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This document describes how ImageMagickHperforms color reduction on an image. To fully understand this document,Iyou should have a knowledge of basic imaging techniques and the tree datastructure and terminology.

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>Description

GFor purposes of color allocation, an image is a set of n pixels,Fwhere each pixel is a point in RGB space. RGB space is aF3-dimensional vector space, and each pixel, p(i), is defined byban ordered triple of red, green, and blue coordinates, (r(i),g(i),b(i)).
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Each primary color component (red, green, or blue)Hrepresents an intensity which varies linearly from 0 to a maximum value, Cmax,Gwhich corresponds to full saturation of that color. Color allocation isNdefined 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.K

The algorithm maps this domain onto a tree in which each node representsGa cube within that domain. In the following discussion, these cubes areFdefined by the coordinate of two opposite vertices: The vertex nearestGthe origin in RGB space and the vertex farthest from the origin.I

The tree's root node represents the the entire domain, (0,0,0) throughF(Cmax,Cmax,Cmax). Each lower level in the tree isJgenerated by subdividing one node's cube into eight smaller cubes of equalGsize. This corresponds to bisecting the parent cube with planes passing#through the midpoints of each edge.0

The basic algorithm operates in three phases:

UClassification builds a color description tree for the image. ReductionIcollapses the tree until the number it represents, at most, is the number&of colors desired in the output image.$Assignment defines the outputHimage's color map and sets each pixel's color by reclassification in the reduced tree.>Our goal is to minimize the numerical discrepancies betweenNthe original colors and quantized colors. To learn more about quantizationIerror, see Measuring Color Reduction Error later in this document.H

Classification begins by initializing a color description treeNof sufficient depth to represent each possible input color in a leaf. However,Fit is impractical to generate a fully-formed color description tree inFthe classification phase for realistic values of Cmax. If colorFcomponents in the input image are quantized to k-bit precision,so that?Cmax = 2^k-1, the tree would need k levelsFbelow the root node to allow representing each possible input color in;a leaf. This becomes prohibitive because the tree's:M

      total number of nodes = 1+Sum(8^i), i=1,k-      For k=8, F      Number of nodes= 1 + (8^1+8^2+....+8^8)                               8^8 - 1 Œ                     = 1 + 8.-----------º                               8 - 1‹                     = 19,173,961
HTherefore, to avoid building a fully populated tree, ImageMagick:
    
  1. CInitializes data structures for nodes only as they are needed;
  2. 
  3. HChooses a maximum depth for the tree as a function of the desired numberFof colors in the output image (currently based-two logarithmof Cmax).
  4. 
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      For Cmax=255, œ                          I      Maximum tree depth = log (255)  °                              2¦                         = log (255) / log (2)î                              e           e¡                         =7.99 ~= 8
KA tree of this depth generally allows the best representation of the sourceJimage with the fastest computational speed and the least amount of memory.GHowever, the default depth is inappropriate for some images. Therefore,-the caller can request a specific tree depth.H

For each pixel in the input image, classification scans downward fromPthe root of the color description tree. At each level of the tree, it identifiesFthe single node which represents a cube in RGB space containingDthe pixel's color. It updates the following data for each such node:



n1:

FNumber of pixels whose color is contained in the RGB cube whichthis node represents;

n2:

HNumber of pixels whose color is not represented in a node at lower depthFin the tree; initially, n2=0 for all nodes except leaves of the tree.

Sr,Sg,Sb:

FSums of the red, green, and blue component valuesHfor all pixels not classified at a lower depth. The combination of theseGsums and n2 will ultimately characterize the mean color of a set(of pixels represented by this node.

E:

LThe distance squared in RGB space between each pixel contained withinHa node and the nodes' center. This represents the quantization error for a node.

