magick/resize.c

/*
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%                 R   R  E      SS       I       ZZ  E                        %
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%                 R R    E         SS    I    ZZ     E                        %
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%                      MagickCore Image Resize Methods                        %
%                                                                             %
%                              Software Design                                %
%                                John Cristy                                  %
%                                 July 1992                                   %
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%                                                                             %
%  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.                                             %
%                                                                             %
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%
%
*/

/*
  Include declarations.
*/
#include "magick/studio.h"
#include "magick/artifact.h"
#include "magick/blob.h"
#include "magick/cache.h"
#include "magick/cache-view.h"
#include "magick/color.h"
#include "magick/color-private.h"
#include "magick/draw.h"
#include "magick/exception.h"
#include "magick/exception-private.h"
#include "magick/gem.h"
#include "magick/image.h"
#include "magick/image-private.h"
#include "magick/list.h"
#include "magick/memory_.h"
#include "magick/magick.h"
#include "magick/pixel-private.h"
#include "magick/property.h"
#include "magick/monitor.h"
#include "magick/monitor-private.h"
#include "magick/pixel.h"
#include "magick/option.h"
#include "magick/resample.h"
#include "magick/resize.h"
#include "magick/resize-private.h"
#include "magick/string_.h"
#include "magick/string-private.h"
#include "magick/thread-private.h"
#include "magick/utility.h"
#include "magick/version.h"
#if defined(MAGICKCORE_LQR_DELEGATE)
#include <lqr.h>
#endif

/*
  Typedef declarations.
*/
struct _ResizeFilter
{
  MagickRealType
    (*filter)(const MagickRealType,const ResizeFilter *),
    (*window)(const MagickRealType,const ResizeFilter *),
    support,        /* filter region of support - the filter support limit */
    window_support, /* window support, usally equal to support (expert only) */
    scale,          /* dimension scaling to fit window support (usally 1.0) */
    blur,           /* x-scale (blur-sharpen) */
    coeff[8];       /* cubic coefficents for smooth Cubic filters */

  size_t
    signature;
};

/*
  Forward declaractions.
*/
static MagickRealType
  I0(MagickRealType x),
  BesselOrderOne(MagickRealType),
  Sinc(const MagickRealType, const ResizeFilter *),
  SincFast(const MagickRealType, const ResizeFilter *);

/*
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+   F i l t e r F u n c t i o n s                                             %
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%
%  These are the various filter and windowing functions that are provided.
%
%  They are internal to this module only.  See AcquireResizeFilterInfo() for
%  details of the access to these functions, via the GetResizeFilterSupport()
%  and GetResizeFilterWeight() API interface.
%
%  The individual filter functions have this format...
%
%     static MagickRealtype *FilterName(const MagickRealType x,
%        const MagickRealType support)
%
%  A description of each parameter follows:
%
%    o x: the distance from the sampling point generally in the range of 0 to
%      support.  The GetResizeFilterWeight() ensures this a positive value.
%
%    o resize_filter: current filter information.  This allows function to
%      access support, and possibly other pre-calculated information defining
%      the functions.
%
*/

#define MagickPIL ((MagickRealType) 3.14159265358979323846264338327950288420L)

static MagickRealType Jinc(const MagickRealType x,
  const ResizeFilter *magick_unused(resize_filter))
{
  /*
    See Pratt "Digital Image Processing" p.97 for Jinc/Bessel functions.
    http://mathworld.wolfram.com/JincFunction.html and page 11 of
    http://www.ph.ed.ac.uk/%7ewjh/teaching/mo/slides/lens/lens.pdf

    The original "zoom" program by Paul Heckbert called this "Bessel".
    But really it is more accurately named "Jinc".
  */
  if (x == 0.0)
    return(0.5*MagickPIL);
  return(BesselOrderOne(MagickPIL*x)/x);
}

static MagickRealType Blackman(const MagickRealType x,
  const ResizeFilter *magick_unused(resize_filter))
{
  /*
    Blackman: 2nd order cosine windowing function:
      0.42 + 0.5 cos(pi x) + 0.08 cos(2pi x)
    Refactored by Chantal Racette and Nicolas Robidoux to one trig
    call and five flops.
  */
  const MagickRealType cospix = cos((double) (MagickPIL*x));
  return(0.34+cospix*(0.5+cospix*0.16));
}

static MagickRealType Bohman(const MagickRealType x,
  const ResizeFilter *magick_unused(resize_filter))
{
  /*
    Bohman: 2rd Order cosine windowing function:
      (1-x) cos(pi x) + sin(pi x) / pi.
    Refactored by Nicolas Robidoux to one trig call, one sqrt call,
    and 7 flops, taking advantage of the fact that the support of
    Bohman is 1 (so that we know that sin(pi x) >= 0).
  */
  const double cospix = cos((double) (MagickPIL*x));
  const double sinpix = sqrt(1.0-cospix*cospix);
  return((1.0-x)*cospix+(1.0/MagickPIL)*sinpix);
}

static MagickRealType Box(const MagickRealType magick_unused(x),
  const ResizeFilter *magick_unused(resize_filter))
{
  /*
    A Box filter is a equal weighting function (all weights equal).
    DO NOT LIMIT results by support or resize point sampling will work
    as it requests points beyond its normal 0.0 support size.
  */
  return(1.0);
}

static MagickRealType CubicBC(const MagickRealType x,
  const ResizeFilter *resize_filter)
{
  /*
    Cubic Filters using B,C determined values:
       Mitchell-Netravali  B=1/3 C=1/3   Qualitively ideal Cubic Filter
       Catmull-Rom         B= 0  C=1/2   Cublic Interpolation Function
       Cubic B-Spline      B= 1  C= 0    Spline Approximation of Gaussian
       Hermite             B= 0  C= 0    Quadratic Spline (support = 1)

    See paper by Mitchell and Netravali, Reconstruction Filters in Computer
    Graphics Computer Graphics, Volume 22, Number 4, August 1988
    http://www.cs.utexas.edu/users/fussell/courses/cs384g/lectures/mitchell/
    Mitchell.pdf.

    Coefficents are determined from B,C values:
       P0 = (  6 - 2*B       )/6
       P1 =         0
       P2 = (-18 +12*B + 6*C )/6
       P3 = ( 12 - 9*B - 6*C )/6
       Q0 = (      8*B +24*C )/6
       Q1 = (    -12*B -48*C )/6
       Q2 = (      6*B +30*C )/6
       Q3 = (    - 1*B - 6*C )/6

    which are used to define the filter:

       P0 + P1*x + P2*x^2 + P3*x^3      0 <= x < 1
       Q0 + Q1*x + Q2*x^2 + Q3*x^3      1 <= x <= 2

    which ensures function is continuous in value and derivative (slope).
  */
  if (x < 1.0)
    return(resize_filter->coeff[0]+x*(resize_filter->coeff[1]+x*
      (resize_filter->coeff[2]+x*resize_filter->coeff[3])));
  if (x < 2.0)
    return(resize_filter->coeff[4]+x*(resize_filter->coeff[5]+x*
      (resize_filter->coeff[6]+x*resize_filter->coeff[7])));
  return(0.0);
}

static MagickRealType Gaussian(const MagickRealType x,
  const ResizeFilter *resize_filter)
{
  /*
    Gaussian with a fixed sigma = 1/2

    Gaussian Formula...
        exp( -(x^2)/((2.0*sigma^2) ) / sqrt(2*PI*sigma^2)))
    The constants are pre-calculated...
        exp( -coeff[0]*(x^2)) ) * coeff[1]
    However the multiplier coefficent is not needed and not used.

    This separates the gaussian 'sigma' value from the 'blur/support' settings
    allows for its use in special 'small sigma' gaussians, without the filter
    'missing' pixels when blur and thus support becomes too small.
  */
  return(exp((double)(-resize_filter->coeff[0]*x*x))); }

static MagickRealType Hanning(const MagickRealType x,
  const ResizeFilter *magick_unused(resize_filter))
{
  /*
    Cosine window function:
      .5+.5cos(pi x).
  */
  const MagickRealType cospix = cos((double) (MagickPIL*x));
  return(0.5+0.5*cospix);
}

static MagickRealType Hamming(const MagickRealType x,
  const ResizeFilter *magick_unused(resize_filter))
{
  /*
    Offset cosine window function:
     .54 + .46 cos(pi x).
  */
  const MagickRealType cospix = cos((double) (MagickPIL*x));
  return(0.54+0.46*cospix);
}

static MagickRealType Kaiser(const MagickRealType x,
  const ResizeFilter *magick_unused(resize_filter))
{
#define Alpha  6.5
#define I0A  (1.0/I0(Alpha))

  /*
    Kaiser Windowing Function (bessel windowing): Alpha is a free
    value from 5 to 8 (currently hardcoded to 6.5).
    Future: make alpha the IOA pre-calculation, an 'expert' setting.
  */
  return(I0A*I0(Alpha*sqrt((double) (1.0-x*x))));
}

static MagickRealType Lagrange(const MagickRealType x,
  const ResizeFilter *resize_filter)
{
  MagickRealType
    value;

  register ssize_t
    i;

  ssize_t
    n,
    order;

  /*
    Lagrange piecewise polynomial fit of sinc: N is the 'order' of the
    lagrange function and depends on the overall support window size
    of the filter. That is: for a support of 2, it gives a lagrange-4
    (piecewise cubic function).

    "n" identifies the piece of the piecewise polynomial.

    See Survey: Interpolation Methods, IEEE Transactions on Medical
    Imaging, Vol 18, No 11, November 1999, p1049-1075, -- Equation 27
    on p1064.
  */
  if (x > resize_filter->support)
    return(0.0);
  order=(ssize_t) (2.0*resize_filter->window_support);  /* number of pieces */
  /*n=(ssize_t)((1.0*order)/2.0+x);   --  which piece does x belong to */
  n = (ssize_t)(resize_filter->window_support + x);
  value=1.0f;
  for (i=0; i < order; i++)
    if (i != n)
      value*=(n-i-x)/(n-i);
  return(value);
}

static MagickRealType Quadratic(const MagickRealType x,
  const ResizeFilter *magick_unused(resize_filter))
{
  /*
    2rd order (quadratic) B-Spline approximation of Gaussian.
  */
  if (x < 0.5)
    return(0.75-x*x);
  if (x < 1.5)
    return(0.5*(x-1.5)*(x-1.5));
  return(0.0);
}

static MagickRealType Sinc(const MagickRealType x,
  const ResizeFilter *magick_unused(resize_filter))
{
  /*
    Scaled sinc(x) function using a trig call:
      sinc(x) == sin(pi x)/(pi x).
  */
  if (x != 0.0)
  {
    const MagickRealType pix = (MagickRealType) (MagickPIL*x);
    return(sin((double) pix)/pix);
  }
  return((MagickRealType) 1.0);
}

static MagickRealType SincFast(const MagickRealType x,
  const ResizeFilter *magick_unused(resize_filter))
{
  /*
    Approximations of the sinc function sin(pi x)/(pi x) over the
    interval [-4,4] constructed by Nicolas Robidoux and Chantal
    Racette with funding from the Natural Sciences and Engineering
    Research Council of Canada.

    Although the approximations are polynomials (for low order of
    approximation) and quotients of polynomials (for higher order of
    approximation) and consequently are similar in form to Taylor
    polynomials/Pade approximants, the approximations are computed
    with a completely different technique.

    Summary: These approximations are "the best" in terms of bang
    (accuracy) for the buck (flops). More specifically: Among the
    polynomial quotients that can be computed using a fixed number of
    flops (with a given "+ - * / budget"), the chosen polynomial
    quotient is the one closest to the approximated function with
    respect to maximum absolute relative error over the given
    interval.

