xref: /external/ImageMagick/MagickCore/resample.c

/*
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%           RRRR    EEEEE   SSSSS   AAA   M   M  PPPP   L      EEEEE          %
%           R   R   E       SS     A   A  MM MM  P   P  L      E              %
%           RRRR    EEE      SSS   AAAAA  M M M  PPPP   L      EEE            %
%           R R     E          SS  A   A  M   M  P      L      E              %
%           R  R    EEEEE   SSSSS  A   A  M   M  P      LLLLL  EEEEE          %
%                                                                             %
%                                                                             %
%                      MagickCore Pixel Resampling Methods                    %
%                                                                             %
%                              Software Design                                %
%                                John Cristy                                  %
%                              Anthony Thyssen                                %
%                                August 2007                                  %
%                                                                             %
%                                                                             %
%  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.                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
*/

/*
  Include declarations.
*/
#include "magick/studio.h"
#include "magick/artifact.h"
#include "magick/color-private.h"
#include "magick/cache.h"
#include "magick/draw.h"
#include "magick/exception-private.h"
#include "magick/gem.h"
#include "magick/image.h"
#include "magick/image-private.h"
#include "magick/log.h"
#include "magick/magick.h"
#include "magick/memory_.h"
#include "magick/pixel-private.h"
#include "magick/quantum.h"
#include "magick/random_.h"
#include "magick/resample.h"
#include "magick/resize.h"
#include "magick/resize-private.h"
#include "magick/transform.h"
#include "magick/signature-private.h"
/*
  EWA Resampling Options
*/
#define WLUT_WIDTH 1024       /* size of the filter cache */

/* select ONE resampling method */
#define EWA 1                 /* Normal EWA handling - raw or clamped */
                              /* if 0 then use "High Quality EWA" */
#define EWA_CLAMP 1           /* EWA Clamping from Nicolas Robidoux */

/* output debugging information */
#define DEBUG_ELLIPSE 0       /* output ellipse info for debug */
#define DEBUG_HIT_MISS 0      /* output hit/miss pixels (as gnuplot commands) */
#define DEBUG_NO_PIXEL_HIT 0  /* Make pixels that fail to hit anything - RED */

/*
  Typedef declarations.
*/
struct _ResampleFilter
{
  CacheView
    *view;

  Image
    *image;

  ExceptionInfo
    *exception;

  MagickBooleanType
    debug;

  /* Information about image being resampled */
  ssize_t
    image_area;

  InterpolatePixelMethod
    interpolate;

  VirtualPixelMethod
    virtual_pixel;

  FilterTypes
    filter;

  /* processing settings needed */
  MagickBooleanType
    limit_reached,
    do_interpolate,
    average_defined;

  MagickPixelPacket
    average_pixel;

  /* current ellipitical area being resampled around center point */
  double
    A, B, C,
    Vlimit, Ulimit, Uwidth, slope;

  /* LUT of weights for filtered average in elliptical area */
  double
    filter_lut[WLUT_WIDTH],
    support;

  size_t
    signature;
};

/*
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%   A c q u i r e R e s a m p l e I n f o                                     %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  AcquireResampleFilter() initializes the information resample needs do to a
%  scaled lookup of a color from an image, using area sampling.
%
%  The algorithm is based on a Elliptical Weighted Average, where the pixels
%  found in a large elliptical area is averaged together according to a
%  weighting (filter) function.  For more details see "Fundamentals of Texture
%  Mapping and Image Warping" a master's thesis by Paul.S.Heckbert, June 17,
%  1989.  Available for free from, http://www.cs.cmu.edu/~ph/
%
%  As EWA resampling (or any sort of resampling) can require a lot of
%  calculations to produce a distorted scaling of the source image for each
%  output pixel, the ResampleFilter structure generated holds that information
%  between individual image resampling.
%
%  This function will make the appropriate AcquireCacheView() calls
%  to view the image, calling functions do not need to open a cache view.
%
%  Usage Example...
%      resample_filter=AcquireResampleFilter(image,exception);
%      SetResampleFilter(resample_filter, GaussianFilter, 1.0);
%      for (y=0; y < (ssize_t) image->rows; y++) {
%        for (x=0; x < (ssize_t) image->columns; x++) {
%          u= ....;   v= ....;
%          ScaleResampleFilter(resample_filter, ... scaling vectors ...);
%          (void) ResamplePixelColor(resample_filter,u,v,&pixel);
%          ... assign resampled pixel value ...
%        }
%      }
%      DestroyResampleFilter(resample_filter);
%
%  The format of the AcquireResampleFilter method is:
%
%     ResampleFilter *AcquireResampleFilter(const 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 ResampleFilter *AcquireResampleFilter(const Image *image,
  ExceptionInfo *exception)
{
  register ResampleFilter
    *resample_filter;

  assert(image != (Image *) NULL);
  assert(image->signature == MagickSignature);
  if (image->debug != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",image->filename);
  assert(exception != (ExceptionInfo *) NULL);
  assert(exception->signature == MagickSignature);

  resample_filter=(ResampleFilter *) AcquireMagickMemory(
    sizeof(*resample_filter));
  if (resample_filter == (ResampleFilter *) NULL)
    ThrowFatalException(ResourceLimitFatalError,"MemoryAllocationFailed");
  (void) ResetMagickMemory(resample_filter,0,sizeof(*resample_filter));

  resample_filter->image=ReferenceImage((Image *) image);
  resample_filter->view=AcquireCacheView(resample_filter->image);
  resample_filter->exception=exception;

  resample_filter->debug=IsEventLogging();
  resample_filter->signature=MagickSignature;

  resample_filter->image_area=(ssize_t) (resample_filter->image->columns*
    resample_filter->image->rows);
  resample_filter->average_defined = MagickFalse;

  /* initialise the resampling filter settings */
  SetResampleFilter(resample_filter, resample_filter->image->filter,
    resample_filter->image->blur);
  resample_filter->interpolate = resample_filter->image->interpolate;
  resample_filter->virtual_pixel=GetImageVirtualPixelMethod(image);

  return(resample_filter);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
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%   D e s t r o y R e s a m p l e I n f o                                     %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  DestroyResampleFilter() finalizes and cleans up the resampling
%  resample_filter as returned by AcquireResampleFilter(), freeing any memory
%  or other information as needed.
%
%  The format of the DestroyResampleFilter method is:
%
%      ResampleFilter *DestroyResampleFilter(ResampleFilter *resample_filter)
%
%  A description of each parameter follows:
%
%    o resample_filter: resampling information structure
%
*/
MagickExport ResampleFilter *DestroyResampleFilter(
  ResampleFilter *resample_filter)
{
  assert(resample_filter != (ResampleFilter *) NULL);
  assert(resample_filter->signature == MagickSignature);
  assert(resample_filter->image != (Image *) NULL);
  if (resample_filter->debug != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",
      resample_filter->image->filename);
  resample_filter->view=DestroyCacheView(resample_filter->view);
  resample_filter->image=DestroyImage(resample_filter->image);
  resample_filter->signature=(~MagickSignature);
  resample_filter=(ResampleFilter *) RelinquishMagickMemory(resample_filter);
  return(resample_filter);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   I n t e r p o l a t e R e s a m p l e F i l t e r                         %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  InterpolateResampleFilter() applies bi-linear or tri-linear interpolation
%  between a floating point coordinate and the pixels surrounding that
%  coordinate.  No pixel area resampling, or scaling of the result is
%  performed.
%
%  The format of the InterpolateResampleFilter method is:
%
%      MagickBooleanType InterpolateResampleFilter(
%        ResampleInfo *resample_filter,const InterpolatePixelMethod method,
%        const double x,const double y,MagickPixelPacket *pixel)
%
%  A description of each parameter follows:
%
%    o resample_filter: the resample filter.
%
%    o method: the pixel clor interpolation method.
%
%    o x,y: A double representing the current (x,y) position of the pixel.
%
%    o pixel: return the interpolated pixel here.
%
*/

static inline double MagickMax(const double x,const double y)
{
  if (x > y)
    return(x);
  return(y);
}

static void BicubicInterpolate(const MagickPixelPacket *pixels,const double dx,
  MagickPixelPacket *pixel)
{
  MagickRealType
    dx2,
    p,
    q,
    r,
    s;

