lbb.cpp

/* lbb (locally bounded bicubic) resampler
 *
 * N. Robidoux, C. Racette and J. Cupitt, 23-28/03/2010
 *
 * N. Robidoux, 16-19/05/2010
 *
 * N. Robidoux, 22/11/2011
 */

/*

    This file is part of VIPS.

    VIPS is free software; you can redistribute it and/or modify it
    under the terms of the GNU Lesser General Public License as
    published by the Free Software Foundation; either version 2 of the
    License, or (at your option) any later version.

    This program is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
    Lesser General Public License for more details.

    You should have received a copy of the GNU Lesser General Public
    License along with this program; if not, write to the Free Software
    Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
    02110-1301 USA

 */

/*

    These files are distributed with VIPS - http://www.vips.ecs.soton.ac.uk

 */

/*
 * 2010 (c) Nicolas Robidoux, Chantal Racette, John Cupitt.
 *
 * N. Robidoux thanks Adam Turcotte, Geert Jordaens, Ralf Meyer,
 * Øyvind Kolås, Minglun Gong, Eric Daoust and Sven Neumann for useful
 * comments and code.
 *
 * C. Racette's image resampling research and programming funded in
 * part by an NSERC (National Science and Engineering Research Council
 * of Canada) Alexander Graham Bell Canada Graduate Scholarship, by an
 * NSERC Discovery Grant awarded to Julien Dompierre (grant number
 * 20-61098) and by N. Robidoux's Laurentian University professional
 * allowance.
 */

/*
 * LBB has two versions:
 *
 *   A "soft" version, which shows a little less staircasing and a
 *   little more haloing, and which is a little more expensive to
 *   compute. We recommend this as the default.
 *
 *   A "sharp" version, which shows a little more staircasing and a
 *   little less haloing, which is a little cheaper (it uses 6 less
 *   comparisons and 12 less "? :").
 *
 * The only difference between the two is that the "soft" versions
 * uses local minima and maxima computed over 3x3 square blocks, and
 * the "sharp" version uses local minima and maxima computed over 3x3
 * crosses.
 *
 * If you want to use the "sharp" version, comment out the following
 * three pre-processor code lines:
 */
/*
#ifndef __LBB_CHEAP_H__
#define __LBB_CHEAP_H__
#endif
*/

/*
 * LBB (Locally Bounded Bicubic) is a high quality nonlinear variant
 * of Catmull-Rom. Images resampled with LBB have much smaller halos
 * than images resampled with windowed sincs or other interpolatory
 * cubic spline filters. Specifically, LBB halos are narrower and the
 * over/undershoot amplitude is smaller. This is accomplished without
 * significantly affecting the smoothness of the result (compared to
 * Catmull-Rom).
 *
 * Another important property is that the resampled values are
 * contained within the range of nearby input values. Consequently, no
 * final clamping is needed to stay "in range" (e.g., 0-255 for
 * standard 8-bit images).
 *
 * LBB was developed by N. Robidoux and C. Racette at the Department
 * of Mathematics and Computer Science of Laurentian University in the
 * course of C. Racette's Masters thesis in Computational
 * Sciences. Preliminary work directly leading to the LBB method and
 * code was performed by C. Racette and N. Robidoux in the course of
 * her honours thesis, and by N. Robidoux, A. Turcotte and E. Daoust
 * during Google Summer of Code 2009 (through two awards made to GIMP
 * to improve GEGL).
 *
 * LBB is a novel method with the following properties:
 *
 * --LBB is a Hermite bicubic method: The bicubic surface is defined,
 *   one convex hull of four nearby input points at a time, using four
 *   point values, four x-derivatives, four y-derivatives, and four
 *   cross-derivatives.
 *
 * --The stencil for values in a square patch is the usual 4x4.
 *
 * --LBB is interpolatory.
 *
 * --It is C^1 with continuous cross derivatives.
 *
 * --When the limiters are inactive, LBB gives the same result as
 *   Catmull-Rom.
 *
 * --When used on binary images, LBB gives results similar to bicubic
 *   Hermite with all first derivatives---but not necessarily the
 *   cross derivatives (this last assertion needs to be double
 *   checked)--at input pixel locations set to zero.
 *
 * --The LBB reconstruction is locally bounded: Over each square
 *   patch, the surface is contained between the minimum and the
 *   maximum of the 16 nearest input pixel values.
 *
 * --Consequently, the LBB reconstruction is globally bounded between
 *   the very smallest input pixel value and the very largest input
 *   pixel value. It is not necessary to clamp results.
 *
 * The LBB method is based on the method of Ken Brodlie, Petros
 * Mashwama and Sohail Butt for constraining Hermite interpolants
 * between globally defined planes:
 *
 *   Visualization of surface data to preserve positivity and other
 *   simple constraints. Computer & Graphics, Vol. 19, Number 4, pages
 *   585-594, 1995. DOI: 10.1016/0097-8493(95)00036-C.
 *
 * Instead of forcing the reconstructed surface to lie between two
 * GLOBALLY defined planes, LBB constrains one patch at a time to lie
 * between LOCALLY defined planes. This is accomplished by
 * constraining the derivatives (x, y and cross) at each input pixel
 * location so that if the constraint was applied everywhere the
 * surface would fit between the min and max of the values at the 9
 * closest pixel locations. Because this is done with each of the four
 * pixel locations which define the bicubic patch, this forces the
 * reconstructed surface to lie between the min and max of the values
 * at the 16 closest values pixel locations. (Each corner defines its
 * own 3x3 subgroup of the 4x4 stencil. Consequently, the surface is
 * necessarily above the minimum of the four minima, which happens to
 * be the minimum over the 4x4. Similarly with the maxima.)
 *
 * The above paragraph described the "soft" version of LBB. The
 * "sharp" version is similar.
 */