    The Remez algorithm, as implemented in the boost library's minimax
    package, is the key to the construction:
    http://www.boost.org/doc/libs/1_36_0/libs/math/doc/...
    ...sf_and_dist/html/math_toolkit/backgrounders/remez.html
  */
  /*
    If outside of the interval of approximation, use the standard trig
    formula.
  */
  if (x > 4.0)
    {
      const MagickRealType pix = (MagickRealType) (MagickPIL*x);
      return(sin((double) pix)/pix);
    }
  {
    /*
      The approximations only depend on x^2 (sinc is an even
      function).
    */
    const MagickRealType xx = x*x;
#if MAGICKCORE_QUANTUM_DEPTH <= 8
    /*
      Maximum absolute relative error 6.3e-6 < 1/2^17.
    */
    const MagickRealType c0 = 0.173610016489197553621906385078711564924e-2L;
    const MagickRealType c1 = -0.384186115075660162081071290162149315834e-3L;
    const MagickRealType c2 = 0.393684603287860108352720146121813443561e-4L;
    const MagickRealType c3 = -0.248947210682259168029030370205389323899e-5L;
    const MagickRealType c4 = 0.107791837839662283066379987646635416692e-6L;
    const MagickRealType c5 = -0.324874073895735800961260474028013982211e-8L;
    const MagickRealType c6 = 0.628155216606695311524920882748052490116e-10L;
    const MagickRealType c7 = -0.586110644039348333520104379959307242711e-12L;
    const MagickRealType p =
      c0+xx*(c1+xx*(c2+xx*(c3+xx*(c4+xx*(c5+xx*(c6+xx*c7))))));
    return((xx-1.0)*(xx-4.0)*(xx-9.0)*(xx-16.0)*p);
#elif MAGICKCORE_QUANTUM_DEPTH <= 16
    /*
      Max. abs. rel. error 2.2e-8 < 1/2^25.
    */
    const MagickRealType c0 = 0.173611107357320220183368594093166520811e-2L;
    const MagickRealType c1 = -0.384240921114946632192116762889211361285e-3L;
    const MagickRealType c2 = 0.394201182359318128221229891724947048771e-4L;
    const MagickRealType c3 = -0.250963301609117217660068889165550534856e-5L;
    const MagickRealType c4 = 0.111902032818095784414237782071368805120e-6L;
    const MagickRealType c5 = -0.372895101408779549368465614321137048875e-8L;
    const MagickRealType c6 = 0.957694196677572570319816780188718518330e-10L;
    const MagickRealType c7 = -0.187208577776590710853865174371617338991e-11L;
    const MagickRealType c8 = 0.253524321426864752676094495396308636823e-13L;
    const MagickRealType c9 = -0.177084805010701112639035485248501049364e-15L;
    const MagickRealType p =
      c0+xx*(c1+xx*(c2+xx*(c3+xx*(c4+xx*(c5+xx*(c6+xx*(c7+xx*(c8+xx*c9))))))));
    return((xx-1.0)*(xx-4.0)*(xx-9.0)*(xx-16.0)*p);
#else
    /*
      Max. abs. rel. error 1.2e-12 < 1/2^39.
    */
    const MagickRealType c0 = 0.173611111110910715186413700076827593074e-2L;
    const MagickRealType c1 = -0.289105544717893415815859968653611245425e-3L;
    const MagickRealType c2 = 0.206952161241815727624413291940849294025e-4L;
    const MagickRealType c3 = -0.834446180169727178193268528095341741698e-6L;
    const MagickRealType c4 = 0.207010104171026718629622453275917944941e-7L;
    const MagickRealType c5 = -0.319724784938507108101517564300855542655e-9L;
    const MagickRealType c6 = 0.288101675249103266147006509214934493930e-11L;
    const MagickRealType c7 = -0.118218971804934245819960233886876537953e-13L;
    const MagickRealType p =
      c0+xx*(c1+xx*(c2+xx*(c3+xx*(c4+xx*(c5+xx*(c6+xx*c7))))));
    const MagickRealType d0 = 1.0L;
    const MagickRealType d1 = 0.547981619622284827495856984100563583948e-1L;
    const MagickRealType d2 = 0.134226268835357312626304688047086921806e-2L;
    const MagickRealType d3 = 0.178994697503371051002463656833597608689e-4L;
    const MagickRealType d4 = 0.114633394140438168641246022557689759090e-6L;
    const MagickRealType q = d0+xx*(d1+xx*(d2+xx*(d3+xx*d4)));
    return((xx-1.0)*(xx-4.0)*(xx-9.0)*(xx-16.0)/q*p);
#endif
  }
}

static MagickRealType Triangle(const MagickRealType x,
  const ResizeFilter *magick_unused(resize_filter))
{
  /*
    1st order (linear) B-Spline, bilinear interpolation, Tent 1D
    filter, or a Bartlett 2D Cone filter.
  */
  if (x < 1.0)
    return(1.0-x);
  return(0.0);
}

static MagickRealType Welsh(const MagickRealType x,
  const ResizeFilter *magick_unused(resize_filter))
{
  /*
    Welsh parabolic windowing filter.
  */
  if (x < 1.0)
    return(1.0-x*x);
  return(0.0);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   A c q u i r e R e s i z e F i l t e r                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  AcquireResizeFilter() allocates the ResizeFilter structure.  Choose
%  from these filters:
%
%  FIR (Finite impulse Response) Filters
%      Box         Triangle   Quadratic
%      Cubic       Hermite    Catrom
%      Mitchell
%
%  IIR (Infinite impulse Response) Filters
%      Gaussian     Sinc        Jinc (Bessel)
%
%  Windowed Sinc/Jinc Filters
%      Blackman     Hanning     Hamming
%      Kaiser       Lanczos
%
%  Special purpose Filters
%      SincFast  Lanczos2D  Robidoux
%
%  The users "-filter" selection is used to lookup the default 'expert'
%  settings for that filter from a internal table.  However any provided
%  'expert' settings (see below) may override this selection.
%
%  FIR filters are used as is, and are limited to that filters support
%  window (unless over-ridden).  'Gaussian' while classed as an IIR
%  filter, is also simply clipped by its support size (currently 1.5
%  or approximatally 3*sigma as recommended by many references)
%
%  The selection is typically either a windowed Sinc, or interpolated
%  filter, for use by functions such as ResizeImage().  However if a
%  'cylindrical' filter flag is requested, any default Sinc weighting
%  and windowing functions will be promoted to cylindrical Jinc form of
%  function.
%
%  Directly requesting 'Sinc' or 'Jinc' will force the use of that
%  filter function without any windowing. This is not recommended,
%  except by image processing experts or in expert options.  Selecting a
%  window filtering version of these functions is better.
%
%  Lanczos is a special case of a Sinc-windowed Sinc, (or Jinc-Jinc for
%  the cylindrical case) but defaulting to 3-lobe support, rather that
%  the default 4 lobe support of the other windowed sinc/jinc filters.
%
%  Two forms of the 'Sinc' function are available: Sinc and SincFast.
%  Sinc is computed using the traditional sin(pi*x)/(pi*x); it is
%  selected if the user specifically specifies the use of a Sinc
%  filter. SincFast uses highly accurate (and fast) polynomial (low Q)
%  and rational (high Q) approximations, and will be used by default in
%  most cases.
%
%  The Lanczos2D and Robidoux filters are tuned for cylindrical
%  (radial) EWA (Elliptical Weighted Average) distortion.  Lanczos2D
%  is a 2 lobe Lanczos-like filter using Jinc (for EWA) or Sinc.
%  Robidoux used to be a sharpened version of Lanczos2D (with
%  blur=0.958033808). Now, it is the unique Cubic 'Keys' filter that
%  exactly preserves images with only vertical or horizontal features
%  when performing 'no-op" with EWA distortion. It turns out to be
%  close to both plain Mitchell and "sharpened" Lanczos2D.
%
%  Special 'expert' options can be used to override any and all filter
%  settings. This is not advised unless you have expert knowledge of
%  the use of resampling filtered techniques.  Check on the results of
%  your selections using the "filter:verbose" setting to make sure you
%  get the exact filter that you are tring to achieve.
%
%    "filter:filter"    Select the main function associated with
%        this filter name, as the weighting function of the filter.
%        This can be used to set a windowing function as a weighting
%        function, for special purposes, such as graphing.
%
%        If a "filter:window" operation has not been provided, then a 'Box'
%        windowing function will be set to denote that no windowing function
%        is being used.
%
%    "filter:window"   Select this windowing function for the filter.
%        While any filter could be used as a windowing function, using the
%        'first lobe' of that filter over the whole support window, using a
%        non-windowing function is not advisible. If no weighting filter
%        function is specifed a 'SincFast' filter will be used.
%
%    "filter:lobes"    Number of lobes to use for the Sinc/Jinc filter.
%        This a simpler method of setting filter support size that will
%        correctly handle the Sinc/Jinc switch for an operators filtering
%        requirements.  Only integers should be given.
%
%    "filter:support"  Set the support size for filtering to the size given
%        This not recommended for Sinc/Jinc windowed filters (lobes should
%        be used instead).  This will override any 'filter:lobes' option.
%
%    "filter:win-support"  Scale windowing function to this size instead.
%        This causes the windowing (or self-windowing Lagrange filter) to act
%        is if the support window it much much larger than what is actually
%        supplied to the calling operator.  The filter however is still
%        clipped to the real support size given, by the support range suppiled
%        to the caller.  If unset this will equal the normal filter support
%        size.
%
%    "filter:blur"     Scale the filter and support window by this amount.
%        A value >1 will generally result in a more burred image with
%        more ringing effects, while a value <1 will sharpen the
%        resulting image with more aliasing and Morie effects.
%
%    "filter:sigma"    The sigma value to use for the Gaussian filter only.
%        Defaults to '1/2' for orthogonal and 'sqrt(2)/2' for cylindrical
%        usage. It effectially provides a alturnative to 'blur' for Gaussians
%        without it also effecting the final 'practical support' size.
%
%    "filter:b"
%    "filter:c"    Override the preset B,C values for a Cubic type of filter
%         If only one of these are given it is assumes to be a 'Keys'
%         type of filter such that B+2C=1, where Keys 'alpha' value = C
%
%    "filter:verbose"   Output the exact results of the filter selections
%         made, as well as plotting data for graphing the resulting filter
%         over support range (blur adjusted).
%
%  Set a true un-windowed Sinc filter with 10 lobes (very slow)
%     -define filter:filter=Sinc
%     -define filter:lobes=8
%
%  For example force an 8 lobe Lanczos (Sinc or Jinc) filter...
%     -filter Lanczos
%     -define filter:lobes=8
%
%  The format of the AcquireResizeFilter method is:
%
%      ResizeFilter *AcquireResizeFilter(const Image *image,
%        const FilterTypes filter_type, const MagickBooleanType radial,
%        ExceptionInfo *exception)
%
%  A description of each parameter follows:
%
%    o image: the image.
%
%    o filter: the filter type, defining a preset filter, window and
%      support.  The artifact settings listed above will override
%      those selections.
%
%    o blur: blur the filter by this amount, use 1.0 if unknown.  Image
%      artifact "filter:blur" will override this API call usage, including
%      any internal change (such as for cylindrical usage).
%
%    o radial: use a 1D orthogonal filter (Sinc) or 2D cylindrical
%      (radial) filter (Jinc)
%
%    o exception: return any errors or warnings in this structure.
%
*/
MagickExport ResizeFilter *AcquireResizeFilter(const Image *image,
  const FilterTypes filter,const MagickRealType blur,
  const MagickBooleanType cylindrical,ExceptionInfo *exception)
{
  const char
    *artifact;

  FilterTypes
    filter_type,
    window_type;

  MagickRealType
    B,
    C,
    sigma;

  register ResizeFilter
    *resize_filter;

  ssize_t
    option;

  /*
    Table Mapping given Filter, into Weighting and Windowing functions.
    A 'Box' windowing function means its a simble non-windowed filter.
    An 'SincFast' filter function could be upgraded to a 'Jinc' filter
    if a "cylindrical", unless a 'Sinc' or 'SincFast' filter was
    specifically requested.

    WARNING: The order of this tabel must match the order of the
    FilterTypes enumeration specified in "resample.h", or the filter
    names will not match the filter being setup.

    You can check filter setups with the "filter:verbose" setting.
  */
  static struct
  {
    FilterTypes
      filter,
      window;
  } const mapping[SentinelFilter] =
  {
    { UndefinedFilter, BoxFilter },      /* Undefined (default to Box)       */
    { PointFilter,     BoxFilter },      /* SPECIAL: Nearest neighbour       */
    { BoxFilter,       BoxFilter },      /* Box averaging filter             */
    { TriangleFilter,  BoxFilter },      /* Linear interpolation filter      */
    { HermiteFilter,   BoxFilter },      /* Hermite interpolation filter     */
    { SincFastFilter,  HanningFilter },  /* Hanning -- cosine-sinc           */
    { SincFastFilter,  HammingFilter },  /* Hamming --      ''    variation  */
    { SincFastFilter,  BlackmanFilter }, /* Blackman -- 2*cosine-sinc        */
    { GaussianFilter,  BoxFilter },      /* Gaussian blur filter             */
    { QuadraticFilter, BoxFilter },      /* Quadratic Gaussian approximation */
    { CubicFilter,     BoxFilter },      /* Cubic B-Spline                   */
    { CatromFilter,    BoxFilter },      /* Cubic interpolator               */
    { MitchellFilter,  BoxFilter },      /* 'Ideal' cubic filter             */
    { LanczosFilter,   SincFastFilter }, /* SPECIAL: 3-lobed sinc-sinc       */
    { JincFilter,      BoxFilter },      /* Raw 3-lobed Jinc function        */
    { SincFilter,      BoxFilter },      /* Raw 4-lobed Sinc function        */
    { SincFastFilter,  KaiserFilter },   /* Kaiser -- square root-sinc       */
    { SincFastFilter,  WelshFilter },    /* Welsh -- parabolic-sinc          */
    { SincFastFilter,  CubicFilter },    /* Parzen -- cubic-sinc             */
    { LagrangeFilter,  BoxFilter },      /* Lagrange self-windowing filter   */
    { SincFastFilter,  BohmanFilter },   /* Bohman -- 2*cosine-sinc          */
    { SincFastFilter,  TriangleFilter }, /* Bartlett -- triangle-sinc        */
    { SincFastFilter,  BoxFilter },      /* Raw fast sinc ("Pade"-type)      */
    { Lanczos2DFilter, JincFilter },     /* SPECIAL: 2-lobed jinc-jinc       */
    { Lanczos2DSharpFilter, JincFilter },/* SPECIAL: ditto sharpened         */
    { RobidouxFilter,  BoxFilter },      /* SPECIAL: Keys cubic tuned for EWA */
  };
  /*
    Table mapping the filter/window from the above table to an actual
    function.  The default support size for that filter as a weighting
    function, the range to scale with to use that function as a sinc
    windowing function, (typ 1.0).

    Note that the filter_type -> function is 1 to 1 except for Sinc(),
    SincFast(), and CubicBC() functions, which may have multiple
    filter to function associations.

