magick/morphology.c

/*
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%    M   M    OOO    RRRR   PPPP   H   H   OOO   L       OOO    GGGG  Y   Y   %
%    MM MM   O   O   R   R  P   P  H   H  O   O  L      O   O  G       Y Y    %
%    M M M   O   O   RRRR   PPPP   HHHHH  O   O  L      O   O  G GGG    Y     %
%    M   M   O   O   R R    P      H   H  O   O  L      O   O  G   G    Y     %
%    M   M    OOO    R  R   P      H   H   OOO   LLLLL   OOO    GGG     Y     %
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%                        MagickCore Morphology Methods                        %
%                                                                             %
%                              Software Design                                %
%                              Anthony Thyssen                                %
%                               January 2010                                  %
%                                                                             %
%                                                                             %
%  Copyright 1999-2010 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.                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Morpology is the the application of various kernals, of any size and even
% shape, to a image in various ways (typically binary, but not always).
%
% Convolution (weighted sum or average) is just one specific type of
% morphology. Just one that is very common for image bluring and sharpening
% effects.  Not only 2D Gaussian blurring, but also 2-pass 1D Blurring.
%
% This module provides not only a general morphology function, and the ability
% to apply more advanced or iterative morphologies, but also functions for the
% generation of many different types of kernel arrays from user supplied
% arguments. Prehaps even the generation of a kernel from a small image.
*/

/*
  Include declarations.
*/
#include "magick/studio.h"
#include "magick/artifact.h"
#include "magick/cache-view.h"
#include "magick/color-private.h"
#include "magick/enhance.h"
#include "magick/exception.h"
#include "magick/exception-private.h"
#include "magick/gem.h"
#include "magick/hashmap.h"
#include "magick/image.h"
#include "magick/image-private.h"
#include "magick/list.h"
#include "magick/magick.h"
#include "magick/memory_.h"
#include "magick/monitor-private.h"
#include "magick/morphology.h"
#include "magick/option.h"
#include "magick/pixel-private.h"
#include "magick/prepress.h"
#include "magick/quantize.h"
#include "magick/registry.h"
#include "magick/semaphore.h"
#include "magick/splay-tree.h"
#include "magick/statistic.h"
#include "magick/string_.h"
#include "magick/string-private.h"
#include "magick/token.h"


/*
 * The following test is for special floating point numbers of value NaN (not
 * a number), that may be used within a Kernel Definition.  NaN's are defined
 * as part of the IEEE standard for floating point number representation.
 *
 * These are used a Kernel value of NaN means that that kernal position is not
 * part of the normal convolution or morphology process, and thus allowing the
 * use of 'shaped' kernels.
 *
 * Special Properities Two NaN's are never equal, even if they are from the
 * same variable That is the IsNaN() macro is only true if the value is NaN.
 */
#define IsNan(a)   ((a)!=(a))

/*
 * Other global definitions used by module
 */
static inline double MagickMin(const double x,const double y)
{
  return( x < y ? x : y);
}
static inline double MagickMax(const double x,const double y)
{
  return( x > y ? x : y);
}
#define Minimize(assign,value) assign=MagickMin(assign,value)
#define Maximize(assign,value) assign=MagickMax(assign,value)

/* Currently these are internal to this module
 * Eventually these may become 'private' to library
 * OR may become externally available to API's
 */
static MagickExport void
  RotateKernel(KernelInfo *, double),
  ScaleKernel(KernelInfo *, double),
  ShowKernel(KernelInfo *),
  ZeroKernelNans(KernelInfo *);


/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%     A c q u i r e K e r n e l I n f o                                       %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  AcquireKernelInfo() takes the given string (generally supplied by the
%  user) and converts it into a Morphology/Convolution Kernel.  This allows
%  users to specify a kernel from a number of pre-defined kernels, or to fully
%  specify their own kernel for a specific Convolution or Morphology
%  Operation.
%
%  The kernel so generated can be any rectangular array of floating point
%  values (doubles) with the 'control point' or 'pixel being affected'
%  anywhere within that array of values.
%
%  Previously IM was restricted to a square of odd size using the exact
%  center as origin, this is no longer the case, and any rectangular kernel
%  with any value being declared the origin. This in turn allows the use of
%  highly asymmetrical kernels.
%
%  The floating point values in the kernel can also include a special value
%  known as 'nan' or 'not a number' to indicate that this value is not part
%  of the kernel array. This allows you to shaped the kernel within its
%  rectangular area. That is 'nan' values provide a 'mask' for the kernel
%  shape.  However at least one non-nan value must be provided for correct
%  working of a kernel.
%
%  The returned kernel should be free using the DestroyKernelInfo() when you
%  are finished with it.
%
%  Input kernel defintion strings can consist of any of three types.
%
%    "name:args"
%         Select from one of the built in kernels, using the name and
%         geometry arguments supplied.  See AcquireKernelBuiltIn()
%
%    "WxH[+X+Y]:num, num, num ..."
%         a kernal of size W by H, with W*H floating point numbers following.
%         the 'center' can be optionally be defined at +X+Y (such that +0+0
%         is top left corner). If not defined the pixel in the center, for
%         odd sizes, or to the immediate top or left of center for even sizes
%         is automatically selected.
%
%    "num, num, num, num, ..."
%         list of floating point numbers defining an 'old style' odd sized
%         square kernel.  At least 9 values should be provided for a 3x3
%         square kernel, 25 for a 5x5 square kernel, 49 for 7x7, etc.
%         Values can be space or comma separated.  This is not recommended.
%
%  Note that 'name' kernels will start with an alphabetic character while the
%  new kernel specification has a ':' character in its specification string.
%  If neither is the case, it is assumed an old style of a simple list of
%  numbers generating a odd-sized square kernel has been given.
%
%  The format of the AcquireKernal method is:
%
%      KernelInfo *AcquireKernelInfo(const char *kernel_string)
%
%  A description of each parameter follows:
%
%    o kernel_string: the Morphology/Convolution kernel wanted.
%
*/

MagickExport KernelInfo *AcquireKernelInfo(const char *kernel_string)
{
  KernelInfo
    *kernel;

  char
    token[MaxTextExtent];

  register unsigned long
    i;

  const char
    *p;

  MagickStatusType
    flags;

  GeometryInfo
    args;

  double
    nan = sqrt((double)-1.0);  /* Special Value : Not A Number */

  assert(kernel_string != (const char *) NULL);
  SetGeometryInfo(&args);

  /* does it start with an alpha - Return a builtin kernel */
  GetMagickToken(kernel_string,&p,token);
  if ( isalpha((int)token[0]) )
  {
    long
      type;

    type=ParseMagickOption(MagickKernelOptions,MagickFalse,token);
    if ( type < 0 || type == UserDefinedKernel )
      return((KernelInfo *)NULL);

    while (((isspace((int) ((unsigned char) *p)) != 0) ||
           (*p == ',') || (*p == ':' )) && (*p != '\0'))
      p++;
    flags = ParseGeometry(p, &args);

    /* special handling of missing values in input string */
    if ( type == RectangleKernel ) {
      if ( (flags & WidthValue) == 0 ) /* if no width then */
        args.rho = args.sigma;         /* then  width = height */
      if ( args.rho < 1.0 )            /* if width too small */
         args.rho = 3;                 /* then  width = 3 */
      if ( args.sigma < 1.0 )          /* if height too small */
        args.sigma = args.rho;         /* then  height = width */
      if ( (flags & XValue) == 0 )     /* center offset if not defined */
        args.xi = (double)(((long)args.rho-1)/2);
      if ( (flags & YValue) == 0 )
        args.psi = (double)(((long)args.sigma-1)/2);
    }

    return(AcquireKernelBuiltIn((KernelInfoType)type, &args));
  }

  kernel=(KernelInfo *) AcquireMagickMemory(sizeof(*kernel));
  if (kernel == (KernelInfo *)NULL)
    return(kernel);
  (void) ResetMagickMemory(kernel,0,sizeof(*kernel));
  kernel->type = UserDefinedKernel;
  kernel->signature = MagickSignature;