#ifdef HAVE_CONFIG_H
#include <config.h>
#endif /*HAVE_CONFIG_H*/
#include <vips/intl.h>

#include <stdio.h>
#include <stdlib.h>

#include <vips/vips.h>
#include <vips/internal.h>

#include "templates.h"

#define VIPS_TYPE_INTERPOLATE_LBB \
	(vips_interpolate_lbb_get_type())
#define VIPS_INTERPOLATE_LBB( obj ) \
	(G_TYPE_CHECK_INSTANCE_CAST( (obj), \
	VIPS_TYPE_INTERPOLATE_LBB, VipsInterpolateLbb ))
#define VIPS_INTERPOLATE_LBB_CLASS( klass ) \
	(G_TYPE_CHECK_CLASS_CAST( (klass), \
	VIPS_TYPE_INTERPOLATE_LBB, VipsInterpolateLbbClass))
#define VIPS_IS_INTERPOLATE_LBB( obj ) \
	(G_TYPE_CHECK_INSTANCE_TYPE( (obj), VIPS_TYPE_INTERPOLATE_LBB ))
#define VIPS_IS_INTERPOLATE_LBB_CLASS( klass ) \
	(G_TYPE_CHECK_CLASS_TYPE( (klass), VIPS_TYPE_INTERPOLATE_LBB ))
#define VIPS_INTERPOLATE_LBB_GET_CLASS( obj ) \
	(G_TYPE_INSTANCE_GET_CLASS( (obj), \
	VIPS_TYPE_INTERPOLATE_LBB, VipsInterpolateLbbClass ))

typedef struct _VipsInterpolateLbb {
	VipsInterpolate parent_object;

} VipsInterpolateLbb;

typedef struct _VipsInterpolateLbbClass {
	VipsInterpolateClass parent_class;

} VipsInterpolateLbbClass;

/*
 * Absolute value and sign macros:
 */
#define LBB_ABS(x)  ( ((x)>=0.) ? (x) : -(x) )
#define LBB_SIGN(x) ( ((x)>=0.) ? 1.0 : -1.0 )
/*
 * MIN and MAX macros set up so that I can put the likely winner in
 * the first argument (forward branch likely blah blah blah):
 */
#define LBB_MIN(x,y) ( ((x)<=(y)) ? (x) : (y) )
#define LBB_MAX(x,y) ( ((x)>=(y)) ? (x) : (y) )

