quantize.c

/*
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%          Q   Q  U   U  A   A  NN  N    T      I        ZZ  E                %
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%           QQQQ   UUU   A   A  N   N    T    IIIII   ZZZZZ  EEEEE            %
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%                                                                             %
%    MagickCore Methods to Reduce the Number of Unique Colors in an Image     %
%                                                                             %
%                           Software Design                                   %
%                                Cristy                                       %
%                              July 1992                                      %
%                                                                             %
%                                                                             %
%  Copyright 1999 ImageMagick Studio LLC, a non-profit organization           %
%  dedicated to making software imaging solutions freely available.           %
%                                                                             %
%  You may not use this file except in compliance with the License.  You may  %
%  obtain a copy of the License at                                            %
%                                                                             %
%    https://imagemagick.org/license/                                         %
%                                                                             %
%  Unless required by applicable law or agreed to in writing, software        %
%  distributed under the License is distributed on an "AS IS" BASIS,          %
%  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.   %
%  See the License for the specific language governing permissions and        %
%  limitations under the License.                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  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 methods 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.
%
%  QuantizeImage() 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 (vertex
%  nearest the origin in RGB space and the vertex farthest from the origin).
%
%  The tree's root node represents 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 restorage_class 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 storage_class 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, storage_class 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.
%
%  This method is based on a similar algorithm written by Paul Raveling.
%
*/

/*
  Include declarations.
*/
#include "magick/studio.h"
#include "magick/artifact.h"
#include "magick/attribute.h"
#include "magick/cache-view.h"
#include "magick/color.h"
#include "magick/color-private.h"
#include "magick/colormap.h"
#include "magick/colorspace.h"
#include "magick/colorspace-private.h"
#include "magick/enhance.h"
#include "magick/exception.h"
#include "magick/exception-private.h"
#include "magick/histogram.h"
#include "magick/image.h"
#include "magick/image-private.h"
#include "magick/list.h"
#include "magick/memory_.h"
#include "magick/monitor.h"
#include "magick/monitor-private.h"
#include "magick/option.h"
#include "magick/pixel-private.h"
#include "magick/quantize.h"
#include "magick/quantum.h"
#include "magick/resource_.h"
#include "magick/string_.h"
#include "magick/string-private.h"
#include "magick/thread-private.h"

/*
  Define declarations.
*/
#if !defined(__APPLE__) && !defined(TARGET_OS_IPHONE)
#define CacheShift  2
#else
#define CacheShift  3
#endif
#define ErrorQueueLength  16
#define ErrorRelativeWeight  MagickSafeReciprocal(16)
#define MaxQNodes  266817
#define MaxTreeDepth  8
#define QNodesInAList  1920

/*
  Typedef declarations.
*/
typedef struct _QNodeInfo
{
  struct _QNodeInfo
    *parent,
    *child[16];

  MagickSizeType
    number_unique;

  DoublePixelPacket
    total_color;

  MagickRealType
    quantize_error;

  size_t
    color_number,
    id,
    level;
} QNodeInfo;

typedef struct _QNodes
{
  QNodeInfo
    *nodes;

  struct _QNodes
    *next;
} QNodes;

typedef struct _QCubeInfo
{
  QNodeInfo
    *root;

  size_t
    colors,
    maximum_colors;

  ssize_t
    transparent_index;

  MagickSizeType
    transparent_pixels;

  DoublePixelPacket
    target;

  MagickRealType
    distance,
    pruning_threshold,
    next_threshold;

  size_t
    nodes,
    free_nodes,
    color_number;

  QNodeInfo
    *next_node;

  QNodes
    *node_queue;

  MemoryInfo
    *memory_info;

  ssize_t
    *cache;

  DoublePixelPacket
    error[ErrorQueueLength];

  MagickRealType
    diffusion,
    weights[ErrorQueueLength];

  QuantizeInfo
    *quantize_info;

  MagickBooleanType
    associate_alpha;

  ssize_t
    x,
    y;

  size_t
    depth;

  MagickOffsetType
    offset;

  MagickSizeType
    span;
} QCubeInfo;

/*
  Method prototypes.
*/
static QCubeInfo
  *GetQCubeInfo(const QuantizeInfo *,const size_t,const size_t);

static QNodeInfo
  *GetQNodeInfo(QCubeInfo *,const size_t,const size_t,QNodeInfo *);

static MagickBooleanType
  AssignImageColors(Image *,QCubeInfo *),
  ClassifyImageColors(QCubeInfo *,const Image *,ExceptionInfo *),
  DitherImage(Image *,QCubeInfo *),
  SetGrayscaleImage(Image *);

static void
  ClosestColor(const Image *,QCubeInfo *,const QNodeInfo *),
  DefineImageColormap(Image *,QCubeInfo *,QNodeInfo *),
  DestroyQCubeInfo(QCubeInfo *),
  PruneLevel(QCubeInfo *,const QNodeInfo *),
  PruneToCubeDepth(QCubeInfo *,const QNodeInfo *),
  ReduceImageColors(const Image *,QCubeInfo *);

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   A c q u i r e Q u a n t i z e I n f o                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  AcquireQuantizeInfo() allocates the QuantizeInfo structure.
%
%  The format of the AcquireQuantizeInfo method is:
%
%      QuantizeInfo *AcquireQuantizeInfo(const ImageInfo *image_info)
%
%  A description of each parameter follows:
%
%    o image_info: the image info.
%
*/
MagickExport QuantizeInfo *AcquireQuantizeInfo(const ImageInfo *image_info)
{
  QuantizeInfo
    *quantize_info;

  quantize_info=(QuantizeInfo *) AcquireMagickMemory(sizeof(*quantize_info));
  if (quantize_info == (QuantizeInfo *) NULL)
    ThrowFatalException(ResourceLimitFatalError,"MemoryAllocationFailed");
  GetQuantizeInfo(quantize_info);
  if (image_info != (ImageInfo *) NULL)
    {
      const char
        *option;

      quantize_info->dither=image_info->dither;
      option=GetImageOption(image_info,"dither");
      if (option != (const char *) NULL)
        quantize_info->dither_method=(DitherMethod) ParseCommandOption(
          MagickDitherOptions,MagickFalse,option);
      quantize_info->measure_error=image_info->verbose;
    }
  return(quantize_info);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   A s s i g n I m a g e C o l o r s                                         %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  AssignImageColors() 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.
%
%  The format of the AssignImageColors() method is:
%
%      MagickBooleanType AssignImageColors(Image *image,QCubeInfo *cube_info)
%
%  A description of each parameter follows.
%
%    o image: the image.
%
%    o cube_info: A pointer to the Cube structure.
%
*/

static inline void AssociateAlphaPixel(const QCubeInfo *cube_info,
  const PixelPacket *pixel,DoublePixelPacket *alpha_pixel)
{
  MagickRealType
    alpha;

  alpha_pixel->index=0;
  if ((cube_info->associate_alpha == MagickFalse) ||
      (pixel->opacity == OpaqueOpacity))
    {
      alpha_pixel->red=(MagickRealType) GetPixelRed(pixel);
      alpha_pixel->green=(MagickRealType) GetPixelGreen(pixel);
      alpha_pixel->blue=(MagickRealType) GetPixelBlue(pixel);
      alpha_pixel->opacity=(MagickRealType) GetPixelOpacity(pixel);
      return;
    }
  alpha=(MagickRealType) (QuantumScale*((MagickRealType) QuantumRange-
    (MagickRealType) GetPixelOpacity(pixel)));
  alpha_pixel->red=alpha*(MagickRealType) GetPixelRed(pixel);
  alpha_pixel->green=alpha*(MagickRealType) GetPixelGreen(pixel);
  alpha_pixel->blue=alpha*(MagickRealType) GetPixelBlue(pixel);
  alpha_pixel->opacity=(MagickRealType) GetPixelOpacity(pixel);
}

static inline size_t ColorToQNodeId(const QCubeInfo *cube_info,
  const DoublePixelPacket *pixel,size_t index)
{
  size_t
    id;

  id=(size_t) (((ScaleQuantumToChar(ClampPixel(GetPixelRed(pixel))) >> index) &
    0x01) | ((ScaleQuantumToChar(ClampPixel(GetPixelGreen(pixel))) >> index) &
    0x01) << 1 | ((ScaleQuantumToChar(ClampPixel(GetPixelBlue(pixel))) >>
    index) & 0x01) << 2);
  if (cube_info->associate_alpha != MagickFalse)
    id|=((ScaleQuantumToChar(ClampPixel(GetPixelOpacity(pixel))) >> index) &
      0x1) << 3;
  return(id);
}

static inline MagickBooleanType IsSameColor(const Image *image,
  const PixelPacket *p,const PixelPacket *q)
{
  if ((GetPixelRed(p) != GetPixelRed(q)) ||
      (GetPixelGreen(p) != GetPixelGreen(q)) ||
      (GetPixelBlue(p) != GetPixelBlue(q)))
    return(MagickFalse);
  if ((image->matte != MagickFalse) &&
      (GetPixelOpacity(p) != GetPixelOpacity(q)))
    return(MagickFalse);
  return(MagickTrue);
}

static MagickBooleanType AssignImageColors(Image *image,QCubeInfo *cube_info)
{
#define AssignImageTag  "Assign/Image"

  ColorspaceType
    colorspace;

  size_t
    number_colors;

  ssize_t
    y;

  /*
    Allocate image colormap.
  */
  colorspace=image->colorspace;
  if (cube_info->quantize_info->colorspace != UndefinedColorspace)
    (void) TransformImageColorspace(image,cube_info->quantize_info->colorspace);
  number_colors=MagickMax(cube_info->colors,cube_info->maximum_colors);
  if (AcquireImageColormap(image,number_colors) == MagickFalse)
    ThrowBinaryImageException(ResourceLimitError,"MemoryAllocationFailed",
      image->filename);
  image->colors=0;
  cube_info->transparent_pixels=0;
  cube_info->transparent_index=(-1);
  DefineImageColormap(image,cube_info,cube_info->root);
  /*
    Create a reduced color image.
  */
  if ((cube_info->quantize_info->dither != MagickFalse) &&
      (cube_info->quantize_info->dither_method != NoDitherMethod))
    (void) DitherImage(image,cube_info);
  else
    {
      CacheView
        *image_view;

      ExceptionInfo
        *exception;

      MagickBooleanType
        status;

      status=MagickTrue;
      exception=(&image->exception);
      image_view=AcquireAuthenticCacheView(image,exception);
#if defined(MAGICKCORE_OPENMP_SUPPORT)
      #pragma omp parallel for schedule(static) shared(status) \
        magick_number_threads(image,image,image->rows,1)
#endif
      for (y=0; y < (ssize_t) image->rows; y++)
      {
        IndexPacket
          *magick_restrict indexes;

        PixelPacket
          *magick_restrict q;

        QCubeInfo
          cube;

        ssize_t
          x;

        ssize_t
          count;

        if (status == MagickFalse)
          continue;
        q=GetCacheViewAuthenticPixels(image_view,0,y,image->columns,1,
          exception);
        if (q == (PixelPacket *) NULL)
          {
            status=MagickFalse;
            continue;
          }
        indexes=GetCacheViewAuthenticIndexQueue(image_view);
        cube=(*cube_info);
        for (x=0; x < (ssize_t) image->columns; x+=count)
        {
          DoublePixelPacket
            pixel;

          const QNodeInfo
            *node_info;

          ssize_t
            i;

          size_t
            id,
            index;