  /* Has a ':' in argument - New user kernel specification */
  p = strchr(kernel_string, ':');
  if ( p != (char *) NULL)
    {
      /* ParseGeometry() needs the geometry separated! -- Arrgghh */
      memcpy(token, kernel_string, p-kernel_string);
      token[p-kernel_string] = '\0';
      flags = ParseGeometry(token, &args);

      /* Size handling and checks of geometry settings */
      if ( (flags & WidthValue) == 0 ) /* if no width then */
        args.rho = args.sigma;         /* then  width = height */
      if ( args.rho < 1.0 )            /* if width too small */
         args.rho = 1.0;               /* then  width = 1 */
      if ( args.sigma < 1.0 )          /* if height too small */
        args.sigma = args.rho;         /* then  height = width */
      kernel->width = (unsigned long)args.rho;
      kernel->height = (unsigned long)args.sigma;

      /* Offset Handling and Checks */
      if ( args.xi  < 0.0 || args.psi < 0.0 )
        return(DestroyKernelInfo(kernel));
      kernel->offset_x = ((flags & XValue)!=0) ? (unsigned long)args.xi
                                               : (kernel->width-1)/2;
      kernel->offset_y = ((flags & YValue)!=0) ? (unsigned long)args.psi
                                               : (kernel->height-1)/2;
      if ( kernel->offset_x >= kernel->width ||
           kernel->offset_y >= kernel->height )
        return(DestroyKernelInfo(kernel));

      p++; /* advance beyond the ':' */
    }
  else
    { /* ELSE - Old old kernel specification, forming odd-square kernel */
      /* count up number of values given */
      p=(const char *) kernel_string;
      while ((isspace((int) ((unsigned char) *p)) != 0) || (*p == '\''))
        p++;  /* ignore "'" chars for convolve filter usage - Cristy */
      for (i=0; *p != '\0'; i++)
      {
        GetMagickToken(p,&p,token);
        if (*token == ',')
          GetMagickToken(p,&p,token);
      }
      /* set the size of the kernel - old sized square */
      kernel->width = kernel->height= (unsigned long) sqrt((double) i+1.0);
      kernel->offset_x = kernel->offset_y = (kernel->width-1)/2;
      p=(const char *) kernel_string;
      while ((isspace((int) ((unsigned char) *p)) != 0) || (*p == '\''))
        p++;  /* ignore "'" chars for convolve filter usage - Cristy */
    }

  /* Read in the kernel values from rest of input string argument */
  kernel->values=(double *) AcquireQuantumMemory(kernel->width,
                        kernel->height*sizeof(double));
  if (kernel->values == (double *) NULL)
    return(DestroyKernelInfo(kernel));

  kernel->value_min = +MagickHuge;
  kernel->value_max = -MagickHuge;
  kernel->range_neg = kernel->range_pos = 0.0;
  for (i=0; (i < kernel->width*kernel->height) && (*p != '\0'); i++)
  {
    GetMagickToken(p,&p,token);
    if (*token == ',')
      GetMagickToken(p,&p,token);
    if (    LocaleCompare("nan",token) == 0
         || LocaleCompare("-",token) == 0 ) {
      kernel->values[i] = nan; /* do not include this value in kernel */
    }
    else {
      kernel->values[i] = StringToDouble(token);
      ( kernel->values[i] < 0)
          ?  ( kernel->range_neg += kernel->values[i] )
          :  ( kernel->range_pos += kernel->values[i] );
      Minimize(kernel->value_min, kernel->values[i]);
      Maximize(kernel->value_max, kernel->values[i]);
    }
  }
  /* check that we recieved at least one real (non-nan) value! */
  if ( kernel->value_min == MagickHuge )
    return(DestroyKernelInfo(kernel));

  /* This should not be needed for a fully defined kernel
   * Perhaps an error should be reported instead!
   * Kept for backward compatibility.
   */
  if ( i < kernel->width*kernel->height ) {
    Minimize(kernel->value_min, kernel->values[i]);
    Maximize(kernel->value_max, kernel->values[i]);
    for ( ; i < kernel->width*kernel->height; i++)
      kernel->values[i]=0.0;
  }

  return(kernel);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%     A c q u i r e K e r n e l B u i l t I n                                 %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  AcquireKernelBuiltIn() returned one of the 'named' built-in types of
%  kernels used for special purposes such as gaussian blurring, skeleton
%  pruning, and edge distance determination.
%
%  They take a KernelType, and a set of geometry style arguments, which were
%  typically decoded from a user supplied string, or from a more complex
%  Morphology Method that was requested.
%
%  The format of the AcquireKernalBuiltIn method is:
%
%      KernelInfo *AcquireKernelBuiltIn(const KernelInfoType type,
%           const GeometryInfo args)
%
%  A description of each parameter follows:
%
%    o type: the pre-defined type of kernel wanted
%
%    o args: arguments defining or modifying the kernel
%
%  Convolution Kernels
%
%    Gaussian  "[{radius}]x{sigma}"
%       Generate a two-dimentional gaussian kernel, as used by -gaussian
%       A sigma is required, (with the 'x'), due to historical reasons.
%
%       NOTE: that the 'radius' is optional, but if provided can limit (clip)
%       the final size of the resulting kernel to a square 2*radius+1 in size.
%       The radius should be at least 2 times that of the sigma value, or
%       sever clipping and aliasing may result.  If not given or set to 0 the
%       radius will be determined so as to produce the best minimal error
%       result, which is usally much larger than is normally needed.
%
%    Blur  "[{radius}]x{sigma}[+angle]"
%       As per Gaussian, but generates a 1 dimensional or linear gaussian
%       blur, at the angle given (current restricted to orthogonal angles).
%       If a 'radius' is given the kernel is clipped to a width of 2*radius+1.
%
%       NOTE that two such blurs perpendicular to each other is equivelent to
%       -blur and the previous gaussian, but is often 10 or more times faster.
%
%    Comet  "[{width}]x{sigma}[+angle]"
%       Blur in one direction only, mush like how a bright object leaves
%       a comet like trail.  The Kernel is actually half a gaussian curve,
%       Adding two such blurs in oppiste directions produces a Linear Blur.
%
%       NOTE: that the first argument is the width of the kernel and not the
%       radius of the kernel.
%
%    # Still to be implemented...
%    #
%    # Laplacian  "{radius}x{sigma}"
%    #    Laplacian (a mexican hat like) Function
%    #
%    # LOG  "{radius},{sigma1},{sigma2}
%    #    Laplacian of Gaussian
%    #
%    # DOG  "{radius},{sigma1},{sigma2}
%    #    Difference of Gaussians
%
%  Boolean Kernels
%
%    Rectangle "{geometry}"
%       Simply generate a rectangle of 1's with the size given. You can also
%       specify the location of the 'control point', otherwise the closest
%       pixel to the center of the rectangle is selected.
%
%       Properly centered and odd sized rectangles work the best.
%
%    Diamond  "[{radius}]"
%       Generate a diamond shaped kernal with given radius to the points.
%       Kernel size will again be radius*2+1 square and defaults to radius 1,
%       generating a 3x3 kernel that is slightly larger than a square.
%
%    Square  "[{radius}]"
%       Generate a square shaped kernel of size radius*2+1, and defaulting
%       to a 3x3 (radius 1).
%
%       Note that using a larger radius for the "Square" or the "Diamond"
%       is also equivelent to iterating the basic morphological method
%       that many times. However However iterating with the smaller radius 1
%       default is actually faster than using a larger kernel radius.
%
%    Disk   "[{radius}]
%       Generate a binary disk of the radius given, radius may be a float.
%       Kernel size will be ceil(radius)*2+1 square.
%       NOTE: Here are some disk shapes of specific interest
%          "disk:1"    => "diamond" or "cross:1"
%          "disk:1.5"  => "square"
%          "disk:2"    => "diamond:2"
%          "disk:2.5"  => a general disk shape of radius 2
%          "disk:2.9"  => "square:2"
%          "disk:3.5"  => default - octagonal/disk shape of radius 3
%          "disk:4.2"  => roughly octagonal shape of radius 4
%          "disk:4.3"  => a general disk shape of radius 4
%       After this all the kernel shape becomes more and more circular.
%
%       Because a "disk" is more circular when using a larger radius, using a
%       larger radius is preferred over iterating the morphological operation.
%
%    Plus  "[{radius}]"
%       Generate a kernel in the shape of a 'plus' sign. The length of each
%       arm is also the radius, which defaults to 2.
%
%       This kernel is not a good general morphological kernel, but is used
%       more for highlighting and marking any single pixels in an image using,
%       a "Dilate" or "Erode" method as appropriate.
%
%       NOTE: "plus:1" is equivelent to a "Diamond" kernel.
%
%       Note that unlike other kernels iterating a plus does not produce the
%       same result as using a larger radius for the cross.
%
%  Distance Measuring Kernels
%
%    Chebyshev "[{radius}][x{scale}]"   largest x or y distance (default r=1)
%    Manhatten "[{radius}][x{scale}]"   square grid distance    (default r=1)
%    Euclidean "[{radius}][x{scale}]"   direct distance         (default r=1)
%
%       Different types of distance measuring methods, which are used with the
%       a 'Distance' morphology method for generating a gradient based on
%       distance from an edge of a binary shape, though there is a technique
%       for handling a anti-aliased shape.
%
%       Chebyshev Distance (also known as Tchebychev Distance) is a value of
%       one to any neighbour, orthogonal or diagonal. One why of thinking of
%       it is the number of squares a 'King' or 'Queen' in chess needs to
%       traverse reach any other position on a chess board.  It results in a
%       'square' like distance function, but one where diagonals are closer
%       than expected.
%
%       Manhatten Distance (also known as Rectilinear Distance, or the Taxi
%       Cab metric), is the distance needed when you can only travel in
%       orthogonal (horizontal or vertical) only.  It is the distance a 'Rook'
%       in chess would travel. It results in a diamond like distances, where
%       diagonals are further than expected.
%
%       Euclidean Distance is the 'direct' or 'as the crow flys distance.
%       However by default the kernel size only has a radius of 1, which
%       limits the distance to 'Knight' like moves, with only orthogonal and
%       diagonal measurements being correct.  As such for the default kernel
%       you will get octagonal like distance function, which is reasonally
%       accurate.
%
%       However if you use a larger radius such as "Euclidean:4" you will
%       get a much smoother distance gradient from the edge of the shape.
%       Of course a larger kernel is slower to use, and generally not needed.
%
%       To allow the use of fractional distances that you get with diagonals
%       the actual distance is scaled by a fixed value which the user can
%       provide.  This is not actually nessary for either ""Chebyshev" or
%       "Manhatten" distance kernels, but is done for all three distance
%       kernels.  If no scale is provided it is set to a value of 100,
%       allowing for a maximum distance measurement of 655 pixels using a Q16
%       version of IM, from any edge.  However for small images this can
%       result in quite a dark gradient.
%
%       See the 'Distance' Morphological Method, for information of how it is
%       applied.
%
*/

