MagickCore/quantize.c

/*
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%    MagickCore Methods to Reduce the Number of Unique Colors in an Image     %
%                                                                             %
%                           Software Design                                   %
%                             John Cristy                                     %
%                              July 1992                                      %
%                                                                             %
%                                                                             %
%  Copyright 1999-2012 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                                            %
%                                                                             %
%    http://www.imagemagick.org/script/license.php                            %
%                                                                             %
%  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:
%  The 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 "MagickCore/studio.h"
#include "MagickCore/attribute.h"
#include "MagickCore/cache-view.h"
#include "MagickCore/color.h"
#include "MagickCore/color-private.h"
#include "MagickCore/colormap.h"
#include "MagickCore/colorspace.h"
#include "MagickCore/colorspace-private.h"
#include "MagickCore/enhance.h"
#include "MagickCore/exception.h"
#include "MagickCore/exception-private.h"
#include "MagickCore/histogram.h"
#include "MagickCore/image.h"
#include "MagickCore/image-private.h"
#include "MagickCore/list.h"
#include "MagickCore/memory_.h"
#include "MagickCore/monitor.h"
#include "MagickCore/monitor-private.h"
#include "MagickCore/option.h"
#include "MagickCore/pixel-accessor.h"
#include "MagickCore/quantize.h"
#include "MagickCore/quantum.h"
#include "MagickCore/quantum-private.h"
#include "MagickCore/string_.h"
#include "MagickCore/thread-private.h"

/*
  Define declarations.
*/
#if !defined(__APPLE__) && !defined(TARGET_OS_IPHONE)
#define CacheShift  2
#else
#define CacheShift  3
#endif
#define ErrorQueueLength  16
#define MaxNodes  266817
#define MaxTreeDepth  8
#define NodesInAList  1920

/*
  Typdef declarations.
*/
typedef struct _RealPixelInfo
{
  MagickRealType
    red,
    green,
    blue,
    alpha;
} RealPixelInfo;

typedef struct _NodeInfo
{
  struct _NodeInfo
    *parent,
    *child[16];

  MagickSizeType
    number_unique;

  RealPixelInfo
    total_color;

  MagickRealType
    quantize_error;

  size_t
    color_number,
    id,
    level;
} NodeInfo;

typedef struct _Nodes
{
  NodeInfo
    *nodes;

  struct _Nodes
    *next;
} Nodes;

typedef struct _CubeInfo
{
  NodeInfo
    *root;

  size_t
    colors,
    maximum_colors;

  ssize_t
    transparent_index;

  MagickSizeType
    transparent_pixels;

  RealPixelInfo
    target;

  MagickRealType
    distance,
    pruning_threshold,
    next_threshold;

  size_t
    nodes,
    free_nodes,
    color_number;

  NodeInfo
    *next_node;

  Nodes
    *node_queue;

  ssize_t
    *cache;

  RealPixelInfo
    error[ErrorQueueLength];

  MagickRealType
    weights[ErrorQueueLength];

  QuantizeInfo
    *quantize_info;

  MagickBooleanType
    associate_alpha;

  ssize_t
    x,
    y;

  size_t
    depth;

  MagickOffsetType
    offset;

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

static NodeInfo
  *GetNodeInfo(CubeInfo *,const size_t,const size_t,NodeInfo *);

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

static size_t
  DefineImageColormap(Image *,CubeInfo *,NodeInfo *);

static void
  ClosestColor(const Image *,CubeInfo *,const NodeInfo *),
  DestroyCubeInfo(CubeInfo *),
  PruneLevel(const Image *,CubeInfo *,const NodeInfo *),
  PruneToCubeDepth(const Image *,CubeInfo *,const NodeInfo *),
  ReduceImageColors(const Image *,CubeInfo *);

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   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,CubeInfo *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 Image *image,
  const CubeInfo *cube_info,const Quantum *pixel,RealPixelInfo *alpha_pixel)
{
  MagickRealType
    alpha;

  if ((cube_info->associate_alpha == MagickFalse) ||
      (GetPixelAlpha(image,pixel)== OpaqueAlpha))
    {
      alpha_pixel->red=(MagickRealType) GetPixelRed(image,pixel);
      alpha_pixel->green=(MagickRealType) GetPixelGreen(image,pixel);
      alpha_pixel->blue=(MagickRealType) GetPixelBlue(image,pixel);
      alpha_pixel->alpha=(MagickRealType) GetPixelAlpha(image,pixel);
      return;
    }
  alpha=(MagickRealType) (QuantumScale*GetPixelAlpha(image,pixel));
  alpha_pixel->red=alpha*GetPixelRed(image,pixel);
  alpha_pixel->green=alpha*GetPixelGreen(image,pixel);
  alpha_pixel->blue=alpha*GetPixelBlue(image,pixel);
  alpha_pixel->alpha=(MagickRealType) GetPixelAlpha(image,pixel);
}

static inline void AssociateAlphaPixelInfo(const Image *image,
  const CubeInfo *cube_info,const PixelInfo *pixel,
  RealPixelInfo *alpha_pixel)
{
  MagickRealType
    alpha;

  if ((cube_info->associate_alpha == MagickFalse) ||
      (pixel->alpha == OpaqueAlpha))
    {
      alpha_pixel->red=(MagickRealType) pixel->red;
      alpha_pixel->green=(MagickRealType) pixel->green;
      alpha_pixel->blue=(MagickRealType) pixel->blue;
      alpha_pixel->alpha=(MagickRealType) pixel->alpha;
      return;
    }
  alpha=(MagickRealType) (QuantumScale*pixel->alpha);
  alpha_pixel->red=alpha*pixel->red;
  alpha_pixel->green=alpha*pixel->green;
  alpha_pixel->blue=alpha*pixel->blue;
  alpha_pixel->alpha=(MagickRealType) pixel->alpha;
}

static inline Quantum ClampToUnsignedQuantum(const MagickRealType value)
{
  if (value <= 0.0)
    return((Quantum) 0);
  if (value >= QuantumRange)
    return((Quantum) QuantumRange);
  return((Quantum) (value+0.5));
}

static inline size_t ColorToNodeId(const CubeInfo *cube_info,
  const RealPixelInfo *pixel,size_t index)
{
  size_t
    id;

  id=(size_t) (((ScaleQuantumToChar(ClampToUnsignedQuantum(pixel->red)) >> index) & 0x01) |
    ((ScaleQuantumToChar(ClampToUnsignedQuantum(pixel->green)) >> index) & 0x01) << 1 |
    ((ScaleQuantumToChar(ClampToUnsignedQuantum(pixel->blue)) >> index) & 0x01) << 2);
  if (cube_info->associate_alpha != MagickFalse)
    id|=((ScaleQuantumToChar(ClampToUnsignedQuantum(pixel->alpha)) >> index) & 0x1) << 3;
  return(id);
}

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

  ssize_t
    y;

  /*
    Allocate image colormap.
  */
  if ((cube_info->quantize_info->colorspace != UndefinedColorspace) &&
      (cube_info->quantize_info->colorspace != CMYKColorspace))
    (void) TransformImageColorspace((Image *) image,
      cube_info->quantize_info->colorspace,exception);
  else
    if ((image->colorspace != GRAYColorspace) &&
        (IssRGBColorspace(image->colorspace) == MagickFalse) &&
        (image->colorspace != CMYColorspace))
      (void) TransformImageColorspace((Image *) image,sRGBColorspace,exception);
  if (AcquireImageColormap(image,cube_info->colors,exception) == MagickFalse)
    ThrowBinaryException(ResourceLimitError,"MemoryAllocationFailed",
      image->filename);
  image->colors=0;
  cube_info->transparent_pixels=0;
  cube_info->transparent_index=(-1);
  (void) 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,exception);
  else
    {
      CacheView
        *image_view;