  dx2=dx*dx;
  p=(pixels[3].red-pixels[2].red)-(pixels[0].red-pixels[1].red);
  q=(pixels[0].red-pixels[1].red)-p;
  r=pixels[2].red-pixels[0].red;
  s=pixels[1].red;
  pixel->red=(dx*dx2*p)+(dx2*q)+(dx*r)+s;
  p=(pixels[3].green-pixels[2].green)-(pixels[0].green-pixels[1].green);
  q=(pixels[0].green-pixels[1].green)-p;
  r=pixels[2].green-pixels[0].green;
  s=pixels[1].green;
  pixel->green=(dx*dx2*p)+(dx2*q)+(dx*r)+s;
  p=(pixels[3].blue-pixels[2].blue)-(pixels[0].blue-pixels[1].blue);
  q=(pixels[0].blue-pixels[1].blue)-p;
  r=pixels[2].blue-pixels[0].blue;
  s=pixels[1].blue;
  pixel->blue=(dx*dx2*p)+(dx2*q)+(dx*r)+s;
  p=(pixels[3].opacity-pixels[2].opacity)-(pixels[0].opacity-pixels[1].opacity);
  q=(pixels[0].opacity-pixels[1].opacity)-p;
  r=pixels[2].opacity-pixels[0].opacity;
  s=pixels[1].opacity;
  pixel->opacity=(dx*dx2*p)+(dx2*q)+(dx*r)+s;
  if (pixel->colorspace == CMYKColorspace)
    {
      p=(pixels[3].index-pixels[2].index)-(pixels[0].index-pixels[1].index);
      q=(pixels[0].index-pixels[1].index)-p;
      r=pixels[2].index-pixels[0].index;
      s=pixels[1].index;
      pixel->index=(dx*dx2*p)+(dx2*q)+(dx*r)+s;
    }
}

static inline MagickRealType CubicWeightingFunction(const MagickRealType x)
{
  MagickRealType
    alpha,
    gamma;

  alpha=MagickMax(x+2.0,0.0);
  gamma=1.0*alpha*alpha*alpha;
  alpha=MagickMax(x+1.0,0.0);
  gamma-=4.0*alpha*alpha*alpha;
  alpha=MagickMax(x+0.0,0.0);
  gamma+=6.0*alpha*alpha*alpha;
  alpha=MagickMax(x-1.0,0.0);
  gamma-=4.0*alpha*alpha*alpha;
  return(gamma/6.0);
}

static inline double MeshInterpolate(const PointInfo *delta,const double p,
  const double x,const double y)
{
  return(delta->x*x+delta->y*y+(1.0-delta->x-delta->y)*p);
}

static inline ssize_t NearestNeighbor(MagickRealType x)
{
  if (x >= 0.0)
    return((ssize_t) (x+0.5));
  return((ssize_t) (x-0.5));
}

static MagickBooleanType InterpolateResampleFilter(
  ResampleFilter *resample_filter,const InterpolatePixelMethod method,
  const double x,const double y,MagickPixelPacket *pixel)
{
  MagickBooleanType
    status;

  register const IndexPacket
    *indexes;

  register const PixelPacket
    *p;

  register ssize_t
    i;

  assert(resample_filter != (ResampleFilter *) NULL);
  assert(resample_filter->signature == MagickSignature);
  status=MagickTrue;

  switch (method)
  {
    case AverageInterpolatePixel:
    {
      MagickPixelPacket
        pixels[16];

      MagickRealType
        alpha[16],
        gamma;

      p=GetCacheViewVirtualPixels(resample_filter->view,(ssize_t) floor(x)-1,
        (ssize_t) floor(y)-1,4,4,resample_filter->exception);
      if (p == (const PixelPacket *) NULL)
        {
          status=MagickFalse;
          break;
        }
      indexes=GetCacheViewVirtualIndexQueue(resample_filter->view);
      for (i=0; i < 16L; i++)
      {
        GetMagickPixelPacket(resample_filter->image,pixels+i);
        SetMagickPixelPacket(resample_filter->image,p,indexes+i,pixels+i);
        alpha[i]=1.0;
        if (resample_filter->image->matte != MagickFalse)
          {
            alpha[i]=QuantumScale*((MagickRealType) GetAlphaPixelComponent(p));
            pixels[i].red*=alpha[i];
            pixels[i].green*=alpha[i];
            pixels[i].blue*=alpha[i];
            if (resample_filter->image->colorspace == CMYKColorspace)
              pixels[i].index*=alpha[i];
          }
        gamma=alpha[i];
        gamma=1.0/(fabs((double) gamma) <= MagickEpsilon ? 1.0 : gamma);
        pixel->red+=gamma*0.0625*pixels[i].red;
        pixel->green+=gamma*0.0625*pixels[i].green;
        pixel->blue+=gamma*0.0625*pixels[i].blue;
        pixel->opacity+=0.0625*pixels[i].opacity;
        if (resample_filter->image->colorspace == CMYKColorspace)
          pixel->index+=gamma*0.0625*pixels[i].index;
        p++;
      }
      break;
    }
    case BicubicInterpolatePixel:
    {
      MagickPixelPacket
        pixels[16],
        u[4];

      MagickRealType
        alpha[16];

      PointInfo
        delta;

      p=GetCacheViewVirtualPixels(resample_filter->view,(ssize_t) floor(x)-1,
        (ssize_t) floor(y)-1,4,4,resample_filter->exception);
      if (p == (const PixelPacket *) NULL)
        {
          status=MagickFalse;
          break;
        }
      indexes=GetCacheViewVirtualIndexQueue(resample_filter->view);
      for (i=0; i < 16L; i++)
      {
        GetMagickPixelPacket(resample_filter->image,pixels+i);
        SetMagickPixelPacket(resample_filter->image,p,indexes+i,pixels+i);
        alpha[i]=1.0;
        if (resample_filter->image->matte != MagickFalse)
          {
            alpha[i]=QuantumScale*((MagickRealType) GetAlphaPixelComponent(p));
            pixels[i].red*=alpha[i];
            pixels[i].green*=alpha[i];
            pixels[i].blue*=alpha[i];
            if (resample_filter->image->colorspace == CMYKColorspace)
              pixels[i].index*=alpha[i];
          }
        p++;
      }
      delta.x=x-floor(x);
      for (i=0; i < 4L; i++)
        BicubicInterpolate(pixels+4*i,delta.x,u+i);
      delta.y=y-floor(y);
      BicubicInterpolate(u,delta.y,pixel);
      break;
    }
    case BilinearInterpolatePixel:
    default:
    {
      MagickPixelPacket
        pixels[4];

      MagickRealType
        alpha[4],
        gamma;

      PointInfo
        delta,
        epsilon;

      p=GetCacheViewVirtualPixels(resample_filter->view,(ssize_t) floor(x),
        (ssize_t) floor(y),2,2,resample_filter->exception);
      if (p == (const PixelPacket *) NULL)
        {
          status=MagickFalse;
          break;
        }
      indexes=GetCacheViewVirtualIndexQueue(resample_filter->view);
      for (i=0; i < 4L; i++)
      {
        pixels[i].red=(MagickRealType) p[i].red;
        pixels[i].green=(MagickRealType) p[i].green;
        pixels[i].blue=(MagickRealType) p[i].blue;
        pixels[i].opacity=(MagickRealType) p[i].opacity;
        alpha[i]=1.0;
      }
      if (resample_filter->image->matte != MagickFalse)
        for (i=0; i < 4L; i++)
        {
          alpha[i]=QuantumScale*((MagickRealType) QuantumRange-p[i].opacity);
          pixels[i].red*=alpha[i];
          pixels[i].green*=alpha[i];
          pixels[i].blue*=alpha[i];
        }
      if (indexes != (IndexPacket *) NULL)
        for (i=0; i < 4L; i++)
        {
          pixels[i].index=(MagickRealType) indexes[i];
          if (resample_filter->image->colorspace == CMYKColorspace)
            pixels[i].index*=alpha[i];
        }
      delta.x=x-floor(x);
      delta.y=y-floor(y);
      epsilon.x=1.0-delta.x;
      epsilon.y=1.0-delta.y;
      gamma=((epsilon.y*(epsilon.x*alpha[0]+delta.x*alpha[1])+delta.y*
        (epsilon.x*alpha[2]+delta.x*alpha[3])));
      gamma=1.0/(fabs((double) gamma) <= MagickEpsilon ? 1.0 : gamma);
      pixel->red=gamma*(epsilon.y*(epsilon.x*pixels[0].red+delta.x*
        pixels[1].red)+delta.y*(epsilon.x*pixels[2].red+delta.x*pixels[3].red));
      pixel->green=gamma*(epsilon.y*(epsilon.x*pixels[0].green+delta.x*
        pixels[1].green)+delta.y*(epsilon.x*pixels[2].green+delta.x*
        pixels[3].green));
      pixel->blue=gamma*(epsilon.y*(epsilon.x*pixels[0].blue+delta.x*
        pixels[1].blue)+delta.y*(epsilon.x*pixels[2].blue+delta.x*
        pixels[3].blue));
      pixel->opacity=(epsilon.y*(epsilon.x*pixels[0].opacity+delta.x*
        pixels[1].opacity)+delta.y*(epsilon.x*pixels[2].opacity+delta.x*
        pixels[3].opacity));
      if (resample_filter->image->colorspace == CMYKColorspace)
        pixel->index=gamma*(epsilon.y*(epsilon.x*pixels[0].index+delta.x*
          pixels[1].index)+delta.y*(epsilon.x*pixels[2].index+delta.x*
          pixels[3].index));
      break;
    }
    case FilterInterpolatePixel:
    {
      CacheView
        *filter_view;