MagickExport KernelInfo *AcquireKernelBuiltIn(const KernelInfoType type,
   const GeometryInfo *args)
{
  KernelInfo
    *kernel;

  register unsigned long
    i;

  register long
    u,
    v;

  double
    nan = sqrt((double)-1.0);  /* Special Value : Not A Number */

  kernel=(KernelInfo *) AcquireMagickMemory(sizeof(*kernel));
  if (kernel == (KernelInfo *) NULL)
    return(kernel);
  (void) ResetMagickMemory(kernel,0,sizeof(*kernel));
  kernel->value_min = kernel->value_max = 0.0;
  kernel->range_neg = kernel->range_pos = 0.0;
  kernel->type = type;
  kernel->signature = MagickSignature;

  switch(type) {
    /* Convolution Kernels */
    case GaussianKernel:
      { double
          sigma = fabs(args->sigma);

        sigma = (sigma <= MagickEpsilon) ? 1.0 : sigma;

        kernel->width = kernel->height =
                            GetOptimalKernelWidth2D(args->rho,sigma);
        kernel->offset_x = kernel->offset_y = (kernel->width-1)/2;
        kernel->range_neg = kernel->range_pos = 0.0;
        kernel->values=(double *) AcquireQuantumMemory(kernel->width,
                              kernel->height*sizeof(double));
        if (kernel->values == (double *) NULL)
          return(DestroyKernelInfo(kernel));

        sigma = 2.0*sigma*sigma; /* simplify the expression */
        for ( i=0, v=-kernel->offset_y; v <= (long)kernel->offset_y; v++)
          for ( u=-kernel->offset_x; u <= (long)kernel->offset_x; u++, i++)
            kernel->range_pos += (
              kernel->values[i] =
                 exp(-((double)(u*u+v*v))/sigma)
                       /*  / (MagickPI*sigma)  */ );
        kernel->value_min = 0;
        kernel->value_max = kernel->values[
                         kernel->offset_y*kernel->width+kernel->offset_x ];

        ScaleKernel(kernel, 0.0); /* Normalize Kernel Values */

        break;
      }
    case BlurKernel:
      { double
          sigma = fabs(args->sigma);

        sigma = (sigma <= MagickEpsilon) ? 1.0 : sigma;

        kernel->width = GetOptimalKernelWidth1D(args->rho,sigma);
        kernel->offset_x = (kernel->width-1)/2;
        kernel->height = 1;
        kernel->offset_y = 0;
        kernel->range_neg = kernel->range_pos = 0.0;
        kernel->values=(double *) AcquireQuantumMemory(kernel->width,
                              kernel->height*sizeof(double));
        if (kernel->values == (double *) NULL)
          return(DestroyKernelInfo(kernel));

#if 1
#define KernelRank 3
        /* Formula derived from GetBlurKernel() in "effect.c" (plus bug fix).
        ** It generates a gaussian 3 times the width, and compresses it into
        ** the expected range.  This produces a closer normalization of the
        ** resulting kernel, especially for very low sigma values.
        ** As such while wierd it is prefered.
        **
        ** I am told this method originally came from Photoshop.
        */
        sigma *= KernelRank;                /* simplify expanded curve */
        v = (kernel->width*KernelRank-1)/2; /* start/end points to fit range */
        (void) ResetMagickMemory(kernel->values,0, (size_t)
                       kernel->width*sizeof(double));
        for ( u=-v; u <= v; u++) {
          kernel->values[(u+v)/KernelRank] +=
                exp(-((double)(u*u))/(2.0*sigma*sigma))
                       /*   / (MagickSQ2PI*sigma/KernelRank)  */ ;
        }
        for (i=0; i < kernel->width; i++)
          kernel->range_pos += kernel->values[i];
#else
        for ( i=0, u=-kernel->offset_x; i < kernel->width; i++, u++)
          kernel->range_pos += (
              kernel->values[i] =
                exp(-((double)(u*u))/(2.0*sigma*sigma))
                       /*  / (MagickSQ2PI*sigma)  */ );
#endif
        kernel->value_min = 0;
        kernel->value_max = kernel->values[ kernel->offset_x ];
        /* Note that neither methods above generate a normalized kernel,
        ** though it gets close. The kernel may be 'clipped' by a user defined
        ** radius, producing a smaller (darker) kernel.  Also for very small
        ** sigma's (> 0.1) the central value becomes larger than one, and thus
        ** producing a very bright kernel.
        */

        /* Normalize the 1D Gaussian Kernel
        **
        ** Because of this the divisor in the above kernel generator is
        ** not needed, so is not done above.
        */
        ScaleKernel(kernel, 0.0); /* Normalize Kernel Values */