    See "filter:verbose" handling below for the function -> filter
    mapping.
  */
  static struct
  {
    MagickRealType
      (*function)(const MagickRealType, const ResizeFilter*),
      lobes,    /* Default lobes/support size of the weighting filter */
      scale,    /* Support when used as a windowing function, equal to
		   the scaling needed to match the support of the
		   windowed function */
      B,C;      /* Cubic spline coefficients, ignored if not a CubicBC
		   filter*/
  } const filters[SentinelFilter] =
  {
    { Box,       0.5, 0.5,     0.0, 0.0 }, /* Undefined (default to Box)  */
    { Box,       0.0, 0.5,     0.0, 0.0 }, /* Point (special handling)    */
    { Box,       0.5, 0.5,     0.0, 0.0 }, /* Box                         */
    { Triangle,  1.0, 1.0,     0.0, 0.0 }, /* Triangle                    */
    { CubicBC,   1.0, 1.0,     0.0, 0.0 }, /* Hermite (cubic  B=C=0)      */
    { Hanning,   1.0, 1.0,     0.0, 0.0 }, /* Hanning, cosine window      */
    { Hamming,   1.0, 1.0,     0.0, 0.0 }, /* Hamming, '' variation       */
    { Blackman,  1.0, 1.0,     0.0, 0.0 }, /* Blackman, 2*cosine window   */
    { Gaussian,  2.0, 1.5,     0.0, 0.0 }, /* Gaussian                    */
    { Quadratic, 1.5, 1.5,     0.0, 0.0 }, /* Quadratic gaussian          */
    { CubicBC,   2.0, 2.0,     1.0, 0.0 }, /* Cubic B-Spline (B=1,C=0)    */
    { CubicBC,   2.0, 1.0,     0.0, 0.5 }, /* Catmull-Rom    (B=0,C=1/2)  */
    { CubicBC,   2.0, 1.0, 1./3., 1./3. }, /* Mitchell       (B=C=1/3)    */
    { SincFast,  3.0, 1.0,     0.0, 0.0 }, /* Lanczos, 3-lobed Sinc-Sinc  */
    { Jinc,      3.0, 1.21967, 0.0, 0.0 }, /* Raw 3-lobed Jinc            */
    { Sinc,      4.0, 1.0,     0.0, 0.0 }, /* Raw 4-lobed Sinc            */
    { Kaiser,    1.0, 1.0,     0.0, 0.0 }, /* Kaiser (square root window) */
    { Welsh,     1.0, 1.0,     0.0, 0.0 }, /* Welsh (parabolic window)    */
    { CubicBC,   2.0, 2.0,     1.0, 0.0 }, /* Parzen (B-Spline window)    */
    { Lagrange,  2.0, 1.0,     0.0, 0.0 }, /* Lagrange sinc approximation */
    { Bohman,    1.0, 1.0,     0.0, 0.0 }, /* Bohman, 2*Cosine window     */
    { Triangle,  1.0, 1.0,     0.0, 0.0 }, /* Bartlett (triangle window)  */
    { SincFast,  4.0, 1.0,     0.0, 0.0 }, /* Raw fast sinc ("Pade"-type) */
    { Jinc,      2.0, 1.21966989, 0.0, 0.0 }, /* Lanczos2D (Jinc-Jinc)    */
    { Jinc,      2.0, 1.16848499, 0.0, 0.0 }, /* Lanczos2D Sharpened      */
    { CubicBC,   2.0, 1.16848499, 0.37821575509399862, 0.31089212245300069 }
         /* Robidoux: Keys cubic close to Lanczos2D with blur=0.958033808 */
  };
  /*
    The known zero crossings of the Jinc() or more accurately the Jinc(x*PI)
    function being used as a filter. It is used by the "filter:lobes" and for
    the 'lobes' number in the above, the for support selection, so users do
    not have to deal with the highly irrational sizes of the 'lobes' of the
    Jinc filter.

    Values taken from
      http://cose.math.bas.bg/webMathematica/webComputing/BesselZeros.jsp
    using Jv-function with v=1, then dividing by PI.
  */
  static MagickRealType
    jinc_zeros[16] =
    {
      1.21966989126651,
      2.23313059438153,
      3.23831548416624,
      4.24106286379607,
      5.24276437687019,
      6.24392168986449,
      7.24475986871996,
      8.24539491395205,
      9.24589268494948,
      10.2462933487549,
      11.2466227948779,
      12.2468984611381,
      13.2471325221811,
      14.2473337358069,
      15.2475085630373,
      16.247661874701
   };

  /*
    Allocate resize filter.
  */
  assert(image != (const Image *) NULL);
  assert(image->signature == MagickSignature);
  if (image->debug != MagickFalse)
    (void) LogMagickEvent(TraceEvent,GetMagickModule(),"%s",image->filename);
  assert(UndefinedFilter < filter && filter < SentinelFilter);
  assert(exception != (ExceptionInfo *) NULL);
  assert(exception->signature == MagickSignature);
  resize_filter=(ResizeFilter *) AcquireMagickMemory(sizeof(*resize_filter));
  if (resize_filter == (ResizeFilter *) NULL)
    ThrowFatalException(ResourceLimitFatalError,"MemoryAllocationFailed");
  /*
    Defaults for the requested filter.
  */
  filter_type=mapping[filter].filter;
  window_type=mapping[filter].window;
  resize_filter->blur = blur;
  sigma = 0.5;
  /* Cylindrical Filters should use Jinc instead of Sinc */
  if (cylindrical != MagickFalse)
    switch (filter_type)
    {
      case SincFilter:
        /* Promote 1D Sinc Filter to a 2D Jinc filter. */
        if ( filter != SincFilter )
          filter_type=JincFilter;
        break;
      case SincFastFilter:
        /* Ditto for SincFast variant */
        if ( filter != SincFastFilter )
          filter_type=JincFilter;
        break;
      case LanczosFilter:
        /* Promote Lanczos from a Sinc-Sinc to a Jinc-Jinc. */
        filter_type=JincFilter;
        window_type=JincFilter;
        break;
      case Lanczos2DSharpFilter:
        /* Sharpened by Nicholas Robidoux so as to optimize for
         * minimal blurring of orthogonal lines
         */
        resize_filter->blur *= 0.958033808;
        break;
      case GaussianFilter:
        sigma = (MagickRealType) (MagickSQ2/2.0);  /* Cylindrical Gaussian sigma is sqrt(2)/2 */
        break;
      default:
        break;
    }
  else
    switch (filter_type)
    {
      case Lanczos2DFilter:
      case Lanczos2DSharpFilter:
        /* Demote to a 2-lobe Sinc-Sinc for orthogonal use. */
        window_type=SincFastFilter;
        break;
      default:
        break;
    }

  artifact=GetImageArtifact(image,"filter:filter");
  if (artifact != (const char *) NULL)
    {
      option=ParseMagickOption(MagickFilterOptions,MagickFalse,artifact);
      if ((UndefinedFilter < option) && (option < SentinelFilter))
        { /* Raw filter request - no window function. */
          filter_type=(FilterTypes) option;
          window_type=BoxFilter;
        }
      if (option == LanczosFilter)
        { /* Lanczos is not a real filter but a self windowing Sinc/Jinc. */
          filter_type=cylindrical != MagickFalse ? JincFilter : LanczosFilter;
          window_type=cylindrical != MagickFalse ? JincFilter : SincFastFilter;
        }
      /* Filter override with a specific window function. */
      artifact=GetImageArtifact(image,"filter:window");
      if (artifact != (const char *) NULL)
        {
          option=ParseMagickOption(MagickFilterOptions,MagickFalse,artifact);
          if ((UndefinedFilter < option) && (option < SentinelFilter))
            {
              if (option != LanczosFilter)
                window_type=(FilterTypes) option;
              else
                window_type=cylindrical != MagickFalse ? JincFilter :
                  SincFastFilter;
            }
        }
    }
  else
    {
      /* Window specified, but no filter function?  Assume Sinc/Jinc. */
      artifact=GetImageArtifact(image,"filter:window");
      if (artifact != (const char *) NULL)
        {
          option=ParseMagickOption(MagickFilterOptions,MagickFalse,
            artifact);
          if ((UndefinedFilter < option) && (option < SentinelFilter))
            {
              filter_type=cylindrical != MagickFalse ?
                         JincFilter : SincFastFilter;
              window_type=(FilterTypes) option;
            }
        }
    }
  /* Assign the real functions to use for the filters selected. */
  resize_filter->filter=filters[filter_type].function;
  resize_filter->support=filters[filter_type].lobes;
  resize_filter->window=filters[window_type].function;
  resize_filter->scale=filters[window_type].scale;
  resize_filter->signature=MagickSignature;

  /* Filter Modifications for orthogonal/cylindrical usage */
  if (cylindrical != MagickFalse)
    switch (filter_type)
    {
      case PointFilter:
      case BoxFilter:
        /* Support for Cylindrical Box should be sqrt(2)/2 */
        resize_filter->support=(MagickRealType) MagickSQ1_2;
        break;
      default:
        break;
    }
  else
    switch (filter_type)
    {
      case Lanczos2DFilter:
      case Lanczos2DSharpFilter:
        /* Demote to a 2-lobe Lanczos (Sinc-Sinc) for orthogonal use. */
        resize_filter->filter=SincFast;
        break;
      default:
        break;
    }

  /*
  ** More Expert Option Modifications
  */

  /* User Sigma Override - no support change */
  artifact=GetImageArtifact(image,"filter:sigma");
  if (artifact != (const char *) NULL)
    sigma=StringToDouble(artifact);
  /* Define coefficents for Gaussian (assumes no cubic window) */
  if ( GaussianFilter ) {
    resize_filter->coeff[0] = 1.0/(2.0*sigma*sigma);
    resize_filter->coeff[1] = (MagickRealType) (1.0/(Magick2PI*sigma*sigma)); /* unused */
  }

  /* Blur Override */
  artifact=GetImageArtifact(image,"filter:blur");
  if (artifact != (const char *) NULL)
    resize_filter->blur=StringToDouble(artifact);
  if (resize_filter->blur < MagickEpsilon)
    resize_filter->blur=(MagickRealType) MagickEpsilon;

  /* Support Overrides */
  artifact=GetImageArtifact(image,"filter:lobes");
  if (artifact != (const char *) NULL)
    {
      ssize_t
        lobes;

      lobes=(ssize_t) StringToLong(artifact);
      if (lobes < 1)
        lobes=1;
      resize_filter->support=(MagickRealType) lobes;
    }
  /* convert Jinc lobes to a real support value */
  if (resize_filter->filter == Jinc)
    {
      if (resize_filter->support > 16)
        resize_filter->support=jinc_zeros[15];  /* largest entry in table */
      else
        resize_filter->support = jinc_zeros[((long)resize_filter->support)-1];
    }
  /* expert override of the support setting */
  artifact=GetImageArtifact(image,"filter:support");
  if (artifact != (const char *) NULL)
    resize_filter->support=fabs(StringToDouble(artifact));
  /*
    Scale windowing function separatally to the support 'clipping'
    window that calling operator is planning to actually use. (Expert
    override)
  */
  resize_filter->window_support=resize_filter->support; /* default */
  artifact=GetImageArtifact(image,"filter:win-support");
  if (artifact != (const char *) NULL)
    resize_filter->window_support=fabs(StringToDouble(artifact));
  /*
    Adjust window function scaling to the windowing support for
    weighting function.  This avoids a division on every filter call.
  */
  resize_filter->scale /= resize_filter->window_support;

  /*
   * Set Cubic Spline B,C values, calculate Cubic coefficients.
  */
  B=0.0;
  C=0.0;
  if ((filters[filter_type].function == CubicBC) ||
      (filters[window_type].function == CubicBC))
    {
      B=filters[filter_type].B;
      C=filters[filter_type].C;
      if (filters[window_type].function == CubicBC)
        {
          B=filters[window_type].B;
          C=filters[window_type].C;
        }
      artifact=GetImageArtifact(image,"filter:b");
      if (artifact != (const char *) NULL)
        {
          B=StringToDouble(artifact);
          C=(1.0-B)/2.0; /* Calculate C as if it is a Keys cubic filter. */
          artifact=GetImageArtifact(image,"filter:c"); /* user C override */
          if (artifact != (const char *) NULL)
            C=StringToDouble(artifact);
        }
      else
        {
          artifact=GetImageArtifact(image,"filter:c");
          if (artifact != (const char *) NULL)
            {
              C=StringToDouble(artifact);
              B=1.0-2.0*C;  /* Calculate B as if it is a Keys cubic filter. */
            }
        }
    /* Convert B,C values into Cubic Coefficents.  See CubicBC().  */
    resize_filter->coeff[0]=(6.0-2.0*B)/6.0;
    resize_filter->coeff[1]=0.0;
    resize_filter->coeff[2]=(-18.0+12.0*B+6.0*C)/6.0;
    resize_filter->coeff[3]=(12.0-9.0*B-6.0*C)/6.0;
    resize_filter->coeff[4]=(8.0*B+24.0*C)/6.0;
    resize_filter->coeff[5]=(-12.0*B-48.0*C)/6.0;
    resize_filter->coeff[6]=(6.0*B+30.0*C)/6.0;
    resize_filter->coeff[7]=(-B-6.0*C)/6.0;
  }

  /*
    Expert Option Request for verbose details of the resulting filter.
  */
#if defined(MAGICKCORE_OPENMP_SUPPORT)
  #pragma omp master
  {
#endif
    artifact=GetImageArtifact(image,"filter:verbose");
    if (artifact != (const char *) NULL)
      {
        double
          support,
          x;

        /*
          Set the weighting function properly when the weighting
          function may not exactly match the filter of the same name.
          EG: a Point filter really uses a Box weighting function
          with a different support than is typically used.

