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<P>This document describes how <A HREF="../index.html">GraphicsMagick</A>
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.
</P>
<P STYLE="margin-bottom: 0in">
</P>
<TABLE WIDTH=100% BORDER=0 CELLPADDING=2 CELLSPACING=2>
<TR>
<TD BGCOLOR="#52799e">
<P ALIGN=LEFT><IMG SRC="../images/right_triangle.png" NAME="Graphic2" ALT=">" ALIGN=BOTTOM WIDTH=15 HEIGHT=14 BORDER=0><B><FONT SIZE=4><FONT FACE="Helvetica, Arial"><FONT COLOR="#ffffff">Description</FONT></FONT></FONT></B></P>
</TD>
</TR>
</TABLE>
<DL>
<DD STYLE="margin-bottom: 0.2in">For purposes of color allocation,
an image is a set of <I>n</I> pixels, where each pixel is a point in
<B>RGB</B> space. <B>RGB</B> space is a 3-dimensional vector space,
and each pixel, <I>p(i)</I>, is defined by an ordered triple of red,
green, and blue coordinates, (<I>r(i)</I>,<I>g(i)</I>,<I>b(i)</I>).
</DD></DL>
<P>
Each primary color component (<I>red</I>, <I>green</I>, or <I>blue</I>)
represents an intensity which varies linearly from 0 to a maximum
value, <I>Cmax</I>, which corresponds to full saturation of that
color. Color allocation is defined over a domain consisting of the
cube in <B>RGB</B> space with opposite vertices at (0,0,0) and
(<I>Cmax</I>,<I>Cmax</I>,<I>Cmax</I>). <B>GraphicsMagick</B> requires
<I>Cmax</I>= <I>255</I>.
</P>
<P>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 <B>RGB</B> space and the vertex
farthest from the origin.
</P>
<P>The tree's root node represents the the entire domain, (0,0,0)
through (<I>Cmax</I>,<I>Cmax</I>,<I>Cmax</I>). 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.
</P>
<P STYLE="margin-bottom: 0in">The basic algorithm operates in three
phases:
</P>
<UL>
<LI><P STYLE="margin-bottom: 0in"><B>Classification,</B>
</P>
<LI><P STYLE="margin-bottom: 0in"><B>Reduction</B>, and
</P>
<LI><P><B>Assignment</B>.
</P>
</UL>
<P><B>Classification</B> builds a color description tree for the
image. <B>Reduction</B> collapses the tree until the number it
represents, at most, is the number of colors desired in the output
image. <B>Assignment</B> defines the output image's color map and
sets each pixel's color by reclassification in the reduced tree. <I>Our
goal is to minimize the numerical discrepancies between the original
colors and quantized colors</I>. To learn more about quantization
error, see <B>Measuring Color Reduction Error</B> later in this
document.
</P>
<P><B>Classification</B> 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
<I>Cmax</I>. If color components in the input image are quantized to
<I>k</I>-bit precision, so that <I>Cmax</I> = <I>2^k-1</I>, the tree
would need <I>k</I> levels below the root node to allow representing
each possible input color in a leaf. This becomes <B>prohibitive</B>
because the tree's:
</P>
<PRE> total number of nodes = 1+Sum(8^i), i=1,k
For k=8,
Number of nodes= 1 + (8^1+8^2+....+8^8)
8^8 - 1
= 1 + 8.-----------
8 - 1
= 19,173,961</PRE><P STYLE="margin-bottom: 0in">
Therefore, to avoid building a fully populated tree, <B>GraphicsMagick</B>:
</P>
<OL>
<LI><P STYLE="margin-bottom: 0in">Initializes data structures for
nodes only as they are needed;
</P>
<LI><P>Chooses a maximum depth for the tree as a function of the
desired number of colors in the output image (<B>currently <I>based-two</I>
logarithm of <I>Cmax</I></B>).
</P>
</OL>
<PRE> For Cmax=255,
Maximum tree depth = log (256)
2
= log (256) / log (2)
e e
= 8</PRE><P>
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.
</P>
<P>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 <B>RGB</B>
space containing the pixel's color. It updates the following data for
each such node:
</P>
<DL>
<DL>
<DL>
<DT><B>n1:</B>
</DT><DD>
Number of pixels whose color is contained in the <B>RGB</B> cube
which this node represents;
</DD><DT>
<B>n2:</B>
</DT><DD>
Number of pixels whose color is not represented in a node at lower
depth in the tree; initially, <B>n2=0</B> for all nodes except
leaves of the tree.
</DD><DT>
<B>Sr,Sg,Sb:</B>
</DT><DD>
Sums of the <I>red</I>, <I>green</I>, and <I>blue</I> component
values for all pixels not classified at a lower depth. The
combination of these sums and <I>n2</I> will ultimately
characterize the mean color of a set of pixels represented by this
node.
</DD><DT>
<B>E:</B>
</DT><DD STYLE="margin-bottom: 0.2in">
The distance squared in <B>RGB</B> space between each pixel
contained within a node and the nodes' center. This represents the
quantization error for a node.
</DD></DL>
</DL>
</DL>
<P>
<B>Reduction</B> repeatedly prunes the tree until the number of nodes
with <I>n2</I> > <I>0</I> 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 <I>E</I> value is minimal
for pruning and merges their color statistics upward. It uses a
pruning threshold, <I>Ep</I>, to govern node selection as follows:
</P>
<PRE> 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</PRE><P>
This has the effect of minimizing any quantization error when merging
two nodes together.