static inline double
lbbicubic( const double c00,
           const double c10,
           const double c01,
           const double c11,
           const double c00dx,
           const double c10dx,
           const double c01dx,
           const double c11dx,
           const double c00dy,
           const double c10dy,
           const double c01dy,
           const double c11dy,
           const double c00dxdy,
           const double c10dxdy,
           const double c01dxdy,
           const double c11dxdy,
           const double uno_one,
           const double uno_two,
           const double uno_thr,
           const double uno_fou,
           const double dos_one,
           const double dos_two,
           const double dos_thr,
           const double dos_fou,
           const double tre_one,
           const double tre_two,
           const double tre_thr,
           const double tre_fou,
           const double qua_one,
           const double qua_two,
           const double qua_thr,
           const double qua_fou )
{
  /*
   * STENCIL (FOOTPRINT) OF INPUT VALUES:
   *
   * The stencil of LBB is the same as for any standard Hermite
   * bicubic (e.g., Catmull-Rom):
   *
   *  (ix-1,iy-1)  (ix,iy-1)    (ix+1,iy-1)  (ix+2,iy-1)
   *  = uno_one    = uno_two    = uno_thr    = uno_fou
   *
   *  (ix-1,iy)    (ix,iy)      (ix+1,iy)    (ix+2,iy)
   *  = dos_one    = dos_two    = dos_thr    = dos_fou
   *                        X
   *  (ix-1,iy+1)  (ix,iy+1)    (ix+1,iy+1)  (ix+2,iy+1)
   *  = tre_one    = tre_two    = tre_thr    = tre_fou
   *
   *  (ix-1,iy+2)  (ix,iy+2)    (ix+1,iy+2)  (ix+2,iy+2)
   *  = qua_one    = qua_two    = qua_thr    = qua_fou
   *
   * where ix is the (pseudo-)floor of the requested left-to-right
   * location ("X"), and iy is the floor of the requested up-to-down
   * location.
   */

#if defined (__LBB_CHEAP_H__)
  /*
   * Computation of the four min and four max over 3x3 input data
   * sub-crosses of the 4x4 input stencil, performed with only 22
   * comparisons and 28 "? :". If you can figure out how to do this
   * more efficiently, let us know.
   *
   * This is the cheaper (but arguably less desirable in terms of
   * quality) version of the computation.
   */
  const double m1    = (dos_two <= dos_thr) ? dos_two : dos_thr  ;
  const double M1    = (dos_two <= dos_thr) ? dos_thr : dos_two  ;
  const double m2    = (tre_two <= tre_thr) ? tre_two : tre_thr  ;
  const double M2    = (tre_two <= tre_thr) ? tre_thr : tre_two  ;
  const double m3    = (uno_two <= dos_one) ? uno_two : dos_one  ;
  const double M3    = (uno_two <= dos_one) ? dos_one : uno_two  ;
  const double m4    = (uno_thr <= dos_fou) ? uno_thr : dos_fou  ;
  const double M4    = (uno_thr <= dos_fou) ? dos_fou : uno_thr  ;
  const double m5    = (tre_one <= qua_two) ? tre_one : qua_two  ;
  const double M5    = (tre_one <= qua_two) ? qua_two : tre_one  ;
  const double m6    = (tre_fou <= qua_thr) ? tre_fou : qua_thr  ;
  const double M6    = (tre_fou <= qua_thr) ? qua_thr : tre_fou  ;
  const double m7    = LBB_MIN(               m1,       tre_two );
  const double M7    = LBB_MAX(               M1,       tre_two );
  const double m8    = LBB_MIN(               m1,       tre_thr );
  const double M8    = LBB_MAX(               M1,       tre_thr );
  const double m9    = LBB_MIN(               m2,       dos_two );
  const double M9    = LBB_MAX(               M2,       dos_two );
  const double m10   = LBB_MIN(               m2,       dos_thr );
  const double M10   = LBB_MAX(               M2,       dos_thr );
  const double min00 = LBB_MIN(               m7,       m3      );
  const double max00 = LBB_MAX(               M7,       M3      );
  const double min10 = LBB_MIN(               m8,       m4      );
  const double max10 = LBB_MAX(               M8,       M4      );
  const double min01 = LBB_MIN(               m9,       m5      );
  const double max01 = LBB_MAX(               M9,       M5      );
  const double min11 = LBB_MIN(              m10,       m6      );
  const double max11 = LBB_MAX(              M10,       M6      );
#else
  /*
   * Computation of the four min and four max over 3x3 input data
   * sub-blocks of the 4x4 input stencil, performed with only 28
   * comparisons and 34 "? :". If you can figure how to do this more
   * efficiently, let us know.
   */
  const double m1    = (dos_two <= dos_thr) ? dos_two : dos_thr  ;
  const double M1    = (dos_two <= dos_thr) ? dos_thr : dos_two  ;
  const double m2    = (tre_two <= tre_thr) ? tre_two : tre_thr  ;
  const double M2    = (tre_two <= tre_thr) ? tre_thr : tre_two  ;
  const double m6    = (dos_one <= tre_one) ? dos_one : tre_one  ;
  const double M6    = (dos_one <= tre_one) ? tre_one : dos_one  ;
  const double m7    = (dos_fou <= tre_fou) ? dos_fou : tre_fou  ;
  const double M7    = (dos_fou <= tre_fou) ? tre_fou : dos_fou  ;
  const double m3    = (uno_two <= uno_thr) ? uno_two : uno_thr  ;
  const double M3    = (uno_two <= uno_thr) ? uno_thr : uno_two  ;
  const double m4    = (qua_two <= qua_thr) ? qua_two : qua_thr  ;
  const double M4    = (qua_two <= qua_thr) ? qua_thr : qua_two  ;
  const double m5    = LBB_MIN(               m1,       m2      );
  const double M5    = LBB_MAX(               M1,       M2      );
  const double m10   = LBB_MIN(               m6,       uno_one );
  const double M10   = LBB_MAX(               M6,       uno_one );
  const double m11   = LBB_MIN(               m6,       qua_one );
  const double M11   = LBB_MAX(               M6,       qua_one );
  const double m12   = LBB_MIN(               m7,       uno_fou );
  const double M12   = LBB_MAX(               M7,       uno_fou );
  const double m13   = LBB_MIN(               m7,       qua_fou );
  const double M13   = LBB_MAX(               M7,       qua_fou );
  const double m8    = LBB_MIN(               m5,       m3      );
  const double M8    = LBB_MAX(               M5,       M3      );
  const double m9    = LBB_MIN(               m5,       m4      );
  const double M9    = LBB_MAX(               M5,       M4      );
  const double min00 = LBB_MIN(               m8,       m10     );
  const double max00 = LBB_MAX(               M8,       M10     );
  const double min10 = LBB_MIN(               m8,       m12     );
  const double max10 = LBB_MAX(               M8,       M12     );
  const double min01 = LBB_MIN(               m9,       m11     );
  const double max01 = LBB_MAX(               M9,       M11     );
  const double min11 = LBB_MIN(               m9,       m13     );
  const double max11 = LBB_MAX(               M9,       M13     );
#endif

