealings in ImageMagick without prior %
% written authorization from the E. I. du Pont de Nemours and Company. %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
</pre>
*
*
* @version 0.90 19 Sep 2000
* @author <a href="http://www.gurge.com/amd/">Adam Doppelt</a>
*/
public class Quantize {
/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% %
% %
% QQQ U U AAA N N TTTTT IIIII ZZZZZ EEEEE %
% Q Q U U A A NN N T I ZZ E %
% Q Q U U AAAAA N N N T I ZZZ EEEEE %
% Q QQ U U A A N NN T I ZZ E %
% QQQQ UUU A A N N T IIIII ZZZZZ EEEEE %
% %
% %
% Reduce the Number of Unique Colors in an Image %
% %
% %
% Software Design %
% John Cristy %
% July 1992 %
% %
% %
% Copyright 1998 E. I. du Pont de Nemours and Company %
% %
% 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 %
% E. I. du Pont de Nemours and Company 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 E. I. du Pont de %
% Nemours and Company shall not be used in advertising or otherwise to %
% promote the sale, use or other dealings in ImageMagick without prior %
% written authorization from the E. I. du Pont de Nemours and Company. %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Realism in computer graphics typically requires using 24 bits/pixel to
% generate an image. Yet many graphic display devices do not contain
% the amount of memory necessary to match the spatial and color
% resolution of the human eye. The QUANTIZE program takes a 24 bit
% image and reduces the number of colors so it can be displayed on
% raster device with less bits per pixel. In most instances, the
% quantized image closely resembles the original reference image.
%
% A reduction of colors in an image is also desirable for image
% transmission and real-time animation.
%
% Function Quantize takes a standard RGB or monochrome images and quantizes
% them down to some fixed number of colors.
%
% 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). QUANTIZE 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, 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 (quantization error).
%
% 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 colors 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
% 1 + sum(i=1,k,8k).
%
% A complete tree would require 19,173,961 nodes for k = 8, cmax = 255.
% Therefore, to avoid building a fully populated tree, QUANTIZE: (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 log2(colormap size)).
%
% 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 count 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.
% 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 a 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.
%
% With the permission of USC Information Sciences Institute, 4676 Admiralty
% Way, Marina del Rey, California 90292, this code was adapted from module
% ALCOLS written by Paul Raveling.
%
% The names of ISI and USC are not used in advertising or publicity
% pertaining to distribution of the software without prior specific
% written permission from ISI.
%
*/
final static boolean QUICK = true;
final static int MAX_RGB = 255;
final static int MAX_NODES = 266817;
final static int MAX_TREE_DEPTH = 8;
// these are precomputed in advance
static int SQUARES[];
static int SHIFT[];
static {
SQUARES = new int[MAX_RGB + MAX_RGB + 1];
for (int i= -MAX_RGB; i <= MAX_RGB; i++) {
SQUARES[i + MAX_RGB] = i * i;
}
SHIFT = new int[MAX_TREE_DEPTH + 1];
for (int i = 0; i < MAX_TREE_DEPTH + 1; ++i) {
SHIFT[i] = 1 << (15 - i);
}
}
/**
* Reduce the image to the given number of colors. The pixels are
* reduced in place.
* @return The new color palette.
*/
public static int[] quantizeImage(int pixels[][], int max_colors) {
Cube cube = new Cube(pixels, max_colors);
cube.classification();
cube.reduction();
cube.assignment();
return cube.colormap;
}
static class Cube {
int pixels[][];
int max_colors;
int colormap[];
Node root;
int depth;
// counter for the number of colors in the cube. this gets
// recalculated often.
int colors;
// counter for the number of nodes in the tree
int nodes;
Cube(int pixels[][], int max_colors) {
this.pixels = pixels;
this.max_colors = max_colors;
int i = max_colors;
// tree_depth = log max_colors
// 4
for (depth = 1; i != 0; depth++) {
i /= 4;
}
if (depth > 1) {
--depth;
}
if (depth > MAX_TREE_DEPTH) {
depth = MAX_TREE_DEPTH;
} else if (depth < 2) {
depth = 2;
}
root = new Node(this);
}
/*
* Procedure 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
* colors 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 1 + sum(i=1,k,8k).