        */
        if (resize_filter->filter == Box)       filter_type=BoxFilter;
        if (resize_filter->filter == Sinc)      filter_type=SincFilter;
        if (resize_filter->filter == SincFast)  filter_type=SincFastFilter;
        if (resize_filter->filter == Jinc)      filter_type=JincFilter;
        if (resize_filter->filter == CubicBC)   filter_type=CubicFilter;
        /*
          Report Filter Details.
        */
        support=GetResizeFilterSupport(resize_filter); /* practical_support */
        (void) fprintf(stdout,"# Resize Filter (for graphing)\n#\n");
        (void) fprintf(stdout,"# filter = %s\n",MagickOptionToMnemonic(
           MagickFilterOptions,filter_type));
        (void) fprintf(stdout,"# window = %s\n",MagickOptionToMnemonic(
           MagickFilterOptions, window_type));
        (void) fprintf(stdout,"# support = %.*g\n",GetMagickPrecision(),
           (double) resize_filter->support);
        (void) fprintf(stdout,"# win-support = %.*g\n",GetMagickPrecision(),
           (double) resize_filter->window_support);
        (void) fprintf(stdout,"# scale_blur = %.*g\n",GetMagickPrecision(),
           (double) resize_filter->blur);
        if ( filter_type == GaussianFilter )
          (void) fprintf(stdout,"# gaussian_sigma = %.*g\n",GetMagickPrecision(),
             (double) sigma);
        (void) fprintf(stdout,"# practical_support = %.*g\n",GetMagickPrecision(),
           (double) support);
        if ( filter_type == CubicFilter || window_type == CubicFilter )
          (void) fprintf(stdout,"# B,C = %.*g,%.*g\n",GetMagickPrecision(),
             (double) B,GetMagickPrecision(),(double) C);
        (void) fprintf(stdout,"\n");
        /*
          Output values of resulting filter graph -- for graphing
          filter result.
        */
        for (x=0.0; x <= support; x+=0.01f)
          (void) fprintf(stdout,"%5.2lf\t%.*g\n",x,GetMagickPrecision(),
            (double) GetResizeFilterWeight(resize_filter,x));
        /* A final value so gnuplot can graph the 'stop' properly. */
        (void) fprintf(stdout,"%5.2lf\t%.*g\n",support,GetMagickPrecision(),
          0.0);
      }
      /* Output the above once only for each image - remove setting */
    (void) DeleteImageArtifact((Image *) image,"filter:verbose");
#if defined(MAGICKCORE_OPENMP_SUPPORT)
  }
#endif
  return(resize_filter);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   A d a p t i v e R e s i z e I m a g e                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  AdaptiveResizeImage() adaptively resize image with pixel resampling.
%
%  The format of the AdaptiveResizeImage method is:
%
%      Image *AdaptiveResizeImage(const Image *image,const size_t columns,
%        const size_t rows,ExceptionInfo *exception)
%
%  A description of each parameter follows:
%
%    o image: the image.
%
%    o columns: the number of columns in the resized image.
%
%    o rows: the number of rows in the resized image.
%
%    o exception: return any errors or warnings in this structure.
%
*/
MagickExport Image *AdaptiveResizeImage(const Image *image,
  const size_t columns,const size_t rows,ExceptionInfo *exception)
{
#define AdaptiveResizeImageTag  "Resize/Image"

  CacheView
    *resize_view;

  Image
    *resize_image;

  MagickBooleanType
    proceed;

  MagickPixelPacket
    pixel;

  PointInfo
    offset;

  ResampleFilter
    *resample_filter;

  ssize_t
    y;

  /*
    Adaptively resize image.
  */
  assert(image != (const 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);
  if ((columns == 0) || (rows == 0))
    return((Image *) NULL);
  if ((columns == image->columns) && (rows == image->rows))
    return(CloneImage(image,0,0,MagickTrue,exception));
  resize_image=CloneImage(image,columns,rows,MagickTrue,exception);
  if (resize_image == (Image *) NULL)
    return((Image *) NULL);
  if (SetImageStorageClass(resize_image,DirectClass) == MagickFalse)
    {
      InheritException(exception,&resize_image->exception);
      resize_image=DestroyImage(resize_image);
      return((Image *) NULL);
    }
  GetMagickPixelPacket(image,&pixel);
  resample_filter=AcquireResampleFilter(image,exception);
  (void) SetResampleFilter(resample_filter,PointFilter,1.0);
  (void) SetResampleFilterInterpolateMethod(resample_filter,
    MeshInterpolatePixel);
  resize_view=AcquireCacheView(resize_image);
  for (y=0; y < (ssize_t) resize_image->rows; y++)
  {
    register IndexPacket
      *restrict resize_indexes;

    register ssize_t
      x;

    register PixelPacket
      *restrict q;

    q=QueueCacheViewAuthenticPixels(resize_view,0,y,resize_image->columns,1,
      exception);
    if (q == (PixelPacket *) NULL)
      break;
    resize_indexes=GetCacheViewAuthenticIndexQueue(resize_view);
    offset.y=((MagickRealType) y*image->rows/resize_image->rows);
    for (x=0; x < (ssize_t) resize_image->columns; x++)
    {
      offset.x=((MagickRealType) x*image->columns/resize_image->columns);
      (void) ResamplePixelColor(resample_filter,offset.x-0.5,offset.y-0.5,
        &pixel);
      SetPixelPacket(resize_image,&pixel,q,resize_indexes+x);
      q++;
    }
    if (SyncCacheViewAuthenticPixels(resize_view,exception) == MagickFalse)
      break;
    proceed=SetImageProgress(image,AdaptiveResizeImageTag,(MagickOffsetType) y,
      image->rows);
    if (proceed == MagickFalse)
      break;
  }
  resample_filter=DestroyResampleFilter(resample_filter);
  resize_view=DestroyCacheView(resize_view);
  return(resize_image);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   B e s s e l O r d e r O n e                                               %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  BesselOrderOne() computes the Bessel function of x of the first kind of
%  order 0.  This is used to create the Jinc() filter function below.
%
%    Reduce x to |x| since j1(x)= -j1(-x), and for x in (0,8]
%
%       j1(x) = x*j1(x);
%
%    For x in (8,inf)
%
%       j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
%
%    where x1 = x-3*pi/4. Compute sin(x1) and cos(x1) as follow:
%
%       cos(x1) =  cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
%               =  1/sqrt(2) * (sin(x) - cos(x))
%       sin(x1) =  sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
%               = -1/sqrt(2) * (sin(x) + cos(x))
%
%  The format of the BesselOrderOne method is:
%
%      MagickRealType BesselOrderOne(MagickRealType x)
%
%  A description of each parameter follows:
%
%    o x: MagickRealType value.
%
*/

#undef I0
static MagickRealType I0(MagickRealType x)
{
  MagickRealType
    sum,
    t,
    y;

  register ssize_t
    i;

  /*
    Zeroth order Bessel function of the first kind.
  */
  sum=1.0;
  y=x*x/4.0;
  t=y;
  for (i=2; t > MagickEpsilon; i++)
  {
    sum+=t;
    t*=y/((MagickRealType) i*i);
  }
  return(sum);
}

#undef J1
static MagickRealType J1(MagickRealType x)
{
  MagickRealType
    p,
    q;

  register ssize_t
    i;

  static const double
    Pone[] =
    {
       0.581199354001606143928050809e+21,
      -0.6672106568924916298020941484e+20,
       0.2316433580634002297931815435e+19,
      -0.3588817569910106050743641413e+17,
       0.2908795263834775409737601689e+15,
      -0.1322983480332126453125473247e+13,
       0.3413234182301700539091292655e+10,
      -0.4695753530642995859767162166e+7,
       0.270112271089232341485679099e+4
    },
    Qone[] =
    {
      0.11623987080032122878585294e+22,
      0.1185770712190320999837113348e+20,
      0.6092061398917521746105196863e+17,
      0.2081661221307607351240184229e+15,
      0.5243710262167649715406728642e+12,
      0.1013863514358673989967045588e+10,
      0.1501793594998585505921097578e+7,
      0.1606931573481487801970916749e+4,
      0.1e+1
    };

  p=Pone[8];
  q=Qone[8];
  for (i=7; i >= 0; i--)
  {
    p=p*x*x+Pone[i];
    q=q*x*x+Qone[i];
  }
  return(p/q);
}

#undef P1
static MagickRealType P1(MagickRealType x)
{
  MagickRealType
    p,
    q;

  register ssize_t
    i;

  static const double
    Pone[] =
    {
      0.352246649133679798341724373e+5,
      0.62758845247161281269005675e+5,
      0.313539631109159574238669888e+5,
      0.49854832060594338434500455e+4,
      0.2111529182853962382105718e+3,
      0.12571716929145341558495e+1
    },
    Qone[] =
    {
      0.352246649133679798068390431e+5,
      0.626943469593560511888833731e+5,
      0.312404063819041039923015703e+5,
      0.4930396490181088979386097e+4,
      0.2030775189134759322293574e+3,
      0.1e+1
    };

  p=Pone[5];
  q=Qone[5];
  for (i=4; i >= 0; i--)
  {
    p=p*(8.0/x)*(8.0/x)+Pone[i];
    q=q*(8.0/x)*(8.0/x)+Qone[i];
  }
  return(p/q);
}

#undef Q1
static MagickRealType Q1(MagickRealType x)
{
  MagickRealType
    p,
    q;

  register ssize_t
    i;

  static const double
    Pone[] =
    {
      0.3511751914303552822533318e+3,
      0.7210391804904475039280863e+3,
      0.4259873011654442389886993e+3,
      0.831898957673850827325226e+2,
      0.45681716295512267064405e+1,
      0.3532840052740123642735e-1
    },
    Qone[] =
    {
      0.74917374171809127714519505e+4,
      0.154141773392650970499848051e+5,
      0.91522317015169922705904727e+4,
      0.18111867005523513506724158e+4,
      0.1038187585462133728776636e+3,
      0.1e+1
    };

  p=Pone[5];
  q=Qone[5];
  for (i=4; i >= 0; i--)
  {
    p=p*(8.0/x)*(8.0/x)+Pone[i];
    q=q*(8.0/x)*(8.0/x)+Qone[i];
  }
  return(p/q);
}

static MagickRealType BesselOrderOne(MagickRealType x)
{
  MagickRealType
    p,
    q;

  if (x == 0.0)
    return(0.0);
  p=x;
  if (x < 0.0)
    x=(-x);
  if (x < 8.0)
    return(p*J1(x));
  q=sqrt((double) (2.0/(MagickPI*x)))*(P1(x)*(1.0/sqrt(2.0)*(sin((double) x)-
    cos((double) x)))-8.0/x*Q1(x)*(-1.0/sqrt(2.0)*(sin((double) x)+
    cos((double) x))));
  if (p < 0.0)
    q=(-q);
  return(q);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   D e s t r o y R e s i z e F i l t e r                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  DestroyResizeFilter() destroy the resize filter.
%
%  The format of the DestroyResizeFilter method is:
%
%      ResizeFilter *DestroyResizeFilter(ResizeFilter *resize_filter)
%
%  A description of each parameter follows:
%
%    o resize_filter: the resize filter.
%
*/
MagickExport ResizeFilter *DestroyResizeFilter(ResizeFilter *resize_filter)
{
  assert(resize_filter != (ResizeFilter *) NULL);
  assert(resize_filter->signature == MagickSignature);
  resize_filter->signature=(~MagickSignature);
  resize_filter=(ResizeFilter *) RelinquishMagickMemory(resize_filter);
  return(resize_filter);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   G e t R e s i z e F i l t e r S u p p o r t                               %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  GetResizeFilterSupport() return the current support window size for this
%  filter.  Note that this may have been enlarged by filter:blur factor.
%
%  The format of the GetResizeFilterSupport method is:
%
%      MagickRealType GetResizeFilterSupport(const ResizeFilter *resize_filter)
%
%  A description of each parameter follows:
%
%    o filter: Image filter to use.
%
*/
MagickExport MagickRealType GetResizeFilterSupport(
  const ResizeFilter *resize_filter)
{
  assert(resize_filter != (ResizeFilter *) NULL);
  assert(resize_filter->signature == MagickSignature);
  return(resize_filter->support*resize_filter->blur);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   G e t R e s i z e F i l t e r W e i g h t                                 %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  GetResizeFilterWeight evaluates the specified resize filter at the point x
%  which usally lies between zero and the filters current 'support' and
%  returns the weight of the filter function at that point.
%
%  The format of the GetResizeFilterWeight method is:
%
%      MagickRealType GetResizeFilterWeight(const ResizeFilter *resize_filter,
%        const MagickRealType x)
%
%  A description of each parameter follows:
%
%    o filter: the filter type.
%
%    o x: the point.
%
*/
MagickExport MagickRealType GetResizeFilterWeight(
  const ResizeFilter *resize_filter,const MagickRealType x)
{
  MagickRealType
    scale,
    x_blur;