  /*
   * The remainder of the "per channel" computation involves the
   * computation of:
   *
   * --8 conditional moves,
   *
   * --8 signs (in which the sign of zero is unimportant),
   *
   * --12 minima of two values,
   *
   * --8 maxima of two values,
   *
   * --8 absolute values,
   *
   * for a grand total of 29 minima, 25 maxima, 8 conditional moves, 8
   * signs, and 8 absolute values. If everything is done with
   * conditional moves, "only" 28+8+8+12+8+8=72 flags are involved
   * (because initial min and max can be computed with one flag).
   *
   * The "per channel" part of the computation also involves 107
   * arithmetic operations (54 *, 21 +, 42 -).
   */

  /*
   * Distances to the local min and max:
   */
  const double u00 = dos_two - min00;
  const double v00 = max00 - dos_two;
  const double u10 = dos_thr - min10;
  const double v10 = max10 - dos_thr;
  const double u01 = tre_two - min01;
  const double v01 = max01 - tre_two;
  const double u11 = tre_thr - min11;
  const double v11 = max11 - tre_thr;

  /*
   * Initial values of the derivatives computed with centered
   * differences. Factors of 1/2 are left out because they are folded
   * in later:
   */
  const double dble_dzdx00i = dos_thr - dos_one;
  const double dble_dzdy11i = qua_thr - dos_thr;
  const double dble_dzdx10i = dos_fou - dos_two;
  const double dble_dzdy01i = qua_two - dos_two;
  const double dble_dzdx01i = tre_thr - tre_one;
  const double dble_dzdy10i = tre_thr - uno_thr;
  const double dble_dzdx11i = tre_fou - tre_two;
  const double dble_dzdy00i = tre_two - uno_two;

  /*
   * Signs of the derivatives. The upcoming clamping does not change
   * them (except if the clamping sends a negative derivative to 0, in
   * which case the sign does not matter anyway).
   */
  const double sign_dzdx00 = LBB_SIGN( dble_dzdx00i );
  const double sign_dzdx10 = LBB_SIGN( dble_dzdx10i );
  const double sign_dzdx01 = LBB_SIGN( dble_dzdx01i );
  const double sign_dzdx11 = LBB_SIGN( dble_dzdx11i );

  const double sign_dzdy00 = LBB_SIGN( dble_dzdy00i );
  const double sign_dzdy10 = LBB_SIGN( dble_dzdy10i );
  const double sign_dzdy01 = LBB_SIGN( dble_dzdy01i );
  const double sign_dzdy11 = LBB_SIGN( dble_dzdy11i );