      Image
        *excerpt_image,
        *filter_image;

      MagickPixelPacket
        pixels[1];

      RectangleInfo
        geometry;

      geometry.width=4L;
      geometry.height=4L;
      geometry.x=(ssize_t) floor(x)-1L;
      geometry.y=(ssize_t) floor(y)-1L;
      excerpt_image=ExcerptImage(resample_filter->image,&geometry,
        resample_filter->exception);
      if (excerpt_image == (Image *) NULL)
        {
          status=MagickFalse;
          break;
        }
      filter_image=ResizeImage(excerpt_image,1,1,resample_filter->image->filter,
        resample_filter->image->blur,resample_filter->exception);
      excerpt_image=DestroyImage(excerpt_image);
      if (filter_image == (Image *) NULL)
        break;
      filter_view=AcquireCacheView(filter_image);
      p=GetCacheViewVirtualPixels(filter_view,0,0,1,1,
        resample_filter->exception);
      if (p != (const PixelPacket *) NULL)
        {
          indexes=GetVirtualIndexQueue(filter_image);
          GetMagickPixelPacket(resample_filter->image,pixels);
          SetMagickPixelPacket(resample_filter->image,p,indexes,pixel);
        }
      filter_view=DestroyCacheView(filter_view);
      filter_image=DestroyImage(filter_image);
      break;
    }
    case IntegerInterpolatePixel:
    {
      MagickPixelPacket
        pixels[1];

      p=GetCacheViewVirtualPixels(resample_filter->view,(ssize_t) floor(x),
        (ssize_t) floor(y),1,1,resample_filter->exception);
      if (p == (const PixelPacket *) NULL)
        {
          status=MagickFalse;
          break;
        }
      indexes=GetCacheViewVirtualIndexQueue(resample_filter->view);
      GetMagickPixelPacket(resample_filter->image,pixels);
      SetMagickPixelPacket(resample_filter->image,p,indexes,pixel);
      break;
    }
    case MeshInterpolatePixel:
    {
      MagickPixelPacket
        pixels[4];

      MagickRealType
        alpha[4],
        gamma;

      PointInfo
        delta,
        luminance;

      p=GetCacheViewVirtualPixels(resample_filter->view,(ssize_t) floor(x),
        (ssize_t) floor(y),2,2,resample_filter->exception);
      if (p == (const PixelPacket *) NULL)
        {
          status=MagickFalse;
          break;
        }
      indexes=GetCacheViewVirtualIndexQueue(resample_filter->view);
      for (i=0; i < 4L; i++)
      {
        GetMagickPixelPacket(resample_filter->image,pixels+i);
        SetMagickPixelPacket(resample_filter->image,p,indexes+i,pixels+i);
        alpha[i]=1.0;
        if (resample_filter->image->matte != MagickFalse)
          {
            alpha[i]=QuantumScale*((MagickRealType) GetAlphaPixelComponent(p));
            pixels[i].red*=alpha[i];
            pixels[i].green*=alpha[i];
            pixels[i].blue*=alpha[i];
            if (resample_filter->image->colorspace == CMYKColorspace)
              pixels[i].index*=alpha[i];
          }
        p++;
      }
      delta.x=x-floor(x);
      delta.y=y-floor(y);
      luminance.x=MagickPixelLuminance(pixels+0)-MagickPixelLuminance(pixels+3);
      luminance.y=MagickPixelLuminance(pixels+1)-MagickPixelLuminance(pixels+2);
      if (fabs(luminance.x) < fabs(luminance.y))
        {
          /*
            Diagonal 0-3 NW-SE.
          */
          if (delta.x <= delta.y)
            {
              /*
                Bottom-left triangle  (pixel:2, diagonal: 0-3).
              */
              delta.y=1.0-delta.y;
              gamma=MeshInterpolate(&delta,alpha[2],alpha[3],alpha[0]);
              gamma=1.0/(fabs((double) gamma) <= MagickEpsilon ? 1.0 : gamma);
              pixel->red=gamma*MeshInterpolate(&delta,pixels[2].red,
                pixels[3].red,pixels[0].red);
              pixel->green=gamma*MeshInterpolate(&delta,pixels[2].green,
                pixels[3].green,pixels[0].green);
              pixel->blue=gamma*MeshInterpolate(&delta,pixels[2].blue,
                pixels[3].blue,pixels[0].blue);
              pixel->opacity=gamma*MeshInterpolate(&delta,pixels[2].opacity,
                pixels[3].opacity,pixels[0].opacity);
              if (resample_filter->image->colorspace == CMYKColorspace)
                pixel->index=gamma*MeshInterpolate(&delta,pixels[2].index,
                  pixels[3].index,pixels[0].index);
            }
          else
            {
              /*
                Top-right triangle (pixel:1, diagonal: 0-3).
              */
              delta.x=1.0-delta.x;
              gamma=MeshInterpolate(&delta,alpha[1],alpha[0],alpha[3]);
              gamma=1.0/(fabs((double) gamma) <= MagickEpsilon ? 1.0 : gamma);
              pixel->red=gamma*MeshInterpolate(&delta,pixels[1].red,
                pixels[0].red,pixels[3].red);
              pixel->green=gamma*MeshInterpolate(&delta,pixels[1].green,
                pixels[0].green,pixels[3].green);
              pixel->blue=gamma*MeshInterpolate(&delta,pixels[1].blue,
                pixels[0].blue,pixels[3].blue);
              pixel->opacity=gamma*MeshInterpolate(&delta,pixels[1].opacity,
                pixels[0].opacity,pixels[3].opacity);
              if (resample_filter->image->colorspace == CMYKColorspace)
                pixel->index=gamma*MeshInterpolate(&delta,pixels[1].index,
                  pixels[0].index,pixels[3].index);
            }
        }
      else
        {
          /*
            Diagonal 1-2 NE-SW.
          */
          if (delta.x <= (1.0-delta.y))
            {
              /*
                Top-left triangle (pixel 0, diagonal: 1-2).
              */
              gamma=MeshInterpolate(&delta,alpha[0],alpha[1],alpha[2]);
              gamma=1.0/(fabs((double) gamma) <= MagickEpsilon ? 1.0 : gamma);
              pixel->red=gamma*MeshInterpolate(&delta,pixels[0].red,
                pixels[1].red,pixels[2].red);
              pixel->green=gamma*MeshInterpolate(&delta,pixels[0].green,
                pixels[1].green,pixels[2].green);
              pixel->blue=gamma*MeshInterpolate(&delta,pixels[0].blue,
                pixels[1].blue,pixels[2].blue);
              pixel->opacity=gamma*MeshInterpolate(&delta,pixels[0].opacity,
                pixels[1].opacity,pixels[2].opacity);
              if (resample_filter->image->colorspace == CMYKColorspace)
                pixel->index=gamma*MeshInterpolate(&delta,pixels[0].index,
                  pixels[1].index,pixels[2].index);
            }
          else
            {
              /*
                Bottom-right triangle (pixel: 3, diagonal: 1-2).
              */
              delta.x=1.0-delta.x;
              delta.y=1.0-delta.y;
              gamma=MeshInterpolate(&delta,alpha[3],alpha[2],alpha[1]);
              gamma=1.0/(fabs((double) gamma) <= MagickEpsilon ? 1.0 : gamma);
              pixel->red=gamma*MeshInterpolate(&delta,pixels[3].red,
                pixels[2].red,pixels[1].red);
              pixel->green=gamma*MeshInterpolate(&delta,pixels[3].green,
                pixels[2].green,pixels[1].green);
              pixel->blue=gamma*MeshInterpolate(&delta,pixels[3].blue,
                pixels[2].blue,pixels[1].blue);
              pixel->opacity=gamma*MeshInterpolate(&delta,pixels[3].opacity,
                pixels[2].opacity,pixels[1].opacity);
              if (resample_filter->image->colorspace == CMYKColorspace)
                pixel->index=gamma*MeshInterpolate(&delta,pixels[3].index,
                  pixels[2].index,pixels[1].index);
            }
        }
      break;
    }
    case NearestNeighborInterpolatePixel:
    {
      MagickPixelPacket
        pixels[1];

      p=GetCacheViewVirtualPixels(resample_filter->view,NearestNeighbor(x),
        NearestNeighbor(y),1,1,resample_filter->exception);
      if (p == (const PixelPacket *) NULL)
        {
          status=MagickFalse;
          break;
        }
      indexes=GetCacheViewVirtualIndexQueue(resample_filter->view);
      GetMagickPixelPacket(resample_filter->image,pixels);
      SetMagickPixelPacket(resample_filter->image,p,indexes,pixel);
      break;
    }
    case SplineInterpolatePixel:
    {
      MagickPixelPacket
        pixels[16];