        /* rotate the kernel by given angle */
        RotateKernel(kernel, args->xi);
        break;
      }
    case CometKernel:
      { double
          sigma = fabs(args->sigma);

        sigma = (sigma <= MagickEpsilon) ? 1.0 : sigma;

        if ( args->rho < 1.0 )
          kernel->width = GetOptimalKernelWidth1D(args->rho,sigma);
        else
          kernel->width = (unsigned long)args->rho;
        kernel->offset_x = kernel->offset_y = 0;
        kernel->height = 1;
        kernel->range_neg = kernel->range_pos = 0.0;
        kernel->values=(double *) AcquireQuantumMemory(kernel->width,
                              kernel->height*sizeof(double));
        if (kernel->values == (double *) NULL)
          return(DestroyKernelInfo(kernel));

        /* A comet blur is half a gaussian curve, so that the object is
        ** blurred in one direction only.  This may not be quite the right
        ** curve so may change in the future. The function must be normalised.
        */
#if 1
#define KernelRank 3
        sigma *= KernelRank;                /* simplify expanded curve */
        v = kernel->width*KernelRank; /* start/end points to fit range */
        (void) ResetMagickMemory(kernel->values,0, (size_t)
                       kernel->width*sizeof(double));
        for ( u=0; u < v; u++) {
          kernel->values[u/KernelRank] +=
               exp(-((double)(u*u))/(2.0*sigma*sigma))
                       /*   / (MagickSQ2PI*sigma/KernelRank)  */ ;
        }
        for (i=0; i < kernel->width; i++)
          kernel->range_pos += kernel->values[i];
#else
        for ( i=0; i < kernel->width; i++)
          kernel->range_pos += (
            kernel->values[i] =
               exp(-((double)(i*i))/(2.0*sigma*sigma))
                       /*  / (MagickSQ2PI*sigma)  */ );
#endif
        kernel->value_min = 0;
        kernel->value_max = kernel->values[0];

        ScaleKernel(kernel, 0.0); /* Normalize Kernel Values */
        RotateKernel(kernel, args->xi);
        break;
      }
    /* Boolean Kernels */
    case RectangleKernel:
    case SquareKernel:
      {
        if ( type == SquareKernel )
          {
            if (args->rho < 1.0)
              kernel->width = kernel->height = 3;  /* default radius = 1 */
            else
              kernel->width = kernel->height = 2*(long)args->rho+1;
            kernel->offset_x = kernel->offset_y = (kernel->width-1)/2;
          }
        else {
            /* NOTE: user defaults set in "AcquireKernelInfo()" */
            if ( args->rho < 1.0 || args->sigma < 1.0 )
              return(DestroyKernelInfo(kernel));    /* invalid args given */
            kernel->width = (unsigned long)args->rho;
            kernel->height = (unsigned long)args->sigma;
            if ( args->xi  < 0.0 || args->xi  > (double)kernel->width ||
                 args->psi < 0.0 || args->psi > (double)kernel->height )
              return(DestroyKernelInfo(kernel));    /* invalid args given */
            kernel->offset_x = (unsigned long)args->xi;
            kernel->offset_y = (unsigned long)args->psi;
          }
        kernel->values=(double *) AcquireQuantumMemory(kernel->width,
                              kernel->height*sizeof(double));
        if (kernel->values == (double *) NULL)
          return(DestroyKernelInfo(kernel));

        /* set all kernel values to 1.0 */
        u=kernel->width*kernel->height;
        for ( i=0; i < (unsigned long)u; i++)
            kernel->values[i] = 1.0;
        kernel->value_min = kernel->value_max = 1.0; /* a flat shape */
        kernel->range_pos = (double) u;
        break;
      }
    case DiamondKernel:
      {
        if (args->rho < 1.0)
          kernel->width = kernel->height = 3;  /* default radius = 1 */
        else
          kernel->width = kernel->height = ((unsigned long)args->rho)*2+1;
        kernel->offset_x = kernel->offset_y = (kernel->width-1)/2;

        kernel->values=(double *) AcquireQuantumMemory(kernel->width,
                              kernel->height*sizeof(double));
        if (kernel->values == (double *) NULL)
          return(DestroyKernelInfo(kernel));

        /* set all kernel values within diamond area to 1.0 */
        for ( i=0, v=-kernel->offset_y; v <= (long)kernel->offset_y; v++)
          for ( u=-kernel->offset_x; u <= (long)kernel->offset_x; u++, i++)
            if ((labs(u)+labs(v)) <= (long)kernel->offset_x)
              kernel->range_pos += kernel->values[i] = 1.0;
            else
              kernel->values[i] = nan;
        kernel->value_min = kernel->value_max = 1.0; /* a flat shape */
        break;
      }
    case DiskKernel:
      {
        long
          limit;

        limit = (long)(args->rho*args->rho);
        if (args->rho < 0.1)             /* default radius approx 3.5 */
          kernel->width = kernel->height = 7L, limit = 10L;
        else
           kernel->width = kernel->height = ((unsigned long)args->rho)*2+1;
        kernel->offset_x = kernel->offset_y = (kernel->width-1)/2;

        kernel->values=(double *) AcquireQuantumMemory(kernel->width,
                              kernel->height*sizeof(double));
        if (kernel->values == (double *) NULL)
          return(DestroyKernelInfo(kernel));

        /* set all kernel values within disk area to 1.0 */
        for ( i=0, v=-kernel->offset_y; v <= (long)kernel->offset_y; v++)
          for ( u=-kernel->offset_x; u <= (long)kernel->offset_x; u++, i++)
            if ((u*u+v*v) <= limit)
              kernel->range_pos += kernel->values[i] = 1.0;
            else
              kernel->values[i] = nan;
        kernel->value_min = kernel->value_max = 1.0; /* a flat shape */
        break;
      }
    case PlusKernel:
      {
        if (args->rho < 1.0)
          kernel->width = kernel->height = 5;  /* default radius 2 */
        else
           kernel->width = kernel->height = ((unsigned long)args->rho)*2+1;
        kernel->offset_x = kernel->offset_y = (kernel->width-1)/2;

        kernel->values=(double *) AcquireQuantumMemory(kernel->width,
                              kernel->height*sizeof(double));
        if (kernel->values == (double *) NULL)
          return(DestroyKernelInfo(kernel));

        /* set all kernel values along axises to 1.0 */
        for ( i=0, v=-kernel->offset_y; v <= (long)kernel->offset_y; v++)
          for ( u=-kernel->offset_x; u <= (long)kernel->offset_x; u++, i++)
            kernel->values[i] = (u == 0 || v == 0) ? 1.0 : nan;
        kernel->value_min = kernel->value_max = 1.0; /* a flat shape */
        kernel->range_pos = kernel->width*2.0 - 1.0;
        break;
      }
    /* Distance Measuring Kernels */
    case ChebyshevKernel:
      {
        double
          scale;

        if (args->rho < 1.0)
          kernel->width = kernel->height = 3;  /* default radius = 1 */
        else
          kernel->width = kernel->height = ((unsigned long)args->rho)*2+1;
        kernel->offset_x = kernel->offset_y = (kernel->width-1)/2;

        kernel->values=(double *) AcquireQuantumMemory(kernel->width,
                              kernel->height*sizeof(double));
        if (kernel->values == (double *) NULL)
          return(DestroyKernelInfo(kernel));

        scale = (args->sigma < 1.0) ? 100.0 : args->sigma;
        for ( i=0, v=-kernel->offset_y; v <= (long)kernel->offset_y; v++)
          for ( u=-kernel->offset_x; u <= (long)kernel->offset_x; u++, i++)
            kernel->range_pos += ( kernel->values[i] =
                 scale*((labs(u)>labs(v)) ? labs(u) : labs(v)) );
        kernel->value_max = kernel->values[0];
        break;
      }
    case ManhattenKernel:
      {
        double
          scale;

        if (args->rho < 1.0)
          kernel->width = kernel->height = 3;  /* default radius = 1 */
        else
           kernel->width = kernel->height = ((unsigned long)args->rho)*2+1;
        kernel->offset_x = kernel->offset_y = (kernel->width-1)/2;