  /*
   * Initial values of the cross-derivatives. Factors of 1/4 are left
   * out because folded in later:
   */
  const double quad_d2zdxdy00i = uno_one - uno_thr + dble_dzdx01i;
  const double quad_d2zdxdy10i = uno_two - uno_fou + dble_dzdx11i;
  const double quad_d2zdxdy01i = qua_thr - qua_one - dble_dzdx00i;
  const double quad_d2zdxdy11i = qua_fou - qua_two - dble_dzdx10i;

  /*
   * Slope limiters. The key multiplier is 3 but we fold a factor of
   * 2, hence 6:
   */
  const double dble_slopelimit_00 = 6.0 * LBB_MIN( u00, v00 );
  const double dble_slopelimit_10 = 6.0 * LBB_MIN( u10, v10 );
  const double dble_slopelimit_01 = 6.0 * LBB_MIN( u01, v01 );
  const double dble_slopelimit_11 = 6.0 * LBB_MIN( u11, v11 );

  /*
   * Clamped first derivatives:
   */
  const double dble_dzdx00 =
    ( sign_dzdx00 * dble_dzdx00i <= dble_slopelimit_00 )
    ? dble_dzdx00i :  sign_dzdx00 * dble_slopelimit_00;
  const double dble_dzdy00 =
    ( sign_dzdy00 * dble_dzdy00i <= dble_slopelimit_00 )
    ? dble_dzdy00i :  sign_dzdy00 * dble_slopelimit_00;
  const double dble_dzdx10 =
    ( sign_dzdx10 * dble_dzdx10i <= dble_slopelimit_10 )
    ? dble_dzdx10i :  sign_dzdx10 * dble_slopelimit_10;
  const double dble_dzdy10 =
    ( sign_dzdy10 * dble_dzdy10i <= dble_slopelimit_10 )
    ? dble_dzdy10i :  sign_dzdy10 * dble_slopelimit_10;
  const double dble_dzdx01 =
    ( sign_dzdx01 * dble_dzdx01i <= dble_slopelimit_01 )
    ? dble_dzdx01i :  sign_dzdx01 * dble_slopelimit_01;
  const double dble_dzdy01 =
    ( sign_dzdy01 * dble_dzdy01i <= dble_slopelimit_01 )
    ? dble_dzdy01i :  sign_dzdy01 * dble_slopelimit_01;
  const double dble_dzdx11 =
    ( sign_dzdx11 * dble_dzdx11i <= dble_slopelimit_11 )
    ? dble_dzdx11i :  sign_dzdx11 * dble_slopelimit_11;
  const double dble_dzdy11 =
    ( sign_dzdy11 * dble_dzdy11i <= dble_slopelimit_11 )
    ? dble_dzdy11i :  sign_dzdy11 * dble_slopelimit_11;

  /*
   * Sums and differences of first derivatives:
   */
  const double twelve_sum00 = 6.0 * ( dble_dzdx00 + dble_dzdy00 );
  const double twelve_dif00 = 6.0 * ( dble_dzdx00 - dble_dzdy00 );
  const double twelve_sum10 = 6.0 * ( dble_dzdx10 + dble_dzdy10 );
  const double twelve_dif10 = 6.0 * ( dble_dzdx10 - dble_dzdy10 );
  const double twelve_sum01 = 6.0 * ( dble_dzdx01 + dble_dzdy01 );
  const double twelve_dif01 = 6.0 * ( dble_dzdx01 - dble_dzdy01 );
  const double twelve_sum11 = 6.0 * ( dble_dzdx11 + dble_dzdy11 );
  const double twelve_dif11 = 6.0 * ( dble_dzdx11 - dble_dzdy11 );

  /*
   * Absolute values of the sums:
   */
  const double twelve_abs_sum00 = LBB_ABS( twelve_sum00 );
  const double twelve_abs_sum10 = LBB_ABS( twelve_sum10 );
  const double twelve_abs_sum01 = LBB_ABS( twelve_sum01 );
  const double twelve_abs_sum11 = LBB_ABS( twelve_sum11 );