      MagickRealType
        alpha[16],
        dx,
        dy,
        gamma;

      PointInfo
        delta;

      ssize_t
        j,
        n;

      p=GetCacheViewVirtualPixels(resample_filter->view,(ssize_t) floor(x)-1,
        (ssize_t) floor(y)-1,4,4,resample_filter->exception);
      if (p == (const PixelPacket *) NULL)
        {
          status=MagickFalse;
          break;
        }
      indexes=GetCacheViewVirtualIndexQueue(resample_filter->view);
      n=0;
      delta.x=x-floor(x);
      delta.y=y-floor(y);
      for (i=(-1); i < 3L; i++)
      {
        dy=CubicWeightingFunction((MagickRealType) i-delta.y);
        for (j=(-1); j < 3L; j++)
        {
          GetMagickPixelPacket(resample_filter->image,pixels+n);
          SetMagickPixelPacket(resample_filter->image,p,indexes+n,pixels+n);
          alpha[n]=1.0;
          if (resample_filter->image->matte != MagickFalse)
            {
              alpha[n]=QuantumScale*((MagickRealType)
                GetAlphaPixelComponent(p));
              pixels[n].red*=alpha[n];
              pixels[n].green*=alpha[n];
              pixels[n].blue*=alpha[n];
              if (resample_filter->image->colorspace == CMYKColorspace)
                pixels[n].index*=alpha[n];
            }
          dx=CubicWeightingFunction(delta.x-(MagickRealType) j);
          gamma=alpha[n];
          gamma=1.0/(fabs((double) gamma) <= MagickEpsilon ? 1.0 : gamma);
          pixel->red+=gamma*dx*dy*pixels[n].red;
          pixel->green+=gamma*dx*dy*pixels[n].green;
          pixel->blue+=gamma*dx*dy*pixels[n].blue;
          if (resample_filter->image->matte != MagickFalse)
            pixel->opacity+=dx*dy*pixels[n].opacity;
          if (resample_filter->image->colorspace == CMYKColorspace)
            pixel->index+=gamma*dx*dy*pixels[n].index;
          n++;
          p++;
        }
      }
      break;
    }
  }
  return(status);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   R e s a m p l e P i x e l C o l o r                                       %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  ResamplePixelColor() samples the pixel values surrounding the location
%  given using an elliptical weighted average, at the scale previously
%  calculated, and in the most efficent manner possible for the
%  VirtualPixelMethod setting.
%
%  The format of the ResamplePixelColor method is:
%
%     MagickBooleanType ResamplePixelColor(ResampleFilter *resample_filter,
%       const double u0,const double v0,MagickPixelPacket *pixel)
%
%  A description of each parameter follows:
%
%    o resample_filter: the resample filter.
%
%    o u0,v0: A double representing the center of the area to resample,
%        The distortion transformed transformed x,y coordinate.
%
%    o pixel: the resampled pixel is returned here.
%
*/
MagickExport MagickBooleanType ResamplePixelColor(
  ResampleFilter *resample_filter,const double u0,const double v0,
  MagickPixelPacket *pixel)
{
  MagickBooleanType
    status;

  ssize_t u,v, v1, v2, uw, hit;
  double u1;
  double U,V,Q,DQ,DDQ;
  double divisor_c,divisor_m;
  register double weight;
  register const PixelPacket *pixels;
  register const IndexPacket *indexes;
  assert(resample_filter != (ResampleFilter *) NULL);
  assert(resample_filter->signature == MagickSignature);

  status=MagickTrue;
  GetMagickPixelPacket(resample_filter->image,pixel);
  if ( resample_filter->do_interpolate ) {
    status=InterpolateResampleFilter(resample_filter,
      resample_filter->interpolate,u0,v0,pixel);
    return(status);
  }

#if DEBUG_ELLIPSE
  fprintf(stderr, "u0=%lf; v0=%lf;\n", u0, v0);
#endif

  /*
    Does resample area Miss the image?
    And is that area a simple solid color - then return that color
  */
  hit = 0;
  switch ( resample_filter->virtual_pixel ) {
    case BackgroundVirtualPixelMethod:
    case ConstantVirtualPixelMethod:
    case TransparentVirtualPixelMethod:
    case BlackVirtualPixelMethod:
    case GrayVirtualPixelMethod:
    case WhiteVirtualPixelMethod:
    case MaskVirtualPixelMethod:
      if ( resample_filter->limit_reached
           || u0 + resample_filter->Ulimit < 0.0
           || u0 - resample_filter->Ulimit > (double) resample_filter->image->columns
           || v0 + resample_filter->Vlimit < 0.0
           || v0 - resample_filter->Vlimit > (double) resample_filter->image->rows
           )
        hit++;
      break;

    case UndefinedVirtualPixelMethod:
    case EdgeVirtualPixelMethod:
      if (    ( u0 + resample_filter->Ulimit < 0.0 && v0 + resample_filter->Vlimit < 0.0 )
           || ( u0 + resample_filter->Ulimit < 0.0
                && v0 - resample_filter->Vlimit > (double) resample_filter->image->rows )
           || ( u0 - resample_filter->Ulimit > (double) resample_filter->image->columns
                && v0 + resample_filter->Vlimit < 0.0 )
           || ( u0 - resample_filter->Ulimit > (double) resample_filter->image->columns
                && v0 - resample_filter->Vlimit > (double) resample_filter->image->rows )
           )
        hit++;
      break;
    case HorizontalTileVirtualPixelMethod:
      if (    v0 + resample_filter->Vlimit < 0.0
           || v0 - resample_filter->Vlimit > (double) resample_filter->image->rows
           )
        hit++;  /* outside the horizontally tiled images. */
      break;
    case VerticalTileVirtualPixelMethod:
      if (    u0 + resample_filter->Ulimit < 0.0
           || u0 - resample_filter->Ulimit > (double) resample_filter->image->columns
           )
        hit++;  /* outside the vertically tiled images. */
      break;
    case DitherVirtualPixelMethod:
      if (    ( u0 + resample_filter->Ulimit < -32.0 && v0 + resample_filter->Vlimit < -32.0 )
           || ( u0 + resample_filter->Ulimit < -32.0
                && v0 - resample_filter->Vlimit > (double) resample_filter->image->rows+32.0 )
           || ( u0 - resample_filter->Ulimit > (double) resample_filter->image->columns+32.0
                && v0 + resample_filter->Vlimit < -32.0 )
           || ( u0 - resample_filter->Ulimit > (double) resample_filter->image->columns+32.0
                && v0 - resample_filter->Vlimit > (double) resample_filter->image->rows+32.0 )
           )
        hit++;
      break;
    case TileVirtualPixelMethod:
    case MirrorVirtualPixelMethod:
    case RandomVirtualPixelMethod:
    case HorizontalTileEdgeVirtualPixelMethod:
    case VerticalTileEdgeVirtualPixelMethod:
    case CheckerTileVirtualPixelMethod:
      /* resampling of area is always needed - no VP limits */
      break;
  }
  if ( hit ) {
    /* whole area is a solid color -- just return that color */
    status=InterpolateResampleFilter(resample_filter,IntegerInterpolatePixel,
      u0,v0,pixel);
    return(status);
  }

  /*
    Scaling limits reached, return an 'averaged' result.
  */
  if ( resample_filter->limit_reached ) {
    switch ( resample_filter->virtual_pixel ) {
      /*  This is always handled by the above, so no need.
        case BackgroundVirtualPixelMethod:
        case ConstantVirtualPixelMethod:
        case TransparentVirtualPixelMethod:
        case GrayVirtualPixelMethod,
        case WhiteVirtualPixelMethod
        case MaskVirtualPixelMethod:
      */
      case UndefinedVirtualPixelMethod:
      case EdgeVirtualPixelMethod:
      case DitherVirtualPixelMethod:
      case HorizontalTileEdgeVirtualPixelMethod:
      case VerticalTileEdgeVirtualPixelMethod:
        /* We need an average edge pixel, from the correct edge!
           How should I calculate an average edge color?
           Just returning an averaged neighbourhood,
           works well in general, but falls down for TileEdge methods.
           This needs to be done properly!!!!!!
        */
        status=InterpolateResampleFilter(resample_filter,
          AverageInterpolatePixel,u0,v0,pixel);
        break;
      case HorizontalTileVirtualPixelMethod:
      case VerticalTileVirtualPixelMethod:
        /* just return the background pixel - Is there more direct way? */
        status=InterpolateResampleFilter(resample_filter,
           IntegerInterpolatePixel,(double)-1,(double)-1,pixel);
        break;
      case TileVirtualPixelMethod:
      case MirrorVirtualPixelMethod:
      case RandomVirtualPixelMethod:
      case CheckerTileVirtualPixelMethod:
      default:
        /* generate a average color of the WHOLE image */
        if ( resample_filter->average_defined == MagickFalse ) {
          Image
            *average_image;