        kernel->values=(double *) AcquireQuantumMemory(kernel->width,
                              kernel->height*sizeof(double));
        if (kernel->values == (double *) NULL)
          return(DestroyKernelInfo(kernel));

        scale = (args->sigma < 1.0) ? 100.0 : args->sigma;
        for ( i=0, v=-kernel->offset_y; v <= (long)kernel->offset_y; v++)
          for ( u=-kernel->offset_x; u <= (long)kernel->offset_x; u++, i++)
            kernel->range_pos += ( kernel->values[i] =
                 scale*(labs(u)+labs(v)) );
        kernel->value_max = kernel->values[0];
        break;
      }
    case EuclideanKernel:
      {
        double
          scale;

        if (args->rho < 1.0)
          kernel->width = kernel->height = 3;  /* default radius = 1 */
        else
           kernel->width = kernel->height = ((unsigned long)args->rho)*2+1;
        kernel->offset_x = kernel->offset_y = (kernel->width-1)/2;

        kernel->values=(double *) AcquireQuantumMemory(kernel->width,
                              kernel->height*sizeof(double));
        if (kernel->values == (double *) NULL)
          return(DestroyKernelInfo(kernel));

        scale = (args->sigma < 1.0) ? 100.0 : args->sigma;
        for ( i=0, v=-kernel->offset_y; v <= (long)kernel->offset_y; v++)
          for ( u=-kernel->offset_x; u <= (long)kernel->offset_x; u++, i++)
            kernel->range_pos += ( kernel->values[i] =
                 scale*sqrt((double)(u*u+v*v)) );
        kernel->value_max = kernel->values[0];
        break;
      }
    /* Undefined Kernels */
    case LaplacianKernel:
    case LOGKernel:
    case DOGKernel:
      assert("Kernel Type has not been defined yet");
      /* FALL THRU */
    default:
      /* Generate a No-Op minimal kernel - 1x1 pixel */
      kernel->values=(double *)AcquireQuantumMemory((size_t)1,sizeof(double));
      if (kernel->values == (double *) NULL)
        return(DestroyKernelInfo(kernel));
      kernel->width = kernel->height = 1;
      kernel->offset_x = kernel->offset_x = 0;
      kernel->type = UndefinedKernel;
      kernel->value_max =
        kernel->range_pos =
          kernel->values[0] = 1.0;  /* a flat single-point no-op kernel! */
      break;
  }

  return(kernel);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%     D e s t r o y K e r n e l I n f o                                       %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  DestroyKernelInfo() frees the memory used by a Convolution/Morphology
%  kernel.
%
%  The format of the DestroyKernelInfo method is:
%
%      KernelInfo *DestroyKernelInfo(KernelInfo *kernel)
%
%  A description of each parameter follows:
%
%    o kernel: the Morphology/Convolution kernel to be destroyed
%
*/

MagickExport KernelInfo *DestroyKernelInfo(KernelInfo *kernel)
{
  assert(kernel != (KernelInfo *) NULL);
  kernel->values=(double *)RelinquishMagickMemory(kernel->values);
  kernel=(KernelInfo *) RelinquishMagickMemory(kernel);
  return(kernel);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%     M o r p h o l o g y I m a g e C h a n n e l                             %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  MorphologyImageChannel() applies a user supplied kernel to the image
%  according to the given mophology method.
%
%  The given kernel is assumed to have been pre-scaled appropriatally, usally
%  by the kernel generator.
%
%  The format of the MorphologyImage method is:
%
%      Image *MorphologyImage(const Image *image, MorphologyMethod method,
%        const long iterations, KernelInfo *kernel, ExceptionInfo *exception)
%      Image *MorphologyImageChannel(const Image *image, const ChannelType
%        channel, MorphologyMethod method, const long iterations, KernelInfo
%        *kernel, ExceptionInfo *exception)
%
%  A description of each parameter follows:
%
%    o image: the image.
%
%    o method: the morphology method to be applied.
%
%    o iterations: apply the operation this many times (or no change).
%                  A value of -1 means loop until no change found.
%                  How this is applied may depend on the morphology method.
%                  Typically this is a value of 1.
%
%    o channel: the channel type.
%
%    o kernel: An array of double representing the morphology kernel.
%              Warning: kernel may be normalized for the Convolve method.
%
%    o exception: return any errors or warnings in this structure.
%
%
% TODO: bias and auto-scale handling of the kernel for convolution
%     The given kernel is assumed to have been pre-scaled appropriatally, usally
%     by the kernel generator.
%
*/

/* Internal function
 * Apply the Morphology method with the given Kernel
 * And return the number of pixels that changed.
 */
static unsigned long MorphologyApply(const Image *image, Image
     *result_image, const MorphologyMethod method, const ChannelType channel,
     const KernelInfo *kernel, ExceptionInfo *exception)
{
#define MorphologyTag  "Morphology/Image"

  long
    progress;

  unsigned long
    y, offx, offy,
    changed;

  MagickBooleanType
    status;

  MagickPixelPacket
    bias;

  CacheView
    *p_view,
    *q_view;

  /*
    Apply Morphology to Image.
  */
  status=MagickTrue;
  changed=0;
  progress=0;

  GetMagickPixelPacket(image,&bias);
  SetMagickPixelPacketBias(image,&bias);
  /* Future: handle auto-bias from user, based on kernel input */

  p_view=AcquireCacheView(image);
  q_view=AcquireCacheView(result_image);

  /* Some methods (including convolve) needs use a reflected kernel.
   * Adjust 'origin' offsets for this reflected kernel.
   */
  offx = kernel->offset_x;
  offy = kernel->offset_y;
  switch(method) {
    case ErodeMorphology:
    case ErodeIntensityMorphology:
      /* kernel is not reflected */
      break;
    default:
      /* kernel needs to be reflected */
      offx = kernel->width-offx-1;
      offy = kernel->height-offy-1;
      break;
  }

#if defined(MAGICKCORE_OPENMP_SUPPORT)
  #pragma omp parallel for schedule(dynamic,4) shared(progress,status)
#endif
  for (y=0; y < image->rows; y++)
  {
    MagickBooleanType
      sync;

    register const PixelPacket
      *restrict p;

    register const IndexPacket
      *restrict p_indexes;

    register PixelPacket
      *restrict q;

    register IndexPacket
      *restrict q_indexes;

    register unsigned long
      x;

    unsigned long
      r;

    if (status == MagickFalse)
      continue;
    p=GetCacheViewVirtualPixels(p_view, -offx,  y-offy,
         image->columns+kernel->width,  kernel->height,  exception);
    q=GetCacheViewAuthenticPixels(q_view,0,y,result_image->columns,1,
         exception);
    if ((p == (const PixelPacket *) NULL) || (q == (PixelPacket *) NULL))
      {
        status=MagickFalse;
        continue;
      }
    p_indexes=GetCacheViewVirtualIndexQueue(p_view);
    q_indexes=GetCacheViewAuthenticIndexQueue(q_view);
    r = (image->columns+kernel->width)*offy+offx; /* constant */

    for (x=0; x < image->columns; x++)
    {
       unsigned long
        v;

      register unsigned long
        u;

      register const double
        *restrict k;

      register const PixelPacket
        *restrict k_pixels;

      register const IndexPacket
        *restrict k_indexes;

      MagickPixelPacket
        result;