  /*
    Windowing function - scale the weighting filter by this amount.
  */
  assert(resize_filter != (ResizeFilter *) NULL);
  assert(resize_filter->signature == MagickSignature);
  x_blur=fabs((double) x)/resize_filter->blur;  /* X offset with blur scaling */
  if ((resize_filter->window_support < MagickEpsilon) ||
      (resize_filter->window == Box))
    scale=1.0;  /* Point or Box Filter -- avoid division by zero */
  else
    {
      scale=resize_filter->scale;
      scale=resize_filter->window(x_blur*scale,resize_filter);
    }
  return(scale*resize_filter->filter(x_blur,resize_filter));
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   M a g n i f y I m a g e                                                   %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  MagnifyImage() is a convenience method that scales an image proportionally
%  to twice its size.
%
%  The format of the MagnifyImage method is:
%
%      Image *MagnifyImage(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 Image *MagnifyImage(const Image *image,ExceptionInfo *exception)
{
  Image
    *magnify_image;

  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);
  magnify_image=ResizeImage(image,2*image->columns,2*image->rows,CubicFilter,
    1.0,exception);
  return(magnify_image);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   M i n i f y I m a g e                                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  MinifyImage() is a convenience method that scales an image proportionally
%  to half its size.
%
%  The format of the MinifyImage method is:
%
%      Image *MinifyImage(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 Image *MinifyImage(const Image *image,ExceptionInfo *exception)
{
  Image
    *minify_image;

  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);
  minify_image=ResizeImage(image,image->columns/2,image->rows/2,CubicFilter,
    1.0,exception);
  return(minify_image);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   R e s a m p l e I m a g e                                                 %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  ResampleImage() resize image in terms of its pixel size, so that when
%  displayed at the given resolution it will be the same size in terms of
%  real world units as the original image at the original resolution.
%
%  The format of the ResampleImage method is:
%
%      Image *ResampleImage(Image *image,const double x_resolution,
%        const double y_resolution,const FilterTypes filter,const double blur,
%        ExceptionInfo *exception)
%
%  A description of each parameter follows:
%
%    o image: the image to be resized to fit the given resolution.
%
%    o x_resolution: the new image x resolution.
%
%    o y_resolution: the new image y resolution.
%
%    o filter: Image filter to use.
%
%    o blur: the blur factor where > 1 is blurry, < 1 is sharp.
%
*/
MagickExport Image *ResampleImage(const Image *image,const double x_resolution,
  const double y_resolution,const FilterTypes filter,const double blur,
  ExceptionInfo *exception)
{
#define ResampleImageTag  "Resample/Image"

  Image
    *resample_image;

  size_t
    height,
    width;

  /*
    Initialize sampled image attributes.
  */
  assert(image != (const 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);
  width=(size_t) (x_resolution*image->columns/(image->x_resolution == 0.0 ?
    72.0 : image->x_resolution)+0.5);
  height=(size_t) (y_resolution*image->rows/(image->y_resolution == 0.0 ?
    72.0 : image->y_resolution)+0.5);
  resample_image=ResizeImage(image,width,height,filter,blur,exception);
  if (resample_image != (Image *) NULL)
    {
      resample_image->x_resolution=x_resolution;
      resample_image->y_resolution=y_resolution;
    }
  return(resample_image);
}
#if defined(MAGICKCORE_LQR_DELEGATE)

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   L i q u i d R e s c a l e I m a g e                                       %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  LiquidRescaleImage() rescales image with seam carving.
%
%  The format of the LiquidRescaleImage method is:
%
%      Image *LiquidRescaleImage(const Image *image,
%        const size_t columns,const size_t rows,
%        const double delta_x,const double rigidity,ExceptionInfo *exception)
%
%  A description of each parameter follows:
%
%    o image: the image.
%
%    o columns: the number of columns in the rescaled image.
%
%    o rows: the number of rows in the rescaled image.
%
%    o delta_x: maximum seam transversal step (0 means straight seams).
%
%    o rigidity: introduce a bias for non-straight seams (typically 0).
%
%    o exception: return any errors or warnings in this structure.
%
*/
MagickExport Image *LiquidRescaleImage(const Image *image,const size_t columns,
  const size_t rows,const double delta_x,const double rigidity,
  ExceptionInfo *exception)
{
#define LiquidRescaleImageTag  "Rescale/Image"

  CacheView
    *rescale_view;

  const char
    *map;

  guchar
    *packet;

  Image
    *rescale_image;

  int
    x,
    y;

  LqrCarver
    *carver;

  LqrRetVal
    lqr_status;

  MagickBooleanType
    status;

  MagickPixelPacket
    pixel;

  unsigned char
    *pixels;

  /*
    Liquid rescale image.
  */
  assert(image != (const 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);
  if ((columns == 0) || (rows == 0))
    return((Image *) NULL);
  if ((columns == image->columns) && (rows == image->rows))
    return(CloneImage(image,0,0,MagickTrue,exception));
  if ((columns <= 2) || (rows <= 2))
    return(ResizeImage(image,columns,rows,image->filter,image->blur,exception));
  if ((columns >= (2*image->columns)) || (rows >= (2*image->rows)))
    {
      Image
        *resize_image;

      size_t
        height,
        width;

      /*
        Honor liquid resize size limitations.
      */
      for (width=image->columns; columns >= (2*width-1); width*=2);
      for (height=image->rows; rows >= (2*height-1); height*=2);
      resize_image=ResizeImage(image,width,height,image->filter,image->blur,
        exception);
      if (resize_image == (Image *) NULL)
        return((Image *) NULL);
      rescale_image=LiquidRescaleImage(resize_image,columns,rows,delta_x,
        rigidity,exception);
      resize_image=DestroyImage(resize_image);
      return(rescale_image);
    }
  map="RGB";
  if (image->matte == MagickFalse)
    map="RGBA";
  if (image->colorspace == CMYKColorspace)
    {
      map="CMYK";
      if (image->matte == MagickFalse)
        map="CMYKA";
    }
  pixels=(unsigned char *) AcquireQuantumMemory(image->columns,image->rows*
    strlen(map)*sizeof(*pixels));
  if (pixels == (unsigned char *) NULL)
    return((Image *) NULL);
  status=ExportImagePixels(image,0,0,image->columns,image->rows,map,CharPixel,
    pixels,exception);
  if (status == MagickFalse)
    {
      pixels=(unsigned char *) RelinquishMagickMemory(pixels);
      ThrowImageException(ResourceLimitError,"MemoryAllocationFailed");
    }
  carver=lqr_carver_new(pixels,image->columns,image->rows,strlen(map));
  if (carver == (LqrCarver *) NULL)
    {
      pixels=(unsigned char *) RelinquishMagickMemory(pixels);
      ThrowImageException(ResourceLimitError,"MemoryAllocationFailed");
    }
  lqr_status=lqr_carver_init(carver,(int) delta_x,rigidity);
  lqr_status=lqr_carver_resize(carver,columns,rows);
  rescale_image=CloneImage(image,lqr_carver_get_width(carver),
    lqr_carver_get_height(carver),MagickTrue,exception);
  if (rescale_image == (Image *) NULL)
    {
      pixels=(unsigned char *) RelinquishMagickMemory(pixels);
      return((Image *) NULL);
    }
  if (SetImageStorageClass(rescale_image,DirectClass) == MagickFalse)
    {
      InheritException(exception,&rescale_image->exception);
      rescale_image=DestroyImage(rescale_image);
      return((Image *) NULL);
    }
  GetMagickPixelPacket(rescale_image,&pixel);
  (void) lqr_carver_scan_reset(carver);
  rescale_view=AcquireCacheView(rescale_image);
  while (lqr_carver_scan(carver,&x,&y,&packet) != 0)
  {
    register IndexPacket
      *restrict rescale_indexes;

    register PixelPacket
      *restrict q;

    q=QueueCacheViewAuthenticPixels(rescale_view,x,y,1,1,exception);
    if (q == (PixelPacket *) NULL)
      break;
    rescale_indexes=GetCacheViewAuthenticIndexQueue(rescale_view);
    pixel.red=QuantumRange*(packet[0]/255.0);
    pixel.green=QuantumRange*(packet[1]/255.0);
    pixel.blue=QuantumRange*(packet[2]/255.0);
    if (image->colorspace != CMYKColorspace)
      {
        if (image->matte == MagickFalse)
          pixel.opacity=QuantumRange*(packet[3]/255.0);
      }
    else
      {
        pixel.index=QuantumRange*(packet[3]/255.0);
        if (image->matte == MagickFalse)
          pixel.opacity=QuantumRange*(packet[4]/255.0);
      }
    SetPixelPacket(rescale_image,&pixel,q,rescale_indexes);
    if (SyncCacheViewAuthenticPixels(rescale_view,exception) == MagickFalse)
      break;
  }
  rescale_view=DestroyCacheView(rescale_view);
  /*
    Relinquish resources.
  */
  lqr_carver_destroy(carver);
  return(rescale_image);
}
#else
MagickExport Image *LiquidRescaleImage(const Image *image,
  const size_t magick_unused(columns),const size_t magick_unused(rows),
  const double magick_unused(delta_x),const double magick_unused(rigidity),
  ExceptionInfo *exception)
{
  assert(image != (const 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);
  (void) ThrowMagickException(exception,GetMagickModule(),MissingDelegateError,
    "DelegateLibrarySupportNotBuiltIn","`%s' (LQR)",image->filename);
  return((Image *) NULL);
}
#endif

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   R e s i z e I m a g e                                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  ResizeImage() scales an image to the desired dimensions, using the given
%  filter (see AcquireFilterInfo()).
%
%  If an undefined filter is given the filter defaults to Mitchell for a
%  colormapped image, a image with a matte channel, or if the image is
%  enlarged.  Otherwise the filter defaults to a Lanczos.
%
%  ResizeImage() was inspired by Paul Heckbert's "zoom" program.
%
%  The format of the ResizeImage method is:
%
%      Image *ResizeImage(Image *image,const size_t columns,
%        const size_t rows,const FilterTypes filter,const double blur,
%        ExceptionInfo *exception)
%
%  A description of each parameter follows:
%
%    o image: the image.
%
%    o columns: the number of columns in the scaled image.
%
%    o rows: the number of rows in the scaled image.
%
%    o filter: Image filter to use.
%
%    o blur: the blur factor where > 1 is blurry, < 1 is sharp.  Typically set
%      this to 1.0.
%
%    o exception: return any errors or warnings in this structure.
%
*/

typedef struct _ContributionInfo
{
  MagickRealType
    weight;

  ssize_t
    pixel;
} ContributionInfo;

static ContributionInfo **DestroyContributionThreadSet(
  ContributionInfo **contribution)
{
  register ssize_t
    i;

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

static ContributionInfo **AcquireContributionThreadSet(const size_t count)
{
  register ssize_t
    i;

  ContributionInfo
    **contribution;

  size_t
    number_threads;

  number_threads=GetOpenMPMaximumThreads();
  contribution=(ContributionInfo **) AcquireQuantumMemory(number_threads,
    sizeof(*contribution));
  if (contribution == (ContributionInfo **) NULL)
    return((ContributionInfo **) NULL);
  (void) ResetMagickMemory(contribution,0,number_threads*sizeof(*contribution));
  for (i=0; i < (ssize_t) number_threads; i++)
  {
    contribution[i]=(ContributionInfo *) AcquireQuantumMemory(count,
      sizeof(**contribution));
    if (contribution[i] == (ContributionInfo *) NULL)
      return(DestroyContributionThreadSet(contribution));
  }
  return(contribution);
}

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

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

static MagickBooleanType HorizontalFilter(const ResizeFilter *resize_filter,
  const Image *image,Image *resize_image,const MagickRealType x_factor,
  const MagickSizeType span,MagickOffsetType *offset,ExceptionInfo *exception)
{
#define ResizeImageTag  "Resize/Image"

  CacheView
    *image_view,
    *resize_view;

  ClassType
    storage_class;

  ContributionInfo
    **restrict contributions;

  MagickBooleanType
    status;

  MagickPixelPacket
    zero;

  MagickRealType
    scale,
    support;

  ssize_t
    x;

  /*
    Apply filter to resize horizontally from image to resize image.
  */
  scale=MagickMax(1.0/x_factor+MagickEpsilon,1.0);
  support=scale*GetResizeFilterSupport(resize_filter);
  storage_class=support > 0.5 ? DirectClass : image->storage_class;
  if (SetImageStorageClass(resize_image,storage_class) == MagickFalse)
    {
      InheritException(exception,&resize_image->exception);
      return(MagickFalse);
    }
  if (support < 0.5)
    {
      /*
        Support too small even for nearest neighbour: Reduce to point
        sampling.
      */
      support=(MagickRealType) 0.5;
      scale=1.0;
    }
  contributions=AcquireContributionThreadSet((size_t) (2.0*support+3.0));
  if (contributions == (ContributionInfo **) NULL)
    {
      (void) ThrowMagickException(exception,GetMagickModule(),
        ResourceLimitError,"MemoryAllocationFailed","`%s'",image->filename);
      return(MagickFalse);
    }
  status=MagickTrue;
  scale=1.0/scale;
  (void) ResetMagickMemory(&zero,0,sizeof(zero));
  image_view=AcquireCacheView(image);
  resize_view=AcquireCacheView(resize_image);
#if defined(MAGICKCORE_OPENMP_SUPPORT)
  #pragma omp parallel for shared(status)
#endif
  for (x=0; x < (ssize_t) resize_image->columns; x++)
  {
    MagickRealType
      center,
      density;

    register const IndexPacket
      *restrict indexes;

    register const PixelPacket
      *restrict p;

    register ContributionInfo
      *restrict contribution;

    register IndexPacket
      *restrict resize_indexes;

    register PixelPacket
      *restrict q;

    register ssize_t
      y;

    ssize_t
      n,
      start,
      stop;

    if (status == MagickFalse)
      continue;
    center=(MagickRealType) (x+0.5)/x_factor;
    start=(ssize_t) MagickMax(center-support+0.5,0.0);
    stop=(ssize_t) MagickMin(center+support+0.5,(double) image->columns);
    density=0.0;
    contribution=contributions[GetOpenMPThreadId()];
    for (n=0; n < (stop-start); n++)
    {
      contribution[n].pixel=start+n;
      contribution[n].weight=GetResizeFilterWeight(resize_filter,scale*
        ((MagickRealType) (start+n)-center+0.5));
      density+=contribution[n].weight;
    }
    if ((density != 0.0) && (density != 1.0))
      {
        register ssize_t
          i;