          CacheView
            *average_view;

          GetMagickPixelPacket(resample_filter->image,
                (MagickPixelPacket *)&(resample_filter->average_pixel));
          resample_filter->average_defined = MagickTrue;

          /* Try to get an averaged pixel color of whole image */
          average_image=ResizeImage(resample_filter->image,1,1,BoxFilter,1.0,
           resample_filter->exception);
          if (average_image == (Image *) NULL)
            {
              *pixel=resample_filter->average_pixel; /* FAILED */
              break;
            }
          average_view=AcquireCacheView(average_image);
          pixels=(PixelPacket *)GetCacheViewVirtualPixels(average_view,0,0,1,1,
            resample_filter->exception);
          if (pixels == (const PixelPacket *) NULL) {
            average_view=DestroyCacheView(average_view);
            average_image=DestroyImage(average_image);
            *pixel=resample_filter->average_pixel; /* FAILED */
            break;
          }
          indexes=(IndexPacket *) GetCacheViewAuthenticIndexQueue(average_view);
          SetMagickPixelPacket(resample_filter->image,pixels,indexes,
            &(resample_filter->average_pixel));
          average_view=DestroyCacheView(average_view);
          average_image=DestroyImage(average_image);

          if ( resample_filter->virtual_pixel == CheckerTileVirtualPixelMethod )
            {
              /* CheckerTile is avergae of image average half background */
              /* FUTURE: replace with a 50% blend of both pixels */

              weight = QuantumScale*((MagickRealType)(QuantumRange-
                          resample_filter->average_pixel.opacity));
              resample_filter->average_pixel.red *= weight;
              resample_filter->average_pixel.green *= weight;
              resample_filter->average_pixel.blue *= weight;
              divisor_c = weight;

              weight = QuantumScale*((MagickRealType)(QuantumRange-
                          resample_filter->image->background_color.opacity));
              resample_filter->average_pixel.red +=
                      weight*resample_filter->image->background_color.red;
              resample_filter->average_pixel.green +=
                      weight*resample_filter->image->background_color.green;
              resample_filter->average_pixel.blue +=
                      weight*resample_filter->image->background_color.blue;
              resample_filter->average_pixel.matte +=
                      resample_filter->image->background_color.opacity;
              divisor_c += weight;

              resample_filter->average_pixel.red /= divisor_c;
              resample_filter->average_pixel.green /= divisor_c;
              resample_filter->average_pixel.blue /= divisor_c;
              resample_filter->average_pixel.matte /= 2;

            }
        }
        *pixel=resample_filter->average_pixel;
        break;
    }
    return(status);
  }

  /*
    Initialize weighted average data collection
  */
  hit = 0;
  divisor_c = 0.0;
  divisor_m = 0.0;
  pixel->red = pixel->green = pixel->blue = 0.0;
  if (resample_filter->image->matte != MagickFalse) pixel->opacity = 0.0;
  if (resample_filter->image->colorspace == CMYKColorspace) pixel->index = 0.0;

  /*
    Determine the parellelogram bounding box fitted to the ellipse
    centered at u0,v0.  This area is bounding by the lines...
  */
  v1 = (ssize_t)ceil(v0 - resample_filter->Vlimit);  /* range of scan lines */
  v2 = (ssize_t)floor(v0 + resample_filter->Vlimit);

  /* scan line start and width accross the parallelogram */
  u1 = u0 + (v1-v0)*resample_filter->slope - resample_filter->Uwidth;
  uw = (ssize_t)(2.0*resample_filter->Uwidth)+1;

#if DEBUG_ELLIPSE
  fprintf(stderr, "v1=%ld; v2=%ld\n", (long)v1, (long)v2);
  fprintf(stderr, "u1=%ld; uw=%ld\n", (long)u1, (long)uw);
#else
# define DEBUG_HIT_MISS 0 /* only valid if DEBUG_ELLIPSE is enabled */
#endif

  /*
    Do weighted resampling of all pixels,  within the scaled ellipse,
    bound by a Parellelogram fitted to the ellipse.
  */
  DDQ = 2*resample_filter->A;
  for( v=v1; v<=v2;  v++ ) {
#if DEBUG_HIT_MISS
    long uu = ceil(u1);   /* actual pixel location (for debug only) */
    fprintf(stderr, "# scan line from pixel %ld, %ld\n", (long)uu, (long)v);
#endif
    u = (ssize_t)ceil(u1);        /* first pixel in scanline */
    u1 += resample_filter->slope; /* start of next scan line */


    /* location of this first pixel, relative to u0,v0 */
    U = (double)u-u0;
    V = (double)v-v0;

    /* Q = ellipse quotent ( if Q<F then pixel is inside ellipse) */
    Q = (resample_filter->A*U + resample_filter->B*V)*U + resample_filter->C*V*V;
    DQ = resample_filter->A*(2.0*U+1) + resample_filter->B*V;

    /* get the scanline of pixels for this v */
    pixels=GetCacheViewVirtualPixels(resample_filter->view,u,v,(size_t) uw,
      1,resample_filter->exception);
    if (pixels == (const PixelPacket *) NULL)
      return(MagickFalse);
    indexes=GetCacheViewVirtualIndexQueue(resample_filter->view);

    /* count up the weighted pixel colors */
    for( u=0; u<uw; u++ ) {
      /* Note that the ellipse has been pre-scaled so F = WLUT_WIDTH */
      if ( Q < (double)WLUT_WIDTH ) {
        weight = resample_filter->filter_lut[(int)Q];

        pixel->opacity  += weight*pixels->opacity;
        divisor_m += weight;

        if (resample_filter->image->matte != MagickFalse)
          weight *= QuantumScale*((MagickRealType)(QuantumRange-pixels->opacity));
        pixel->red   += weight*pixels->red;
        pixel->green += weight*pixels->green;
        pixel->blue  += weight*pixels->blue;
        if (resample_filter->image->colorspace == CMYKColorspace)
          pixel->index += weight*(*indexes);
        divisor_c += weight;

        hit++;
#if DEBUG_HIT_MISS
        /* mark the pixel according to hit/miss of the ellipse */
        fprintf(stderr, "set arrow from %lf,%lf to %lf,%lf nohead ls 3\n",
                     (long)uu-.1,(double)v-.1,(long)uu+.1,(long)v+.1);
        fprintf(stderr, "set arrow from %lf,%lf to %lf,%lf nohead ls 3\n",
                     (long)uu+.1,(double)v-.1,(long)uu-.1,(long)v+.1);
      } else {
        fprintf(stderr, "set arrow from %lf,%lf to %lf,%lf nohead ls 1\n",
                     (long)uu-.1,(double)v-.1,(long)uu+.1,(long)v+.1);
        fprintf(stderr, "set arrow from %lf,%lf to %lf,%lf nohead ls 1\n",
                     (long)uu+.1,(double)v-.1,(long)uu-.1,(long)v+.1);
      }
      uu++;
#else
      }
#endif
      pixels++;
      indexes++;
      Q += DQ;
      DQ += DDQ;
    }
  }
#if DEBUG_ELLIPSE
  fprintf(stderr, "Hit=%ld;  Total=%ld;\n", (long)hit, (long)uw*(v2-v1) );
#endif

  /*
    Result sanity check -- this should NOT happen
  */
  if ( hit == 0 ) {
    /* not enough pixels in resampling, resort to direct interpolation */
#if DEBUG_NO_PIXEL_HIT
    pixel->opacity = pixel->red = pixel->green = pixel->blue = 0;
    pixel->red = QuantumRange; /* show pixels for which EWA fails */
#else
    status=InterpolateResampleFilter(resample_filter,
      resample_filter->interpolate,u0,v0,pixel);
#endif
    return status;
  }

  /*
    Finialize results of resampling
  */
  divisor_m = 1.0/divisor_m;
  pixel->opacity = (MagickRealType) ClampToQuantum(divisor_m*pixel->opacity);
  divisor_c = 1.0/divisor_c;
  pixel->red   = (MagickRealType) ClampToQuantum(divisor_c*pixel->red);
  pixel->green = (MagickRealType) ClampToQuantum(divisor_c*pixel->green);
  pixel->blue  = (MagickRealType) ClampToQuantum(divisor_c*pixel->blue);
  if (resample_filter->image->colorspace == CMYKColorspace)
    pixel->index = (MagickRealType) ClampToQuantum(divisor_c*pixel->index);
  return(MagickTrue);
}