      MagickBooleanType
        status;

      status=MagickTrue;
      image_view=AcquireAuthenticCacheView(image,exception);
#if defined(MAGICKCORE_OPENMP_SUPPORT)
      #pragma omp parallel for schedule(static,4) shared(status)
#endif
      for (y=0; y < (ssize_t) image->rows; y++)
      {
        CubeInfo
          cube;

        register Quantum
          *restrict q;

        register ssize_t
          x;

        ssize_t
          count;

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

          register const NodeInfo
            *node_info;

          register 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++)
          {
            PixelInfo
              packet;

            GetPixelInfoPixel(image,q+count*GetPixelChannels(image),&packet);
            if (IsPixelEquivalent(image,q,&packet) == MagickFalse)
              break;
          }
          AssociateAlphaPixel(image,&cube,q,&pixel);
          node_info=cube.root;
          for (index=MaxTreeDepth-1; (ssize_t) index > 0; index--)
          {
            id=ColorToNodeId(&cube,&pixel,index);
            if (node_info->child[id] == (NodeInfo *) NULL)
              break;
            node_info=node_info->child[id];
          }
          /*
            Find closest color among siblings and their children.
          */
          cube.target=pixel;
          cube.distance=(MagickRealType) (4.0*(QuantumRange+1.0)*
            (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(image,(Quantum) index,q);
            if (cube.quantize_info->measure_error == MagickFalse)
              {
                SetPixelRed(image,ClampToQuantum(
                  image->colormap[index].red),q);
                SetPixelGreen(image,ClampToQuantum(
                  image->colormap[index].green),q);
                SetPixelBlue(image,ClampToQuantum(
                  image->colormap[index].blue),q);
                if (cube.associate_alpha != MagickFalse)
                  SetPixelAlpha(image,ClampToQuantum(
                    image->colormap[index].alpha),q);
              }
            q+=GetPixelChannels(image);
          }
        }
        if (SyncCacheViewAuthenticPixels(image_view,exception) == MagickFalse)
          status=MagickFalse;
        if (image->progress_monitor != (MagickProgressMonitor) NULL)
          {
            MagickBooleanType
              proceed;

#if defined(MAGICKCORE_OPENMP_SUPPORT)
            #pragma omp critical (MagickCore_AssignImageColors)
#endif
            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,exception);
  if ((cube_info->quantize_info->number_colors == 2) &&
      (cube_info->quantize_info->colorspace == GRAYColorspace))
    {
      double
        intensity;

      register PixelInfo
        *restrict q;

      register ssize_t
        i;

      /*
        Monochrome image.
      */
      q=image->colormap;
      for (i=0; i < (ssize_t) image->colors; i++)
      {
        intensity=(double) ((MagickRealType) GetPixelInfoIntensity(q) <
          ((MagickRealType) QuantumRange/2.0) ? 0 : QuantumRange);
        q->red=intensity;
        q->green=intensity;
        q->blue=intensity;
        q++;
      }
    }
  (void) SyncImage(image,exception);
  if ((cube_info->quantize_info->colorspace != UndefinedColorspace) &&
      (cube_info->quantize_info->colorspace != CMYKColorspace))
    (void) TransformImageColorspace((Image *) image,sRGBColorspace,exception);
  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(CubeInfo *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,CubeInfo *cube_info)
{
  MagickBooleanType
    associate_alpha;

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

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

  CacheView
    *image_view;

  MagickBooleanType
    proceed;

  MagickRealType
    bisect;

  NodeInfo
    *node_info;

  RealPixelInfo
    error,
    mid,
    midpoint,
    pixel;

  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 != UndefinedColorspace) &&
      (cube_info->quantize_info->colorspace != CMYKColorspace))
    (void) TransformImageColorspace((Image *) image,
      cube_info->quantize_info->colorspace,exception);
  else
    if ((image->colorspace != GRAYColorspace) &&
        (image->colorspace != CMYColorspace) &&
        (IssRGBColorspace(image->colorspace) == MagickFalse))
      (void) TransformImageColorspace((Image *) image,sRGBColorspace,exception);
  midpoint.red=(MagickRealType) QuantumRange/2.0;
  midpoint.green=(MagickRealType) QuantumRange/2.0;
  midpoint.blue=(MagickRealType) QuantumRange/2.0;
  midpoint.alpha=(MagickRealType) QuantumRange/2.0;
  error.alpha=0.0;
  image_view=AcquireVirtualCacheView(image,exception);
  for (y=0; y < (ssize_t) image->rows; y++)
  {
    register const Quantum
      *restrict p;

    register ssize_t
      x;

    p=GetCacheViewVirtualPixels(image_view,0,y,image->columns,1,exception);
    if (p == (const Quantum *) NULL)
      break;
    if (cube_info->nodes > MaxNodes)
      {
        /*
          Prune one level if the color tree is too large.
        */
        PruneLevel(image,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++)
      {
        PixelInfo
          packet;

        GetPixelInfoPixel(image,p+count*GetPixelChannels(image),&packet);
        if (IsPixelEquivalent(image,p,&packet) == MagickFalse)
          break;
      }
      AssociateAlphaPixel(image,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++)
      {
        bisect*=0.5;
        id=ColorToNodeId(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.alpha+=(id & 8) != 0 ? bisect : -bisect;
        if (node_info->child[id] == (NodeInfo *) NULL)
          {
            /*
              Set colors of new node to contain pixel.
            */
            node_info->child[id]=GetNodeInfo(cube_info,id,level,node_info);
            if (node_info->child[id] == (NodeInfo *) NULL)
              (void) ThrowMagickException(exception,GetMagickModule(),
                ResourceLimitError,"MemoryAllocationFailed","'%s'",
                image->filename);
            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.alpha=QuantumScale*(pixel.alpha-mid.alpha);
        node_info->quantize_error+=sqrt((double) (count*error.red*error.red+
          count*error.green*error.green+count*error.blue*error.blue+
          count*error.alpha*error.alpha));
        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*pixel.red;
      node_info->total_color.green+=count*QuantumScale*pixel.green;
      node_info->total_color.blue+=count*QuantumScale*pixel.blue;
      if (cube_info->associate_alpha != MagickFalse)
        node_info->total_color.alpha+=count*QuantumScale*pixel.alpha;
      p+=count*GetPixelChannels(image);
    }
    if (cube_info->colors > cube_info->maximum_colors)
      {
        PruneToCubeDepth(image,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++)
  {
    register const Quantum
      *restrict p;

    register ssize_t
      x;

    p=GetCacheViewVirtualPixels(image_view,0,y,image->columns,1,exception);
    if (p == (const Quantum *) NULL)
      break;
    if (cube_info->nodes > MaxNodes)
      {
        /*
          Prune one level if the color tree is too large.
        */
        PruneLevel(image,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++)
      {
        PixelInfo
          packet;

