

n_{1}:  Number of pixels whose color is contained in the RGB cube which this node represents; 
n_{2}:  Number of pixels whose color is not represented in a node at lower depth in the tree; initially, n_{2} = 0 for all nodes except leaves of the tree. 
S_{r}, S_{g}, S_{b}:  
Sums of the red, green, and blue component values for all pixels not classified at a lower depth. The combination of these sums and n_{2} 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. 
E_{p} = 0
while number of nodes with (n_{2} > 0) > required maximum number of colors
prune all nodes such that E <= E_{p}
Set E_{p} 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 n_{2} S_{r}, S_{g}, and S_{b} 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, n_{2} pixels exist for which that node represents the smallest volume in RGB space containing those pixel’s colors. When n_{2} > 0 the node will uniquely define a color in the output image. At the beginning of reduction, n_{2} = 0 for all nodes except the leaves of the tree which represent colors present in the input image.
The other pixel count, n_{1}, indicates the total number of colors within the cubic volume which the node represents. This includes n_{1}  n_{2} 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 n_{2} > 0, it divides S_{r}, S_{g}, and S_{b} by n_{2}. 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, c_{max}. 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 highfrequency 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 perceptuallybased, 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
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