#if EWA && EWA_CLAMP
/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
-   C l a m p U p A x e s                                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% ClampUpAxes() function converts the input vectors into a major and
% minor axis unit vectors, and their magnatude.  This form allows us
% to ensure that the ellipse generated is never smaller than the unit
% circle and thus never too small for use in EWA resampling.
%
% This purely mathematical 'magic' was provided by Professor Nicolas
% Robidoux and his Masters student Chantal Racette.
%
% See Reference: "We Recommend Singular Value Decomposition", David Austin
%   http://www.ams.org/samplings/feature-column/fcarc-svd
%
% By generating Major and Minor Axis vectors, we can actually use the
% ellipse in its "canonical form", by remapping the dx,dy of the
% sampled point into distances along the major and minor axis unit
% vectors.
%    http://en.wikipedia.org/wiki/Ellipse#Canonical_form
*/
static void ClampUpAxes(const double dux,
                        const double dvx,
                        const double duy,
                        const double dvy,
                        double *major_mag,
                        double *minor_mag,
                        double *major_unit_x,
                        double *major_unit_y,
                        double *minor_unit_x,
                        double *minor_unit_y)
{
  /*
   * ClampUpAxes takes an input 2x2 matrix
   *
   * [ a b ] = [ dux duy ]
   * [ c d ] = [ dvx dvy ]
   *
   * and computes from it the major and minor axis vectors [major_x,
   * major_y] and [minor_x,minor_y] of the smallest ellipse containing
   * both the unit disk and the ellipse which is the image of the unit
   * disk by the linear transformation
   *
   * [ dux duy ] [S] = [s]
   * [ dvx dvy ] [T] = [t]
   *
   * (The vector [S,T] is the difference between a position in output
   * space and [X,Y]; the vector [s,t] is the difference between a
   * position in input space and [x,y].)
   */
  /*
   * Outputs:
   *
   * major_mag is the half-length of the major axis of the "new"
   * ellipse (in input space).
   *
   * minor_mag is the half-length of the minor axis of the "new"
   * ellipse (in input space).
   *
   * major_unit_x is the x-coordinate of the major axis direction vector
   * of both the "old" and "new" ellipses.
   *
   * major_unit_y is the y-coordinate of the major axis direction vector.
   *
   * minor_unit_x is the x-coordinate of the minor axis direction vector.
   *
   * minor_unit_y is the y-coordinate of the minor axis direction vector.
   *
   * Unit vectors are useful for computing projections, in particular,
   * to compute the distance between a point in output space and the
   * center (of a disk) from the position of the corresponding point
   * in input space.
   *
   * Now, if you want to modify the input pair of tangent vectors so
   * that it defines the modified ellipse, all you have to do is set
   *
   * newdux = major_mag * major_unit_x
   * newdvx = major_mag * major_unit_y
   * newduy = minor_mag * minor_unit_x = minor_mag * -major_unit_y
   * newdvy = minor_mag * minor_unit_y = minor_mag *  major_unit_x
   *
   * and use these new tangent vectors "as if" they were the original
   * ones.  Most of the time this is a rather drastic change in the
   * tangent vectors (even if the singular values are large enough not
   * to be clampled). A technical explanation of why things still work
   * is found at the end of the discussion below.
   *
   */
  /*
   * Discussion:
   *
   * GOAL: Fix things so that the pullback, in input space, of a disk
   * of radius r in output space is an ellipse which contains, at
   * least, a disc of radius r. (Make this hold for any r>0.)
   *
   * METHOD: Find the singular values and (unit) left singular vectors
   * of Jinv, clampling up the singular values to 1, and multiplying
   * the unit left singular vectors by the new singular values in
   * order to get the minor and major ellipse axis vectors.
   *
   * Inputs:
   *
   * The Jacobian matrix of the transformation at the output point
   * under consideration is defined as follows:
   *
   * Consider the transformation (x,y) -> (X,Y) from input locations
   * to output locations. (Anthony Thyssen, elsewhere in resample.c,
   * uses the notation (u,v) -> (x,y) instead of (x,y) -> (X,Y).)
   *
   * The Jacobian matrix J is equal to
   *
   *   [ A, B ] = [ dX/dx, dX/dy ]
   *   [ C, D ] = [ dY/dx, dY/dy ]
   *
   * Consequently, the vector [A,C] is the tangent vector
   * corresponding to input changes in the horizontal direction, and
   * the vector [B,D] is the tangent vector corresponding to input
   * changes in the vertical direction.
   *
   * In the context of resampling, it is more natural to use the
   * inverse Jacobian matrix Jinv. Jinv is
   *
   *   [ a, b ] = [ dx/dX, dx/dY ]
   *   [ c, d ] = [ dy/dX, dy/dY ]
   *
   * Note: Jinv can be computed from J with the following matrix
   * formula:
   *
   *   Jinv = 1/(A*D-B*C) [  D, -B ]
   *                      [ -C,  A ]
   *
   * What we (implicitly) want to do is replace Jinv by a new Jinv
   * which generates an ellipse which is as close as possible to the
   * original but which contains the unit disk. This is accomplished
   * as follows:
   *
   * Let
   *
   *   Jinv = U Sigma V^T
   *
   * be an SVD decomposition of Jinv. (The SVD is not unique. The
   * final ellipse does not depend on the particular SVD.) In
   * principle, what we want is to clamp up the entries of the
   * diagonal matrix Sigma so that they are at least 1, and then set
   *
   *   Jinv = U newSigma V^T.
   *
   * However, we do not need to compute V^T for the following reason:
   * V is an orthogonal matrix (that is, it represents a combination
   * of a rotation and a reflexion). Consequently, V maps the unit
   * circle to itself. For this reason, the exact value of V does not
   * affect the final ellipse, and we choose the identity matrix.
   * That is, we simply set
   *
   *   Jinv = U newSigma,
   *
   * omitting the V^T factor altogether. In the end, we return the two
   * diagonal entries of newSigma together with the two columns of U,
   * for a total of six returned quantities.
   */
  /*
   * ClampUpAxes was written by Nicolas Robidoux and Chantal Racette
   * of Laurentian University with funding from the National Science
   * and Engineering Research Council of Canada.
   *
   * The idea of using the SVD to clamp the singular values of the
   * linear part of the affine approximation of the pullback
   * transformation comes from the astrophysicist Craig DeForest, who
   * implemented it for use with (approximate) Gaussian filtering in
   * his PDL::Transform code (PDL = Perl Data Language).
   *
   * The only (possibly) new math in the following is the selection of
   * the largest row of the eigen matrix system in order to stabilize
   * the computation in near rank-deficient cases, and the
   * corresponding efficient repair of degenerate cases using the norm
   * of this largest row. Omitting the "V^T" factor of the SVD may
   * also be a new "trick."
   */
  const double a = dux;
  const double b = duy;
  const double c = dvx;
  const double d = dvy;
  /*
   * n is the matrix Jinv * transpose(Jinv). Eigenvalues of n are the
   * squares of the singular values of Jinv.
   */
  const double aa = a*a;
  const double bb = b*b;
  const double cc = c*c;
  const double dd = d*d;
  /*
   * Eigenvectors of n are left singular vectors of Jinv.
   */
  const double n11 = aa+bb;
  const double n12 = a*c+b*d;
  const double n21 = n12;
  const double n22 = cc+dd;
  const double det = a*d-b*c;
  const double twice_det = det+det;
  const double frobenius_squared = n11+n22;
  const double discriminant =
    (frobenius_squared+twice_det)*(frobenius_squared-twice_det);
  const double sqrt_discriminant = sqrt(discriminant);
  /*
   * s1 is the largest singular value of the inverse Jacobian
   * matrix. In other words, its reciprocal is the smallest singular
   * value of the Jacobian matrix itself.
   * If s1 = 0, both singular values are 0, and any orthogonal pair of
   * left and right factors produces a singular decomposition of Jinv.
   */
  /*
   * Initially, we only compute the squares of the singular values.
   */
  const double s1s1 = 0.5*(frobenius_squared+sqrt_discriminant);
  /*
   * s2 the smallest singular value of the inverse Jacobian
   * matrix. Its reciprocal is the largest singular value of the
   * Jacobian matrix itself.
   */
  const double s2s2 = 0.5*(frobenius_squared-sqrt_discriminant);
  const double s1s1minusn11 = s1s1-n11;
  const double s1s1minusn22 = s1s1-n22;
  /*
   * u1, the first column of the U factor of a singular decomposition
   * of Jinv, is a (non-normalized) left singular vector corresponding
   * to s1. It has entries u11 and u21. We compute u1 from the fact
   * that it is an eigenvector of n corresponding to the eigenvalue
   * s1^2.
   */
  const double s1s1minusn11_squared = s1s1minusn11*s1s1minusn11;
  const double s1s1minusn22_squared = s1s1minusn22*s1s1minusn22;
  /*
   * The following selects the largest row of n-s1^2 I as the one
   * which is used to find the eigenvector. If both s1^2-n11 and
   * s1^2-n22 are zero, n-s1^2 I is the zero matrix.  In that case,
   * any vector is an eigenvector; in addition, norm below is equal to
   * zero, and, in exact arithmetic, this is the only case in which
   * norm = 0. So, setting u1 to the simple but arbitrary vector [1,0]
   * if norm = 0 safely takes care of all cases.
   */
  const double temp_u11 =
    ( (s1s1minusn11_squared>=s1s1minusn22_squared) ? n12 : s1s1minusn22 );
  const double temp_u21 =
    ( (s1s1minusn11_squared>=s1s1minusn22_squared) ? s1s1minusn11 : n21 );
  const double norm = sqrt(temp_u11*temp_u11+temp_u21*temp_u21);
  /*
   * Finalize the entries of first left singular vector (associated
   * with the largest singular value).
   */
  const double u11 = ( (norm>0.0) ? temp_u11/norm : 1.0 );
  const double u21 = ( (norm>0.0) ? temp_u21/norm : 0.0 );
  /*
   * Clamp the singular values up to 1.
   */
  *major_mag = ( (s1s1<=1.0) ? 1.0 : sqrt(s1s1) );
  *minor_mag = ( (s2s2<=1.0) ? 1.0 : sqrt(s2s2) );
  /*
   * Return the unit major and minor axis direction vectors.
   */
  *major_unit_x = u11;
  *major_unit_y = u21;
  *minor_unit_x = -u21;
  *minor_unit_y = u11;
}