        GetPixelInfoPixel(image,p+count*GetPixelChannels(image),&packet);
        if (IsPixelEquivalent(image,p,&packet) == MagickFalse)
          break;
      }
      AssociateAlphaPixel(image,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++)
      {
        bisect*=0.5;
        id=ColorToNodeId(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.alpha+=(id & 8) != 0 ? bisect : -bisect;
        if (node_info->child[id] == (NodeInfo *) NULL)
          {
            /*
              Set colors of new node to contain pixel.
            */
            node_info->child[id]=GetNodeInfo(cube_info,id,level,node_info);
            if (node_info->child[id] == (NodeInfo *) NULL)
              (void) ThrowMagickException(exception,GetMagickModule(),
                ResourceLimitError,"MemoryAllocationFailed","%s",
                image->filename);
            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.alpha=QuantumScale*(pixel.alpha-mid.alpha);
        node_info->quantize_error+=sqrt((double) (count*error.red*error.red+
          count*error.green*error.green+count*error.blue*error.blue+
          count*error.alpha*error.alpha));
        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*pixel.red;
      node_info->total_color.green+=count*QuantumScale*pixel.green;
      node_info->total_color.blue+=count*QuantumScale*pixel.blue;
      if (cube_info->associate_alpha != MagickFalse)
        node_info->total_color.alpha+=count*QuantumScale*pixel.alpha;
      p+=count*GetPixelChannels(image);
    }
    proceed=SetImageProgress(image,ClassifyImageTag,(MagickOffsetType) y,
      image->rows);
    if (proceed == MagickFalse)
      break;
  }
  image_view=DestroyCacheView(image_view);
  if ((cube_info->quantize_info->colorspace != UndefinedColorspace) &&
      (cube_info->quantize_info->colorspace != CMYKColorspace))
    (void) TransformImageColorspace((Image *) image,sRGBColorspace,exception);
  return(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,CubeInfo *cube_info,
%        const NodeInfo *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 NodeInfo which points to a
%      node in the color cube tree that is to be pruned.
%
*/
static void ClosestColor(const Image *image,CubeInfo *cube_info,
  const NodeInfo *node_info)
{
  register ssize_t
    i;

  size_t
    number_children;

  /*
    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] != (NodeInfo *) NULL)
      ClosestColor(image,cube_info,node_info->child[i]);
  if (node_info->number_unique != 0)
    {
      MagickRealType
        pixel;

      register MagickRealType
        alpha,
        beta,
        distance;

      register PixelInfo
        *restrict p;

      register RealPixelInfo
        *restrict q;

      /*
        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*p->alpha);
          beta=(MagickRealType) (QuantumScale*q->alpha);
        }
      pixel=alpha*p->red-beta*q->red;
      distance=pixel*pixel;
      if (distance <= cube_info->distance)
        {
          pixel=alpha*p->green-beta*q->green;
          distance+=pixel*pixel;
          if (distance <= cube_info->distance)
            {
              pixel=alpha*p->blue-beta*q->blue;
              distance+=pixel*pixel;
              if (distance <= cube_info->distance)
                {
                  pixel=alpha-beta;
                  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,
%        ExceptionInfo *exception)
%
%  A description of each parameter follows:
%
%    o image: the image.
%
%    o exception: return any errors or warnings in this structure.
%
*/
MagickExport MagickBooleanType CompressImageColormap(Image *image,
  ExceptionInfo *exception)
{
  QuantizeInfo
    quantize_info;

  assert(image != (Image *) NULL);
  assert(image->signature == MagickSignature);
  if (image->debug != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",image->filename);
  if (IsPaletteImage(image,exception) == MagickFalse)
    return(MagickFalse);
  GetQuantizeInfo(&quantize_info);
  quantize_info.number_colors=image->colors;
  quantize_info.tree_depth=MaxTreeDepth;
  return(QuantizeImage(&quantize_info,image,exception));
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   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.  DefineImageColormap() returns the number of
%  colors in the image colormap.
%
%  The format of the DefineImageColormap method is:
%
%      size_t DefineImageColormap(Image *image,CubeInfo *cube_info,
%        NodeInfo *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 NodeInfo which points to a
%      node in the color cube tree that is to be pruned.
%
*/
static size_t DefineImageColormap(Image *image,CubeInfo *cube_info,
  NodeInfo *node_info)
{
  register ssize_t
    i;

  size_t
    number_children;

  /*
    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] != (NodeInfo *) NULL)
      (void) DefineImageColormap(image,cube_info,node_info->child[i]);
  if (node_info->number_unique != 0)
    {
      register MagickRealType
        alpha;

      register PixelInfo
        *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=1.0/(fabs(alpha) <= MagickEpsilon ? 1.0 : alpha);
      if (cube_info->associate_alpha == MagickFalse)
        {
          q->red=(double) ClampToQuantum((MagickRealType)
            (alpha*QuantumRange*node_info->total_color.red));
          q->green=(double) ClampToQuantum((MagickRealType)
            (alpha*QuantumRange*node_info->total_color.green));
          q->blue=(double) ClampToQuantum((MagickRealType)
            (alpha*(double) QuantumRange*node_info->total_color.blue));
          q->alpha=OpaqueAlpha;
        }
      else
        {
          MagickRealType
            opacity;

          opacity=(MagickRealType) (alpha*QuantumRange*
            node_info->total_color.alpha);
          q->alpha=(double) ClampToQuantum(opacity);
          if (q->alpha == OpaqueAlpha)
            {
              q->red=(double) ClampToQuantum((MagickRealType)
                (alpha*QuantumRange*node_info->total_color.red));
              q->green=(double) ClampToQuantum((MagickRealType)
                (alpha*QuantumRange*node_info->total_color.green));
              q->blue=(double) ClampToQuantum((MagickRealType)
                (alpha*QuantumRange*node_info->total_color.blue));
            }
          else
            {
              MagickRealType
                gamma;

              gamma=(MagickRealType) (QuantumScale*q->alpha);
              gamma=1.0/(fabs(gamma) <= MagickEpsilon ? 1.0 : gamma);
              q->red=(double) ClampToQuantum((MagickRealType)
                (alpha*gamma*QuantumRange*node_info->total_color.red));
              q->green=(double) ClampToQuantum((MagickRealType)
                (alpha*gamma*QuantumRange*node_info->total_color.green));
              q->blue=(double) ClampToQuantum((MagickRealType)
                (alpha*gamma*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++;
    }
  return(image->colors);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   D e s t r o y C u b e I n f o                                             %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  DestroyCubeInfo() deallocates memory associated with an image.
%
%  The format of the DestroyCubeInfo method is:
%
%      DestroyCubeInfo(CubeInfo *cube_info)
%
%  A description of each parameter follows:
%
%    o cube_info: the address of a structure of type CubeInfo.
%
*/
static void DestroyCubeInfo(CubeInfo *cube_info)
{
  register Nodes
    *nodes;

  /*
    Release color cube tree storage.
  */
  do
  {
    nodes=cube_info->node_queue->next;
    cube_info->node_queue->nodes=(NodeInfo *) RelinquishMagickMemory(
      cube_info->node_queue->nodes);
    cube_info->node_queue=(Nodes *) RelinquishMagickMemory(
      cube_info->node_queue);
    cube_info->node_queue=nodes;
  } while (cube_info->node_queue != (Nodes *) NULL);
  if (cube_info->cache != (ssize_t *) NULL)
    cube_info->cache=(ssize_t *) RelinquishMagickMemory(cube_info->cache);
  cube_info->quantize_info=DestroyQuantizeInfo(cube_info->quantize_info);
  cube_info=(CubeInfo *) 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)
{
  (void) LogMagickEvent(TraceEvent,GetMagickModule(),"...");
  assert(quantize_info != (QuantizeInfo *) NULL);
  assert(quantize_info->signature == MagickSignature);
  quantize_info->signature=(~MagickSignature);
  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,CubeInfo *cube_info,
%        ExceptionInfo *exception)
%
%  A description of each parameter follows.
%
%    o image: the image.
%
%    o cube_info: A pointer to the Cube structure.
%
%    o exception: return any errors or warnings in this structure.
%
*/

static RealPixelInfo **DestroyPixelThreadSet(RealPixelInfo **pixels)
{
  register ssize_t
    i;