          /*
            Identify the deepest node containing the pixel's color.
          */
          for (count=1; (x+count) < (ssize_t) image->columns; count++)
            if (IsSameColor(image,q,q+count) == MagickFalse)
              break;
          AssociateAlphaPixel(&cube,q,&pixel);
          node_info=cube.root;
          for (index=MaxTreeDepth-1; (ssize_t) index > 0; index--)
          {
            id=ColorToQNodeId(&cube,&pixel,index);
            if (node_info->child[id] == (QNodeInfo *) NULL)
              break;
            node_info=node_info->child[id];
          }
          /*
            Find closest color among siblings and their children.
          */
          cube.target=pixel;
          cube.distance=(MagickRealType) (4.0*((MagickRealType) QuantumRange+
            1.0)*((MagickRealType) QuantumRange+1.0)+1.0);
          ClosestColor(image,&cube,node_info->parent);
          index=cube.color_number;
          for (i=0; i < (ssize_t) count; i++)
          {
            if (image->storage_class == PseudoClass)
              SetPixelIndex(indexes+x+i,index);
            if (cube.quantize_info->measure_error == MagickFalse)
              {
                SetPixelRgb(q,image->colormap+index);
                if (cube.associate_alpha != MagickFalse)
                  SetPixelOpacity(q,image->colormap[index].opacity);
              }
            q++;
          }
        }
        if (SyncCacheViewAuthenticPixels(image_view,exception) == MagickFalse)
          status=MagickFalse;
        if (image->progress_monitor != (MagickProgressMonitor) NULL)
          {
            MagickBooleanType
              proceed;

            proceed=SetImageProgress(image,AssignImageTag,(MagickOffsetType) y,
              image->rows);
            if (proceed == MagickFalse)
              status=MagickFalse;
          }
      }
      image_view=DestroyCacheView(image_view);
    }
  if (cube_info->quantize_info->measure_error != MagickFalse)
    (void) GetImageQuantizeError(image);
  if ((cube_info->quantize_info->number_colors == 2) &&
      (IsGrayColorspace(cube_info->quantize_info->colorspace)))
    {
      double
        intensity;

      /*
        Monochrome image.
      */
      intensity=GetPixelLuma(image,image->colormap+0) <
        (MagickRealType) QuantumRange/2.0 ? 0.0 : (double) QuantumRange;
      if ((image->colors > 1) &&
          (GetPixelLuma(image,image->colormap+0) > 
           GetPixelLuma(image,image->colormap+1)))
        intensity=(double) QuantumRange;
      image->colormap[0].red=intensity;
      image->colormap[0].green=intensity;
      image->colormap[0].blue=intensity;
      if (image->colors > 1)
        {
          image->colormap[1].red=(double) QuantumRange-intensity;
          image->colormap[1].green=(double) QuantumRange-intensity;
          image->colormap[1].blue=(double) QuantumRange-intensity;
        }
    }
  (void) SyncImage(image);
  if ((cube_info->quantize_info->colorspace != UndefinedColorspace) &&
      (IssRGBCompatibleColorspace(colorspace) == MagickFalse))
    (void) TransformImageColorspace(image,colorspace);
  return(MagickTrue);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   C l a s s i f y I m a g e C o l o r s                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  ClassifyImageColors() 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 storage_class 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, storage_class 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:
%
%    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.
%
%  The format of the ClassifyImageColors() method is:
%
%      MagickBooleanType ClassifyImageColors(QCubeInfo *cube_info,
%        const Image *image,ExceptionInfo *exception)
%
%  A description of each parameter follows.
%
%    o cube_info: A pointer to the Cube structure.
%
%    o image: the image.
%
*/

static inline void SetAssociatedAlpha(const Image *image,QCubeInfo *cube_info)
{
  MagickBooleanType
    associate_alpha;

  associate_alpha=image->matte;
  if ((cube_info->quantize_info->number_colors == 2) &&
      ((cube_info->quantize_info->colorspace == LinearGRAYColorspace) ||
       (cube_info->quantize_info->colorspace == GRAYColorspace)))
    associate_alpha=MagickFalse;
  cube_info->associate_alpha=associate_alpha;
}

static MagickBooleanType ClassifyImageColors(QCubeInfo *cube_info,
  const Image *image,ExceptionInfo *exception)
{
#define ClassifyImageTag  "Classify/Image"

  CacheView
    *image_view;

  DoublePixelPacket
    error,
    mid,
    midpoint,
    pixel;

  MagickBooleanType
    proceed;

  MagickRealType
    bisect;

  QNodeInfo
    *node_info;

  size_t
    count,
    id,
    index,
    level;

  ssize_t
    y;

  /*
    Classify the first cube_info->maximum_colors colors to a tree depth of 8.
  */
  SetAssociatedAlpha(image,cube_info);
  if (cube_info->quantize_info->colorspace != image->colorspace)
    {
      if ((cube_info->quantize_info->colorspace != UndefinedColorspace) &&
          (cube_info->quantize_info->colorspace != CMYKColorspace))
        (void) TransformImageColorspace((Image *) image,
          cube_info->quantize_info->colorspace);
      else
        if (IssRGBCompatibleColorspace(image->colorspace) == MagickFalse)
          (void) TransformImageColorspace((Image *) image,sRGBColorspace);
    }
  midpoint.red=(MagickRealType) QuantumRange/2.0;
  midpoint.green=(MagickRealType) QuantumRange/2.0;
  midpoint.blue=(MagickRealType) QuantumRange/2.0;
  midpoint.opacity=(MagickRealType) QuantumRange/2.0;
  midpoint.index=(MagickRealType) QuantumRange/2.0;
  error.opacity=0.0;
  image_view=AcquireVirtualCacheView(image,exception);
  for (y=0; y < (ssize_t) image->rows; y++)
  {
    const PixelPacket
      *magick_restrict p;

    ssize_t
      x;

    p=GetCacheViewVirtualPixels(image_view,0,y,image->columns,1,exception);
    if (p == (const PixelPacket *) NULL)
      break;
    if (cube_info->nodes > MaxQNodes)
      {
        /*
          Prune one level if the color tree is too large.
        */
        PruneLevel(cube_info,cube_info->root);
        cube_info->depth--;
      }
    for (x=0; x < (ssize_t) image->columns; x+=(ssize_t) count)
    {
      /*
        Start at the root and descend the color cube tree.
      */
      for (count=1; (x+(ssize_t) count) < (ssize_t) image->columns; count++)
        if (IsSameColor(image,p,p+count) == MagickFalse)
          break;
      AssociateAlphaPixel(cube_info,p,&pixel);
      index=MaxTreeDepth-1;
      bisect=((MagickRealType) QuantumRange+1.0)/2.0;
      mid=midpoint;
      node_info=cube_info->root;
      for (level=1; level <= MaxTreeDepth; level++)
      {
        double
          distance;

        bisect*=0.5;
        id=ColorToQNodeId(cube_info,&pixel,index);
        mid.red+=(id & 1) != 0 ? bisect : -bisect;
        mid.green+=(id & 2) != 0 ? bisect : -bisect;
        mid.blue+=(id & 4) != 0 ? bisect : -bisect;
        mid.opacity+=(id & 8) != 0 ? bisect : -bisect;
        if (node_info->child[id] == (QNodeInfo *) NULL)
          {
            /*
              Set colors of new node to contain pixel.
            */
            node_info->child[id]=GetQNodeInfo(cube_info,id,level,node_info);
            if (node_info->child[id] == (QNodeInfo *) NULL)
              {
                (void) ThrowMagickException(exception,GetMagickModule(),
                  ResourceLimitError,"MemoryAllocationFailed","`%s'",
                  image->filename);
                continue;
              }
            if (level == MaxTreeDepth)
              cube_info->colors++;
          }
        /*
          Approximate the quantization error represented by this node.
        */
        node_info=node_info->child[id];
        error.red=QuantumScale*(pixel.red-mid.red);
        error.green=QuantumScale*(pixel.green-mid.green);
        error.blue=QuantumScale*(pixel.blue-mid.blue);
        if (cube_info->associate_alpha != MagickFalse)
          error.opacity=QuantumScale*(pixel.opacity-mid.opacity);
        distance=(double) (error.red*error.red+error.green*error.green+
          error.blue*error.blue+error.opacity*error.opacity);
        if (IsNaN(distance) != 0)
          distance=0.0;
        node_info->quantize_error+=count*sqrt(distance);
        cube_info->root->quantize_error+=node_info->quantize_error;
        index--;
      }
      /*
        Sum RGB for this leaf for later derivation of the mean cube color.
      */
      node_info->number_unique+=count;
      node_info->total_color.red+=count*QuantumScale*(MagickRealType)
        ClampPixel(pixel.red);
      node_info->total_color.green+=count*QuantumScale*(MagickRealType)
        ClampPixel(pixel.green);
      node_info->total_color.blue+=count*QuantumScale*(MagickRealType)
        ClampPixel(pixel.blue);
      if (cube_info->associate_alpha != MagickFalse)
        node_info->total_color.opacity+=count*QuantumScale*(MagickRealType)
          ClampPixel(pixel.opacity);
      else
        node_info->total_color.opacity+=count*QuantumScale*(MagickRealType)
          ClampPixel(OpaqueOpacity);
      p+=(ptrdiff_t) count;
    }
    if (cube_info->colors > cube_info->maximum_colors)
      {
        PruneToCubeDepth(cube_info,cube_info->root);
        break;
      }
    proceed=SetImageProgress(image,ClassifyImageTag,(MagickOffsetType) y,
      image->rows);
    if (proceed == MagickFalse)
      break;
  }
  for (y++; y < (ssize_t) image->rows; y++)
  {
    const PixelPacket
      *magick_restrict p;

    ssize_t
      x;

    p=GetCacheViewVirtualPixels(image_view,0,y,image->columns,1,exception);
    if (p == (const PixelPacket *) NULL)
      break;
    if (cube_info->nodes > MaxQNodes)
      {
        /*
          Prune one level if the color tree is too large.
        */
        PruneLevel(cube_info,cube_info->root);
        cube_info->depth--;
      }
    for (x=0; x < (ssize_t) image->columns; x+=(ssize_t) count)
    {
      /*
        Start at the root and descend the color cube tree.
      */
      for (count=1; (x+(ssize_t) count) < (ssize_t) image->columns; count++)
        if (IsSameColor(image,p,p+count) == MagickFalse)
          break;
      AssociateAlphaPixel(cube_info,p,&pixel);
      index=MaxTreeDepth-1;
      bisect=((MagickRealType) QuantumRange+1.0)/2.0;
      mid=midpoint;
      node_info=cube_info->root;
      for (level=1; level <= cube_info->depth; level++)
      {
        double
          distance;