#endif
/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   S c a l e R e s a m p l e F i l t e r                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  ScaleResampleFilter() does all the calculations needed to resample an image
%  at a specific scale, defined by two scaling vectors.  This not using
%  a orthogonal scaling, but two distorted scaling vectors, to allow the
%  generation of a angled ellipse.
%
%  As only two deritive scaling vectors are used the center of the ellipse
%  must be the center of the lookup.  That is any curvature that the
%  distortion may produce is discounted.
%
%  The input vectors are produced by either finding the derivitives of the
%  distortion function, or the partial derivitives from a distortion mapping.
%  They do not need to be the orthogonal dx,dy scaling vectors, but can be
%  calculated from other derivatives.  For example you could use  dr,da/r
%  polar coordinate vector scaling vectors
%
%  If   u,v =  DistortEquation(x,y)   OR   u = Fu(x,y); v = Fv(x,y)
%  Then the scaling vectors are determined from the deritives...
%      du/dx, dv/dx     and    du/dy, dv/dy
%  If the resulting scaling vectors is othogonally aligned then...
%      dv/dx = 0   and   du/dy  =  0
%  Producing an othogonally alligned ellipse in source space for the area to
%  be resampled.
%
%  Note that scaling vectors are different to argument order.  Argument order
%  is the general order the deritives are extracted from the distortion
%  equations, and not the scaling vectors. As such the middle two vaules
%  may be swapped from what you expect.  Caution is advised.
%
%  WARNING: It is assumed that any SetResampleFilter() method call will
%  always be performed before the ScaleResampleFilter() method, so that the
%  size of the ellipse will match the support for the resampling filter being
%  used.
%
%  The format of the ScaleResampleFilter method is:
%
%     void ScaleResampleFilter(const ResampleFilter *resample_filter,
%       const double dux,const double duy,const double dvx,const double dvy)
%
%  A description of each parameter follows:
%
%    o resample_filter: the resampling resample_filterrmation defining the
%      image being resampled
%
%    o dux,duy,dvx,dvy:
%         The deritives or scaling vectors defining the EWA ellipse.
%         NOTE: watch the order, which is based on the order deritives
%         are usally determined from distortion equations (see above).
%         The middle two values may need to be swapped if you are thinking
%         in terms of scaling vectors.
%
*/
MagickExport void ScaleResampleFilter(ResampleFilter *resample_filter,
  const double dux,const double duy,const double dvx,const double dvy)
{
  double A,B,C,F;


  assert(resample_filter != (ResampleFilter *) NULL);
  assert(resample_filter->signature == MagickSignature);

  resample_filter->limit_reached = MagickFalse;

  /* A 'point' filter forces use of interpolation instead of area sampling */
  if ( resample_filter->filter == PointFilter )
    return; /* EWA turned off - nothing to do */

#if DEBUG_ELLIPSE
  fprintf(stderr, "# -----\n" );
  fprintf(stderr, "dux=%lf; dvx=%lf;   duy=%lf; dvy=%lf;\n",
       dux, dvx, duy, dvy);
#endif

  /* Find Ellipse Coefficents such that
        A*u^2 + B*u*v + C*v^2 = F
     With u,v relative to point around which we are resampling.
     And the given scaling dx,dy vectors in u,v space
         du/dx,dv/dx   and  du/dy,dv/dy
  */
#if EWA
  /* Direct conversion of derivatives into elliptical coefficients
     However when magnifying images, the scaling vectors will be small
     resulting in a ellipse that is too small to sample properly.
     As such we need to clamp the major/minor axis to a minumum of 1.0
     to prevent it getting too small.
  */
#if EWA_CLAMP
  { double major_mag,
           minor_mag,
           major_x,
           major_y,
           minor_x,
           minor_y;

  ClampUpAxes(dux,dvx,duy,dvy, &major_mag, &minor_mag,
                &major_x, &major_y, &minor_x, &minor_y);
  major_x *= major_mag;  major_y *= major_mag;
  minor_x *= minor_mag;  minor_y *= minor_mag;
#if DEBUG_ELLIPSE
  fprintf(stderr, "major_x=%lf; major_y=%lf;  minor_x=%lf; minor_y=%lf;\n",
        major_x, major_y, minor_x, minor_y);
#endif
  A = major_y*major_y+minor_y*minor_y;
  B = -2.0*(major_x*major_y+minor_x*minor_y);
  C = major_x*major_x+minor_x*minor_x;
  F = major_mag*minor_mag;
  F *= F; /* square it */
  }
#else /* raw EWA */
  A = dvx*dvx+dvy*dvy;
  B = -2.0*(dux*dvx+duy*dvy);
  C = dux*dux+duy*duy;
  F = dux*dvy-duy*dvx;
  F *= F; /* square it */
#endif

#else /* HQ_EWA */
  /*
    This Paul Heckbert's "Higher Quality EWA" formula, from page 60 in his
    thesis, which adds a unit circle to the elliptical area so as to do both
    Reconstruction and Prefiltering of the pixels in the resampling.  It also
    means it is always likely to have at least 4 pixels within the area of the
    ellipse, for weighted averaging.  No scaling will result with F == 4.0 and
    a circle of radius 2.0, and F smaller than this means magnification is
    being used.

    NOTE: This method produces a very blury result at near unity scale while
    producing perfect results for string minitification and magnifications.

    However filter support is fixed to 2.0 (no good for Windowed Sinc filters)
  */
  A = dvx*dvx+dvy*dvy+1;
  B = -2.0*(dux*dvx+duy*dvy);
  C = dux*dux+duy*duy+1;
  F = A*C - B*B/4;
#endif

#if DEBUG_ELLIPSE
  fprintf(stderr, "A=%lf; B=%lf; C=%lf; F=%lf\n", A,B,C,F);

  /* Figure out the various information directly about the ellipse.
     This information currently not needed at this time, but may be
     needed later for better limit determination.

     It is also good to have as a record for future debugging
  */
  { double alpha, beta, gamma, Major, Minor;
    double Eccentricity, Ellipse_Area, Ellipse_Angle;

    alpha = A+C;
    beta  = A-C;
    gamma = sqrt(beta*beta + B*B );

    if ( alpha - gamma <= MagickEpsilon )
      Major = MagickHuge;
    else
      Major = sqrt(2*F/(alpha - gamma));
    Minor = sqrt(2*F/(alpha + gamma));

    fprintf(stderr, "# Major=%lf; Minor=%lf\n", Major, Minor );

    /* other information about ellipse include... */
    Eccentricity = Major/Minor;
    Ellipse_Area = MagickPI*Major*Minor;
    Ellipse_Angle =  atan2(B, A-C);

    fprintf(stderr, "# Angle=%lf   Area=%lf\n",
         RadiansToDegrees(Ellipse_Angle), Ellipse_Area );
  }
#endif

  /* The scaling vectors is impossibly large (producing a very large raw F
     value), we may as well not bother doing any form of resampling, as you
     risk an near infinite resampled area.  In this case some alturnative
     means of pixel sampling, such as the average of the whole image is needed
     to get a reasonable result. Calculate only as needed.
  */
  if ( (4*A*C - B*B) > MagickHuge ) {
    resample_filter->limit_reached = MagickTrue;
    return;
  }