      /* Copy input to ouput image for unused channels
       * This removes need for 'cloning' a new image every iteration
       */
      *q = p[r];
      if (image->colorspace == CMYKColorspace)
        q_indexes[x] = p_indexes[r];

      result.index=0; /* stop compiler warnings */
      switch (method) {
        case ConvolveMorphology:
          result=bias;
          break;  /* default result is the convolution bias */
#if 1
        case DilateMorphology:
          result.red     =
          result.green   =
          result.blue    =
          result.opacity =
          result.index   = -MagickHuge;
          break;
        case ErodeMorphology:
          result.red     =
          result.green   =
          result.blue    =
          result.opacity =
          result.index   = +MagickHuge;
          break;
#endif
        default:
          /* Otherwise just start with the original pixel value */
          result.red     = p[r].red;
          result.green   = p[r].green;
          result.blue    = p[r].blue;
          result.opacity = QuantumRange - p[r].opacity;
          if ( image->colorspace == CMYKColorspace)
             result.index   = p_indexes[r];
          break;
      }

      switch ( method ) {
        case ConvolveMorphology:
            /* Weighted Average of pixels
             *
             * NOTE for correct working of this operation for asymetrical
             * kernels, the kernel needs to be applied in its reflected form.
             * That is its values needs to be reversed.
             */
            if (((channel & OpacityChannel) == 0) ||
                      (image->matte == MagickFalse))
              {
                /* Convolution (no transparency) */
                k = &kernel->values[ kernel->width*kernel->height-1 ];
                k_pixels = p;
                k_indexes = p_indexes;
                for (v=0; v < kernel->height; v++) {
                  for (u=0; u < kernel->width; u++, k--) {
                    if ( IsNan(*k) ) continue;
                    result.red     += (*k)*k_pixels[u].red;
                    result.green   += (*k)*k_pixels[u].green;
                    result.blue    += (*k)*k_pixels[u].blue;
                    /* result.opacity += not involved here */
                    if ( image->colorspace == CMYKColorspace)
                      result.index   += (*k)*k_indexes[u];
                  }
                  k_pixels += image->columns+kernel->width;
                  k_indexes += image->columns+kernel->width;
                }
              }
            else
              { /* Kernel & Alpha weighted Convolution */
                MagickRealType
                  alpha,  /* alpha value * kernel weighting */
                  gamma;  /* weighting divisor */

                gamma=0.0;
                k = &kernel->values[ kernel->width*kernel->height-1 ];
                k_pixels = p;
                k_indexes = p_indexes;
                for (v=0; v < kernel->height; v++) {
                  for (u=0; u < kernel->width; u++, k--) {
                    if ( IsNan(*k) ) continue;
                    alpha=(*k)*(QuantumScale*(QuantumRange-
                                          k_pixels[u].opacity));
                    gamma += alpha;
                    result.red     += alpha*k_pixels[u].red;
                    result.green   += alpha*k_pixels[u].green;
                    result.blue    += alpha*k_pixels[u].blue;
                    result.opacity += (*k)*(QuantumRange-k_pixels[u].opacity);
                    if ( image->colorspace == CMYKColorspace)
                      result.index   += alpha*k_indexes[u];
                  }
                  k_pixels += image->columns+kernel->width;
                  k_indexes += image->columns+kernel->width;
                }
                gamma=1.0/(fabs((double) gamma) <= MagickEpsilon ? 1.0 : gamma);
                result.red *= gamma;
                result.green *= gamma;
                result.blue *= gamma;
                result.opacity *= gamma;
                result.index *= gamma;
              }
            break;

        case DilateMorphology:
            /* Maximize Value within kernel shape
             *
             * NOTE for correct working of this operation for asymetrical
             * kernels, the kernel needs to be applied in its reflected form.
             * That is its values needs to be reversed.
             *
             * NOTE: in normal Greyscale Morphology, the kernel value should
             * be added to the real value, this is currently not done, due to
             * the nature of the boolean kernels being used.
             *
             */
            k = &kernel->values[ kernel->width*kernel->height-1 ];
            k_pixels = p;
            k_indexes = p_indexes;
            for (v=0; v < kernel->height; v++) {
              for (u=0; u < kernel->width; u++, k--) {
                if ( IsNan(*k) || (*k) < 0.5 ) continue;
                Maximize(result.red,     k_pixels[u].red);
                Maximize(result.green,   k_pixels[u].green);
                Maximize(result.blue,    k_pixels[u].blue);
                Maximize(result.opacity, QuantumRange-k_pixels[u].opacity);
                if ( image->colorspace == CMYKColorspace)
                  Maximize(result.index,   k_indexes[u]);
              }
              k_pixels += image->columns+kernel->width;
              k_indexes += image->columns+kernel->width;
            }
            break;

        case ErodeMorphology:
            /* Minimize Value within kernel shape
             *
             * NOTE that the kernel is not reflected for this operation!
             *
             * NOTE: in normal Greyscale Morphology, the kernel value should
             * be added to the real value, this is currently not done, due to
             * the nature of the boolean kernels being used.
             */
            k = kernel->values;
            k_pixels = p;
            k_indexes = p_indexes;
            for (v=0; v < kernel->height; v++) {
              for (u=0; u < kernel->width; u++, k++) {
                if ( IsNan(*k) || (*k) < 0.5 ) continue;
                Minimize(result.red,     k_pixels[u].red);
                Minimize(result.green,   k_pixels[u].green);
                Minimize(result.blue,    k_pixels[u].blue);
                Minimize(result.opacity, QuantumRange-k_pixels[u].opacity);
                if ( image->colorspace == CMYKColorspace)
                  Minimize(result.index,   k_indexes[u]);
              }
              k_pixels += image->columns+kernel->width;
              k_indexes += image->columns+kernel->width;
            }
            break;

        case DilateIntensityMorphology:
            /* Select pixel with maximum intensity within kernel shape
             *
             * WARNING: the intensity test fails for CMYK and does not
             * take into account the moderating effect of teh alpha channel
             * on the intensity.
             *
             * NOTE for correct working of this operation for asymetrical
             * kernels, the kernel needs to be applied in its reflected form.
             * That is its values needs to be reversed.
             */
            k = &kernel->values[ kernel->width*kernel->height-1 ];
            k_pixels = p;
            k_indexes = p_indexes;
            for (v=0; v < kernel->height; v++) {
              for (u=0; u < kernel->width; u++, k--) {
                if ( IsNan(*k) || (*k) < 0.5 ) continue; /* boolean kernel */
                if ( result.red == 0.0 ||
                     PixelIntensity(&(k_pixels[u])) > PixelIntensity(q) ) {
                  /* copy the whole pixel - no channel selection */
                  *q = k_pixels[u];
                  if ( result.red > 0.0 ) changed++;
                  result.red = 1.0;
                }
              }
              k_pixels += image->columns+kernel->width;
              k_indexes += image->columns+kernel->width;
            }
            break;

        case ErodeIntensityMorphology:
            /* Select pixel with mimimum intensity within kernel shape
             *
             * WARNING: the intensity test fails for CMYK and does not
             * take into account the moderating effect of teh alpha channel
             * on the intensity.
             *
             * NOTE that the kernel is not reflected for this operation!
             */
            k = kernel->values;
            k_pixels = p;
            k_indexes = p_indexes;
            for (v=0; v < kernel->height; v++) {
              for (u=0; u < kernel->width; u++, k++) {
                if ( IsNan(*k) || (*k) < 0.5 ) continue;
                if ( result.red == 0.0 ||
                     PixelIntensity(&(k_pixels[u])) < PixelIntensity(q) ) {
                  /* copy the whole pixel - no channel selection */
                  *q = k_pixels[u];
                  if ( result.red > 0.0 ) changed++;
                  result.red = 1.0;
                }
              }
              k_pixels += image->columns+kernel->width;
              k_indexes += image->columns+kernel->width;
            }
            break;