        /*
          Normalize.
        */
        density=1.0/density;
        for (i=0; i < n; i++)
          contribution[i].weight*=density;
      }
    p=GetCacheViewVirtualPixels(image_view,contribution[0].pixel,0,(size_t)
      (contribution[n-1].pixel-contribution[0].pixel+1),image->rows,exception);
    q=QueueCacheViewAuthenticPixels(resize_view,x,0,1,resize_image->rows,
      exception);
    if ((p == (const PixelPacket *) NULL) || (q == (PixelPacket *) NULL))
      {
        status=MagickFalse;
        continue;
      }
    indexes=GetCacheViewVirtualIndexQueue(image_view);
    resize_indexes=GetCacheViewAuthenticIndexQueue(resize_view);
    for (y=0; y < (ssize_t) resize_image->rows; y++)
    {
      MagickPixelPacket
        pixel;

      MagickRealType
        alpha;

      register ssize_t
        i;

      ssize_t
        j;

      pixel=zero;
      if (image->matte == MagickFalse)
        {
          for (i=0; i < n; i++)
          {
            j=y*(contribution[n-1].pixel-contribution[0].pixel+1)+
              (contribution[i].pixel-contribution[0].pixel);
            alpha=contribution[i].weight;
            pixel.red+=alpha*(p+j)->red;
            pixel.green+=alpha*(p+j)->green;
            pixel.blue+=alpha*(p+j)->blue;
            pixel.opacity+=alpha*(p+j)->opacity;
          }
          SetRedPixelComponent(q,ClampRedPixelComponent(&pixel));
          SetGreenPixelComponent(q,ClampGreenPixelComponent(&pixel));
          SetBluePixelComponent(q,ClampBluePixelComponent(&pixel));
          SetOpacityPixelComponent(q,ClampOpacityPixelComponent(&pixel));
          if ((image->colorspace == CMYKColorspace) &&
              (resize_image->colorspace == CMYKColorspace))
            {
              for (i=0; i < n; i++)
              {
                j=y*(contribution[n-1].pixel-contribution[0].pixel+1)+
                  (contribution[i].pixel-contribution[0].pixel);
                alpha=contribution[i].weight;
                pixel.index+=alpha*indexes[j];
              }
              resize_indexes[y]=(IndexPacket) ClampToQuantum(pixel.index);
            }
        }
      else
        {
          MagickRealType
            gamma;

          gamma=0.0;
          for (i=0; i < n; i++)
          {
            j=y*(contribution[n-1].pixel-contribution[0].pixel+1)+
              (contribution[i].pixel-contribution[0].pixel);
            alpha=contribution[i].weight*QuantumScale*
              GetAlphaPixelComponent(p+j);
            pixel.red+=alpha*(p+j)->red;
            pixel.green+=alpha*(p+j)->green;
            pixel.blue+=alpha*(p+j)->blue;
            pixel.opacity+=contribution[i].weight*(p+j)->opacity;
            gamma+=alpha;
          }
          gamma=1.0/(fabs((double) gamma) <= MagickEpsilon ? 1.0 : gamma);
          q->red=ClampToQuantum(gamma*GetRedPixelComponent(&pixel));
          q->green=ClampToQuantum(gamma*GetGreenPixelComponent(&pixel));
          q->blue=ClampToQuantum(gamma*GetBluePixelComponent(&pixel));
          SetOpacityPixelComponent(q,ClampOpacityPixelComponent(&pixel));
          if ((image->colorspace == CMYKColorspace) &&
              (resize_image->colorspace == CMYKColorspace))
            {
              for (i=0; i < n; i++)
              {
                j=y*(contribution[n-1].pixel-contribution[0].pixel+1)+
                  (contribution[i].pixel-contribution[0].pixel);
                alpha=contribution[i].weight*QuantumScale*
                  GetAlphaPixelComponent(p+j);
                pixel.index+=alpha*indexes[j];
              }
              resize_indexes[y]=(IndexPacket) ClampToQuantum(gamma*
               GetIndexPixelComponent(&pixel));
            }
        }
      if ((resize_image->storage_class == PseudoClass) &&
          (image->storage_class == PseudoClass))
        {
          i=(ssize_t) (MagickMin(MagickMax(center,(double) start),(double) stop-
            1.0)+0.5);
          j=y*(contribution[n-1].pixel-contribution[0].pixel+1)+
            (contribution[i-start].pixel-contribution[0].pixel);
          resize_indexes[y]=indexes[j];
        }
      q++;
    }
    if (SyncCacheViewAuthenticPixels(resize_view,exception) == MagickFalse)
      status=MagickFalse;
    if (image->progress_monitor != (MagickProgressMonitor) NULL)
      {
        MagickBooleanType
          proceed;

#if defined(MAGICKCORE_OPENMP_SUPPORT)
  #pragma omp critical (MagickCore_HorizontalFilter)
#endif
        proceed=SetImageProgress(image,ResizeImageTag,(*offset)++,span);
        if (proceed == MagickFalse)
          status=MagickFalse;
      }
  }
  resize_view=DestroyCacheView(resize_view);
  image_view=DestroyCacheView(image_view);
  contributions=DestroyContributionThreadSet(contributions);
  return(status);
}

static MagickBooleanType VerticalFilter(const ResizeFilter *resize_filter,
  const Image *image,Image *resize_image,const MagickRealType y_factor,
  const MagickSizeType span,MagickOffsetType *offset,ExceptionInfo *exception)
{
  CacheView
    *image_view,
    *resize_view;

  ClassType
    storage_class;

  ContributionInfo
    **restrict contributions;

  MagickBooleanType
    status;

  MagickPixelPacket
    zero;

  MagickRealType
    scale,
    support;

  ssize_t
    y;

  /*
    Apply filter to resize vertically from image to resize image.
  */
  scale=MagickMax(1.0/y_factor+MagickEpsilon,1.0);
  support=scale*GetResizeFilterSupport(resize_filter);
  storage_class=support > 0.5 ? DirectClass : image->storage_class;
  if (SetImageStorageClass(resize_image,storage_class) == MagickFalse)
    {
      InheritException(exception,&resize_image->exception);
      return(MagickFalse);
    }
  if (support < 0.5)
    {
      /*
        Support too small even for nearest neighbour: Reduce to point
        sampling.
      */
      support=(MagickRealType) 0.5;
      scale=1.0;
    }
  contributions=AcquireContributionThreadSet((size_t) (2.0*support+3.0));
  if (contributions == (ContributionInfo **) NULL)
    {
      (void) ThrowMagickException(exception,GetMagickModule(),
        ResourceLimitError,"MemoryAllocationFailed","`%s'",image->filename);
      return(MagickFalse);
    }
  status=MagickTrue;
  scale=1.0/scale;
  (void) ResetMagickMemory(&zero,0,sizeof(zero));
  image_view=AcquireCacheView(image);
  resize_view=AcquireCacheView(resize_image);
#if defined(MAGICKCORE_OPENMP_SUPPORT)
  #pragma omp parallel for shared(status)
#endif
  for (y=0; y < (ssize_t) resize_image->rows; y++)
  {
    MagickRealType
      center,
      density;

    register const IndexPacket
      *restrict indexes;

    register const PixelPacket
      *restrict p;

    register ContributionInfo
      *restrict contribution;

    register IndexPacket
      *restrict resize_indexes;

    register PixelPacket
      *restrict q;

    register ssize_t
      x;

    ssize_t
      n,
      start,
      stop;

    if (status == MagickFalse)
      continue;
    center=(MagickRealType) (y+0.5)/y_factor;
    start=(ssize_t) MagickMax(center-support+0.5,0.0);
    stop=(ssize_t) MagickMin(center+support+0.5,(double) image->rows);
    density=0.0;
    contribution=contributions[GetOpenMPThreadId()];
    for (n=0; n < (stop-start); n++)
    {
      contribution[n].pixel=start+n;
      contribution[n].weight=GetResizeFilterWeight(resize_filter,scale*
        ((MagickRealType) (start+n)-center+0.5));
      density+=contribution[n].weight;
    }
    if ((density != 0.0) && (density != 1.0))
      {
        register ssize_t
          i;

        /*
          Normalize.
        */
        density=1.0/density;
        for (i=0; i < n; i++)
          contribution[i].weight*=density;
      }
    p=GetCacheViewVirtualPixels(image_view,0,contribution[0].pixel,
      image->columns,(size_t) (contribution[n-1].pixel-contribution[0].pixel+1),
      exception);
    q=QueueCacheViewAuthenticPixels(resize_view,0,y,resize_image->columns,1,
      exception);
    if ((p == (const PixelPacket *) NULL) || (q == (PixelPacket *) NULL))
      {
        status=MagickFalse;
        continue;
      }
    indexes=GetCacheViewVirtualIndexQueue(image_view);
    resize_indexes=GetCacheViewAuthenticIndexQueue(resize_view);
    for (x=0; x < (ssize_t) resize_image->columns; x++)
    {
      MagickPixelPacket
        pixel;

      MagickRealType
        alpha;

      register ssize_t
        i;

      ssize_t
        j;

      pixel=zero;
      if (image->matte == MagickFalse)
        {
          for (i=0; i < n; i++)
          {
            j=(ssize_t) ((contribution[i].pixel-contribution[0].pixel)*
              image->columns+x);
            alpha=contribution[i].weight;
            pixel.red+=alpha*(p+j)->red;
            pixel.green+=alpha*(p+j)->green;
            pixel.blue+=alpha*(p+j)->blue;
            pixel.opacity+=alpha*(p+j)->opacity;
          }
          SetRedPixelComponent(q,ClampRedPixelComponent(&pixel));
          SetGreenPixelComponent(q,ClampGreenPixelComponent(&pixel));
          SetBluePixelComponent(q,ClampBluePixelComponent(&pixel));
          SetOpacityPixelComponent(q,ClampOpacityPixelComponent(&pixel));
          if ((image->colorspace == CMYKColorspace) &&
              (resize_image->colorspace == CMYKColorspace))
            {
              for (i=0; i < n; i++)
              {
                j=(ssize_t) ((contribution[i].pixel-contribution[0].pixel)*
                  image->columns+x);
                alpha=contribution[i].weight;
                pixel.index+=alpha*indexes[j];
              }
              resize_indexes[x]=(IndexPacket) ClampToQuantum(pixel.index);
            }
        }
      else
        {
          MagickRealType
            gamma;

          gamma=0.0;
          for (i=0; i < n; i++)
          {
            j=(ssize_t) ((contribution[i].pixel-contribution[0].pixel)*
              image->columns+x);
            alpha=contribution[i].weight*QuantumScale*
              GetAlphaPixelComponent(p+j);
            pixel.red+=alpha*(p+j)->red;
            pixel.green+=alpha*(p+j)->green;
            pixel.blue+=alpha*(p+j)->blue;
            pixel.opacity+=contribution[i].weight*(p+j)->opacity;
            gamma+=alpha;
          }
          gamma=1.0/(fabs((double) gamma) <= MagickEpsilon ? 1.0 : gamma);
          q->red=ClampToQuantum(gamma*GetRedPixelComponent(&pixel));
          q->green=ClampToQuantum(gamma*GetGreenPixelComponent(&pixel));
          q->blue=ClampToQuantum(gamma*GetBluePixelComponent(&pixel));
          SetOpacityPixelComponent(q,ClampOpacityPixelComponent(&pixel));
          if ((image->colorspace == CMYKColorspace) &&
              (resize_image->colorspace == CMYKColorspace))
            {
              for (i=0; i < n; i++)
              {
                j=(ssize_t) ((contribution[i].pixel-contribution[0].pixel)*
                  image->columns+x);
                alpha=contribution[i].weight*QuantumScale*
                  GetAlphaPixelComponent(p+j);
                pixel.index+=alpha*indexes[j];
              }
              resize_indexes[x]=(IndexPacket) ClampToQuantum(gamma*
                GetIndexPixelComponent(&pixel));
            }
        }
      if ((resize_image->storage_class == PseudoClass) &&
          (image->storage_class == PseudoClass))
        {
          i=(ssize_t) (MagickMin(MagickMax(center,(double) start),(double) stop-
            1.0)+0.5);
          j=(ssize_t) ((contribution[i-start].pixel-contribution[0].pixel)*
            image->columns+x);
          resize_indexes[x]=indexes[j];
        }
      q++;
    }
    if (SyncCacheViewAuthenticPixels(resize_view,exception) == MagickFalse)
      status=MagickFalse;
    if (image->progress_monitor != (MagickProgressMonitor) NULL)
      {
        MagickBooleanType
          proceed;