  /* Scale ellipse by the appropriate size */
  F *= resample_filter->support;
  F *= resample_filter->support;

  /* Othogonal bounds of the Ellipse */
  resample_filter->Ulimit = sqrt(4*C*F/(4*A*C-B*B));
  resample_filter->Vlimit = sqrt(4*A*F/(4*A*C-B*B));

  /* Horizontally aligned Parallelogram fitted to Ellipse */
  resample_filter->Uwidth = sqrt(F/A);   /* Parallelogram Width / 2 */
  resample_filter->slope = -B/(2*A);     /* Slope of the parallelogram */

#if DEBUG_ELLIPSE
  fprintf(stderr, "Ulimit=%lf; Vlimit=%lf; UWidth=%lf; Slope=%lf;\n",
           resample_filter->Ulimit, resample_filter->Vlimit,
           resample_filter->Uwidth, resample_filter->slope );
#endif

  /* Check the absolute area of the Parallogram involved...
   * This limit needs more work, as it gets too slow for
   * larger images involved with tiled views of the horizon. */
  if ( (resample_filter->Uwidth * resample_filter->Vlimit)
         > (4.0*resample_filter->image_area)) {
    resample_filter->limit_reached = MagickTrue;
    return;
  }

  /* Scale ellipse formula to directly index the Filter Lookup Table */
  { register double scale;
    scale = (double)WLUT_WIDTH/F;
    resample_filter->A = A*scale;
    resample_filter->B = B*scale;
    resample_filter->C = C*scale;
    /* ..ple_filter->F = WLUT_WIDTH; -- hardcoded */
  }
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   S e t R e s a m p l e F i l t e r                                         %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  SetResampleFilter() set the resampling filter lookup table based on a
%  specific filter.  Note that the filter is used as a radial filter not as a
%  two pass othogonally aligned resampling filter.
%
%  The default Filter, is Gaussian, which is the standard filter used by the
%  original paper on the Elliptical Weighted Everage Algorithm. However other
%  filters can also be used.
%
%  The format of the SetResampleFilter method is:
%
%    void SetResampleFilter(ResampleFilter *resample_filter,
%      const FilterTypes filter,const double blur)
%
%  A description of each parameter follows:
%
%    o resample_filter: resampling resample_filterrmation structure
%
%    o filter: the resize filter for elliptical weighting LUT
%
%    o blur: filter blur factor (radial scaling) for elliptical weighting LUT
%
*/
MagickExport void SetResampleFilter(ResampleFilter *resample_filter,
  const FilterTypes filter,const double blur)
{
  register int
     Q;

  double
     r_scale;

  ResizeFilter
     *resize_filter;

  assert(resample_filter != (ResampleFilter *) NULL);
  assert(resample_filter->signature == MagickSignature);

  resample_filter->do_interpolate = MagickFalse;
  resample_filter->filter = filter;

  if ( filter == PointFilter )
    {
      resample_filter->do_interpolate = MagickTrue;
      return;  /* EWA turned off - nothing more to do */
    }

  if ( filter == UndefinedFilter )
    resample_filter->filter = MitchellFilter;  /* a far less blurry filter */

  resize_filter = AcquireResizeFilter(resample_filter->image,
       resample_filter->filter,blur,MagickTrue,resample_filter->exception);
  if (resize_filter == (ResizeFilter *) NULL)
    {
      (void) ThrowMagickException(resample_filter->exception,GetMagickModule(),
           ModuleError, "UnableToSetFilteringValue",
           "Fall back to default EWA gaussian filter");
      resample_filter->filter = PointFilter;
    }

#if EWA
  resample_filter->support = GetResizeFilterSupport(resize_filter);
  if ( resample_filter->filter == GaussianFilter )
    resample_filter->support = 2.0;  /* larger gaussian support */
#else
  resample_filter->support = 2.0;  /* fixed support size for HQ-EWA */
#endif

  /* Scale radius so the filter LUT covers the full support range */
  r_scale = resample_filter->support*sqrt(1.0/(double)WLUT_WIDTH);

  /* Fill the LUT with a 1D resize filter function */
  for(Q=0; Q<WLUT_WIDTH; Q++)
    resample_filter->filter_lut[Q] = (double)
         GetResizeFilterWeight(resize_filter,sqrt((double)Q)*r_scale);

  /* finished with the resize filter */
  resize_filter = DestroyResizeFilter(resize_filter);

  /*
    Adjust the scaling of the default unit circle
    This assumes that any real scaling changes will always
    take place AFTER the filter method has been initialized.
  */

  ScaleResampleFilter(resample_filter, 1.0, 0.0, 0.0, 1.0);

#if 0
  This is old code kept for reference only.  It is very wrong.
  /*
    Create Normal Gaussian 2D Filter Weighted Lookup Table.
    A normal EWA guassual lookup would use   exp(Q*ALPHA)
    where  Q = distance squared from 0.0 (center) to 1.0 (edge)
    and    ALPHA = -4.0*ln(2.0)  ==>  -2.77258872223978123767
    However the table is of length 1024, and equates to a radius of 2px
    thus needs to be scaled by  ALPHA*4/1024 and any blur factor squared

    The above came from some reference code provided by Fred Weinhaus
    and seems to have been a guess that was appropriate for its use
    in a 3d perspective landscape mapping program.
  */
  r_scale = -2.77258872223978123767/(WLUT_WIDTH*blur*blur);
  for(Q=0; Q<WLUT_WIDTH; Q++)
    resample_filter->filter_lut[Q] = exp((double)Q*r_scale);
  resample_filter->support = WLUT_WIDTH;
  break;
#endif

#if defined(MAGICKCORE_OPENMP_SUPPORT)
  /* if( GetOpenMPThreadId() == 0 ) */
#endif
    if (GetImageArtifact(resample_filter->image,"resample:verbose")
          != (const char *) NULL)
      {
        /* Debug output of the filter weighting LUT
          Gnuplot the LUT with hoizontal adjusted to 'r' using...
            plot [0:2][-.2:1] "lut.dat" using (sqrt($0/1024)*2):1 with lines
          The filter values is normalized for comparision
        */
        printf("#\n");
        printf("# Resampling Filter LUT (%d values)\n", WLUT_WIDTH);
        printf("#\n");
        printf("# Note: values in table are using a squared radius lookup.\n");
        printf("# And the whole table represents the filters support.\n");
        printf("#\n");
        for(Q=0; Q<WLUT_WIDTH; Q++)
          printf("%8.*g %.*g\n",
               GetMagickPrecision(),sqrt((double)Q)*r_scale,
               GetMagickPrecision(),resample_filter->filter_lut[Q] );
      }
  return;
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   S e t R e s a m p l e F i l t e r I n t e r p o l a t e M e t h o d       %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  SetResampleFilterInterpolateMethod() changes the interpolation method
%  associated with the specified resample filter.
%
%  The format of the SetResampleFilterInterpolateMethod method is:
%
%      MagickBooleanType SetResampleFilterInterpolateMethod(
%        ResampleFilter *resample_filter,const InterpolateMethod method)
%
%  A description of each parameter follows:
%
%    o resample_filter: the resample filter.
%
%    o method: the interpolation method.
%
*/
MagickExport MagickBooleanType SetResampleFilterInterpolateMethod(
  ResampleFilter *resample_filter,const InterpolatePixelMethod method)
{
  assert(resample_filter != (ResampleFilter *) NULL);
  assert(resample_filter->signature == MagickSignature);
  assert(resample_filter->image != (Image *) NULL);

  if (resample_filter->debug != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",
      resample_filter->image->filename);

  resample_filter->interpolate=method;

  return(MagickTrue);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   S e t R e s a m p l e F i l t e r V i r t u a l P i x e l M e t h o d     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  SetResampleFilterVirtualPixelMethod() changes the virtual pixel method
%  associated with the specified resample filter.
%
%  The format of the SetResampleFilterVirtualPixelMethod method is:
%
%      MagickBooleanType SetResampleFilterVirtualPixelMethod(
%        ResampleFilter *resample_filter,const VirtualPixelMethod method)
%
%  A description of each parameter follows:
%
%    o resample_filter: the resample filter.
%
%    o method: the virtual pixel method.
%
*/
MagickExport MagickBooleanType SetResampleFilterVirtualPixelMethod(
  ResampleFilter *resample_filter,const VirtualPixelMethod method)
{
  assert(resample_filter != (ResampleFilter *) NULL);
  assert(resample_filter->signature == MagickSignature);
  assert(resample_filter->image != (Image *) NULL);
  if (resample_filter->debug != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",
      resample_filter->image->filename);
  resample_filter->virtual_pixel=method;
  if (method != UndefinedVirtualPixelMethod)
    (void) SetCacheViewVirtualPixelMethod(resample_filter->view,method);
  return(MagickTrue);
}