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

static RealPixelInfo **AcquirePixelThreadSet(const size_t count)
{
  RealPixelInfo
    **pixels;

  register ssize_t
    i;

  size_t
    number_threads;

  number_threads=GetOpenMPMaximumThreads();
  pixels=(RealPixelInfo **) AcquireQuantumMemory(number_threads,
    sizeof(*pixels));
  if (pixels == (RealPixelInfo **) NULL)
    return((RealPixelInfo **) NULL);
  (void) ResetMagickMemory(pixels,0,number_threads*sizeof(*pixels));
  for (i=0; i < (ssize_t) number_threads; i++)
  {
    pixels[i]=(RealPixelInfo *) AcquireQuantumMemory(count,
      2*sizeof(**pixels));
    if (pixels[i] == (RealPixelInfo *) NULL)
      return(DestroyPixelThreadSet(pixels));
  }
  return(pixels);
}

static inline ssize_t CacheOffset(CubeInfo *cube_info,
  const RealPixelInfo *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(ClampToUnsignedQuantum(pixel->red))) |
    GreenShift(ScaleQuantumToChar(ClampToUnsignedQuantum(pixel->green))) |
    BlueShift(ScaleQuantumToChar(ClampToUnsignedQuantum(pixel->blue))));
  if (cube_info->associate_alpha != MagickFalse)
    offset|=AlphaShift(ScaleQuantumToChar(ClampToUnsignedQuantum(
      pixel->alpha)));
  return(offset);
}

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

  CacheView
    *image_view;

  MagickBooleanType
    status;

  RealPixelInfo
    **pixels;

  ssize_t
    y;

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

    CubeInfo
      cube;

    RealPixelInfo
      *current,
      *previous;

    register Quantum
      *restrict q;

    register ssize_t
      x;

    size_t
      index;

    ssize_t
      v;

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

      register ssize_t
        i;

      ssize_t
        u;

      q-=(y & 0x01)*GetPixelChannels(image);
      u=(y & 0x01) != 0 ? (ssize_t) image->columns-1-x : x;
      AssociateAlphaPixel(image,&cube,q,&pixel);
      if (x > 0)
        {
          pixel.red+=7*current[u-v].red/16;
          pixel.green+=7*current[u-v].green/16;
          pixel.blue+=7*current[u-v].blue/16;
          if (cube.associate_alpha != MagickFalse)
            pixel.alpha+=7*current[u-v].alpha/16;
        }
      if (y > 0)
        {
          if (x < (ssize_t) (image->columns-1))
            {
              pixel.red+=previous[u+v].red/16;
              pixel.green+=previous[u+v].green/16;
              pixel.blue+=previous[u+v].blue/16;
              if (cube.associate_alpha != MagickFalse)
                pixel.alpha+=previous[u+v].alpha/16;
            }
          pixel.red+=5*previous[u].red/16;
          pixel.green+=5*previous[u].green/16;
          pixel.blue+=5*previous[u].blue/16;
          if (cube.associate_alpha != MagickFalse)
            pixel.alpha+=5*previous[u].alpha/16;
          if (x > 0)
            {
              pixel.red+=3*previous[u-v].red/16;
              pixel.green+=3*previous[u-v].green/16;
              pixel.blue+=3*previous[u-v].blue/16;
              if (cube.associate_alpha != MagickFalse)
                pixel.alpha+=3*previous[u-v].alpha/16;
            }
        }
      pixel.red=(MagickRealType) ClampToUnsignedQuantum(pixel.red);
      pixel.green=(MagickRealType) ClampToUnsignedQuantum(pixel.green);
      pixel.blue=(MagickRealType) ClampToUnsignedQuantum(pixel.blue);
      if (cube.associate_alpha != MagickFalse)
        pixel.alpha=(MagickRealType) ClampToUnsignedQuantum(pixel.alpha);
      i=CacheOffset(&cube,&pixel);
      if (cube.cache[i] < 0)
        {
          register NodeInfo
            *node_info;

          register 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=ColorToNodeId(&cube,&pixel,index);
            if (node_info->child[id] == (NodeInfo *) NULL)
              break;
            node_info=node_info->child[id];
          }
          /*
            Find closest color among siblings and their children.
          */
          cube.target=pixel;
          cube.distance=(MagickRealType) (4.0*(QuantumRange+1.0)*(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(image,(Quantum) index,q);
      if (cube.quantize_info->measure_error == MagickFalse)
        {
          SetPixelRed(image,ClampToQuantum(image->colormap[index].red),q);
          SetPixelGreen(image,ClampToQuantum(image->colormap[index].green),q);
          SetPixelBlue(image,ClampToQuantum(image->colormap[index].blue),q);
          if (cube.associate_alpha != MagickFalse)
            SetPixelAlpha(image,ClampToQuantum(image->colormap[index].alpha),q);
        }
      if (SyncCacheViewAuthenticPixels(image_view,exception) == MagickFalse)
        status=MagickFalse;
      /*
        Store the error.
      */
      AssociateAlphaPixelInfo(image,&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].alpha=pixel.alpha-color.alpha;
      if (image->progress_monitor != (MagickProgressMonitor) NULL)
        {
          MagickBooleanType
            proceed;

#if defined(MAGICKCORE_OPENMP_SUPPORT)
          #pragma omp critical (MagickCore_FloydSteinbergDither)
#endif
          proceed=SetImageProgress(image,DitherImageTag,(MagickOffsetType) y,
            image->rows);
          if (proceed == MagickFalse)
            status=MagickFalse;
        }
      q+=((y+1) & 0x01)*GetPixelChannels(image);
    }
  }
  image_view=DestroyCacheView(image_view);
  pixels=DestroyPixelThreadSet(pixels);
  return(MagickTrue);
}

static MagickBooleanType
  RiemersmaDither(Image *,CacheView *,CubeInfo *,const unsigned int,
    ExceptionInfo *exception);