        bisect*=0.5;
        id=ColorToQNodeId(cube_info,&pixel,index);
        mid.red+=(id & 1) != 0 ? bisect : -bisect;
        mid.green+=(id & 2) != 0 ? bisect : -bisect;
        mid.blue+=(id & 4) != 0 ? bisect : -bisect;
        mid.opacity+=(id & 8) != 0 ? bisect : -bisect;
        if (node_info->child[id] == (QNodeInfo *) NULL)
          {
            /*
              Set colors of new node to contain pixel.
            */
            node_info->child[id]=GetQNodeInfo(cube_info,id,level,node_info);
            if (node_info->child[id] == (QNodeInfo *) NULL)
              {
                (void) ThrowMagickException(exception,GetMagickModule(),
                  ResourceLimitError,"MemoryAllocationFailed","%s",
                  image->filename);
                  continue;
               }
            if (level == cube_info->depth)
              cube_info->colors++;
          }
        /*
          Approximate the quantization error represented by this node.
        */
        node_info=node_info->child[id];
        error.red=QuantumScale*(pixel.red-mid.red);
        error.green=QuantumScale*(pixel.green-mid.green);
        error.blue=QuantumScale*(pixel.blue-mid.blue);
        if (cube_info->associate_alpha != MagickFalse)
          error.opacity=QuantumScale*(pixel.opacity-mid.opacity);
        distance=(double) (error.red*error.red+error.green*error.green+
          error.blue*error.blue+error.opacity*error.opacity);
        if (IsNaN(distance) != 0)
          distance=0.0;
        node_info->quantize_error+=count*sqrt(distance);
        cube_info->root->quantize_error+=node_info->quantize_error;
        index--;
      }
      /*
        Sum RGB for this leaf for later derivation of the mean cube color.
      */
      node_info->number_unique+=count;
      node_info->total_color.red+=count*QuantumScale*(MagickRealType)
        ClampPixel(pixel.red);
      node_info->total_color.green+=count*QuantumScale*(MagickRealType)
        ClampPixel(pixel.green);
      node_info->total_color.blue+=count*QuantumScale*(MagickRealType)
        ClampPixel(pixel.blue);
      if (cube_info->associate_alpha != MagickFalse)
        node_info->total_color.opacity+=count*QuantumScale*(MagickRealType)
          ClampPixel(pixel.opacity);
      else
        node_info->total_color.opacity+=count*QuantumScale*(MagickRealType)
          ClampPixel(OpaqueOpacity);
      p+=(ptrdiff_t) count;
    }
    proceed=SetImageProgress(image,ClassifyImageTag,(MagickOffsetType) y,
      image->rows);
    if (proceed == MagickFalse)
      break;
  }
  image_view=DestroyCacheView(image_view);
  if (cube_info->quantize_info->colorspace != image->colorspace)
    if ((cube_info->quantize_info->colorspace != UndefinedColorspace) &&
        (cube_info->quantize_info->colorspace != CMYKColorspace))
      (void) TransformImageColorspace((Image *) image,sRGBColorspace);
  return(y < (ssize_t) image->rows ? MagickFalse : MagickTrue);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   C l o n e Q u a n t i z e I n f o                                         %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  CloneQuantizeInfo() makes a duplicate of the given quantize info structure,
%  or if quantize info is NULL, a new one.
%
%  The format of the CloneQuantizeInfo method is:
%
%      QuantizeInfo *CloneQuantizeInfo(const QuantizeInfo *quantize_info)
%
%  A description of each parameter follows:
%
%    o clone_info: Method CloneQuantizeInfo returns a duplicate of the given
%      quantize info, or if image info is NULL a new one.
%
%    o quantize_info: a structure of type info.
%
*/
MagickExport QuantizeInfo *CloneQuantizeInfo(const QuantizeInfo *quantize_info)
{
  QuantizeInfo
    *clone_info;

  clone_info=(QuantizeInfo *) AcquireMagickMemory(sizeof(*clone_info));
  if (clone_info == (QuantizeInfo *) NULL)
    ThrowFatalException(ResourceLimitFatalError,"MemoryAllocationFailed");
  GetQuantizeInfo(clone_info);
  if (quantize_info == (QuantizeInfo *) NULL)
    return(clone_info);
  clone_info->number_colors=quantize_info->number_colors;
  clone_info->tree_depth=quantize_info->tree_depth;
  clone_info->dither=quantize_info->dither;
  clone_info->dither_method=quantize_info->dither_method;
  clone_info->colorspace=quantize_info->colorspace;
  clone_info->measure_error=quantize_info->measure_error;
  return(clone_info);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   C l o s e s t C o l o r                                                   %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  ClosestColor() traverses the color cube tree at a particular node and
%  determines which colormap entry best represents the input color.
%
%  The format of the ClosestColor method is:
%
%      void ClosestColor(const Image *image,QCubeInfo *cube_info,
%        const QNodeInfo *node_info)
%
%  A description of each parameter follows.
%
%    o image: the image.
%
%    o cube_info: A pointer to the Cube structure.
%
%    o node_info: the address of a structure of type QNodeInfo which points to a
%      node in the color cube tree that is to be pruned.
%
*/
static void ClosestColor(const Image *image,QCubeInfo *cube_info,
  const QNodeInfo *node_info)
{
  size_t
    number_children;

  ssize_t
    i;

  /*
    Traverse any children.
  */
  number_children=cube_info->associate_alpha == MagickFalse ? 8UL : 16UL;
  for (i=0; i < (ssize_t) number_children; i++)
    if (node_info->child[i] != (QNodeInfo *) NULL)
      ClosestColor(image,cube_info,node_info->child[i]);
  if (node_info->number_unique != 0)
    {
      MagickRealType
        alpha,
        beta,
        distance,
        pixel;

      DoublePixelPacket
        *magick_restrict q;

      PixelPacket
        *magick_restrict p;

      /*
        Determine if this color is "closest".
      */
      p=image->colormap+node_info->color_number;
      q=(&cube_info->target);
      alpha=1.0;
      beta=1.0;
      if (cube_info->associate_alpha != MagickFalse)
        {
          alpha=(MagickRealType) QuantumScale*GetPixelAlpha(p);
          beta=(MagickRealType) QuantumScale*GetPixelAlpha(q);
        }
      pixel=alpha*GetPixelRed(p)-beta*GetPixelRed(q);
      distance=pixel*pixel;
      if (distance <= cube_info->distance)
        {
          pixel=(MagickRealType) alpha*GetPixelGreen(p)-beta*GetPixelGreen(q);
          distance+=pixel*pixel;
          if (distance <= cube_info->distance)
            {
              pixel=(MagickRealType) alpha*GetPixelBlue(p)-beta*GetPixelBlue(q);
              distance+=pixel*pixel;
              if (distance <= cube_info->distance)
                {
                  if (cube_info->associate_alpha != MagickFalse)
                    {
                      pixel=(MagickRealType) GetPixelAlpha(p)-GetPixelAlpha(q);
                      distance+=pixel*pixel;
                    }
                  if (distance <= cube_info->distance)
                    {
                      cube_info->distance=distance;
                      cube_info->color_number=node_info->color_number;
                    }
                }
            }
        }
    }
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   C o m p r e s s I m a g e C o l o r m a p                                 %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  CompressImageColormap() compresses an image colormap by removing any
%  duplicate or unused color entries.
%
%  The format of the CompressImageColormap method is:
%
%      MagickBooleanType CompressImageColormap(Image *image)
%
%  A description of each parameter follows:
%
%    o image: the image.
%
*/
MagickExport MagickBooleanType CompressImageColormap(Image *image)
{
  QuantizeInfo
    quantize_info;

  assert(image != (Image *) NULL);
  assert(image->signature == MagickCoreSignature);
  if (IsEventLogging() != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",image->filename);
  if (IsPaletteImage(image,&image->exception) == MagickFalse)
    return(MagickFalse);
  GetQuantizeInfo(&quantize_info);
  quantize_info.number_colors=image->colors;
  quantize_info.tree_depth=MaxTreeDepth;
  return(QuantizeImage(&quantize_info,image));
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   D e f i n e I m a g e C o l o r m a p                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  DefineImageColormap() traverses the color cube tree and notes each colormap
%  entry.  A colormap entry is any node in the color cube tree where the
%  of unique colors is not zero.
%
%  The format of the DefineImageColormap method is:
%
%      void DefineImageColormap(Image *image,QCubeInfo *cube_info,
%        QNodeInfo *node_info)
%
%  A description of each parameter follows.
%
%    o image: the image.
%
%    o cube_info: A pointer to the Cube structure.
%
%    o node_info: the address of a structure of type QNodeInfo which points to a
%      node in the color cube tree that is to be pruned.
%
*/
static void DefineImageColormap(Image *image,QCubeInfo *cube_info,
  QNodeInfo *node_info)
{
  size_t
    number_children;

  ssize_t
    i;

  /*
    Traverse any children.
  */
  number_children=cube_info->associate_alpha == MagickFalse ? 8UL : 16UL;
  for (i=0; i < (ssize_t) number_children; i++)
    if (node_info->child[i] != (QNodeInfo *) NULL)
      DefineImageColormap(image,cube_info,node_info->child[i]);
  if (node_info->number_unique != 0)
    {
      MagickRealType
        alpha;

      PixelPacket
        *magick_restrict q;

      /*
        Colormap entry is defined by the mean color in this cube.
      */
      q=image->colormap+image->colors;
      alpha=(MagickRealType) ((MagickOffsetType) node_info->number_unique);
      alpha=MagickSafeReciprocal(alpha);
      if (cube_info->associate_alpha == MagickFalse)
        {
          SetPixelRed(q,ClampToQuantum(alpha*(MagickRealType) QuantumRange*
            node_info->total_color.red));
          SetPixelGreen(q,ClampToQuantum(alpha*(MagickRealType) QuantumRange*
            node_info->total_color.green));
          SetPixelBlue(q,ClampToQuantum(alpha*(MagickRealType) QuantumRange*
            node_info->total_color.blue));
          SetPixelOpacity(q,OpaqueOpacity);
        }
      else
        {
          MagickRealType
            opacity;

          opacity=(MagickRealType) (alpha*(MagickRealType) QuantumRange*
            node_info->total_color.opacity);
          SetPixelOpacity(q,ClampToQuantum(opacity));
          if (q->opacity == OpaqueOpacity)
            {
              SetPixelRed(q,ClampToQuantum(alpha*(MagickRealType)
                QuantumRange*node_info->total_color.red));
              SetPixelGreen(q,ClampToQuantum(alpha*(MagickRealType)
                QuantumRange*node_info->total_color.green));
              SetPixelBlue(q,ClampToQuantum(alpha*(MagickRealType)
                QuantumRange*node_info->total_color.blue));
            }
          else
            {
              double
                gamma;

              gamma=(double) (QuantumScale*((double) QuantumRange-(double)
                q->opacity));
              gamma=MagickSafeReciprocal(gamma);
              SetPixelRed(q,ClampToQuantum(alpha*gamma*(MagickRealType)
                QuantumRange*node_info->total_color.red));
              SetPixelGreen(q,ClampToQuantum(alpha*gamma*(MagickRealType)
                QuantumRange*node_info->total_color.green));
              SetPixelBlue(q,ClampToQuantum(alpha*gamma*(MagickRealType)
                QuantumRange*node_info->total_color.blue));
              if (node_info->number_unique > cube_info->transparent_pixels)
                {
                  cube_info->transparent_pixels=node_info->number_unique;
                  cube_info->transparent_index=(ssize_t) image->colors;
                }
            }
        }
      node_info->color_number=image->colors++;
    }
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   D e s t r o y Q C u b e I n f o                                           %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  DestroyQCubeInfo() deallocates memory associated with an image.
%
%  The format of the DestroyQCubeInfo method is:
%
%      DestroyQCubeInfo(QCubeInfo *cube_info)
%
%  A description of each parameter follows:
%
%    o cube_info: the address of a structure of type QCubeInfo.
%
*/
static void DestroyQCubeInfo(QCubeInfo *cube_info)
{
  QNodes
    *nodes;