  /*
   * Scaled distances to the min:
   */
  const double u00_times_36 = 36.0 * u00;
  const double u10_times_36 = 36.0 * u10;
  const double u01_times_36 = 36.0 * u01;
  const double u11_times_36 = 36.0 * u11;

  /*
   * First cross-derivative limiter:
   */
  const double first_limit00 = twelve_abs_sum00 - u00_times_36;
  const double first_limit10 = twelve_abs_sum10 - u10_times_36;
  const double first_limit01 = twelve_abs_sum01 - u01_times_36;
  const double first_limit11 = twelve_abs_sum11 - u11_times_36;

  const double quad_d2zdxdy00ii = LBB_MAX( quad_d2zdxdy00i, first_limit00 );
  const double quad_d2zdxdy10ii = LBB_MAX( quad_d2zdxdy10i, first_limit10 );
  const double quad_d2zdxdy01ii = LBB_MAX( quad_d2zdxdy01i, first_limit01 );
  const double quad_d2zdxdy11ii = LBB_MAX( quad_d2zdxdy11i, first_limit11 );

  /*
   * Scaled distances to the max:
   */
  const double v00_times_36 = 36.0 * v00;
  const double v10_times_36 = 36.0 * v10;
  const double v01_times_36 = 36.0 * v01;
  const double v11_times_36 = 36.0 * v11;

  /*
   * Second cross-derivative limiter:
   */
  const double second_limit00 = v00_times_36 - twelve_abs_sum00;
  const double second_limit10 = v10_times_36 - twelve_abs_sum10;
  const double second_limit01 = v01_times_36 - twelve_abs_sum01;
  const double second_limit11 = v11_times_36 - twelve_abs_sum11;

  const double quad_d2zdxdy00iii = LBB_MIN( quad_d2zdxdy00ii, second_limit00 );
  const double quad_d2zdxdy10iii = LBB_MIN( quad_d2zdxdy10ii, second_limit10 );
  const double quad_d2zdxdy01iii = LBB_MIN( quad_d2zdxdy01ii, second_limit01 );
  const double quad_d2zdxdy11iii = LBB_MIN( quad_d2zdxdy11ii, second_limit11 );

  /*
   * Absolute values of the differences:
   */
  const double twelve_abs_dif00 = LBB_ABS( twelve_dif00 );
  const double twelve_abs_dif10 = LBB_ABS( twelve_dif10 );
  const double twelve_abs_dif01 = LBB_ABS( twelve_dif01 );
  const double twelve_abs_dif11 = LBB_ABS( twelve_dif11 );

  /*
   * Third cross-derivative limiter:
   */
  const double third_limit00 = twelve_abs_dif00 - v00_times_36;
  const double third_limit10 = twelve_abs_dif10 - v10_times_36;
  const double third_limit01 = twelve_abs_dif01 - v01_times_36;
  const double third_limit11 = twelve_abs_dif11 - v11_times_36;

  const double quad_d2zdxdy00iiii = LBB_MAX( quad_d2zdxdy00iii, third_limit00);
  const double quad_d2zdxdy10iiii = LBB_MAX( quad_d2zdxdy10iii, third_limit10);
  const double quad_d2zdxdy01iiii = LBB_MAX( quad_d2zdxdy01iii, third_limit01);
  const double quad_d2zdxdy11iiii = LBB_MAX( quad_d2zdxdy11iii, third_limit11);

  /*
   * Fourth cross-derivative limiter:
   */
  const double fourth_limit00 = u00_times_36 - twelve_abs_dif00;
  const double fourth_limit10 = u10_times_36 - twelve_abs_dif10;
  const double fourth_limit01 = u01_times_36 - twelve_abs_dif01;
  const double fourth_limit11 = u11_times_36 - twelve_abs_dif11;

  const double quad_d2zdxdy00 = LBB_MIN( quad_d2zdxdy00iiii, fourth_limit00);
  const double quad_d2zdxdy10 = LBB_MIN( quad_d2zdxdy10iiii, fourth_limit10);
  const double quad_d2zdxdy01 = LBB_MIN( quad_d2zdxdy01iiii, fourth_limit01);
  const double quad_d2zdxdy11 = LBB_MIN( quad_d2zdxdy11iiii, fourth_limit11);

  /*
   * Part of the result which does not need derivatives:
   */
  const double newval1 = c00 * dos_two + c10 * dos_thr +
                         c01 * tre_two + c11 * tre_thr;