*
* A complete tree would require 19,173,961 nodes for k = 8,
* cmax = 255. Therefore, to avoid building a fully populated
* tree, QUANTIZE: (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 log2(colormap size)).
*
* 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 It updates the
* following data for each such node:
*
* number_pixels : Number of pixels whose color is contained
* in the RGB cube which this node represents;
*
* unique : 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.
*
* total_red/green/blue : 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.
*/
void classification() {
int pixels[][] = this.pixels;
int width = pixels.length;
int height = pixels[0].length;
// convert to indexed color
for (int x = width; x-- > 0; ) {
for (int y = height; y-- > 0; ) {
int pixel = pixels[x][y];
int red = (pixel >> 16) & 0xFF;
int green = (pixel >> 8) & 0xFF;
int blue = (pixel >> 0) & 0xFF;
// a hard limit on the number of nodes in the tree
if (nodes > MAX_NODES) {
System.out.println("pruning");
root.pruneLevel();
--depth;
}
// walk the tree to depth, increasing the
// number_pixels count for each node
Node node = root;
for (int level = 1; level <= depth; ++level) {
int id = (((red > node.mid_red ? 1 : 0) << 0) |
((green > node.mid_green ? 1 : 0) << 1) |
((blue > node.mid_blue ? 1 : 0) << 2));
if (node.child[id] == null) {
new Node(node, id, level);
}
node = node.child[id];
node.number_pixels += SHIFT[level];
}
++node.unique;
node.total_red += red;
node.total_green += green;
node.total_blue += blue;
}
}
}
/*
* reduction repeatedly prunes the tree until the number of
* nodes with unique > 0 is less than or equal to the maximum
* number of colors allowed in the output image.
*
* 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 statistics of the 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.
*/
void reduction() {
int threshold = 1;
while (colors > max_colors) {
colors = 0;
threshold = root.reduce(threshold, Integer.MAX_VALUE);
}
}
/**
* The result of a closest color search.
*/
static class Search {
int distance;
int color_number;
}
/*
* Procedure 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.
*/
void assignment() {
colormap = new int[colors];
colors = 0;
root.colormap();
int pixels[][] = this.pixels;
int width = pixels.length;
int height = pixels[0].length;
Search search = new Search();
// convert to indexed color
for (int x = width; x-- > 0; ) {
for (int y = height; y-- > 0; ) {
int pixel = pixels[x][y];
int red = (pixel >> 16) & 0xFF;
int green = (pixel >> 8) & 0xFF;
int blue = (pixel >> 0) & 0xFF;
// walk the tree to find the cube containing that color
Node node = root;
for ( ; ; ) {
int id = (((red > node.mid_red ? 1 : 0) << 0) |
((green > node.mid_green ? 1 : 0) << 1) |
((blue > node.mid_blue ? 1 : 0) << 2) );
if (node.child[id] == null) {
break;
}
node = node.child[id];
}
if (QUICK) {
// if QUICK is set, just use that
// node. Strictly speaking, this isn't
// necessarily best match.
pixels[x][y] = node.color_number;
} else {
// Find the closest color.
search.distance = Integer.MAX_VALUE;
node.parent.closestColor(red, green, blue, search);
pixels[x][y] = search.color_number;
}
}
}
}
/**
* A single Node in the tree.