  /*
    Release color cube tree storage.
  */
  do
  {
    nodes=cube_info->node_queue->next;
    cube_info->node_queue->nodes=(QNodeInfo *) RelinquishMagickMemory(
      cube_info->node_queue->nodes);
    cube_info->node_queue=(QNodes *) RelinquishMagickMemory(
      cube_info->node_queue);
    cube_info->node_queue=nodes;
  } while (cube_info->node_queue != (QNodes *) NULL);
  if (cube_info->memory_info != (MemoryInfo *) NULL)
    cube_info->memory_info=RelinquishVirtualMemory(cube_info->memory_info);
  cube_info->quantize_info=DestroyQuantizeInfo(cube_info->quantize_info);
  cube_info=(QCubeInfo *) RelinquishMagickMemory(cube_info);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   D e s t r o y Q u a n t i z e I n f o                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  DestroyQuantizeInfo() deallocates memory associated with an QuantizeInfo
%  structure.
%
%  The format of the DestroyQuantizeInfo method is:
%
%      QuantizeInfo *DestroyQuantizeInfo(QuantizeInfo *quantize_info)
%
%  A description of each parameter follows:
%
%    o quantize_info: Specifies a pointer to an QuantizeInfo structure.
%
*/
MagickExport QuantizeInfo *DestroyQuantizeInfo(QuantizeInfo *quantize_info)
{
  assert(quantize_info != (QuantizeInfo *) NULL);
  assert(quantize_info->signature == MagickCoreSignature);
  if (IsEventLogging() != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"...");
  quantize_info->signature=(~MagickCoreSignature);
  quantize_info=(QuantizeInfo *) RelinquishMagickMemory(quantize_info);
  return(quantize_info);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   D i t h e r I m a g e                                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  DitherImage() distributes the difference between an original image and
%  the corresponding color reduced algorithm to neighboring pixels using
%  serpentine-scan Floyd-Steinberg error diffusion. DitherImage returns
%  MagickTrue if the image is dithered otherwise MagickFalse.
%
%  The format of the DitherImage method is:
%
%      MagickBooleanType DitherImage(Image *image,QCubeInfo *cube_info)
%
%  A description of each parameter follows.
%
%    o image: the image.
%
%    o cube_info: A pointer to the Cube structure.
%
*/

static DoublePixelPacket **DestroyPixelTLS(DoublePixelPacket **pixels)
{
  ssize_t
    i;

  assert(pixels != (DoublePixelPacket **) NULL);
  for (i=0; i < (ssize_t) GetMagickResourceLimit(ThreadResource); i++)
    if (pixels[i] != (DoublePixelPacket *) NULL)
      pixels[i]=(DoublePixelPacket *) RelinquishMagickMemory(pixels[i]);
  pixels=(DoublePixelPacket **) RelinquishMagickMemory(pixels);
  return(pixels);
}

static DoublePixelPacket **AcquirePixelTLS(const size_t count)
{
  DoublePixelPacket
    **pixels;

  size_t
    number_threads;

  ssize_t
    i;

  number_threads=(size_t) GetMagickResourceLimit(ThreadResource);
  pixels=(DoublePixelPacket **) AcquireQuantumMemory(number_threads,
    sizeof(*pixels));
  if (pixels == (DoublePixelPacket **) NULL)
    return((DoublePixelPacket **) NULL);
  (void) memset(pixels,0,number_threads*sizeof(*pixels));
  for (i=0; i < (ssize_t) number_threads; i++)
  {
    pixels[i]=(DoublePixelPacket *) AcquireQuantumMemory(count,2*
      sizeof(**pixels));
    if (pixels[i] == (DoublePixelPacket *) NULL)
      return(DestroyPixelTLS(pixels));
  }
  return(pixels);
}

static inline ssize_t CacheOffset(QCubeInfo *cube_info,
  const DoublePixelPacket *pixel)
{
#define RedShift(pixel) (((pixel) >> CacheShift) << (0*(8-CacheShift)))
#define GreenShift(pixel) (((pixel) >> CacheShift) << (1*(8-CacheShift)))
#define BlueShift(pixel) (((pixel) >> CacheShift) << (2*(8-CacheShift)))
#define AlphaShift(pixel) (((pixel) >> CacheShift) << (3*(8-CacheShift)))

  ssize_t
    offset;

  offset=(ssize_t) (RedShift(ScaleQuantumToChar(ClampPixel(pixel->red))) |
    GreenShift(ScaleQuantumToChar(ClampPixel(pixel->green))) |
    BlueShift(ScaleQuantumToChar(ClampPixel(pixel->blue))));
  if (cube_info->associate_alpha != MagickFalse)
    offset|=AlphaShift(ScaleQuantumToChar(ClampPixel(pixel->opacity)));
  return(offset);
}

static MagickBooleanType FloydSteinbergDither(Image *image,QCubeInfo *cube_info)
{
#define DitherImageTag  "Dither/Image"

  CacheView
    *image_view;

  DoublePixelPacket
    **pixels;

  ExceptionInfo
    *exception;

  MagickBooleanType
    status;

  ssize_t
    y;

  /*
    Distribute quantization error using Floyd-Steinberg.
  */
  pixels=AcquirePixelTLS(image->columns);
  if (pixels == (DoublePixelPacket **) NULL)
    return(MagickFalse);
  exception=(&image->exception);
  status=MagickTrue;
  image_view=AcquireAuthenticCacheView(image,exception);
  for (y=0; y < (ssize_t) image->rows; y++)
  {
    const int
      id = GetOpenMPThreadId();

    DoublePixelPacket
      *current,
      *previous;

    IndexPacket
      *magick_restrict indexes;

    PixelPacket
      *magick_restrict q;

    QCubeInfo
      cube;

    size_t
      index;

    ssize_t
      x,
      v;

    if (status == MagickFalse)
      continue;
    q=GetCacheViewAuthenticPixels(image_view,0,y,image->columns,1,exception);
    if (q == (PixelPacket *) NULL)
      {
        status=MagickFalse;
        continue;
      }
    indexes=GetCacheViewAuthenticIndexQueue(image_view);
    cube=(*cube_info);
    current=pixels[id]+(y & 0x01)*image->columns;
    previous=pixels[id]+((y+1) & 0x01)*image->columns;
    v=(ssize_t) ((y & 0x01) ? -1 : 1);
    for (x=0; x < (ssize_t) image->columns; x++)
    {
      DoublePixelPacket
        color,
        pixel;

      ssize_t
        i;

      ssize_t
        u;

      u=(y & 0x01) ? (ssize_t) image->columns-1-x : x;
      AssociateAlphaPixel(&cube,q+u,&pixel);
      if (x > 0)
        {
          pixel.red+=7.0*cube_info->diffusion*current[u-v].red/16;
          pixel.green+=7.0*cube_info->diffusion*current[u-v].green/16;
          pixel.blue+=7.0*cube_info->diffusion*current[u-v].blue/16;
          if (cube.associate_alpha != MagickFalse)
            pixel.opacity+=7.0*cube_info->diffusion*current[u-v].opacity/16;
        }
      if (y > 0)
        {
          if (x < (ssize_t) (image->columns-1))
            {
              pixel.red+=cube_info->diffusion*previous[u+v].red/16;
              pixel.green+=cube_info->diffusion*previous[u+v].green/16;
              pixel.blue+=cube_info->diffusion*previous[u+v].blue/16;
              if (cube.associate_alpha != MagickFalse)
                pixel.opacity+=cube_info->diffusion*previous[u+v].opacity/16;
            }
          pixel.red+=5.0*cube_info->diffusion*previous[u].red/16;
          pixel.green+=5.0*cube_info->diffusion*previous[u].green/16;
          pixel.blue+=5.0*cube_info->diffusion*previous[u].blue/16;
          if (cube.associate_alpha != MagickFalse)
            pixel.opacity+=5.0*cube_info->diffusion*previous[u].opacity/16;
          if (x > 0)
            {
              pixel.red+=3.0*cube_info->diffusion*previous[u-v].red/16;
              pixel.green+=3.0*cube_info->diffusion*previous[u-v].green/16;
              pixel.blue+=3.0*cube_info->diffusion*previous[u-v].blue/16;
              if (cube.associate_alpha != MagickFalse)
                pixel.opacity+=3.0*cube_info->diffusion*
                  previous[u-v].opacity/16;
            }
        }
      pixel.red=(MagickRealType) ClampPixel(pixel.red);
      pixel.green=(MagickRealType) ClampPixel(pixel.green);
      pixel.blue=(MagickRealType) ClampPixel(pixel.blue);
      if (cube.associate_alpha != MagickFalse)
        pixel.opacity=(MagickRealType) ClampPixel(pixel.opacity);
      i=CacheOffset(&cube,&pixel);
      if (cube.cache[i] < 0)
        {
          QNodeInfo
            *node_info;

          size_t
            id;

          /*
            Identify the deepest node containing the pixel's color.
          */
          node_info=cube.root;
          for (index=MaxTreeDepth-1; (ssize_t) index > 0; index--)
          {
            id=ColorToQNodeId(&cube,&pixel,index);
            if (node_info->child[id] == (QNodeInfo *) NULL)
              break;
            node_info=node_info->child[id];
          }
          /*
            Find closest color among siblings and their children.
          */
          cube.target=pixel;
          cube.distance=(MagickRealType) (4.0*((MagickRealType) QuantumRange+
            1.0)*((MagickRealType) QuantumRange+1.0)+1.0);
          ClosestColor(image,&cube,node_info->parent);
          cube.cache[i]=(ssize_t) cube.color_number;
        }
      /*
        Assign pixel to closest colormap entry.
      */
      index=(size_t) cube.cache[i];
      if (image->storage_class == PseudoClass)
        SetPixelIndex(indexes+u,index);
      if (cube.quantize_info->measure_error == MagickFalse)
        {
          SetPixelRgb(q+u,image->colormap+index);
          if (cube.associate_alpha != MagickFalse)
            SetPixelOpacity(q+u,image->colormap[index].opacity);
        }
      if (SyncCacheViewAuthenticPixels(image_view,exception) == MagickFalse)
        status=MagickFalse;
      /*
        Store the error.
      */
      AssociateAlphaPixel(&cube,image->colormap+index,&color);
      current[u].red=pixel.red-color.red;
      current[u].green=pixel.green-color.green;
      current[u].blue=pixel.blue-color.blue;
      if (cube.associate_alpha != MagickFalse)
        current[u].opacity=pixel.opacity-color.opacity;
      if (image->progress_monitor != (MagickProgressMonitor) NULL)
        {
          MagickBooleanType
            proceed;

          proceed=SetImageProgress(image,DitherImageTag,(MagickOffsetType) y,
            image->rows);
          if (proceed == MagickFalse)
            status=MagickFalse;
        }
    }
  }
  image_view=DestroyCacheView(image_view);
  pixels=DestroyPixelTLS(pixels);
  return(MagickTrue);
}

static MagickBooleanType
  RiemersmaDither(Image *,CacheView *,QCubeInfo *,const unsigned int);

static MagickBooleanType Riemersma(Image *image,CacheView *image_view,
  QCubeInfo *cube_info,const size_t level,const unsigned int direction)
{
  MagickStatusType
    status;

  status=MagickTrue;
  if (level == 1)
    switch (direction)
    {
      case WestGravity:
      {
        status=RiemersmaDither(image,image_view,cube_info,EastGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,SouthGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,WestGravity);
        break;
      }
      case EastGravity:
      {
        status=RiemersmaDither(image,image_view,cube_info,WestGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,NorthGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,EastGravity);
        break;
      }
      case NorthGravity:
      {
        status=RiemersmaDither(image,image_view,cube_info,SouthGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,EastGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,NorthGravity);
        break;
      }
      case SouthGravity:
      {
        status=RiemersmaDither(image,image_view,cube_info,NorthGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,WestGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,SouthGravity);
        break;
      }
      default:
        break;
    }
  else
    switch (direction)
    {
      case WestGravity:
      {
        status=Riemersma(image,image_view,cube_info,level-1,NorthGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,EastGravity);
        if (status != MagickFalse)
          status=Riemersma(image,image_view,cube_info,level-1,WestGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,SouthGravity);
        if (status != MagickFalse)
          status=Riemersma(image,image_view,cube_info,level-1,WestGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,WestGravity);
        if (status != MagickFalse)
          status=Riemersma(image,image_view,cube_info,level-1,SouthGravity);
        break;
      }
      case EastGravity:
      {
        status=Riemersma(image,image_view,cube_info,level-1,SouthGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,WestGravity);
        if (status != MagickFalse)
          status=Riemersma(image,image_view,cube_info,level-1,EastGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,NorthGravity);
        if (status != MagickFalse)
          status=Riemersma(image,image_view,cube_info,level-1,EastGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,EastGravity);
        if (status != MagickFalse)
          status=Riemersma(image,image_view,cube_info,level-1,NorthGravity);
        break;
      }
      case NorthGravity:
      {
        status=Riemersma(image,image_view,cube_info,level-1,WestGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,SouthGravity);
        if (status != MagickFalse)
          status=Riemersma(image,image_view,cube_info,level-1,NorthGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,EastGravity);
        if (status != MagickFalse)
          status=Riemersma(image,image_view,cube_info,level-1,NorthGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,NorthGravity);
        if (status != MagickFalse)
          status=Riemersma(image,image_view,cube_info,level-1,EastGravity);
        break;
      }
      case SouthGravity:
      {
        status=Riemersma(image,image_view,cube_info,level-1,EastGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,NorthGravity);
        if (status != MagickFalse)
          status=Riemersma(image,image_view,cube_info,level-1,SouthGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,WestGravity);
        if (status != MagickFalse)
          status=Riemersma(image,image_view,cube_info,level-1,SouthGravity);
        if (status != MagickFalse)
          status=RiemersmaDither(image,image_view,cube_info,SouthGravity);
        if (status != MagickFalse)
          status=Riemersma(image,image_view,cube_info,level-1,WestGravity);
        break;
      }
      default:
        break;
    }
  return(status != 0 ? MagickTrue : MagickFalse);
}

static MagickBooleanType RiemersmaDither(Image *image,CacheView *image_view,
  QCubeInfo *cube_info,const unsigned int direction)
{
#define DitherImageTag  "Dither/Image"