        case DistanceMorphology:
          /* Add kernel value and select the minimum value found.
           * The result is a iterative distance from edge function.
           *
           * All Distance Kernels are symetrical, but that may not always
           * be the case. For example how about a distance from left edges?
           * To make it work correctly for asymetrical kernels the reflected
           * kernel needs to be applied.
           */
#if 0
          /* No need to do distance morphology if original value is zero
           * Unfortunatally I have not been able to get this right
           * when channel selection also becomes involved. -- Arrgghhh
           */
          if (   ((channel & RedChannel) == 0 && p[r].red == 0)
              || ((channel & GreenChannel) == 0 && p[r].green == 0)
              || ((channel & BlueChannel) == 0 && p[r].blue == 0)
              || ((channel & OpacityChannel) == 0 && p[r].opacity == 0)
              || (( (channel & IndexChannel) == 0
                  || image->colorspace != CMYKColorspace
                                               ) && p_indexes[x] ==0 )
             )
            break;
#endif
            k = &kernel->values[ kernel->width*kernel->height-1 ];
            k_pixels = p;
            k_indexes = p_indexes;
            for (v=0; v < kernel->height; v++) {
              for (u=0; u < kernel->width; u++, k--) {
                if ( IsNan(*k) ) continue;
                Minimize(result.red,     (*k)+k_pixels[u].red);
                Minimize(result.green,   (*k)+k_pixels[u].green);
                Minimize(result.blue,    (*k)+k_pixels[u].blue);
                Minimize(result.opacity, (*k)+QuantumRange-k_pixels[u].opacity);
                if ( image->colorspace == CMYKColorspace)
                  Minimize(result.index,   (*k)+k_indexes[u]);
              }
              k_pixels += image->columns+kernel->width;
              k_indexes += image->columns+kernel->width;
            }
            break;

        case UndefinedMorphology:
        default:
            break; /* Do nothing */
      }
      switch ( method ) {
        case UndefinedMorphology:
        case DilateIntensityMorphology:
        case ErodeIntensityMorphology:
          break;  /* full pixel was directly assigned */
        default:
          /* Assign the results */
          if ((channel & RedChannel) != 0)
            q->red = ClampToQuantum(result.red);
          if ((channel & GreenChannel) != 0)
            q->green = ClampToQuantum(result.green);
          if ((channel & BlueChannel) != 0)
            q->blue = ClampToQuantum(result.blue);
          if ((channel & OpacityChannel) != 0
              && image->matte == MagickTrue )
            q->opacity = ClampToQuantum(QuantumRange-result.opacity);
          if ((channel & IndexChannel) != 0
              && image->colorspace == CMYKColorspace)
            q_indexes[x] = ClampToQuantum(result.index);
          break;
      }
      if (   ( p[r].red != q->red )
          || ( p[r].green != q->green )
          || ( p[r].blue != q->blue )
          || ( p[r].opacity != q->opacity )
          || ( image->colorspace == CMYKColorspace &&
                  p_indexes[r] != q_indexes[x] ) )
        changed++;  /* The pixel had some value changed! */
      p++;
      q++;
    } /* x */
    sync=SyncCacheViewAuthenticPixels(q_view,exception);
    if (sync == MagickFalse)
      status=MagickFalse;
    if (image->progress_monitor != (MagickProgressMonitor) NULL)
      {
        MagickBooleanType
          proceed;

#if defined(MAGICKCORE_OPENMP_SUPPORT)
  #pragma omp critical (MagickCore_MorphologyImage)
#endif
        proceed=SetImageProgress(image,MorphologyTag,progress++,image->rows);
        if (proceed == MagickFalse)
          status=MagickFalse;
      }
  } /* y */
  result_image->type=image->type;
  q_view=DestroyCacheView(q_view);
  p_view=DestroyCacheView(p_view);
  return(status ? changed : 0);
}

MagickExport Image *MorphologyImage(const Image *image,MorphologyMethod method,
  const long iterations,KernelInfo *kernel, ExceptionInfo *exception)
{
  Image
    *morphology_image;

  morphology_image=MorphologyImageChannel(image,DefaultChannels,method,
    iterations,kernel,exception);
  return(morphology_image);
}

MagickExport Image *MorphologyImageChannel(const Image *image,
  const ChannelType channel, MorphologyMethod method, const long iterations,
  KernelInfo *kernel, ExceptionInfo *exception)
{
  unsigned long
    count,
    limit,
    changed;

  Image
    *new_image,
    *old_image;

  const char
    *artifact;

  assert(image != (Image *) NULL);
  assert(image->signature == MagickSignature);
  assert(exception != (ExceptionInfo *) NULL);
  assert(exception->signature == MagickSignature);

  if ( GetImageArtifact(image,"showkernel") != (const char *) NULL)
    ShowKernel(kernel);  /* request to display the kernel to stderr */

  if ( iterations == 0 )
    return((Image *)NULL); /* null operation - nothing to do! */

  /* kernel must be valid at this point
   * (except maybe for posible future morphology methods like "Prune"
   */
  assert(kernel != (KernelInfo *)NULL);

  count = 0;
  changed = 1;
  limit = iterations;
  if ( iterations < 0 )
    limit = image->columns > image->rows ? image->columns : image->rows;

  /* Special morphology cases */
  switch( method ) {
    case CloseMorphology:
      new_image = MorphologyImageChannel(image, channel, DilateMorphology,
            iterations, kernel, exception);
      if (new_image == (Image *) NULL)
        return((Image *) NULL);
      method = ErodeMorphology;
      break;
    case OpenMorphology:
      new_image = MorphologyImageChannel(image, channel, ErodeMorphology,
            iterations, kernel, exception);
      if (new_image == (Image *) NULL)
        return((Image *) NULL);
      method = DilateMorphology;
      break;
    case CloseIntensityMorphology:
      new_image = MorphologyImageChannel(image, channel, DilateIntensityMorphology,
            iterations, kernel, exception);
      if (new_image == (Image *) NULL)
        return((Image *) NULL);
      method = ErodeIntensityMorphology;
      break;
    case OpenIntensityMorphology:
      new_image = MorphologyImageChannel(image, channel, ErodeIntensityMorphology,
            iterations, kernel, exception);
      if (new_image == (Image *) NULL)
        return((Image *) NULL);
      method = DilateIntensityMorphology;
      break;

    case ConvolveMorphology:
      /* Scale or Normalize kernel according to user wishes
      ** WARNING: this directly modifies the kernel
      ** which probably should not be done.
      */
      artifact = GetImageArtifact(image,"convolve:scale");
      if ( artifact != (char *)NULL )
        ScaleKernel(kernel, StringToDouble(artifact) );
      /* FALL-THRU */
    default:
      /* Do a morphology iteration just once at this point!
        This ensures a new_image has been generated, but allows us
        to skip the creation of 'old_image' if it isn't needed.
      */
      new_image=CloneImage(image,0,0,MagickTrue,exception);
      if (new_image == (Image *) NULL)
        return((Image *) NULL);
      if (SetImageStorageClass(new_image,DirectClass) == MagickFalse)
        {
          InheritException(exception,&new_image->exception);
          new_image=DestroyImage(new_image);
          return((Image *) NULL);
        }
      changed = MorphologyApply(image,new_image,method,channel,kernel,
            exception);
      count++;
      if ( GetImageArtifact(image,"verbose") != (const char *) NULL )
        fprintf(stderr, "Morphology %s:%lu => Changed %lu\n",
              MagickOptionToMnemonic(MagickMorphologyOptions, method),
              count, changed);
  }

  /* Repeat an interative morphology until count or no change reached */
  if ( count < limit && changed > 0 ) {
    old_image = CloneImage(new_image,0,0,MagickTrue,exception);
    if (old_image == (Image *) NULL)
        return(DestroyImage(new_image));
    if (SetImageStorageClass(old_image,DirectClass) == MagickFalse)
      {
        InheritException(exception,&old_image->exception);
        old_image=DestroyImage(old_image);
        return(DestroyImage(new_image));
      }
    while( count < limit && changed != 0 )
      {
        Image *tmp = old_image;
        old_image = new_image;
        new_image = tmp;
        changed = MorphologyApply(old_image,new_image,method,channel,kernel,
              exception);
        count++;
        if ( GetImageArtifact(image,"verbose") != (const char *) NULL )
          fprintf(stderr, "Morphology %s:%lu => Changed %lu\n",
                MagickOptionToMnemonic(MagickMorphologyOptions, method),
                count, changed);
      }
    DestroyImage(old_image);
  }

  return(new_image);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%     R o t a t e K e r n e l                                                 %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  RotateKernel() rotates the kernel by the angle given.  Currently it is
%  restricted to 90 degree angles, but this may be improved in the future.
%
%  The format of the RotateKernel method is:
%
%      void RotateKernel(KernelInfo *kernel, double angle)
%
%  A description of each parameter follows:
%
%    o kernel: the Morphology/Convolution kernel
%
%    o angle: angle to rotate in degrees
%
*/
static void RotateKernel(KernelInfo *kernel, double angle)
{
  /* WARNING: Currently assumes the kernel (rightly) is horizontally symetrical
  **
  ** TODO: expand beyond simple 90 degree rotates, flips and flops
  */