static void Riemersma(Image *image,CacheView *image_view,CubeInfo *cube_info,
  const size_t level,const unsigned int direction,ExceptionInfo *exception)
{
  if (level == 1)
    switch (direction)
    {
      case WestGravity:
      {
        (void) RiemersmaDither(image,image_view,cube_info,EastGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,SouthGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,WestGravity,
          exception);
        break;
      }
      case EastGravity:
      {
        (void) RiemersmaDither(image,image_view,cube_info,WestGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,NorthGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,EastGravity,
          exception);
        break;
      }
      case NorthGravity:
      {
        (void) RiemersmaDither(image,image_view,cube_info,SouthGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,EastGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,NorthGravity,
          exception);
        break;
      }
      case SouthGravity:
      {
        (void) RiemersmaDither(image,image_view,cube_info,NorthGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,WestGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,SouthGravity,
          exception);
        break;
      }
      default:
        break;
    }
  else
    switch (direction)
    {
      case WestGravity:
      {
        Riemersma(image,image_view,cube_info,level-1,NorthGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,EastGravity,
          exception);
        Riemersma(image,image_view,cube_info,level-1,WestGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,SouthGravity,
          exception);
        Riemersma(image,image_view,cube_info,level-1,WestGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,WestGravity,
          exception);
        Riemersma(image,image_view,cube_info,level-1,SouthGravity,
          exception);
        break;
      }
      case EastGravity:
      {
        Riemersma(image,image_view,cube_info,level-1,SouthGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,WestGravity,
          exception);
        Riemersma(image,image_view,cube_info,level-1,EastGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,NorthGravity,
          exception);
        Riemersma(image,image_view,cube_info,level-1,EastGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,EastGravity,
          exception);
        Riemersma(image,image_view,cube_info,level-1,NorthGravity,
          exception);
        break;
      }
      case NorthGravity:
      {
        Riemersma(image,image_view,cube_info,level-1,WestGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,SouthGravity,
          exception);
        Riemersma(image,image_view,cube_info,level-1,NorthGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,EastGravity,
          exception);
        Riemersma(image,image_view,cube_info,level-1,NorthGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,NorthGravity,
          exception);
        Riemersma(image,image_view,cube_info,level-1,EastGravity,
          exception);
        break;
      }
      case SouthGravity:
      {
        Riemersma(image,image_view,cube_info,level-1,EastGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,NorthGravity,
          exception);
        Riemersma(image,image_view,cube_info,level-1,SouthGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,WestGravity,
          exception);
        Riemersma(image,image_view,cube_info,level-1,SouthGravity,
          exception);
        (void) RiemersmaDither(image,image_view,cube_info,SouthGravity,
          exception);
        Riemersma(image,image_view,cube_info,level-1,WestGravity,
          exception);
        break;
      }
      default:
        break;
    }
}

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

  MagickBooleanType
    proceed;

  RealPixelInfo
    color,
    pixel;

  register CubeInfo
    *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))
    {
      register Quantum
        *restrict q;

      register ssize_t
        i;

      /*
        Distribute error.
      */
      q=GetCacheViewAuthenticPixels(image_view,p->x,p->y,1,1,exception);
      if (q == (Quantum *) NULL)
        return(MagickFalse);
      AssociateAlphaPixel(image,cube_info,q,&pixel);
      for (i=0; i < ErrorQueueLength; i++)
      {
        pixel.red+=p->weights[i]*p->error[i].red;
        pixel.green+=p->weights[i]*p->error[i].green;
        pixel.blue+=p->weights[i]*p->error[i].blue;
        if (cube_info->associate_alpha != MagickFalse)
          pixel.alpha+=p->weights[i]*p->error[i].alpha;
      }
      pixel.red=(MagickRealType) ClampToUnsignedQuantum(pixel.red);
      pixel.green=(MagickRealType) ClampToUnsignedQuantum(pixel.green);
      pixel.blue=(MagickRealType) ClampToUnsignedQuantum(pixel.blue);
      if (cube_info->associate_alpha != MagickFalse)
        pixel.alpha=(MagickRealType) ClampToUnsignedQuantum(pixel.alpha);
      i=CacheOffset(cube_info,&pixel);
      if (p->cache[i] < 0)
        {
          register NodeInfo
            *node_info;

          register 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=ColorToNodeId(cube_info,&pixel,index);
            if (node_info->child[id] == (NodeInfo *) NULL)
              break;
            node_info=node_info->child[id];
          }
          node_info=node_info->parent;
          /*
            Find closest color among siblings and their children.
          */
          p->target=pixel;
          p->distance=(MagickRealType) (4.0*(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) p->cache[i];
      if (image->storage_class == PseudoClass)
        SetPixelIndex(image,(Quantum) index,q);
      if (cube_info->quantize_info->measure_error == MagickFalse)
        {
          SetPixelRed(image,ClampToQuantum(image->colormap[index].red),q);
          SetPixelGreen(image,ClampToQuantum(image->colormap[index].green),q);
          SetPixelBlue(image,ClampToQuantum(image->colormap[index].blue),q);
          if (cube_info->associate_alpha != MagickFalse)
            SetPixelAlpha(image,ClampToQuantum(image->colormap[index].alpha),q);
        }
      if (SyncCacheViewAuthenticPixels(image_view,exception) == MagickFalse)
        return(MagickFalse);
      /*
        Propagate the error as the last entry of the error queue.
      */
      (void) CopyMagickMemory(p->error,p->error+1,(ErrorQueueLength-1)*
        sizeof(p->error[0]));
      AssociateAlphaPixelInfo(image,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].alpha=pixel.alpha-color.alpha;
      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 inline ssize_t MagickMax(const ssize_t x,const ssize_t y)
{
  if (x > y)
    return(x);
  return(y);
}

static inline ssize_t MagickMin(const ssize_t x,const ssize_t y)
{
  if (x < y)
    return(x);
  return(y);
}

static MagickBooleanType DitherImage(Image *image,CubeInfo *cube_info,
  ExceptionInfo *exception)
{
  CacheView
    *image_view;

  MagickBooleanType
    status;

  register ssize_t
    i;

  size_t
    depth;

  if (cube_info->quantize_info->dither_method != RiemersmaDitherMethod)
    return(FloydSteinbergDither(image,cube_info,exception));
  /*
    Distribute quantization error along a Hilbert curve.
  */
  (void) ResetMagickMemory(cube_info->error,0,ErrorQueueLength*
    sizeof(*cube_info->error));
  cube_info->x=0;
  cube_info->y=0;
  i=MagickMax((ssize_t) image->columns,(ssize_t) image->rows);
  for (depth=1; i != 0; depth++)
    i>>=1;
  if ((ssize_t) (1L << depth) < MagickMax((ssize_t) image->columns,(ssize_t) image->rows))
    depth++;
  cube_info->offset=0;
  cube_info->span=(MagickSizeType) image->columns*image->rows;
  image_view=AcquireAuthenticCacheView(image,exception);
  if (depth > 1)
    Riemersma(image,image_view,cube_info,depth-1,NorthGravity,exception);
  status=RiemersmaDither(image,image_view,cube_info,ForgetGravity,exception);
  image_view=DestroyCacheView(image_view);
  return(status);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   G e t C u b e I n f o                                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  GetCubeInfo() initialize the Cube data structure.
%
%  The format of the GetCubeInfo method is:
%
%      CubeInfo GetCubeInfo(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 CubeInfo *GetCubeInfo(const QuantizeInfo *quantize_info,
  const size_t depth,const size_t maximum_colors)
{
  CubeInfo
    *cube_info;

  MagickRealType
    sum,
    weight;

  register ssize_t
    i;

  size_t
    length;