  DoublePixelPacket
    color,
    pixel;

  MagickBooleanType
    proceed;

  QCubeInfo
    *p;

  size_t
    index;

  p=cube_info;
  if ((p->x >= 0) && (p->x < (ssize_t) image->columns) &&
      (p->y >= 0) && (p->y < (ssize_t) image->rows))
    {
      ExceptionInfo
        *exception;

      IndexPacket
        *magick_restrict indexes;

      PixelPacket
        *magick_restrict q;

      ssize_t
        i;

      /*
        Distribute error.
      */
      exception=(&image->exception);
      q=GetCacheViewAuthenticPixels(image_view,p->x,p->y,1,1,exception);
      if (q == (PixelPacket *) NULL)
        return(MagickFalse);
      indexes=GetCacheViewAuthenticIndexQueue(image_view);
      AssociateAlphaPixel(cube_info,q,&pixel);
      for (i=0; i < ErrorQueueLength; i++)
      {
        pixel.red+=ErrorRelativeWeight*cube_info->diffusion*p->weights[i]*
          p->error[i].red;
        pixel.green+=ErrorRelativeWeight*cube_info->diffusion*p->weights[i]*
          p->error[i].green;
        pixel.blue+=ErrorRelativeWeight*cube_info->diffusion*p->weights[i]*
          p->error[i].blue;
        if (cube_info->associate_alpha != MagickFalse)
          pixel.opacity+=ErrorRelativeWeight*cube_info->diffusion*p->weights[i]*
            p->error[i].opacity;
      }
      pixel.red=(MagickRealType) ClampPixel(pixel.red);
      pixel.green=(MagickRealType) ClampPixel(pixel.green);
      pixel.blue=(MagickRealType) ClampPixel(pixel.blue);
      if (cube_info->associate_alpha != MagickFalse)
        pixel.opacity=(MagickRealType) ClampPixel(pixel.opacity);
      i=CacheOffset(cube_info,&pixel);
      if (p->cache[i] < 0)
        {
          QNodeInfo
            *node_info;

          size_t
            id;

          /*
            Identify the deepest node containing the pixel's color.
          */
          node_info=p->root;
          for (index=MaxTreeDepth-1; (ssize_t) index > 0; index--)
          {
            id=ColorToQNodeId(cube_info,&pixel,index);
            if (node_info->child[id] == (QNodeInfo *) NULL)
              break;
            node_info=node_info->child[id];
          }
          /*
            Find closest color among siblings and their children.
          */
          p->target=pixel;
          p->distance=(MagickRealType) (4.0*((MagickRealType) QuantumRange+
           1.0)*((MagickRealType) QuantumRange+1.0)+1.0);
          ClosestColor(image,p,node_info->parent);
          p->cache[i]=(ssize_t) p->color_number;
        }
      /*
        Assign pixel to closest colormap entry.
      */
      index=(size_t) (1*p->cache[i]);
      if (image->storage_class == PseudoClass)
        *indexes=(IndexPacket) index;
      if (cube_info->quantize_info->measure_error == MagickFalse)
        {
          SetPixelRgb(q,image->colormap+index);
          if (cube_info->associate_alpha != MagickFalse)
            SetPixelOpacity(q,image->colormap[index].opacity);
        }
      if (SyncCacheViewAuthenticPixels(image_view,exception) == MagickFalse)
        return(MagickFalse);
      /*
        Propagate the error as the last entry of the error queue.
      */
      (void) memmove(p->error,p->error+1,(ErrorQueueLength-1)*
        sizeof(p->error[0]));
      AssociateAlphaPixel(cube_info,image->colormap+index,&color);
      p->error[ErrorQueueLength-1].red=pixel.red-color.red;
      p->error[ErrorQueueLength-1].green=pixel.green-color.green;
      p->error[ErrorQueueLength-1].blue=pixel.blue-color.blue;
      if (cube_info->associate_alpha != MagickFalse)
        p->error[ErrorQueueLength-1].opacity=pixel.opacity-color.opacity;
      proceed=SetImageProgress(image,DitherImageTag,p->offset,p->span);
      if (proceed == MagickFalse)
        return(MagickFalse);
      p->offset++;
    }
  switch (direction)
  {
    case WestGravity: p->x--; break;
    case EastGravity: p->x++; break;
    case NorthGravity: p->y--; break;
    case SouthGravity: p->y++; break;
  }
  return(MagickTrue);
}

static MagickBooleanType DitherImage(Image *image,QCubeInfo *cube_info)
{
  CacheView
    *image_view;

  const char
    *artifact;

  MagickBooleanType
    status;

  size_t
    extent,
    level;

  artifact=GetImageArtifact(image,"dither:diffusion-amount");
  if (artifact != (const char *) NULL)
    cube_info->diffusion=StringToDoubleInterval(artifact,1.0);
  if (cube_info->quantize_info->dither_method != RiemersmaDitherMethod)
    return(FloydSteinbergDither(image,cube_info));
  /*
    Distribute quantization error along a Hilbert curve.
  */
  (void) memset(cube_info->error,0,ErrorQueueLength*sizeof(*cube_info->error));
  cube_info->x=0;
  cube_info->y=0;
  extent=MagickMax(image->columns,image->rows);
  level=(size_t) log2((double) extent);
  if ((1UL << level) < extent)
    level++;
  cube_info->offset=0;
  cube_info->span=(MagickSizeType) image->columns*image->rows;
  image_view=AcquireAuthenticCacheView(image,&image->exception);
  status=MagickTrue;
  if (level > 0)
    status=Riemersma(image,image_view,cube_info,level,NorthGravity);
  if (status != MagickFalse)
    status=RiemersmaDither(image,image_view,cube_info,ForgetGravity);
  image_view=DestroyCacheView(image_view);
  return(status);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   G e t Q C u b e I n f o                                                   %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  GetQCubeInfo() initialize the Cube data structure.
%
%  The format of the GetQCubeInfo method is:
%
%      QCubeInfo GetQCubeInfo(const QuantizeInfo *quantize_info,
%        const size_t depth,const size_t maximum_colors)
%
%  A description of each parameter follows.
%
%    o quantize_info: Specifies a pointer to an QuantizeInfo structure.
%
%    o depth: Normally, this integer value is zero or one.  A zero or
%      one tells Quantize to choose a optimal tree depth of Log4(number_colors).
%      A tree of this depth generally allows the best representation of the
%      reference image with the least amount of memory and the fastest
%      computational speed.  In some cases, such as an image with low color
%      dispersion (a few number of colors), a value other than
%      Log4(number_colors) is required.  To expand the color tree completely,
%      use a value of 8.
%
%    o maximum_colors: maximum colors.
%
*/
static QCubeInfo *GetQCubeInfo(const QuantizeInfo *quantize_info,
  const size_t depth,const size_t maximum_colors)
{
  MagickRealType
    weight;

  QCubeInfo
    *cube_info;

  size_t
    length;

  ssize_t
    i;

  /*
    Initialize tree to describe color cube_info.
  */
  cube_info=(QCubeInfo *) AcquireMagickMemory(sizeof(*cube_info));
  if (cube_info == (QCubeInfo *) NULL)
    return((QCubeInfo *) NULL);
  (void) memset(cube_info,0,sizeof(*cube_info));
  cube_info->depth=depth;
  if (cube_info->depth > MaxTreeDepth)
    cube_info->depth=MaxTreeDepth;
  if (cube_info->depth < 2)
    cube_info->depth=2;
  cube_info->maximum_colors=maximum_colors;
  /*
    Initialize root node.
  */
  cube_info->root=GetQNodeInfo(cube_info,0,0,(QNodeInfo *) NULL);
  if (cube_info->root == (QNodeInfo *) NULL)
    return((QCubeInfo *) NULL);
  cube_info->root->parent=cube_info->root;
  cube_info->quantize_info=CloneQuantizeInfo(quantize_info);
  if (cube_info->quantize_info->dither == MagickFalse)
    return(cube_info);
  /*
    Initialize dither resources.
  */
  length=(size_t) (1UL << (4*(8-CacheShift)));
  cube_info->memory_info=AcquireVirtualMemory(length,sizeof(*cube_info->cache));
  if (cube_info->memory_info == (MemoryInfo *) NULL)
    return((QCubeInfo *) NULL);
  cube_info->cache=(ssize_t *) GetVirtualMemoryBlob(cube_info->memory_info);
  /*
    Initialize color cache.
  */
  (void) memset(cube_info->cache,(-1),sizeof(*cube_info->cache)*length);
  /*
    Distribute weights along a curve of exponential decay.
  */
  weight=1.0;
  for (i=0; i < ErrorQueueLength; i++)
  {
    cube_info->weights[i]=MagickSafeReciprocal(weight);
    weight*=exp(log(1.0/ErrorRelativeWeight)/(ErrorQueueLength-1.0));
  }
  cube_info->diffusion=1.0;
  return(cube_info);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   G e t N o d e I n f o                                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  GetQNodeInfo() allocates memory for a new node in the color cube tree and
%  presets all fields to zero.
%
%  The format of the GetQNodeInfo method is:
%
%      QNodeInfo *GetQNodeInfo(QCubeInfo *cube_info,const size_t id,
%        const size_t level,QNodeInfo *parent)
%
%  A description of each parameter follows.
%
%    o node: The GetQNodeInfo method returns a pointer to a queue of nodes.
%
%    o id: Specifies the child number of the node.
%
%    o level: Specifies the level in the storage_class the node resides.
%
*/
static QNodeInfo *GetQNodeInfo(QCubeInfo *cube_info,const size_t id,
  const size_t level,QNodeInfo *parent)
{
  QNodeInfo
    *node_info;

  if (cube_info->free_nodes == 0)
    {
      QNodes
        *nodes;

      /*
        Allocate a new queue of nodes.
      */
      nodes=(QNodes *) AcquireMagickMemory(sizeof(*nodes));
      if (nodes == (QNodes *) NULL)
        return((QNodeInfo *) NULL);
      nodes->nodes=(QNodeInfo *) AcquireQuantumMemory(QNodesInAList,
        sizeof(*nodes->nodes));
      if (nodes->nodes == (QNodeInfo *) NULL)
        return((QNodeInfo *) NULL);
      nodes->next=cube_info->node_queue;
      cube_info->node_queue=nodes;
      cube_info->next_node=nodes->nodes;
      cube_info->free_nodes=QNodesInAList;
    }
  cube_info->nodes++;
  cube_info->free_nodes--;
  node_info=cube_info->next_node++;
  (void) memset(node_info,0,sizeof(*node_info));
  node_info->parent=parent;
  node_info->id=id;
  node_info->level=level;
  return(node_info);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%  G e t I m a g e Q u a n t i z e E r r o r                                  %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  GetImageQuantizeError() measures the difference between the original
%  and quantized images.  This difference is the total quantization error.
%  The error is computed by summing over all pixels in an image the distance
%  squared in RGB space between each reference pixel value and its quantized
%  value.  These values are computed:
%
%    o mean_error_per_pixel:  This value is the mean error for any single
%      pixel in the image.
%
%    o normalized_mean_square_error:  This value is the normalized mean
%      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.
%
%    o normalized_maximum_square_error:  This value is the normalized
%      maximum 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 your image.
%
%  The format of the GetImageQuantizeError method is:
%
%      MagickBooleanType GetImageQuantizeError(Image *image)
%
%  A description of each parameter follows.
%
%    o image: the image.
%
*/
MagickExport MagickBooleanType GetImageQuantizeError(Image *image)
{
  CacheView
    *image_view;