  /* Modulus the angle */
  angle = fmod(angle, 360.0);
  if ( angle < 0 )
    angle += 360.0;

  if ( 315.0 < angle || angle <= 45.0 )
    return;   /* no change! - At least at this time */

  switch (kernel->type) {
    /* These built-in kernels are cylindrical kernels, rotating is useless */
    case GaussianKernel:
    case LaplacianKernel:
    case LOGKernel:
    case DOGKernel:
    case DiskKernel:
    case ChebyshevKernel:
    case ManhattenKernel:
    case EuclideanKernel:
      return;

    /* These may be rotatable at non-90 angles in the future */
    /* but simply rotating them in multiples of 90 degrees is useless */
    case SquareKernel:
    case DiamondKernel:
    case PlusKernel:
      return;

    /* These only allows a +/-90 degree rotation (by transpose) */
    /* A 180 degree rotation is useless */
    case BlurKernel:
    case RectangleKernel:
      if ( 135.0 < angle && angle <= 225.0 )
        return;
      if ( 225.0 < angle && angle <= 315.0 )
        angle -= 180;
      break;

    /* these are freely rotatable in 90 degree units */
    case CometKernel:
    case UndefinedKernel:
    case UserDefinedKernel:
      break;
  }
  if ( 135.0 < angle && angle <= 225.0 )
    {
      /* For a 180 degree rotation - also know as a reflection */
      /* This is actually a very very common operation! */
      /* Basically all that is needed is a reversal of the kernel data! */
      unsigned long
        i,j;
      register double
        *k,t;

      k=kernel->values;
      for ( i=0, j=kernel->width*kernel->height-1;  i<j;  i++, j--)
        t=k[i],  k[i]=k[j],  k[j]=t;

      kernel->offset_x = kernel->width - kernel->offset_x - 1;
      kernel->offset_y = kernel->width - kernel->offset_y - 1;
      angle = fmod(angle+180.0, 360.0);
    }
  if ( 45.0 < angle && angle <= 135.0 )
    { /* Do a transpose and a flop, of the image, which results in a 90
       * degree rotation using two mirror operations.
       *
       * WARNING: this assumes the original image was a 1 dimentional image
       * but currently that is the only built-ins it is applied to.
       */
      unsigned long
        t;
      t = kernel->width;
      kernel->width = kernel->height;
      kernel->height = t;
      t = kernel->offset_x;
      kernel->offset_x = kernel->offset_y;
      kernel->offset_y = t;
      angle = fmod(450.0 - angle, 360.0);
    }
  /* At this point angle should be between -45 (315) and +45 degrees
   * In the future some form of non-orthogonal angled rotates could be
   * performed here, posibily with a linear kernel restriction.
   */

#if 0
    Not currently in use!
    { /* Do a flop, this assumes kernel is horizontally symetrical.
       * Each row of the kernel needs to be reversed!
       */
      unsigned long
        y;
      register unsigned long
        x,r;
      register double
        *k,t;

      for ( y=0, k=kernel->values; y < kernel->height; y++, k+=kernel->width)
        for ( x=0, r=kernel->width-1; x<kernel->width/2; x++, r--)
          t=k[x],  k[x]=k[r],  k[r]=t;

      kernel->offset_x = kernel->width - kernel->offset_x - 1;
      angle = fmod(angle+180.0, 360.0);
    }
#endif
  return;
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+     S c a l e K e r n e l                                                   %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  ScaleKernel() scales the kernel by the given amount.  Scaling by value of
%  zero will result in a normalization of the kernel.
%
%  For positive kernels normalization scales the kernel so the addition os all
%  values is 1.0.  While for kernels where values add to zero it is scaled
%  so that the convolution output range covers 1.0.  In such 'zero kernels'
%  it is generally recomended that the user also provides a 50% bias to the
%  output results.
%
%  Correct normalization assumes the 'range_*' attributes of the kernel
%  structure have been correctly set during the kernel creation.
%
%  The format of the ScaleKernel method is:
%
%      void ScaleKernel(KernelInfo *kernel)
%
%  A description of each parameter follows:
%
%    o kernel: the Morphology/Convolution kernel
%
%    o scale: multiple all values by this, if zero normalize instead.
%
*/
static void ScaleKernel(KernelInfo *kernel, double scale)
{
  register unsigned long
    i;

  if ( fabs(scale) < MagickEpsilon ) {
    if ( fabs(kernel->range_pos + kernel->range_neg) < MagickEpsilon )
      scale = 1/(kernel->range_pos - kernel->range_neg); /* zero kernels */
    else
      scale = 1/(kernel->range_pos + kernel->range_neg); /* non-zero kernel */
  }

  for (i=0; i < kernel->width*kernel->height; i++)
    if ( ! IsNan(kernel->values[i]) )
      kernel->values[i] *= scale;

  kernel->range_pos *= scale; /* convolution output range */
  kernel->range_neg *= scale;
  kernel->value_max *= scale; /* maximum and minimum values in kernel */
  kernel->value_min *= scale;

  return;
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+     S h o w K e r n e l                                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  ShowKernel() Output the details of the given kernel defination to
%  standard error, as per a users 'showkernel' option request.
%
%  The format of the ShowKernel method is:
%
%      void ShowKernel(KernelInfo *kernel)
%
%  A description of each parameter follows:
%
%    o kernel: the Morphology/Convolution kernel
%
% FUTURE: return the information in a string for API usage.
*/
static void ShowKernel(KernelInfo *kernel)
{
  unsigned long
    i, u, v;

  fprintf(stderr,
        "Kernel \"%s\" of size %lux%lu%+ld%+ld with values from %.*lg to %.*lg\n",
        MagickOptionToMnemonic(MagickKernelOptions, kernel->type),
        kernel->width, kernel->height,
        kernel->offset_x, kernel->offset_y,
        GetMagickPrecision(), kernel->value_min,
        GetMagickPrecision(), kernel->value_max);
  fprintf(stderr, "Forming convolution output range from %.*lg to %.*lg%s\n",
        GetMagickPrecision(), kernel->range_neg,
        GetMagickPrecision(), kernel->range_pos,
        /*kernel->normalized == MagickTrue ? " (normalized)" : */ "" );
  for (i=v=0; v < kernel->height; v++) {
    fprintf(stderr,"%2ld:",v);
    for (u=0; u < kernel->width; u++, i++)
      if ( IsNan(kernel->values[i]) )
        fprintf(stderr," %*s", GetMagickPrecision()+2, "nan");
      else
        fprintf(stderr," %*.*lg", GetMagickPrecision()+2,
             GetMagickPrecision(), kernel->values[i]);
    fprintf(stderr,"\n");
  }
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+     Z e r o K e r n e l N a n s                                             % 
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  ZeroKernelNans() replaces any special 'nan' value that may be present in
%  the kernel with a zero value.  This is typically done when the kernel will
%  be used in special hardware (GPU) convolution processors, to simply
%  matters.
%
%  The format of the ZeroKernelNans method is:
%
%      voidZeroKernelNans (KernelInfo *kernel)
%
%  A description of each parameter follows:
%
%    o kernel: the Morphology/Convolution kernel
%
% FUTURE: return the information in a string for API usage.
*/
static void ZeroKernelNans(KernelInfo *kernel)
{
  register unsigned long
    i;

  for (i=0; i < kernel->width*kernel->height; i++)
    if ( IsNan(kernel->values[i]) )
      kernel->values[i] = 0.0;

  return;
}