 

 


JReduction repeatedly prunes the tree until the number of nodes with n2H> 0 is less than or equal to the maximum number of colors allowedKin the output image. On any given iteration over the tree, it selects thoseHnodes whose E value is minimal for pruning and merges their colorIstatistics upward. It uses a pruning threshold, Ep, to govern nodeselection as follows:*
      Ep = 0f      while number of nodes with (n2 > 0) > required maximum number of colorsU         prune all nodes such that E <= Epc         Set Ep  to minimum E in remaining nodes
IThis has the effect of minimizing any quantization error when merging twonodes together.H

When a node to be pruned has offspring, the pruning procedure invokesIitself recursively in order to prune the tree from the leaves upward. The,values of n2,Sr, Sg andSb in a node beingHpruned are always added to the corresponding data in that node's parent.IThis retains the pruned node's color characteristics for later averaging.G

For each node, n2 pixels exist for which that node representsHthe smallest volume in RGB space containing those pixel's colors.FWhen n2 > 0 the node will uniquely define a color in the,output image. At the beginning of reduction,n2 = 0 for allInodes except the leaves of the tree which represent colors present in the input image.I

The other pixel count, n1, indicates the total number of colorsJwithin the cubic volume which the node represents. This includes n1K- n2 pixels whose colors should be defined by nodes at a lower level in the tree.I

Assignment generates the output image from the pruned tree. The#output image consists of two parts:

    
  1. IA color map, which is an array of color descriptions (RGB triples)0for each color present in the output image.
  2. 
  3. IA pixel array, which represents each pixel as an index into the color map array.
  4. 
LFirst, the assignment phase makes one pass over the pruned color descriptionGtree to establish the image's color map. For each node with n2 > 0,it divides Sr,4Sg, and Sb by n2. This producesHthe mean color of all pixels that classify no lower than this node. Each2of these colors becomes an entry in the color map.F

Finally, the assignment phase reclassifies each pixel in the prunedKtree to identify the deepest node containing the pixel's color. The pixel'sGvalue in the pixel array becomes the index of this node's mean color inthe color map.F

Empirical evidence suggests that the distances in color spaces suchGas YUV, or YIQ correspond to perceptual color differencesFmore closely than do distances in RGB space. These color spacesHmay give better results when color reducing an image. Here the algorithmJis as described except each pixel is a point in the alternate color space.FFor convenience, the color components are normalized to the range 0 toUa maximum value, Cmax. The color reduction can then proceed as described.

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>Measuring3Color Reduction Error

NDepending on the image, the color reduction error may be obvious or invisible.FImages with high spatial frequencies (such as hair or grass) will showGerror much less than pictures with large smoothly shaded areas (such asFfaces). This is because the high-frequency contour edges introduced byQthe color reduction process are masked by the high frequencies in the image.
G


To measure the difference between the original and color reduced)images (the total color reduction error),ImageMagick sums overGall pixels in an image the distance squared in RGB space betweenIeach original pixel value and its color reduced value. ImageMagickIprints several error measurements including the mean error per pixel, the8normalized mean error, and the normalized maximum error.N

The normalized error measurement can be used to compare images. In general,Kthe closer the mean error is to zero the more the quantized image resemblesHthe source image. Ideally, the error should be perceptually-based, since9the human eye is the final judge of quantization quality.A

These errors are measured and printed when -verbose and-colorsare"specified on the command line:



!mean error per pixel:

9is the mean error for any single pixel in the image.

)normalized mean square error:

His the normalized mean square quantization error for any single pixel inthe image.

QThis 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:

His the largest normalized square quantization error for any single pixelin the image.

FThis distance measure is normalized to a range between and blue valuesin the image.



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>Authors

rJohn Cristy, magick@wizards.dupont.com,ImageMagickStudio.

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>Acknowledgements

HPaul Raveling, USC Information Sciences Institute, for theKoriginal idea of using space subdivision for the color reduction algorithm.FWith Paul's permission, this document is an adaptation from a documenthe wrote.

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>Copyright

1Copyright (C) 2000 ImageMagick Studio

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JThe above copyright notice and this permission notice shall be included>in all copies or substantial portions of ImageMagick.

JThe software is provided "as is", without warranty of any kind, expressKor implied, including but not limited to the warranties of merchantability,Ffitness for a particular purpose and noninfringement.In no event shallGImageMagick Studio be liable for any claim, damages or other liability,Fwhether in an action of contract, tort or otherwise, arising from, outZof or in connection with ImageMagick or the use or other dealings in ImageMagick.

GExcept as contained in this notice, the name of the E. I. du Pont deLNemours and Company shall not be used in advertising or otherwise to promoteRthe sale, use or other dealings in ImageMagick without prior written authorization%from the ImageMagick Studio.


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