  ExceptionInfo
    *exception;

  IndexPacket
    *indexes;

  MagickRealType
    alpha,
    area,
    beta,
    distance,
    gamma,
    maximum_error,
    mean_error,
    mean_error_per_pixel;

  ssize_t
    index,
    y;

  assert(image != (Image *) NULL);
  assert(image->signature == MagickCoreSignature);
  if (IsEventLogging() != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",image->filename);
  image->total_colors=GetNumberColors(image,(FILE *) NULL,&image->exception);
  (void) memset(&image->error,0,sizeof(image->error));
  if (image->storage_class == DirectClass)
    return(MagickTrue);
  alpha=1.0;
  beta=1.0;
  area=3.0*image->columns*image->rows;
  maximum_error=0.0;
  mean_error_per_pixel=0.0;
  mean_error=0.0;
  exception=(&image->exception);
  image_view=AcquireVirtualCacheView(image,exception);
  for (y=0; y < (ssize_t) image->rows; y++)
  {
    const PixelPacket
      *magick_restrict p;

    ssize_t
      x;

    p=GetCacheViewVirtualPixels(image_view,0,y,image->columns,1,exception);
    if (p == (const PixelPacket *) NULL)
      break;
    indexes=GetCacheViewAuthenticIndexQueue(image_view);
    for (x=0; x < (ssize_t) image->columns; x++)
    {
      index=(ssize_t) GetPixelIndex(indexes+x);
      if (image->matte != MagickFalse)
        {
          alpha=QuantumScale*(MagickRealType) GetPixelAlpha(p);
          beta=QuantumScale*((MagickRealType) QuantumRange-
            (MagickRealType) image->colormap[index].opacity);
        }
      distance=fabs((double) (alpha*(MagickRealType) GetPixelRed(p)-beta*
        (MagickRealType) image->colormap[index].red));
      mean_error_per_pixel+=distance;
      mean_error+=distance*distance;
      if (distance > maximum_error)
        maximum_error=distance;
      distance=fabs((double) (alpha*(MagickRealType) GetPixelGreen(p)-beta*
        (MagickRealType) image->colormap[index].green));
      mean_error_per_pixel+=distance;
      mean_error+=distance*distance;
      if (distance > maximum_error)
        maximum_error=distance;
      distance=fabs((double) (alpha*(MagickRealType) GetPixelBlue(p)-beta*
        (MagickRealType) image->colormap[index].blue));
      mean_error_per_pixel+=distance;
      mean_error+=distance*distance;
      if (distance > maximum_error)
        maximum_error=distance;
      p++;
    }
  }
  image_view=DestroyCacheView(image_view);
  gamma=MagickSafeReciprocal(area);
  image->error.mean_error_per_pixel=gamma*mean_error_per_pixel;
  image->error.normalized_mean_error=gamma*QuantumScale*QuantumScale*mean_error;
  image->error.normalized_maximum_error=QuantumScale*maximum_error;
  return(MagickTrue);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   G e t Q u a n t i z e I n f o                                             %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  GetQuantizeInfo() initializes the QuantizeInfo structure.
%
%  The format of the GetQuantizeInfo method is:
%
%      GetQuantizeInfo(QuantizeInfo *quantize_info)
%
%  A description of each parameter follows:
%
%    o quantize_info: Specifies a pointer to a QuantizeInfo structure.
%
*/
MagickExport void GetQuantizeInfo(QuantizeInfo *quantize_info)
{
  assert(quantize_info != (QuantizeInfo *) NULL);
  if (IsEventLogging() != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"...");
  (void) memset(quantize_info,0,sizeof(*quantize_info));
  quantize_info->number_colors=256;
  quantize_info->dither=MagickTrue;
  quantize_info->dither_method=RiemersmaDitherMethod;
  quantize_info->colorspace=UndefinedColorspace;
  quantize_info->measure_error=MagickFalse;
  quantize_info->signature=MagickCoreSignature;
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%     P o s t e r i z e I m a g e C h a n n e l                               %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  PosterizeImage() reduces the image to a limited number of colors for a
%  "poster" effect.
%
%  The format of the PosterizeImage method is:
%
%      MagickBooleanType PosterizeImage(Image *image,const size_t levels,
%        const MagickBooleanType dither)
%      MagickBooleanType PosterizeImageChannel(Image *image,
%        const ChannelType channel,const size_t levels,
%        const MagickBooleanType dither)
%
%  A description of each parameter follows:
%
%    o image: Specifies a pointer to an Image structure.
%
%    o levels: Number of color levels allowed in each channel.  Very low values
%      (2, 3, or 4) have the most visible effect.
%
%    o dither: Set this integer value to something other than zero to dither
%      the mapped image.
%
*/

MagickExport MagickBooleanType PosterizeImage(Image *image,const size_t levels,
  const MagickBooleanType dither)
{
  MagickBooleanType
    status;

  status=PosterizeImageChannel(image,DefaultChannels,levels,dither);
  return(status);
}

static inline Quantum PosterizePixel(const Quantum pixel,const size_t levels)
{ 
  double
    step_size;

  size_t
    level_index;

  if (levels < 2)
    return(pixel);
  step_size=1.0/(levels-1.0);
  level_index=(step_size == 0.0) ? 0 : floor((double) pixel/step_size);
  return(ClampToQuantum(level_index*step_size));
}   

MagickExport MagickBooleanType PosterizeImageChannel(Image *image,
  const ChannelType channel,const size_t levels,const MagickBooleanType dither)
{
#define PosterizeImageTag  "Posterize/Image"

  CacheView
    *image_view;

  ExceptionInfo
    *exception;

  MagickBooleanType
    status;

  MagickOffsetType
    progress;

  QuantizeInfo
    *quantize_info;

  ssize_t
    i,
    y;

  assert(image != (Image *) NULL);
  assert(image->signature == MagickCoreSignature);
  if (IsEventLogging() != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",image->filename);
  if (image->storage_class == PseudoClass)
#if defined(MAGICKCORE_OPENMP_SUPPORT)
    #pragma omp parallel for schedule(static) shared(progress,status) \
      magick_number_threads(image,image,image->colors,1)
#endif
    for (i=0; i < (ssize_t) image->colors; i++)
    {
      /*
        Posterize colormap.
      */
      if ((channel & RedChannel) != 0)
        image->colormap[i].red=(MagickRealType)
          PosterizePixel(image->colormap[i].red,levels);
      if ((channel & GreenChannel) != 0)
        image->colormap[i].green=(MagickRealType)
          PosterizePixel(image->colormap[i].green,levels);
      if ((channel & BlueChannel) != 0)
        image->colormap[i].blue=(MagickRealType)
          PosterizePixel(image->colormap[i].blue,levels);
      if ((channel & OpacityChannel) != 0)
        image->colormap[i].opacity=(MagickRealType)
          PosterizePixel(image->colormap[i].opacity,levels);
    }
  /*
    Posterize image.
  */
  status=MagickTrue;
  progress=0;
  exception=(&image->exception);
  image_view=AcquireAuthenticCacheView(image,exception);
#if defined(MAGICKCORE_OPENMP_SUPPORT)
  #pragma omp parallel for schedule(static) shared(progress,status) \
    magick_number_threads(image,image,image->rows,1)
#endif
  for (y=0; y < (ssize_t) image->rows; y++)
  {
    IndexPacket
      *magick_restrict indexes;

    PixelPacket
      *magick_restrict q;

    ssize_t
      x;

    if (status == MagickFalse)
      continue;
    q=GetCacheViewAuthenticPixels(image_view,0,y,image->columns,1,exception);
    if (q == (PixelPacket *) NULL)
      {
        status=MagickFalse;
        continue;
      }
    indexes=GetCacheViewAuthenticIndexQueue(image_view);
    for (x=0; x < (ssize_t) image->columns; x++)
    {
      if ((channel & RedChannel) != 0)
        SetPixelRed(q,PosterizePixel(GetPixelRed(q),levels));
      if ((channel & GreenChannel) != 0)
        SetPixelGreen(q,PosterizePixel(GetPixelGreen(q),levels));
      if ((channel & BlueChannel) != 0)
        SetPixelBlue(q,PosterizePixel(GetPixelBlue(q),levels));
      if (((channel & OpacityChannel) != 0) &&
          (image->matte != MagickFalse))
        SetPixelOpacity(q,PosterizePixel(GetPixelOpacity(q),levels));
      if (((channel & IndexChannel) != 0) &&
          (image->colorspace == CMYKColorspace))
        SetPixelIndex(indexes+x,PosterizePixel(GetPixelIndex(indexes+x),
          levels));
      q++;
    }
    if (SyncCacheViewAuthenticPixels(image_view,exception) == MagickFalse)
      status=MagickFalse;
    if (image->progress_monitor != (MagickProgressMonitor) NULL)
      {
        MagickBooleanType
          proceed;

#if defined(MAGICKCORE_OPENMP_SUPPORT)
        #pragma omp atomic
#endif
        progress++;
        proceed=SetImageProgress(image,PosterizeImageTag,progress,image->rows);
        if (proceed == MagickFalse)
          status=MagickFalse;
      }
  }
  image_view=DestroyCacheView(image_view);
  quantize_info=AcquireQuantizeInfo((ImageInfo *) NULL);
  quantize_info->number_colors=(size_t) MagickMin((ssize_t) levels*levels*
    levels,MaxColormapSize);
  quantize_info->dither=dither;
  status=QuantizeImage(quantize_info,image);
  quantize_info=DestroyQuantizeInfo(quantize_info);
  return(status);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   P r u n e C h i l d                                                       %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  PruneChild() deletes the given node and merges its statistics into its
%  parent.
%
%  The format of the PruneSubtree method is:
%
%      PruneChild(QCubeInfo *cube_info,const QNodeInfo *node_info)
%
%  A description of each parameter follows.
%
%    o cube_info: A pointer to the Cube structure.
%
%    o node_info: pointer to node in color cube tree that is to be pruned.
%
*/
static void PruneChild(QCubeInfo *cube_info,const QNodeInfo *node_info)
{
  QNodeInfo
    *parent;

  size_t
    number_children;

  ssize_t
    i;

  /*
    Traverse any children.
  */
  number_children=cube_info->associate_alpha == MagickFalse ? 8UL : 16UL;
  for (i=0; i < (ssize_t) number_children; i++)
    if (node_info->child[i] != (QNodeInfo *) NULL)
      PruneChild(cube_info,node_info->child[i]);
  if (cube_info->nodes > cube_info->maximum_colors)
    {
      /*
        Merge color statistics into parent.
      */
      parent=node_info->parent;
      parent->number_unique+=node_info->number_unique;
      parent->total_color.red+=node_info->total_color.red;
      parent->total_color.green+=node_info->total_color.green;
      parent->total_color.blue+=node_info->total_color.blue;
      parent->total_color.opacity+=node_info->total_color.opacity;
      parent->child[node_info->id]=(QNodeInfo *) NULL;
      cube_info->nodes--;
    }
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+  P r u n e L e v e l                                                        %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  PruneLevel() deletes all nodes at the bottom level of the color tree merging
%  their color statistics into their parent node.
%
%  The format of the PruneLevel method is:
%
%      PruneLevel(QCubeInfo *cube_info,const QNodeInfo *node_info)
%
%  A description of each parameter follows.
%
%    o cube_info: A pointer to the Cube structure.
%
%    o node_info: pointer to node in color cube tree that is to be pruned.
%
*/
static void PruneLevel(QCubeInfo *cube_info,const QNodeInfo *node_info)
{
  size_t
    number_children;

  ssize_t
    i;