#if defined(MAGICKCORE_OPENMP_SUPPORT)
  #pragma omp critical (MagickCore_VerticalFilter)
#endif
        proceed=SetImageProgress(image,ResizeImageTag,(*offset)++,span);
        if (proceed == MagickFalse)
          status=MagickFalse;
      }
  }
  resize_view=DestroyCacheView(resize_view);
  image_view=DestroyCacheView(image_view);
  contributions=DestroyContributionThreadSet(contributions);
  return(status);
}

MagickExport Image *ResizeImage(const Image *image,const size_t columns,
  const size_t rows,const FilterTypes filter,const double blur,
  ExceptionInfo *exception)
{
#define WorkLoadFactor  0.265

  FilterTypes
    filter_type;

  Image
    *filter_image,
    *resize_image;

  MagickOffsetType
    offset;

  MagickRealType
    x_factor,
    y_factor;

  MagickSizeType
    span;

  MagickStatusType
    status;

  ResizeFilter
    *resize_filter;

  /*
    Acquire resize image.
  */
  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);
  if ((columns == 0) || (rows == 0))
    ThrowImageException(ImageError,"NegativeOrZeroImageSize");
  if ((columns == image->columns) && (rows == image->rows) &&
      (filter == UndefinedFilter) && (blur == 1.0))
    return(CloneImage(image,0,0,MagickTrue,exception));
  resize_image=CloneImage(image,columns,rows,MagickTrue,exception);
  if (resize_image == (Image *) NULL)
    return(resize_image);
  /*
    Acquire resize filter.
  */
  x_factor=(MagickRealType) columns/(MagickRealType) image->columns;
  y_factor=(MagickRealType) rows/(MagickRealType) image->rows;
  if ((x_factor*y_factor) > WorkLoadFactor)
    filter_image=CloneImage(image,columns,image->rows,MagickTrue,exception);
  else
    filter_image=CloneImage(image,image->columns,rows,MagickTrue,exception);
  if (filter_image == (Image *) NULL)
    return(DestroyImage(resize_image));
  filter_type=LanczosFilter;
  if (filter != UndefinedFilter)
    filter_type=filter;
  else
    if ((x_factor == 1.0) && (y_factor == 1.0))
      filter_type=PointFilter;
    else
      if ((image->storage_class == PseudoClass) ||
          (image->matte != MagickFalse) || ((x_factor*y_factor) > 1.0))
        filter_type=MitchellFilter;
  resize_filter=AcquireResizeFilter(image,filter_type,blur,MagickFalse,
    exception);
  /*
    Resize image.
  */
  offset=0;
  if ((x_factor*y_factor) > WorkLoadFactor)
    {
      span=(MagickSizeType) (filter_image->columns+rows);
      status=HorizontalFilter(resize_filter,image,filter_image,x_factor,span,
        &offset,exception);
      status&=VerticalFilter(resize_filter,filter_image,resize_image,y_factor,
        span,&offset,exception);
    }
  else
    {
      span=(MagickSizeType) (filter_image->rows+columns);
      status=VerticalFilter(resize_filter,image,filter_image,y_factor,span,
        &offset,exception);
      status&=HorizontalFilter(resize_filter,filter_image,resize_image,x_factor,
        span,&offset,exception);
    }
  /*
    Free resources.
  */
  filter_image=DestroyImage(filter_image);
  resize_filter=DestroyResizeFilter(resize_filter);
  if ((status == MagickFalse) || (resize_image == (Image *) NULL))
    return((Image *) NULL);
  resize_image->type=image->type;
  return(resize_image);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   S a m p l e I m a g e                                                     %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  SampleImage() scales an image to the desired dimensions with pixel
%  sampling.  Unlike other scaling methods, this method does not introduce
%  any additional color into the scaled image.
%
%  The format of the SampleImage method is:
%
%      Image *SampleImage(const Image *image,const size_t columns,
%        const size_t rows,ExceptionInfo *exception)
%
%  A description of each parameter follows:
%
%    o image: the image.
%
%    o columns: the number of columns in the sampled image.
%
%    o rows: the number of rows in the sampled image.
%
%    o exception: return any errors or warnings in this structure.
%
*/
MagickExport Image *SampleImage(const Image *image,const size_t columns,
  const size_t rows,ExceptionInfo *exception)
{
#define SampleImageTag  "Sample/Image"

  CacheView
    *image_view,
    *sample_view;

  Image
    *sample_image;

  MagickBooleanType
    status;

  MagickOffsetType
    progress;

  register ssize_t
    x;

  ssize_t
    *x_offset,
    y;

  /*
    Initialize sampled image attributes.
  */
  assert(image != (const 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);
  if ((columns == 0) || (rows == 0))
    ThrowImageException(ImageError,"NegativeOrZeroImageSize");
  if ((columns == image->columns) && (rows == image->rows))
    return(CloneImage(image,0,0,MagickTrue,exception));
  sample_image=CloneImage(image,columns,rows,MagickTrue,exception);
  if (sample_image == (Image *) NULL)
    return((Image *) NULL);
  /*
    Allocate scan line buffer and column offset buffers.
  */
  x_offset=(ssize_t *) AcquireQuantumMemory((size_t) sample_image->columns,
    sizeof(*x_offset));
  if (x_offset == (ssize_t *) NULL)
    {
      sample_image=DestroyImage(sample_image);
      ThrowImageException(ResourceLimitError,"MemoryAllocationFailed");
    }
  for (x=0; x < (ssize_t) sample_image->columns; x++)
    x_offset[x]=(ssize_t) (((MagickRealType) x+0.5)*image->columns/
      sample_image->columns);
  /*
    Sample each row.
  */
  status=MagickTrue;
  progress=0;
  image_view=AcquireCacheView(image);
  sample_view=AcquireCacheView(sample_image);
#if defined(MAGICKCORE_OPENMP_SUPPORT)
  #pragma omp parallel for schedule(dynamic,4) shared(progress,status)
#endif
  for (y=0; y < (ssize_t) sample_image->rows; y++)
  {
    register const IndexPacket
      *restrict indexes;

    register const PixelPacket
      *restrict p;

    register IndexPacket
      *restrict sample_indexes;

    register PixelPacket
      *restrict q;

    register ssize_t
      x;

    ssize_t
      y_offset;

    if (status == MagickFalse)
      continue;
    y_offset=(ssize_t) (((MagickRealType) y+0.5)*image->rows/
      sample_image->rows);
    p=GetCacheViewVirtualPixels(image_view,0,y_offset,image->columns,1,
      exception);
    q=QueueCacheViewAuthenticPixels(sample_view,0,y,sample_image->columns,1,
      exception);
    if ((p == (const PixelPacket *) NULL) || (q == (PixelPacket *) NULL))
      {
        status=MagickFalse;
        continue;
      }
    indexes=GetCacheViewAuthenticIndexQueue(image_view);
    sample_indexes=GetCacheViewAuthenticIndexQueue(sample_view);
    /*
      Sample each column.
    */
    for (x=0; x < (ssize_t) sample_image->columns; x++)
      *q++=p[x_offset[x]];
    if ((image->storage_class == PseudoClass) ||
        (image->colorspace == CMYKColorspace))
      for (x=0; x < (ssize_t) sample_image->columns; x++)
        sample_indexes[x]=indexes[x_offset[x]];
    if (SyncCacheViewAuthenticPixels(sample_view,exception) == MagickFalse)
      status=MagickFalse;
    if (image->progress_monitor != (MagickProgressMonitor) NULL)
      {
        MagickBooleanType
          proceed;

#if defined(MAGICKCORE_OPENMP_SUPPORT)
        #pragma omp critical (MagickCore_SampleImage)
#endif
        proceed=SetImageProgress(image,SampleImageTag,progress++,image->rows);
        if (proceed == MagickFalse)
          status=MagickFalse;
      }
  }
  image_view=DestroyCacheView(image_view);
  sample_view=DestroyCacheView(sample_view);
  x_offset=(ssize_t *) RelinquishMagickMemory(x_offset);
  sample_image->type=image->type;
  return(sample_image);
}

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   S c a l e I m a g e                                                       %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  ScaleImage() changes the size of an image to the given dimensions.
%
%  The format of the ScaleImage method is:
%
%      Image *ScaleImage(const Image *image,const size_t columns,
%        const size_t rows,ExceptionInfo *exception)
%
%  A description of each parameter follows:
%
%    o image: the image.
%
%    o columns: the number of columns in the scaled image.
%
%    o rows: the number of rows in the scaled image.
%
%    o exception: return any errors or warnings in this structure.
%
*/
MagickExport Image *ScaleImage(const Image *image,const size_t columns,
  const size_t rows,ExceptionInfo *exception)
{
#define ScaleImageTag  "Scale/Image"

  CacheView
    *image_view,
    *scale_view;

  Image
    *scale_image;

  MagickBooleanType
    next_column,
    next_row,
    proceed;

  MagickPixelPacket
    pixel,
    *scale_scanline,
    *scanline,
    *x_vector,
    *y_vector,
    zero;

  PointInfo
    scale,
    span;

  register ssize_t
    i;

  ssize_t
    number_rows,
    y;

  /*
    Initialize scaled image attributes.
  */
  assert(image != (const 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);
  if ((columns == 0) || (rows == 0))
    return((Image *) NULL);
  if ((columns == image->columns) && (rows == image->rows))
    return(CloneImage(image,0,0,MagickTrue,exception));
  scale_image=CloneImage(image,columns,rows,MagickTrue,exception);
  if (scale_image == (Image *) NULL)
    return((Image *) NULL);
  if (SetImageStorageClass(scale_image,DirectClass) == MagickFalse)
    {
      InheritException(exception,&scale_image->exception);
      scale_image=DestroyImage(scale_image);
      return((Image *) NULL);
    }
  /*
    Allocate memory.
  */
  x_vector=(MagickPixelPacket *) AcquireQuantumMemory((size_t) image->columns,
    sizeof(*x_vector));
  scanline=x_vector;
  if (image->rows != scale_image->rows)
    scanline=(MagickPixelPacket *) AcquireQuantumMemory((size_t) image->columns,
      sizeof(*scanline));
  scale_scanline=(MagickPixelPacket *) AcquireQuantumMemory((size_t)
    scale_image->columns,sizeof(*scale_scanline));
  y_vector=(MagickPixelPacket *) AcquireQuantumMemory((size_t) image->columns,
    sizeof(*y_vector));
  if ((scanline == (MagickPixelPacket *) NULL) ||
      (scale_scanline == (MagickPixelPacket *) NULL) ||
      (x_vector == (MagickPixelPacket *) NULL) ||
      (y_vector == (MagickPixelPacket *) NULL))
    {
      scale_image=DestroyImage(scale_image);
      ThrowImageException(ResourceLimitError,"MemoryAllocationFailed");
    }
  /*
    Scale image.
  */
  number_rows=0;
  next_row=MagickTrue;
  span.y=1.0;
  scale.y=(double) scale_image->rows/(double) image->rows;
  (void) ResetMagickMemory(y_vector,0,(size_t) image->columns*
    sizeof(*y_vector));
  GetMagickPixelPacket(image,&pixel);
  (void) ResetMagickMemory(&zero,0,sizeof(zero));
  i=0;
  image_view=AcquireCacheView(image);
  scale_view=AcquireCacheView(scale_image);
  for (y=0; y < (ssize_t) scale_image->rows; y++)
  {
    register const IndexPacket
      *restrict indexes;

    register const PixelPacket
      *restrict p;

    register IndexPacket
      *restrict scale_indexes;

    register MagickPixelPacket
      *restrict s,
      *restrict t;

    register PixelPacket
      *restrict q;

    register ssize_t
      x;