  /*
    Initialize tree to describe color cube_info.
  */
  cube_info=(CubeInfo *) AcquireMagickMemory(sizeof(*cube_info));
  if (cube_info == (CubeInfo *) NULL)
    return((CubeInfo *) NULL);
  (void) ResetMagickMemory(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=GetNodeInfo(cube_info,0,0,(NodeInfo *) NULL);
  if (cube_info->root == (NodeInfo *) NULL)
    return((CubeInfo *) 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->cache=(ssize_t *) AcquireQuantumMemory(length,
    sizeof(*cube_info->cache));
  if (cube_info->cache == (ssize_t *) NULL)
    return((CubeInfo *) NULL);
  /*
    Initialize color cache.
  */
  for (i=0; i < (ssize_t) length; i++)
    cube_info->cache[i]=(-1);
  /*
    Distribute weights along a curve of exponential decay.
  */
  weight=1.0;
  for (i=0; i < ErrorQueueLength; i++)
  {
    cube_info->weights[ErrorQueueLength-i-1]=1.0/weight;
    weight*=exp(log(((double) QuantumRange+1.0))/(ErrorQueueLength-1.0));
  }
  /*
    Normalize the weighting factors.
  */
  weight=0.0;
  for (i=0; i < ErrorQueueLength; i++)
    weight+=cube_info->weights[i];
  sum=0.0;
  for (i=0; i < ErrorQueueLength; i++)
  {
    cube_info->weights[i]/=weight;
    sum+=cube_info->weights[i];
  }
  cube_info->weights[0]+=1.0-sum;
  return(cube_info);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   G e t N o d e I n f o                                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  GetNodeInfo() allocates memory for a new node in the color cube tree and
%  presets all fields to zero.
%
%  The format of the GetNodeInfo method is:
%
%      NodeInfo *GetNodeInfo(CubeInfo *cube_info,const size_t id,
%        const size_t level,NodeInfo *parent)
%
%  A description of each parameter follows.
%
%    o node: The GetNodeInfo 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 NodeInfo *GetNodeInfo(CubeInfo *cube_info,const size_t id,
  const size_t level,NodeInfo *parent)
{
  NodeInfo
    *node_info;

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

      /*
        Allocate a new queue of nodes.
      */
      nodes=(Nodes *) AcquireMagickMemory(sizeof(*nodes));
      if (nodes == (Nodes *) NULL)
        return((NodeInfo *) NULL);
      nodes->nodes=(NodeInfo *) AcquireQuantumMemory(NodesInAList,
        sizeof(*nodes->nodes));
      if (nodes->nodes == (NodeInfo *) NULL)
        return((NodeInfo *) NULL);
      nodes->next=cube_info->node_queue;
      cube_info->node_queue=nodes;
      cube_info->next_node=nodes->nodes;
      cube_info->free_nodes=NodesInAList;
    }
  cube_info->nodes++;
  cube_info->free_nodes--;
  node_info=cube_info->next_node++;
  (void) ResetMagickMemory(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:  Thsi 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,
%        ExceptionInfo *exception)
%
%  A description of each parameter follows.
%
%    o image: the image.
%
%    o exception: return any errors or warnings in this structure.
%
*/
MagickExport MagickBooleanType GetImageQuantizeError(Image *image,
  ExceptionInfo *exception)
{
  CacheView
    *image_view;

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

  size_t
    index;

  ssize_t
    y;

  assert(image != (Image *) NULL);
  assert(image->signature == MagickSignature);
  if (image->debug != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",image->filename);
  image->total_colors=GetNumberColors(image,(FILE *) NULL,exception);
  (void) ResetMagickMemory(&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;
  image_view=AcquireVirtualCacheView(image,exception);
  for (y=0; y < (ssize_t) image->rows; y++)
  {
    register const Quantum
      *restrict p;

    register ssize_t
      x;

    p=GetCacheViewVirtualPixels(image_view,0,y,image->columns,1,exception);
    if (p == (const Quantum *) NULL)
      break;
    for (x=0; x < (ssize_t) image->columns; x++)
    {
      index=1UL*GetPixelIndex(image,p);
      if (image->matte != MagickFalse)
        {
          alpha=(MagickRealType) (QuantumScale*GetPixelAlpha(image,p));
          beta=(MagickRealType) (QuantumScale*image->colormap[index].alpha);
        }
      distance=fabs(alpha*GetPixelRed(image,p)-beta*
        image->colormap[index].red);
      mean_error_per_pixel+=distance;
      mean_error+=distance*distance;
      if (distance > maximum_error)
        maximum_error=distance;
      distance=fabs(alpha*GetPixelGreen(image,p)-beta*
        image->colormap[index].green);
      mean_error_per_pixel+=distance;
      mean_error+=distance*distance;
      if (distance > maximum_error)
        maximum_error=distance;
      distance=fabs(alpha*GetPixelBlue(image,p)-beta*
        image->colormap[index].blue);
      mean_error_per_pixel+=distance;
      mean_error+=distance*distance;
      if (distance > maximum_error)
        maximum_error=distance;
      p+=GetPixelChannels(image);
    }
  }
  image_view=DestroyCacheView(image_view);
  image->error.mean_error_per_pixel=(double) mean_error_per_pixel/area;
  image->error.normalized_mean_error=(double) QuantumScale*QuantumScale*
    mean_error/area;
  image->error.normalized_maximum_error=(double) 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)
{
  (void) LogMagickEvent(TraceEvent,GetMagickModule(),"...");
  assert(quantize_info != (QuantizeInfo *) NULL);
  (void) ResetMagickMemory(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=MagickSignature;
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%     P o s t e r i z e I m a g e                                             %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  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,ExceptionInfo *exception)
%
%  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.
%
%    o exception: return any errors or warnings in this structure.
%
*/

static inline ssize_t MagickRound(MagickRealType x)
{
  /*
    Round the fraction to nearest integer.
  */
  if (x >= 0.0)
    return((ssize_t) (x+0.5));
  return((ssize_t) (x-0.5));
}

MagickExport MagickBooleanType PosterizeImage(Image *image,const size_t levels,
  const MagickBooleanType dither,ExceptionInfo *exception)
{
#define PosterizeImageTag  "Posterize/Image"
#define PosterizePixel(pixel) (Quantum) (QuantumRange*(MagickRound( \
  QuantumScale*pixel*(levels-1)))/MagickMax((ssize_t) levels-1,1))

  CacheView
    *image_view;

  MagickBooleanType
    status;

  MagickOffsetType
    progress;

  QuantizeInfo
    *quantize_info;

  register ssize_t
    i;

  ssize_t
    y;

  assert(image != (Image *) NULL);
  assert(image->signature == MagickSignature);
  if (image->debug != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",image->filename);
  if (image->storage_class == PseudoClass)
#if defined(MAGICKCORE_OPENMP_SUPPORT)
    #pragma omp parallel for schedule(static,4) shared(progress,status)
#endif
    for (i=0; i < (ssize_t) image->colors; i++)
    {
      /*
        Posterize colormap.
      */
      if ((GetPixelRedTraits(image) & UpdatePixelTrait) != 0)
        image->colormap[i].red=(double)
          PosterizePixel(image->colormap[i].red);
      if ((GetPixelGreenTraits(image) & UpdatePixelTrait) != 0)
        image->colormap[i].green=(double)
          PosterizePixel(image->colormap[i].green);
      if ((GetPixelBlueTraits(image) & UpdatePixelTrait) != 0)
        image->colormap[i].blue=(double)
          PosterizePixel(image->colormap[i].blue);
      if ((GetPixelAlphaTraits(image) & UpdatePixelTrait) != 0)
        image->colormap[i].alpha=(double)
          PosterizePixel(image->colormap[i].alpha);
    }
  /*
    Posterize image.
  */
  status=MagickTrue;
  progress=0;
  image_view=AcquireAuthenticCacheView(image,exception);
#if defined(MAGICKCORE_OPENMP_SUPPORT)
  #pragma omp parallel for schedule(static,4) shared(progress,status)
#endif
  for (y=0; y < (ssize_t) image->rows; y++)
  {
    register Quantum
      *restrict q;

    register ssize_t
      x;