  /*
    Traverse any children.
  */
  number_children=cube_info->associate_alpha == MagickFalse ? 8UL : 16UL;
  for (i=0; i < (ssize_t) number_children; i++)
    if (node_info->child[i] != (QNodeInfo *) NULL)
      PruneLevel(cube_info,node_info->child[i]);
  if (node_info->level == cube_info->depth)
    PruneChild(cube_info,node_info);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+  P r u n e T o C u b e D e p t h                                            %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  PruneToCubeDepth() deletes any nodes at a depth greater than
%  cube_info->depth while merging their color statistics into their parent
%  node.
%
%  The format of the PruneToCubeDepth method is:
%
%      PruneToCubeDepth(QCubeInfo *cube_info,const QNodeInfo *node_info)
%
%  A description of each parameter follows.
%
%    o cube_info: A pointer to the Cube structure.
%
%    o node_info: pointer to node in color cube tree that is to be pruned.
%
*/
static void PruneToCubeDepth(QCubeInfo *cube_info,const QNodeInfo *node_info)
{
  size_t
    number_children;

  ssize_t
    i;

  /*
    Traverse any children.
  */
  number_children=cube_info->associate_alpha == MagickFalse ? 8UL : 16UL;
  for (i=0; i < (ssize_t) number_children; i++)
    if (node_info->child[i] != (QNodeInfo *) NULL)
      PruneToCubeDepth(cube_info,node_info->child[i]);
  if (node_info->level > cube_info->depth)
    PruneChild(cube_info,node_info);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%  Q u a n t i z e I m a g e                                                  %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  QuantizeImage() analyzes the colors within a reference image and chooses a
%  fixed number of colors to represent the image.  The goal of the algorithm
%  is to minimize the color difference between the input and output image while
%  minimizing the processing time.
%
%  The format of the QuantizeImage method is:
%
%      MagickBooleanType QuantizeImage(const QuantizeInfo *quantize_info,
%        Image *image)
%
%  A description of each parameter follows:
%
%    o quantize_info: Specifies a pointer to an QuantizeInfo structure.
%
%    o image: the image.
%
*/
MagickExport MagickBooleanType QuantizeImage(const QuantizeInfo *quantize_info,
  Image *image)
{
  MagickBooleanType
    status;

  QCubeInfo
    *cube_info;

  size_t
    depth,
    maximum_colors;

  assert(quantize_info != (const QuantizeInfo *) NULL);
  assert(quantize_info->signature == MagickCoreSignature);
  assert(image != (Image *) NULL);
  assert(image->signature == MagickCoreSignature);
  if (IsEventLogging() != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",image->filename);
  maximum_colors=quantize_info->number_colors;
  if (maximum_colors == 0)
    maximum_colors=MaxColormapSize;
  if (maximum_colors > MaxColormapSize)
    maximum_colors=MaxColormapSize;
  if (image->matte == MagickFalse)
    {
      if (SetImageGray(image,&image->exception) != MagickFalse)
        (void) SetGrayscaleImage(image);
    }
  depth=quantize_info->tree_depth;
  if (depth == 0)
    {
      size_t
        colors;

      /*
        Depth of color tree is: Log4(colormap size)+2.
      */
      colors=maximum_colors;
      for (depth=1; colors != 0; depth++)
        colors>>=2;
      if ((quantize_info->dither != MagickFalse) && (depth > 2))
        depth--;
      if ((image->matte != MagickFalse) && (depth > 5))
        depth--;
      if (SetImageGray(image,&image->exception) != MagickFalse)
        depth=MaxTreeDepth;
    }
  /*
    Initialize color cube.
  */
  cube_info=GetQCubeInfo(quantize_info,depth,maximum_colors);
  if (cube_info == (QCubeInfo *) NULL)
    ThrowBinaryImageException(ResourceLimitError,"MemoryAllocationFailed",
      image->filename);
  status=ClassifyImageColors(cube_info,image,&image->exception);
  if (status != MagickFalse)
    {
      /*
        Reduce the number of colors in the image.
      */
      if (cube_info->colors > cube_info->maximum_colors)
        ReduceImageColors(image,cube_info);
      status=AssignImageColors(image,cube_info);
    }
  DestroyQCubeInfo(cube_info);
  return(status);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   Q u a n t i z e I m a g e s                                               %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  QuantizeImages() analyzes the colors within a set of reference images and
%  chooses a fixed number of colors to represent the set.  The goal of the
%  algorithm is to minimize the color difference between the input and output
%  images while minimizing the processing time.
%
%  The format of the QuantizeImages method is:
%
%      MagickBooleanType QuantizeImages(const QuantizeInfo *quantize_info,
%        Image *images)
%
%  A description of each parameter follows:
%
%    o quantize_info: Specifies a pointer to an QuantizeInfo structure.
%
%    o images: Specifies a pointer to a list of Image structures.
%
*/
MagickExport MagickBooleanType QuantizeImages(const QuantizeInfo *quantize_info,
  Image *images)
{
  Image
    *image;

  MagickBooleanType
    proceed,
    status;

  MagickProgressMonitor
    progress_monitor;

  QCubeInfo
    *cube_info;

  size_t
    depth,
    maximum_colors,
    number_images;

  ssize_t
    i;

  assert(quantize_info != (const QuantizeInfo *) NULL);
  assert(quantize_info->signature == MagickCoreSignature);
  assert(images != (Image *) NULL);
  assert(images->signature == MagickCoreSignature);
  if (IsEventLogging() != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",images->filename);
  if (GetNextImageInList(images) == (Image *) NULL)
    {
      /*
        Handle a single image with QuantizeImage.
      */
      status=QuantizeImage(quantize_info,images);
      return(status);
    }
  status=MagickFalse;
  maximum_colors=quantize_info->number_colors;
  if (maximum_colors == 0)
    maximum_colors=MaxColormapSize;
  if (maximum_colors > MaxColormapSize)
    maximum_colors=MaxColormapSize;
  depth=quantize_info->tree_depth;
  if (depth == 0)
    {
      size_t
        colors;

      /*
        Depth of color tree is: Log4(colormap size)+2.
      */
      colors=maximum_colors;
      for (depth=1; colors != 0; depth++)
        colors>>=2;
      if (quantize_info->dither != MagickFalse)
        depth--;
    }
  /*
    Initialize color cube.
  */
  cube_info=GetQCubeInfo(quantize_info,depth,maximum_colors);
  if (cube_info == (QCubeInfo *) NULL)
    {
      (void) ThrowMagickException(&images->exception,GetMagickModule(),
        ResourceLimitError,"MemoryAllocationFailed","`%s'",images->filename);
      return(MagickFalse);
    }
  number_images=GetImageListLength(images);
  image=images;
  for (i=0; image != (Image *) NULL; i++)
  {
    progress_monitor=SetImageProgressMonitor(image,(MagickProgressMonitor) NULL,
      image->client_data);
    status=ClassifyImageColors(cube_info,image,&image->exception);
    if (status == MagickFalse)
      break;
    (void) SetImageProgressMonitor(image,progress_monitor,image->client_data);
    proceed=SetImageProgress(image,AssignImageTag,(MagickOffsetType) i,
      number_images);
    if (proceed == MagickFalse)
      break;
    image=GetNextImageInList(image);
  }
  if (status != MagickFalse)
    {
      /*
        Reduce the number of colors in an image sequence.
      */
      ReduceImageColors(images,cube_info);
      image=images;
      for (i=0; image != (Image *) NULL; i++)
      {
        progress_monitor=SetImageProgressMonitor(image,(MagickProgressMonitor)
          NULL,image->client_data);
        status=AssignImageColors(image,cube_info);
        if (status == MagickFalse)
          break;
        (void) SetImageProgressMonitor(image,progress_monitor,
          image->client_data);
        proceed=SetImageProgress(image,AssignImageTag,(MagickOffsetType) i,
          number_images);
        if (proceed == MagickFalse)
          break;
        image=GetNextImageInList(image);
      }
    }
  DestroyQCubeInfo(cube_info);
  return(status);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   Q u a n t i z e E r r o r F l a t t e n                                   %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  QuantizeErrorFlatten() traverses the color cube and flattens the quantization
%  error into a sorted 1D array.  This accelerates the color reduction process.
%
%  Contributed by Yoya.
%
%  The format of the QuantizeErrorFlatten method is:
%
%      size_t QuantizeErrorFlatten(const QCubeInfo *cube_info,
%        const QNodeInfo *node_info,const ssize_t offset,
%        MagickRealType *quantize_error)
%
%  A description of each parameter follows.
%
%    o cube_info: A pointer to the Cube structure.
%
%    o node_info: pointer to node in color cube tree that is current pointer.
%
%    o offset: quantize error offset.
%
%    o quantize_error: the quantization error vector.
%
*/
static size_t QuantizeErrorFlatten(const QCubeInfo *cube_info,
  const QNodeInfo *node_info,const ssize_t offset,
  MagickRealType *quantize_error)
{
  size_t
    n,
    number_children;

  ssize_t
    i;

  if (offset >= (ssize_t) cube_info->nodes)
    return(0);
  quantize_error[offset]=node_info->quantize_error;
  n=1;
  number_children=cube_info->associate_alpha == MagickFalse ? 8UL : 16UL;
  for (i=0; i < (ssize_t) number_children ; i++)
    if (node_info->child[i] != (QNodeInfo *) NULL)
      n+=QuantizeErrorFlatten(cube_info,node_info->child[i],offset+n,
        quantize_error);
  return(n);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   R e d u c e                                                               %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Reduce() traverses the color cube tree and prunes any node whose
%  quantization error falls below a particular threshold.
%
%  The format of the Reduce method is:
%
%      Reduce(QCubeInfo *cube_info,const QNodeInfo *node_info)
%
%  A description of each parameter follows.
%
%    o cube_info: A pointer to the Cube structure.
%
%    o node_info: pointer to node in color cube tree that is to be pruned.
%
*/
static void Reduce(QCubeInfo *cube_info,const QNodeInfo *node_info)
{
  size_t
    number_children;

  ssize_t
    i;

  /*
    Traverse any children.
  */
  number_children=cube_info->associate_alpha == MagickFalse ? 8UL : 16UL;
  for (i=0; i < (ssize_t) number_children; i++)
    if (node_info->child[i] != (QNodeInfo *) NULL)
      Reduce(cube_info,node_info->child[i]);
  if (node_info->quantize_error <= cube_info->pruning_threshold)
    PruneChild(cube_info,node_info);
  else
    {
      /*
        Find minimum pruning threshold.
      */
      if (node_info->number_unique > 0)
        cube_info->colors++;
      if (node_info->quantize_error < cube_info->next_threshold)
        cube_info->next_threshold=node_info->quantize_error;
    }
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   R e d u c e I m a g e C o l o r s                                         %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  ReduceImageColors() 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.
%  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.
%
%  The format of the ReduceImageColors method is:
%
%      ReduceImageColors(const Image *image,QCubeInfo *cube_info)
%
%  A description of each parameter follows.
%
%    o image: the image.
%
%    o cube_info: A pointer to the Cube structure.
%
*/

static int MagickRealTypeCompare(const void *error_p,const void *error_q)
{
  MagickRealType
    *p,
    *q;

  p=(MagickRealType *) error_p;
  q=(MagickRealType *) error_q;
  if (*p > *q)
    return(1);
  if (fabs((double) (*q-*p)) <= MagickEpsilon)
    return(0);
  return(-1);
}

static void ReduceImageColors(const Image *image,QCubeInfo *cube_info)
{
#define ReduceImageTag  "Reduce/Image"