</P>
<P>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 <I>n2</I>,<I>Sr</I>, <I>Sg</I> and <I>Sb</I> 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.
</P>
<P>For each node, <I>n2</I> pixels exist for which that node
represents the smallest volume in <B>RGB</B> space containing those
pixel's colors. When <I>n2</I> > <I>0</I> the node will uniquely
define a color in the output image. At the beginning of reduction, <I>n2</I>
= <I>0</I> for all nodes except the leaves of the tree which
represent colors present in the input image.
</P>
<P>The other pixel count, <I>n1</I>, indicates the total number of
colors within the cubic volume which the node represents. This
includes <I>n1</I> - <I>n2</I> pixels whose colors should be defined
by nodes at a lower level in the tree.
</P>
<P STYLE="margin-bottom: 0in"><B>Assignment</B> generates the output
image from the pruned tree. The output image consists of two parts:
</P>
<OL>
<LI><P STYLE="margin-bottom: 0in">A color map, which is an array of
color descriptions (<B>RGB</B> triples) for each color present in
the output image.
</P>
<LI><P>A pixel array, which represents each pixel as an index into
the color map array.
</P>
</OL>
<P>First, the assignment phase makes one pass over the pruned color
description tree to establish the image's color map. For each node
with <I>n2</I> > <I>0</I>, it divides <I>Sr</I>, <I>Sg</I>, and <I>Sb</I>
by <I>n2</I>. 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.
</P>
<P>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.
</P>
<P>Empirical evidence suggests that the distances in color spaces
such as <B>YUV</B>, or <B>YIQ</B> correspond to perceptual color
differences more closely than do distances in <B>RGB</B> 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, <I>Cmax</I>. The color
reduction can then proceed as described.</P>
<P STYLE="margin-bottom: 0in">
</P>
<TABLE WIDTH=100% BORDER=0 CELLPADDING=2 CELLSPACING=2>
<TR>
<TD BGCOLOR="#52799e">
<P ALIGN=LEFT><IMG SRC="../images/right_triangle.png" NAME="Graphic3" ALT=">" ALIGN=BOTTOM WIDTH=15 HEIGHT=14 BORDER=0><B><FONT SIZE=4><FONT FACE="Helvetica, Arial"><FONT COLOR="#ffffff">Measuring
Color Reduction Error</FONT></FONT></FONT></B></P>
</TD>
</TR>
</TABLE>
<DL>
<DD STYLE="margin-bottom: 0.2in">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.
</DD></DL>
<P>
To measure the difference between the original and color reduced
images (the total color reduction error), <B>GraphicsMagick</B> sums
over all pixels in an image the distance squared in <B>RGB</B> space
between each original pixel value and its color reduced value.
<B>GraphicsMagick</B> prints several error measurements including the
mean error per pixel, the normalized mean error, and the normalized
maximum error.
</P>
<P>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.
</P>
<P>These errors are measured and printed when <B>-verbose</B> and
<B>-colors</B><I>are specified on the command line</I>:
</P>
<DL>
<DL>
<DL>
<DT><B>mean error per pixel:</B>
</DT><DD>
is the mean error for any single pixel in the image.
</DD><DT>
<B>normalized mean square error:</B>
</DT><DD>
is the normalized mean square quantization error for any single
pixel in the image.
</DD><DD>
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.
</DD><DT>
<B>normalized maximum square error:</B>
</DT><DD>
is the largest normalized square quantization error for any single
pixel in the image.
</DD><DD STYLE="margin-bottom: 0.2in">
This distance measure is normalized to a range between and blue
values in the image.
</DD></DL>
</DL>
</DL>
<P STYLE="margin-bottom: 0in">
</P>
<TABLE WIDTH=100% BORDER=0 CELLPADDING=2 CELLSPACING=2>
<TR>
<TD BGCOLOR="#52799e">
<P ALIGN=LEFT><IMG SRC="../images/right_triangle.png" NAME="Graphic4" ALT=">" ALIGN=BOTTOM WIDTH=15 HEIGHT=14 BORDER=0><B><FONT SIZE=4><FONT FACE="Helvetica, Arial"><FONT COLOR="#ffffff">Authors</FONT></FONT></FONT></B></P>
</TD>
</TR>
</TABLE>
<DL>
<DD STYLE="margin-bottom: 0.2in"><I>John Cristy</I>,
<A HREF="mailto:magick@imagemagick.org"><I>magick@imagemagick.org</I>,</A><B>
ImageMagick Studio</B>.
</DD></DL>
<TABLE WIDTH=100% BORDER=0 CELLPADDING=2 CELLSPACING=2>
<TR>
<TD BGCOLOR="#52799e">
<P ALIGN=LEFT><IMG SRC="../images/right_triangle.png" NAME="Graphic5" ALT=">" ALIGN=BOTTOM WIDTH=15 HEIGHT=14 BORDER=0><B><FONT SIZE=4><FONT FACE="Helvetica, Arial"><FONT COLOR="#ffffff">Acknowledgements</FONT></FONT></FONT></B></P>
</TD>
</TR>
</TABLE>
<DL>
<DD STYLE="margin-bottom: 0.2in"><I>Paul Raveling</I>, <B>USC
Information Sciences Institute</B>, 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.
</DD><HR>
</DL>
<P ALIGN=CENTER><A HREF="Copyright.html">Copyright</A> <FONT FACE="Abadi Mt Condensed Extra Bold">©</FONT>
GraphicsMagick Group 2002, 2003, 2004</P>
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