*/
static class Node {
Cube cube;
// parent node
Node parent;
// child nodes
Node child[];
int nchild;
// our index within our parent
int id;
// our level within the tree
int level;
// our color midpoint
int mid_red;
int mid_green;
int mid_blue;
// the pixel count for this node and all children
int number_pixels;
// the pixel count for this node
int unique;
// the sum of all pixels contained in this node
int total_red;
int total_green;
int total_blue;
// used to build the colormap
int color_number;
Node(Cube cube) {
this.cube = cube;
this.parent = this;
this.child = new Node[8];
this.id = 0;
this.level = 0;
this.number_pixels = Integer.MAX_VALUE;
this.mid_red = (MAX_RGB + 1) >> 1;
this.mid_green = (MAX_RGB + 1) >> 1;
this.mid_blue = (MAX_RGB + 1) >> 1;
}
Node(Node parent, int id, int level) {
this.cube = parent.cube;
this.parent = parent;
this.child = new Node[8];
this.id = id;
this.level = level;
// add to the cube
++cube.nodes;
if (level == cube.depth) {
++cube.colors;
}
// add to the parent
++parent.nchild;
parent.child[id] = this;
// figure out our midpoint
int bi = (1 << (MAX_TREE_DEPTH - level)) >> 1;
mid_red = parent.mid_red + ((id & 1) > 0 ? bi : -bi);
mid_green = parent.mid_green + ((id & 2) > 0 ? bi : -bi);
mid_blue = parent.mid_blue + ((id & 4) > 0 ? bi : -bi);
}
/**
* Remove this child node, and make sure our parent
* absorbs our pixel statistics.
*/
void pruneChild() {
--parent.nchild;
parent.unique += unique;
parent.total_red += total_red;
parent.total_green += total_green;
parent.total_blue += total_blue;
parent.child[id] = null;
--cube.nodes;
cube = null;
parent = null;
}
/**
* Prune the lowest layer of the tree.
*/
void pruneLevel() {
if (nchild != 0) {
for (int id = 0; id < 8; id++) {
if (child[id] != null) {
child[id].pruneLevel();
}
}
}
if (level == cube.depth) {
pruneChild();
}
}
/**
* Remove any nodes that have fewer than threshold
* pixels. Also, as long as we're walking the tree:
*
* - figure out the color with the fewest pixels
* - recalculate the total number of colors in the tree
*/
int reduce(int threshold, int next_threshold) {
if (nchild != 0) {
for (int id = 0; id < 8; id++) {
if (child[id] != null) {
next_threshold = child[id].reduce(threshold, next_threshold);
}
}
}
if (number_pixels <= threshold) {
pruneChild();
} else {
if (unique != 0) {
cube.colors++;
}
if (number_pixels < next_threshold) {
next_threshold = number_pixels;
}
}
return next_threshold;
}
/*
* colormap traverses the color cube tree and notes each
* colormap entry. A colormap entry is any node in the
* color cube tree where the number of unique colors is
* not zero.
*/
void colormap() {
if (nchild != 0) {
for (int id = 0; id < 8; id++) {
if (child[id] != null) {
child[id].colormap();
}
}
}
if (unique != 0) {
int r = ((total_red + (unique >> 1)) / unique);
int g = ((total_green + (unique >> 1)) / unique);
int b = ((total_blue + (unique >> 1)) / unique);
cube.colormap[cube.colors] = ((( 0xFF) << 24) |
((r & 0xFF) << 16) |
((g & 0xFF) << 8) |
((b & 0xFF) << 0));
color_number = cube.colors++;
}
}
/* ClosestColor traverses the color cube tree at a
* particular node and determines which colormap entry
* best represents the input color.
*/
void closestColor(int red, int green, int blue, Search search) {
if (nchild != 0) {
for (int id = 0; id < 8; id++) {
if (child[id] != null) {
child[id].closestColor(red, green, blue, search);
}
}
}
if (unique != 0) {
int color = cube.colormap[color_number];
int distance = distance(color, red, green, blue);
if (distance < search.distance) {
search.distance = distance;
search.color_number = color_number;
}
}
}
/**
* Figure out the distance between this node and som color.
*/
final static int distance(int color, int r, int g, int b) {
return (SQUARES[((color >> 16) & 0xFF) - r + MAX_RGB] +
SQUARES[((color >> 8) & 0xFF) - g + MAX_RGB] +
SQUARES[((color >> 0) & 0xFF) - b + MAX_RGB]);
}
public String toString() {
StringBuffer buf = new StringBuffer();
if (parent == this) {
buf.append("root");
} else {
buf.append("node");
}
buf.append(' ');
buf.append(level);
buf.append(" [");
buf.append(mid_red);
buf.append(',');
buf.append(mid_green);
buf.append(',');
buf.append(mid_blue);
buf.append(']');
return new String(buf);
}
}
}
}
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