  MagickBooleanType
    proceed;

  MagickOffsetType
    offset;

  size_t
    span;

  cube_info->next_threshold=0.0;
  if (cube_info->colors > cube_info->maximum_colors)
    {
      MagickRealType
        *quantize_error;

      /*
        Enable rapid reduction of the number of unique colors.
      */
      quantize_error=(MagickRealType *) AcquireQuantumMemory(cube_info->nodes,
        sizeof(*quantize_error));
      if (quantize_error != (MagickRealType *) NULL)
        {
          (void) QuantizeErrorFlatten(cube_info,cube_info->root,0,
            quantize_error);
          qsort(quantize_error,cube_info->nodes,sizeof(MagickRealType),
            MagickRealTypeCompare);
          if (cube_info->nodes > (110*(cube_info->maximum_colors+1)/100))
            cube_info->next_threshold=quantize_error[cube_info->nodes-110*
              (cube_info->maximum_colors+1)/100];
          quantize_error=(MagickRealType *) RelinquishMagickMemory(
            quantize_error);
        }
  }
  for (span=cube_info->colors; cube_info->colors > cube_info->maximum_colors; )
  {
    cube_info->pruning_threshold=cube_info->next_threshold;
    cube_info->next_threshold=cube_info->root->quantize_error-1;
    cube_info->colors=0;
    Reduce(cube_info,cube_info->root);
    offset=(MagickOffsetType) span-cube_info->colors;
    proceed=SetImageProgress(image,ReduceImageTag,offset,span-
      cube_info->maximum_colors+1);
    if (proceed == MagickFalse)
      break;
  }
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   R e m a p I m a g e                                                       %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  RemapImage() replaces the colors of an image with the closest color from
%  a reference image.
%
%  The format of the RemapImage method is:
%
%      MagickBooleanType RemapImage(const QuantizeInfo *quantize_info,
%        Image *image,const Image *remap_image)
%
%  A description of each parameter follows:
%
%    o quantize_info: Specifies a pointer to an QuantizeInfo structure.
%
%    o image: the image.
%
%    o remap_image: the reference image.
%
*/
MagickExport MagickBooleanType RemapImage(const QuantizeInfo *quantize_info,
  Image *image,const Image *remap_image)
{
  MagickBooleanType
    status;

  QCubeInfo
    *cube_info;

  /*
    Initialize color cube.
  */
  assert(image != (Image *) NULL);
  assert(image->signature == MagickCoreSignature);
  if (IsEventLogging() != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",image->filename);
  assert(remap_image != (Image *) NULL);
  assert(remap_image->signature == MagickCoreSignature);
  cube_info=GetQCubeInfo(quantize_info,MaxTreeDepth,MaxColormapSize);
  if (cube_info == (QCubeInfo *) NULL)
    ThrowBinaryImageException(ResourceLimitError,"MemoryAllocationFailed",
      image->filename);
  cube_info->quantize_info->colorspace=remap_image->colorspace;
  status=ClassifyImageColors(cube_info,remap_image,&image->exception);
  if (status != MagickFalse)
    {
      /*
        Classify image colors from the reference image.
      */
      cube_info->quantize_info->number_colors=cube_info->colors;
      if (cube_info->colors > cube_info->maximum_colors)
        ReduceImageColors(image,cube_info);
      status=AssignImageColors(image,cube_info);
    }
  DestroyQCubeInfo(cube_info);
  return(status);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   R e m a p I m a g e s                                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  RemapImages() replaces the colors of a sequence of images with the
%  closest color from a reference image.
%
%  The format of the RemapImage method is:
%
%      MagickBooleanType RemapImages(const QuantizeInfo *quantize_info,
%        Image *images,Image *remap_image)
%
%  A description of each parameter follows:
%
%    o quantize_info: Specifies a pointer to an QuantizeInfo structure.
%
%    o images: the image sequence.
%
%    o remap_image: the reference image.
%
*/
MagickExport MagickBooleanType RemapImages(const QuantizeInfo *quantize_info,
  Image *images,const Image *remap_image)
{
  Image
    *image;

  MagickBooleanType
    status;

  QCubeInfo
    *cube_info;

  assert(images != (Image *) NULL);
  assert(images->signature == MagickCoreSignature);
  if (IsEventLogging() != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",images->filename);
  image=images;
  if (remap_image == (Image *) NULL)
    {
      /*
        Create a global colormap for an image sequence.
      */
      status=QuantizeImages(quantize_info,images);
      return(status);
    }
  /*
    Classify image colors from the reference image.
  */
  cube_info=GetQCubeInfo(quantize_info,MaxTreeDepth,
    quantize_info->number_colors);
  if (cube_info == (QCubeInfo *) NULL)
    ThrowBinaryImageException(ResourceLimitError,"MemoryAllocationFailed",
      image->filename);
  status=ClassifyImageColors(cube_info,remap_image,&image->exception);
  if (status != MagickFalse)
    {
      /*
        Classify image colors from the reference image.
      */
      cube_info->quantize_info->number_colors=cube_info->colors;
      image=images;
      for ( ; image != (Image *) NULL; image=GetNextImageInList(image))
      {
        status=AssignImageColors(image,cube_info);
        if (status == MagickFalse)
          break;
      }
    }
  DestroyQCubeInfo(cube_info);
  return(status);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   S e t G r a y s c a l e I m a g e                                         %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  SetGrayscaleImage() converts an image to a PseudoClass grayscale image.
%
%  The format of the SetGrayscaleImage method is:
%
%      MagickBooleanType SetGrayscaleImage(Image *image)
%
%  A description of each parameter follows:
%
%    o image: The image.
%
*/

#if defined(__cplusplus) || defined(c_plusplus)
extern "C" {
#endif

static int IntensityCompare(const void *x,const void *y)
{
  double
    intensity;

  PixelPacket
    *color_1,
    *color_2;

  color_1=(PixelPacket *) x;
  color_2=(PixelPacket *) y;
  intensity=PixelPacketIntensity(color_1)-PixelPacketIntensity(color_2);
  if (intensity < (double) INT_MIN)
    intensity=(double) INT_MIN;
  if (intensity > (double) INT_MAX)
    intensity=(double) INT_MAX;
  return((int) intensity);
}

#if defined(__cplusplus) || defined(c_plusplus)
}
#endif

static MagickBooleanType SetGrayscaleImage(Image *image)
{
  CacheView
    *image_view;

  ExceptionInfo
    *exception;

  MagickBooleanType
    status;

  PixelPacket
    *colormap;

  size_t
    extent;

  ssize_t
    *colormap_index,
    i,
    j,
    y;

  assert(image != (Image *) NULL);
  assert(image->signature == MagickCoreSignature);
  exception=(&image->exception);
  if (image->type != GrayscaleType)
    (void) TransformImageColorspace(image,GRAYColorspace);
  extent=MagickMax(image->colors+1,MagickMax(MaxColormapSize,MaxMap+1));
  colormap_index=(ssize_t *) AcquireQuantumMemory(extent,
    sizeof(*colormap_index));
  if (colormap_index == (ssize_t *) NULL)
    ThrowBinaryException(ResourceLimitError,"MemoryAllocationFailed",
      image->filename);
  if (image->storage_class != PseudoClass)
    {
      (void) memset(colormap_index,(-1),extent*sizeof(*colormap_index));
      if (AcquireImageColormap(image,MaxColormapSize) == MagickFalse)
        {
          colormap_index=(ssize_t *) RelinquishMagickMemory(colormap_index);
          ThrowBinaryException(ResourceLimitError,"MemoryAllocationFailed",
            image->filename);
        }
      image->colors=0;
      status=MagickTrue;
      image_view=AcquireAuthenticCacheView(image,exception);
#if defined(MAGICKCORE_OPENMP_SUPPORT)
      #pragma omp parallel for schedule(static) shared(status) \
        magick_number_threads(image,image,image->rows,1)
#endif
      for (y=0; y < (ssize_t) image->rows; y++)
      {
        IndexPacket
          *magick_restrict indexes;

        PixelPacket
          *magick_restrict q;

        ssize_t
          x;

        if (status == MagickFalse)
          continue;
        q=GetCacheViewAuthenticPixels(image_view,0,y,image->columns,1,
          exception);
        if (q == (PixelPacket *) NULL)
          {
            status=MagickFalse;
            continue;
          }
        indexes=GetCacheViewAuthenticIndexQueue(image_view);
        for (x=0; x < (ssize_t) image->columns; x++)
        {
          size_t
            intensity;

          intensity=ScaleQuantumToMap(GetPixelRed(q));
          if (colormap_index[intensity] < 0)
            {
#if defined(MAGICKCORE_OPENMP_SUPPORT)
              #pragma omp critical (MagickCore_SetGrayscaleImage)
#endif
              if (colormap_index[intensity] < 0)
                {
                  colormap_index[intensity]=(ssize_t) image->colors;
                  image->colormap[image->colors].red=GetPixelRed(q);
                  image->colormap[image->colors].green=GetPixelGreen(q);
                  image->colormap[image->colors].blue=GetPixelBlue(q);
                  image->colors++;
               }
            }
          SetPixelIndex(indexes+x,colormap_index[intensity]);
          q++;
        }
        if (SyncCacheViewAuthenticPixels(image_view,exception) == MagickFalse)
          status=MagickFalse;
      }
      image_view=DestroyCacheView(image_view);
    }
  (void) memset(colormap_index,0,extent*sizeof(*colormap_index));
  for (i=0; i < (ssize_t) image->colors; i++)
    image->colormap[i].opacity=(Quantum) i;
  qsort((void *) image->colormap,image->colors,sizeof(PixelPacket),
    IntensityCompare);
  colormap=(PixelPacket *) AcquireQuantumMemory(image->colors,
    sizeof(*colormap));
  if (colormap == (PixelPacket *) NULL)
    {
      colormap_index=(ssize_t *) RelinquishMagickMemory(colormap_index);
      ThrowBinaryException(ResourceLimitError,"MemoryAllocationFailed",
        image->filename);
    }
  j=0;
  colormap[j]=image->colormap[0];
  for (i=0; i < (ssize_t) image->colors; i++)
  {
    if (IsSameColor(image,&colormap[j],&image->colormap[i]) == MagickFalse)
      {
        j++;
        colormap[j]=image->colormap[i];
      }
    colormap_index[(ssize_t) image->colormap[i].opacity]=j;
  }
  image->colors=(size_t) (j+1);
  image->colormap=(PixelPacket *) RelinquishMagickMemory(image->colormap);
  image->colormap=colormap;
  status=MagickTrue;
  image_view=AcquireAuthenticCacheView(image,exception);
#if defined(MAGICKCORE_OPENMP_SUPPORT)
  #pragma omp parallel for schedule(static) shared(status) \
    magick_number_threads(image,image,image->rows,1)
#endif
  for (y=0; y < (ssize_t) image->rows; y++)
  {
    IndexPacket
      *magick_restrict indexes;

    const PixelPacket
      *magick_restrict q;

    ssize_t
      x;

    if (status == MagickFalse)
      continue;
    q=GetCacheViewAuthenticPixels(image_view,0,y,image->columns,1,exception);
    if (q == (PixelPacket *) NULL)
      {
        status=MagickFalse;
        continue;
      }
    indexes=GetCacheViewAuthenticIndexQueue(image_view);
    for (x=0; x < (ssize_t) image->columns; x++)
      SetPixelIndex(indexes+x,colormap_index[ScaleQuantumToMap(GetPixelIndex(
        indexes+x))]);
    if (SyncCacheViewAuthenticPixels(image_view,exception) == MagickFalse)
      status=MagickFalse;
  }
  image_view=DestroyCacheView(image_view);
  colormap_index=(ssize_t *) RelinquishMagickMemory(colormap_index);
  image->type=GrayscaleType;
  if (SetImageMonochrome(image,&image->exception) != MagickFalse)
    image->type=BilevelType;
  return(status);
}