    q=QueueCacheViewAuthenticPixels(scale_view,0,y,scale_image->columns,1,
      exception);
    if (q == (PixelPacket *) NULL)
      break;
    scale_indexes=GetAuthenticIndexQueue(scale_image);
    if (scale_image->rows == image->rows)
      {
        /*
          Read a new scanline.
        */
        p=GetCacheViewVirtualPixels(image_view,0,i++,image->columns,1,
          exception);
        if (p == (const PixelPacket *) NULL)
          break;
        indexes=GetCacheViewVirtualIndexQueue(image_view);
        for (x=0; x < (ssize_t) image->columns; x++)
        {
          x_vector[x].red=(MagickRealType) GetRedPixelComponent(p);
          x_vector[x].green=(MagickRealType) GetGreenPixelComponent(p);
          x_vector[x].blue=(MagickRealType) GetBluePixelComponent(p);
          if (image->matte != MagickFalse)
            x_vector[x].opacity=(MagickRealType) GetOpacityPixelComponent(p);
          if (indexes != (IndexPacket *) NULL)
            x_vector[x].index=(MagickRealType) indexes[x];
          p++;
        }
      }
    else
      {
        /*
          Scale Y direction.
        */
        while (scale.y < span.y)
        {
          if ((next_row != MagickFalse) &&
              (number_rows < (ssize_t) image->rows))
            {
              /*
                Read a new scanline.
              */
              p=GetCacheViewVirtualPixels(image_view,0,i++,image->columns,1,
                exception);
              if (p == (const PixelPacket *) NULL)
                break;
              indexes=GetCacheViewVirtualIndexQueue(image_view);
              for (x=0; x < (ssize_t) image->columns; x++)
              {
                x_vector[x].red=(MagickRealType) GetRedPixelComponent(p);
                x_vector[x].green=(MagickRealType) GetGreenPixelComponent(p);
                x_vector[x].blue=(MagickRealType) GetBluePixelComponent(p);
                if (image->matte != MagickFalse)
                  x_vector[x].opacity=(MagickRealType)
                    GetOpacityPixelComponent(p);
                if (indexes != (IndexPacket *) NULL)
                  x_vector[x].index=(MagickRealType) indexes[x];
                p++;
              }
              number_rows++;
            }
          for (x=0; x < (ssize_t) image->columns; x++)
          {
            y_vector[x].red+=scale.y*x_vector[x].red;
            y_vector[x].green+=scale.y*x_vector[x].green;
            y_vector[x].blue+=scale.y*x_vector[x].blue;
            if (scale_image->matte != MagickFalse)
              y_vector[x].opacity+=scale.y*x_vector[x].opacity;
            if (scale_indexes != (IndexPacket *) NULL)
              y_vector[x].index+=scale.y*x_vector[x].index;
          }
          span.y-=scale.y;
          scale.y=(double) scale_image->rows/(double) image->rows;
          next_row=MagickTrue;
        }
        if ((next_row != MagickFalse) && (number_rows < (ssize_t) image->rows))
          {
            /*
              Read a new scanline.
            */
            p=GetCacheViewVirtualPixels(image_view,0,i++,image->columns,1,
              exception);
            if (p == (const PixelPacket *) NULL)
              break;
            indexes=GetCacheViewVirtualIndexQueue(image_view);
            for (x=0; x < (ssize_t) image->columns; x++)
            {
              x_vector[x].red=(MagickRealType) GetRedPixelComponent(p);
              x_vector[x].green=(MagickRealType) GetGreenPixelComponent(p);
              x_vector[x].blue=(MagickRealType) GetBluePixelComponent(p);
              if (image->matte != MagickFalse)
                x_vector[x].opacity=(MagickRealType)
                  GetOpacityPixelComponent(p);
              if (indexes != (IndexPacket *) NULL)
                x_vector[x].index=(MagickRealType) indexes[x];
              p++;
            }
            number_rows++;
            next_row=MagickFalse;
          }
        s=scanline;
        for (x=0; x < (ssize_t) image->columns; x++)
        {
          pixel.red=y_vector[x].red+span.y*x_vector[x].red;
          pixel.green=y_vector[x].green+span.y*x_vector[x].green;
          pixel.blue=y_vector[x].blue+span.y*x_vector[x].blue;
          if (image->matte != MagickFalse)
            pixel.opacity=y_vector[x].opacity+span.y*x_vector[x].opacity;
          if (scale_indexes != (IndexPacket *) NULL)
            pixel.index=y_vector[x].index+span.y*x_vector[x].index;
          s->red=pixel.red;
          s->green=pixel.green;
          s->blue=pixel.blue;
          if (scale_image->matte != MagickFalse)
            s->opacity=pixel.opacity;
          if (scale_indexes != (IndexPacket *) NULL)
            s->index=pixel.index;
          s++;
          y_vector[x]=zero;
        }
        scale.y-=span.y;
        if (scale.y <= 0)
          {
            scale.y=(double) scale_image->rows/(double) image->rows;
            next_row=MagickTrue;
          }
        span.y=1.0;
      }
    if (scale_image->columns == image->columns)
      {
        /*
          Transfer scanline to scaled image.
        */
        s=scanline;
        for (x=0; x < (ssize_t) scale_image->columns; x++)
        {
          q->red=ClampToQuantum(s->red);
          q->green=ClampToQuantum(s->green);
          q->blue=ClampToQuantum(s->blue);
          if (scale_image->matte != MagickFalse)
            q->opacity=ClampToQuantum(s->opacity);
          if (scale_indexes != (IndexPacket *) NULL)
            scale_indexes[x]=(IndexPacket) ClampToQuantum(s->index);
          q++;
          s++;
        }
      }
    else
      {
        /*
          Scale X direction.
        */
        pixel=zero;
        next_column=MagickFalse;
        span.x=1.0;
        s=scanline;
        t=scale_scanline;
        for (x=0; x < (ssize_t) image->columns; x++)
        {
          scale.x=(double) scale_image->columns/(double) image->columns;
          while (scale.x >= span.x)
          {
            if (next_column != MagickFalse)
              {
                pixel=zero;
                t++;
              }
            pixel.red+=span.x*s->red;
            pixel.green+=span.x*s->green;
            pixel.blue+=span.x*s->blue;
            if (image->matte != MagickFalse)
              pixel.opacity+=span.x*s->opacity;
            if (scale_indexes != (IndexPacket *) NULL)
              pixel.index+=span.x*s->index;
            t->red=pixel.red;
            t->green=pixel.green;
            t->blue=pixel.blue;
            if (scale_image->matte != MagickFalse)
              t->opacity=pixel.opacity;
            if (scale_indexes != (IndexPacket *) NULL)
              t->index=pixel.index;
            scale.x-=span.x;
            span.x=1.0;
            next_column=MagickTrue;
          }
        if (scale.x > 0)
          {
            if (next_column != MagickFalse)
              {
                pixel=zero;
                next_column=MagickFalse;
                t++;
              }
            pixel.red+=scale.x*s->red;
            pixel.green+=scale.x*s->green;
            pixel.blue+=scale.x*s->blue;
            if (scale_image->matte != MagickFalse)
              pixel.opacity+=scale.x*s->opacity;
            if (scale_indexes != (IndexPacket *) NULL)
              pixel.index+=scale.x*s->index;
            span.x-=scale.x;
          }
        s++;
      }
      if (span.x > 0)
        {
          s--;
          pixel.red+=span.x*s->red;
          pixel.green+=span.x*s->green;
          pixel.blue+=span.x*s->blue;
          if (scale_image->matte != MagickFalse)
            pixel.opacity+=span.x*s->opacity;
          if (scale_indexes != (IndexPacket *) NULL)
            pixel.index+=span.x*s->index;
        }
      if ((next_column == MagickFalse) &&
          ((ssize_t) (t-scale_scanline) < (ssize_t) scale_image->columns))
        {
          t->red=pixel.red;
          t->green=pixel.green;
          t->blue=pixel.blue;
          if (scale_image->matte != MagickFalse)
            t->opacity=pixel.opacity;
          if (scale_indexes != (IndexPacket *) NULL)
            t->index=pixel.index;
        }
      /*
        Transfer scanline to scaled image.
      */
      t=scale_scanline;
      for (x=0; x < (ssize_t) scale_image->columns; x++)
      {
        q->red=ClampToQuantum(t->red);
        q->green=ClampToQuantum(t->green);
        q->blue=ClampToQuantum(t->blue);
        if (scale_image->matte != MagickFalse)
          q->opacity=ClampToQuantum(t->opacity);
        if (scale_indexes != (IndexPacket *) NULL)
          scale_indexes[x]=(IndexPacket) ClampToQuantum(t->index);
        t++;
        q++;
      }
    }
    if (SyncCacheViewAuthenticPixels(scale_view,exception) == MagickFalse)
      break;
    proceed=SetImageProgress(image,ScaleImageTag,(MagickOffsetType) y,
      image->rows);
    if (proceed == MagickFalse)
      break;
  }
  scale_view=DestroyCacheView(scale_view);
  image_view=DestroyCacheView(image_view);
  /*
    Free allocated memory.
  */
  y_vector=(MagickPixelPacket *) RelinquishMagickMemory(y_vector);
  scale_scanline=(MagickPixelPacket *) RelinquishMagickMemory(scale_scanline);
  if (scale_image->rows != image->rows)
    scanline=(MagickPixelPacket *) RelinquishMagickMemory(scanline);
  x_vector=(MagickPixelPacket *) RelinquishMagickMemory(x_vector);
  scale_image->type=image->type;
  return(scale_image);
}

#if 0
      THIS IS NOT USED  --  to be removed
/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
+   S e t R e s i z e F i l t e r S u p p o r t                               %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  SetResizeFilterSupport() specifies which IR filter to use to window
%
%  The format of the SetResizeFilterSupport method is:
%
%      void SetResizeFilterSupport(ResizeFilter *resize_filter,
%        const MagickRealType support)
%
%  A description of each parameter follows:
%
%    o resize_filter: the resize filter.
%
%    o support: the filter spport radius.
%
*/
MagickExport void SetResizeFilterSupport(ResizeFilter *resize_filter,
  const MagickRealType support)
{
  assert(resize_filter != (ResizeFilter *) NULL);
  assert(resize_filter->signature == MagickSignature);
  resize_filter->support=support;
}
#endif

/*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                             %
%                                                                             %
%                                                                             %
%   T h u m b n a i l I m a g e                                               %
%                                                                             %
%                                                                             %
%                                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  ThumbnailImage() changes the size of an image to the given dimensions and
%  removes any associated profiles.  The goal is to produce small low cost
%  thumbnail images suited for display on the Web.
%
%  The format of the ThumbnailImage method is:
%
%      Image *ThumbnailImage(const Image *image,const size_t columns,
%        const size_t rows,ExceptionInfo *exception)
%
%  A description of each parameter follows:
%
%    o image: the image.
%
%    o columns: the number of columns in the scaled image.
%
%    o rows: the number of rows in the scaled image.
%
%    o exception: return any errors or warnings in this structure.
%
*/
MagickExport Image *ThumbnailImage(const Image *image,const size_t columns,
  const size_t rows,ExceptionInfo *exception)
{
#define SampleFactor  5

  char
    value[MaxTextExtent];

  const char
    *name;

  Image
    *thumbnail_image;

  MagickRealType
    x_factor,
    y_factor;

  size_t
    version;

  struct stat
    attributes;

  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);
  x_factor=(MagickRealType) columns/(MagickRealType) image->columns;
  y_factor=(MagickRealType) rows/(MagickRealType) image->rows;
  if ((x_factor*y_factor) > 0.1)
    thumbnail_image=ResizeImage(image,columns,rows,image->filter,image->blur,
      exception);
  else
    if (((SampleFactor*columns) < 128) || ((SampleFactor*rows) < 128))
      thumbnail_image=ResizeImage(image,columns,rows,image->filter,
        image->blur,exception);
    else
      {
        Image
          *sample_image;

        sample_image=SampleImage(image,SampleFactor*columns,SampleFactor*rows,
          exception);
        if (sample_image == (Image *) NULL)
          return((Image *) NULL);
        thumbnail_image=ResizeImage(sample_image,columns,rows,image->filter,
          image->blur,exception);
        sample_image=DestroyImage(sample_image);
      }
  if (thumbnail_image == (Image *) NULL)
    return(thumbnail_image);
  (void) ParseAbsoluteGeometry("0x0+0+0",&thumbnail_image->page);
  if (thumbnail_image->matte == MagickFalse)
    (void) SetImageAlphaChannel(thumbnail_image,OpaqueAlphaChannel);
  thumbnail_image->depth=8;
  thumbnail_image->interlace=NoInterlace;
  /*
    Strip all profiles except color profiles.
  */
  ResetImageProfileIterator(thumbnail_image);
  for (name=GetNextImageProfile(thumbnail_image); name != (const char *) NULL; )
  {
    if ((LocaleCompare(name,"icc") != 0) && (LocaleCompare(name,"icm") != 0))
     {
       (void) DeleteImageProfile(thumbnail_image,name);
       ResetImageProfileIterator(thumbnail_image);
     }
    name=GetNextImageProfile(thumbnail_image);
  }
  (void) DeleteImageProperty(thumbnail_image,"comment");
  (void) CopyMagickString(value,image->magick_filename,MaxTextExtent);
  if (strstr(image->magick_filename,"//") == (char *) NULL)
    (void) FormatMagickString(value,MaxTextExtent,"file://%s",
      image->magick_filename);
  (void) SetImageProperty(thumbnail_image,"Thumb::URI",value);
  (void) CopyMagickString(value,image->magick_filename,MaxTextExtent);
  if (GetPathAttributes(image->filename,&attributes) != MagickFalse)
    {
      (void) FormatMagickString(value,MaxTextExtent,"%.20g",(double)
        attributes.st_mtime);
      (void) SetImageProperty(thumbnail_image,"Thumb::MTime",value);
    }
  (void) FormatMagickString(value,MaxTextExtent,"%.20g",(double)
    attributes.st_mtime);
  (void) FormatMagickSize(GetBlobSize(image),MagickFalse,value);
  (void) ConcatenateMagickString(value,"B",MaxTextExtent);
  (void) SetImageProperty(thumbnail_image,"Thumb::Size",value);
  (void) FormatMagickString(value,MaxTextExtent,"image/%s",image->magick);
  LocaleLower(value);
  (void) SetImageProperty(thumbnail_image,"Thumb::Mimetype",value);
  (void) SetImageProperty(thumbnail_image,"software",
    GetMagickVersion(&version));
  (void) FormatMagickString(value,MaxTextExtent,"%.20g",(double)
    image->magick_columns);
  (void) SetImageProperty(thumbnail_image,"Thumb::Image::Width",value);
  (void) FormatMagickString(value,MaxTextExtent,"%.20g",(double)
    image->magick_rows);
  (void) SetImageProperty(thumbnail_image,"Thumb::Image::height",value);
  (void) FormatMagickString(value,MaxTextExtent,"%.20g",(double)
    GetImageListLength(image));
  (void) SetImageProperty(thumbnail_image,"Thumb::Document::Pages",value);
  return(thumbnail_image);
}