    if (status == MagickFalse)
      continue;
    q=GetCacheViewAuthenticPixels(image_view,0,y,image->columns,1,exception);
    if (q == (Quantum *) NULL)
      {
        status=MagickFalse;
        continue;
      }
    for (x=0; x < (ssize_t) image->columns; x++)
    {
      if ((GetPixelRedTraits(image) & UpdatePixelTrait) != 0)
        SetPixelRed(image,PosterizePixel(GetPixelRed(image,q)),q);
      if ((GetPixelGreenTraits(image) & UpdatePixelTrait) != 0)
        SetPixelGreen(image,PosterizePixel(GetPixelGreen(image,q)),q);
      if ((GetPixelBlueTraits(image) & UpdatePixelTrait) != 0)
        SetPixelBlue(image,PosterizePixel(GetPixelBlue(image,q)),q);
      if (((GetPixelBlackTraits(image) & UpdatePixelTrait) != 0) &&
          (image->colorspace == CMYKColorspace))
        SetPixelBlack(image,PosterizePixel(GetPixelBlack(image,q)),q);
      if (((GetPixelAlphaTraits(image) & UpdatePixelTrait) != 0) &&
          (image->matte == MagickTrue))
        SetPixelAlpha(image,PosterizePixel(GetPixelAlpha(image,q)),q);
      q+=GetPixelChannels(image);
    }
    if (SyncCacheViewAuthenticPixels(image_view,exception) == MagickFalse)
      status=MagickFalse;
    if (image->progress_monitor != (MagickProgressMonitor) NULL)
      {
        MagickBooleanType
          proceed;

#if defined(MAGICKCORE_OPENMP_SUPPORT)
        #pragma omp critical (MagickCore_PosterizeImage)
#endif
        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+1);
  quantize_info->dither=dither;
  quantize_info->tree_depth=MaxTreeDepth;
  status=QuantizeImage(quantize_info,image,exception);
  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(const Image *image,CubeInfo *cube_info,
%        const NodeInfo *node_info)
%
%  A description of each parameter follows.
%
%    o image: the image.
%
%    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(const Image *image,CubeInfo *cube_info,
  const NodeInfo *node_info)
{
  NodeInfo
    *parent;

  register ssize_t
    i;

  size_t
    number_children;

  /*
    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] != (NodeInfo *) NULL)
      PruneChild(image,cube_info,node_info->child[i]);
  /*
    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.alpha+=node_info->total_color.alpha;
  parent->child[node_info->id]=(NodeInfo *) 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(const Image *image,CubeInfo *cube_info,
%        const NodeInfo *node_info)
%
%  A description of each parameter follows.
%
%    o image: the image.
%
%    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(const Image *image,CubeInfo *cube_info,
  const NodeInfo *node_info)
{
  register ssize_t
    i;

  size_t
    number_children;

  /*
    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] != (NodeInfo *) NULL)
      PruneLevel(image,cube_info,node_info->child[i]);
  if (node_info->level == cube_info->depth)
    PruneChild(image,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(const Image *image,CubeInfo *cube_info,
%        const NodeInfo *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(const Image *image,CubeInfo *cube_info,
  const NodeInfo *node_info)
{
  register ssize_t
    i;

  size_t
    number_children;

  /*
    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] != (NodeInfo *) NULL)
      PruneToCubeDepth(image,cube_info,node_info->child[i]);
  if (node_info->level > cube_info->depth)
    PruneChild(image,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,ExceptionInfo *exception)
%
%  A description of each parameter follows:
%
%    o quantize_info: Specifies a pointer to an QuantizeInfo structure.
%
%    o image: the image.
%
%    o exception: return any errors or warnings in this structure.
%
*/

static MagickBooleanType DirectToColormapImage(Image *image,
  ExceptionInfo *exception)
{
  CacheView
    *image_view;

  MagickBooleanType
    status;

  register ssize_t
    i;

  size_t
    number_colors;

  ssize_t
    y;

  status=MagickTrue;
  number_colors=(size_t) (image->columns*image->rows);
  if (AcquireImageColormap(image,number_colors,exception) == MagickFalse)
    ThrowBinaryException(ResourceLimitError,"MemoryAllocationFailed",
      image->filename);
  if (image->colors != number_colors)
    return(MagickFalse);
  i=0;
  image_view=AcquireAuthenticCacheView(image,exception);
  for (y=0; y < (ssize_t) image->rows; y++)
  {
    MagickBooleanType
      proceed;

    register Quantum
      *restrict q;

    register ssize_t
      x;

    q=GetCacheViewAuthenticPixels(image_view,0,y,image->columns,1,exception);
    if (q == (Quantum *) NULL)
      break;
    for (x=0; x < (ssize_t) image->columns; x++)
    {
      image->colormap[i].red=(double) GetPixelRed(image,q);
      image->colormap[i].green=(double) GetPixelGreen(image,q);
      image->colormap[i].blue=(double) GetPixelBlue(image,q);
      image->colormap[i].alpha=(double) GetPixelAlpha(image,q);
      SetPixelIndex(image,(Quantum) i,q);
      i++;
      q+=GetPixelChannels(image);
    }
    if (SyncCacheViewAuthenticPixels(image_view,exception) == MagickFalse)
      break;
    proceed=SetImageProgress(image,AssignImageTag,(MagickOffsetType) y,
      image->rows);
    if (proceed == MagickFalse)
      status=MagickFalse;
  }
  image_view=DestroyCacheView(image_view);
  return(status);
}

MagickExport MagickBooleanType QuantizeImage(const QuantizeInfo *quantize_info,
  Image *image,ExceptionInfo *exception)
{
  CubeInfo
    *cube_info;

  MagickBooleanType
    status;

  size_t
    depth,
    maximum_colors;

  assert(quantize_info != (const QuantizeInfo *) NULL);
  assert(quantize_info->signature == MagickSignature);
  assert(image != (Image *) NULL);
  assert(image->signature == MagickSignature);
  if (image->debug != 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->columns*image->rows) <= maximum_colors)
    (void) DirectToColormapImage(image,exception);
  if ((IsImageGray(image,exception) != MagickFalse) &&
      (image->matte == MagickFalse))
    (void) SetGrayscaleImage(image,exception);
  if ((image->storage_class == PseudoClass) &&
      (image->colors <= maximum_colors))
    return(MagickTrue);
  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--;
    }
  /*
    Initialize color cube.
  */
  cube_info=GetCubeInfo(quantize_info,depth,maximum_colors);
  if (cube_info == (CubeInfo *) NULL)
    ThrowBinaryException(ResourceLimitError,"MemoryAllocationFailed",
      image->filename);
  status=ClassifyImageColors(cube_info,image,exception);
  if (status != MagickFalse)
    {
      /*
        Reduce the number of colors in the image.
      */
      ReduceImageColors(image,cube_info);
      status=AssignImageColors(image,cube_info,exception);
    }
  DestroyCubeInfo(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,ExceptionInfo *exception)
%
%  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.
%
%    o exception: return any errors or warnings in this structure.
%
*/
MagickExport MagickBooleanType QuantizeImages(const QuantizeInfo *quantize_info,
  Image *images,ExceptionInfo *exception)
{
  CubeInfo
    *cube_info;

  Image
    *image;

  MagickBooleanType
    proceed,
    status;

  MagickProgressMonitor
    progress_monitor;

  register ssize_t
    i;

  size_t
    depth,
    maximum_colors,
    number_images;

  assert(quantize_info != (const QuantizeInfo *) NULL);
  assert(quantize_info->signature == MagickSignature);
  assert(images != (Image *) NULL);
  assert(images->signature == MagickSignature);
  if (images->debug != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",images->filename);
  if (GetNextImageInList(images) == (Image *) NULL)
    {
      /*
        Handle a single image with QuantizeImage.
      */
      status=QuantizeImage(quantize_info,images,exception);
      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=GetCubeInfo(quantize_info,depth,maximum_colors);
  if (cube_info == (CubeInfo *) NULL)
    {
      (void) ThrowMagickException(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,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,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);
      }
    }
  DestroyCubeInfo(cube_info);
  return(status);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   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(const Image *image,CubeInfo *cube_info,const NodeInfo *node_info)
%
%  A description of each parameter follows.
%
%    o image: the image.
%
%    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(const Image *image,CubeInfo *cube_info,
  const NodeInfo *node_info)
{
  register ssize_t
    i;

  size_t
    number_children;

  /*
    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] != (NodeInfo *) NULL)
      Reduce(image,cube_info,node_info->child[i]);
  if (node_info->quantize_error <= cube_info->pruning_threshold)
    PruneChild(image,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,CubeInfo *cube_info)
%
%  A description of each parameter follows.
%
%    o image: the image.
%
%    o cube_info: A pointer to the Cube structure.
%
*/
static void ReduceImageColors(const Image *image,CubeInfo *cube_info)
{
#define ReduceImageTag  "Reduce/Image"

  MagickBooleanType
    proceed;

  MagickOffsetType
    offset;

  size_t
    span;

  cube_info->next_threshold=0.0;
  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(image,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 a dither of the colors
%  provided.
%
%  The format of the RemapImage method is:
%
%      MagickBooleanType RemapImage(const QuantizeInfo *quantize_info,
%        Image *image,const Image *remap_image,ExceptionInfo *exception)
%
%  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.
%
%    o exception: return any errors or warnings in this structure.
%
*/
MagickExport MagickBooleanType RemapImage(const QuantizeInfo *quantize_info,
  Image *image,const Image *remap_image,ExceptionInfo *exception)
{
  CubeInfo
    *cube_info;

  MagickBooleanType
    status;

  /*
    Initialize color cube.
  */
  assert(image != (Image *) NULL);
  assert(image->signature == MagickSignature);
  if (image->debug != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",image->filename);
  assert(remap_image != (Image *) NULL);
  assert(remap_image->signature == MagickSignature);
  cube_info=GetCubeInfo(quantize_info,MaxTreeDepth,
    quantize_info->number_colors);
  if (cube_info == (CubeInfo *) NULL)
    ThrowBinaryException(ResourceLimitError,"MemoryAllocationFailed",
      image->filename);
  status=ClassifyImageColors(cube_info,remap_image,exception);
  if (status != MagickFalse)
    {
      /*
        Classify image colors from the reference image.
      */
      cube_info->quantize_info->number_colors=cube_info->colors;
      status=AssignImageColors(image,cube_info,exception);
    }
  DestroyCubeInfo(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,ExceptionInfo *exception)
%
%  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.
%
%    o exception: return any errors or warnings in this structure.
%
*/
MagickExport MagickBooleanType RemapImages(const QuantizeInfo *quantize_info,
  Image *images,const Image *remap_image,ExceptionInfo *exception)
{
  CubeInfo
    *cube_info;

  Image
    *image;

  MagickBooleanType
    status;

  assert(images != (Image *) NULL);
  assert(images->signature == MagickSignature);
  if (images->debug != 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,exception);
      return(status);
    }
  /*
    Classify image colors from the reference image.
  */
  cube_info=GetCubeInfo(quantize_info,MaxTreeDepth,
    quantize_info->number_colors);
  if (cube_info == (CubeInfo *) NULL)
    ThrowBinaryException(ResourceLimitError,"MemoryAllocationFailed",
      image->filename);
  status=ClassifyImageColors(cube_info,remap_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,exception);
        if (status == MagickFalse)
          break;
      }
    }
  DestroyCubeInfo(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,ExceptionInfo *exeption)
%
%  A description of each parameter follows:
%
%    o image: The image.
%
%    o exception: return any errors or warnings in this structure.
%
*/

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

static int IntensityCompare(const void *x,const void *y)
{
  PixelInfo
    *color_1,
    *color_2;

  ssize_t
    intensity;

  color_1=(PixelInfo *) x;
  color_2=(PixelInfo *) y;
  intensity=GetPixelInfoIntensity(color_1)-(ssize_t)
    GetPixelInfoIntensity(color_2);
  return((int) intensity);
}

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

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

  MagickBooleanType
    status;

  PixelInfo
    *colormap;

  register ssize_t
    i;

  ssize_t
    *colormap_index,
    j,
    y;

  assert(image != (Image *) NULL);
  assert(image->signature == MagickSignature);
  if (image->type != GrayscaleType)
    (void) TransformImageColorspace(image,GRAYColorspace,exception);
  colormap_index=(ssize_t *) AcquireQuantumMemory(MaxMap+1,
    sizeof(*colormap_index));
  if (colormap_index == (ssize_t *) NULL)
    ThrowBinaryException(ResourceLimitError,"MemoryAllocationFailed",
      image->filename);
  if (image->storage_class != PseudoClass)
    {
      for (i=0; i <= (ssize_t) MaxMap; i++)
        colormap_index[i]=(-1);
      if (AcquireImageColormap(image,MaxMap+1,exception) == MagickFalse)
        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,4) shared(status)
#endif
      for (y=0; y < (ssize_t) image->rows; y++)
      {
        register Quantum
          *restrict q;

        register ssize_t
          x;

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

          intensity=ScaleQuantumToMap(GetPixelRed(image,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=(double)
                    GetPixelRed(image,q);
                  image->colormap[image->colors].green=(double)
                    GetPixelGreen(image,q);
                  image->colormap[image->colors].blue=(double)
                    GetPixelBlue(image,q);
                  image->colors++;
               }
            }
          SetPixelIndex(image,(Quantum) 
            colormap_index[intensity],q);
          q+=GetPixelChannels(image);
        }
        if (SyncCacheViewAuthenticPixels(image_view,exception) == MagickFalse)
          status=MagickFalse;
      }
      image_view=DestroyCacheView(image_view);
    }
  for (i=0; i < (ssize_t) image->colors; i++)
    image->colormap[i].alpha=(double) i;
  qsort((void *) image->colormap,image->colors,sizeof(PixelInfo),
    IntensityCompare);
  colormap=(PixelInfo *) AcquireQuantumMemory(image->colors,
    sizeof(*colormap));
  if (colormap == (PixelInfo *) NULL)
    ThrowBinaryException(ResourceLimitError,"MemoryAllocationFailed",
      image->filename);
  j=0;
  colormap[j]=image->colormap[0];
  for (i=0; i < (ssize_t) image->colors; i++)
  {
    if (IsPixelInfoEquivalent(&colormap[j],&image->colormap[i]) == MagickFalse)
      {
        j++;
        colormap[j]=image->colormap[i];
      }
    colormap_index[(ssize_t) image->colormap[i].alpha]=j;
  }
  image->colors=(size_t) (j+1);
  image->colormap=(PixelInfo *) RelinquishMagickMemory(image->colormap);
  image->colormap=colormap;
  status=MagickTrue;
  image_view=AcquireAuthenticCacheView(image,exception);
#if defined(MAGICKCORE_OPENMP_SUPPORT)
  #pragma omp parallel for schedule(static,4) shared(status)
#endif
  for (y=0; y < (ssize_t) image->rows; y++)
  {
    register Quantum
      *restrict q;

    register ssize_t
      x;

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