  /*
   * Twice the part of the result which only needs first derivatives.
   */
  const double newval2 = c00dx * dble_dzdx00 + c10dx * dble_dzdx10 +
                         c01dx * dble_dzdx01 + c11dx * dble_dzdx11 +
                         c00dy * dble_dzdy00 + c10dy * dble_dzdy10 +
                         c01dy * dble_dzdy01 + c11dy * dble_dzdy11;

  /*
   * Four times the part of the result which only uses cross
   * derivatives:
   */
  const double newval3 = c00dxdy * quad_d2zdxdy00 + c10dxdy * quad_d2zdxdy10 +
                         c01dxdy * quad_d2zdxdy01 + c11dxdy * quad_d2zdxdy11;

  const double newval = newval1 + .5 * newval2 + .25 * newval3;

  return newval;
}

/*
 * Call lbb with a type conversion operator as a parameter.
 *
 * It would be nice to do this with templates but we can't figure out
 * how to do it cleanly. Suggestions welcome!
 */
#define LBB_CONVERSION( conversion )                     \
  template <typename T> static void inline               \
  lbb_ ## conversion(       void*      restrict pout,    \
                      const VipsPel*   restrict pin,     \
                      const int             bands,       \
                      const int             lskip,       \
                      const double          relative_x,  \
                      const double          relative_y ) \
  { \
    T* restrict out = (T *) pout; \
    \
    const T* restrict in = (T *) pin; \
    \
    const int one_shift     =  -bands; \
    const int thr_shift     =   bands; \
    const int fou_shift     = 2*bands; \
    \
    const int uno_two_shift =  -lskip; \
    \
    const int tre_two_shift =   lskip; \
    const int qua_two_shift = 2*lskip; \
    \
    const int uno_one_shift = uno_two_shift + one_shift; \
    const int dos_one_shift =                 one_shift; \
    const int tre_one_shift = tre_two_shift + one_shift; \
    const int qua_one_shift = qua_two_shift + one_shift; \
    \
    const int uno_thr_shift = uno_two_shift + thr_shift; \
    const int dos_thr_shift =                 thr_shift; \
    const int tre_thr_shift = tre_two_shift + thr_shift; \
    const int qua_thr_shift = qua_two_shift + thr_shift; \
    \
    const int uno_fou_shift = uno_two_shift + fou_shift; \
    const int dos_fou_shift =                 fou_shift; \
    const int tre_fou_shift = tre_two_shift + fou_shift; \
    const int qua_fou_shift = qua_two_shift + fou_shift; \
    \
    const double xp1over2   = relative_x; \
    const double xm1over2   = xp1over2 - 1.0; \
    const double onepx      = 0.5 + xp1over2; \
    const double onemx      = 1.5 - xp1over2; \
    const double xp1over2sq = xp1over2 * xp1over2; \
    \
    const double yp1over2   = relative_y; \
    const double ym1over2   = yp1over2 - 1.0; \
    const double onepy      = 0.5 + yp1over2; \
    const double onemy      = 1.5 - yp1over2; \
    const double yp1over2sq = yp1over2 * yp1over2; \
    \
    const double xm1over2sq = xm1over2 * xm1over2; \
    const double ym1over2sq = ym1over2 * ym1over2; \
    \
    const double twice1px = onepx + onepx; \
    const double twice1py = onepy + onepy; \
    const double twice1mx = onemx + onemx; \
    const double twice1my = onemy + onemy; \
    \
    const double xm1over2sq_times_ym1over2sq = xm1over2sq * ym1over2sq; \
    const double xp1over2sq_times_ym1over2sq = xp1over2sq * ym1over2sq; \
    const double xp1over2sq_times_yp1over2sq = xp1over2sq * yp1over2sq; \
    const double xm1over2sq_times_yp1over2sq = xm1over2sq * yp1over2sq; \
    \
    const double four_times_1px_times_1py = twice1px * twice1py; \
    const double four_times_1mx_times_1py = twice1mx * twice1py; \
    const double twice_xp1over2_times_1py = xp1over2 * twice1py; \
    const double twice_xm1over2_times_1py = xm1over2 * twice1py; \
    \
    const double twice_xm1over2_times_1my = xm1over2 * twice1my; \
    const double twice_xp1over2_times_1my = xp1over2 * twice1my; \
    const double four_times_1mx_times_1my = twice1mx * twice1my; \
    const double four_times_1px_times_1my = twice1px * twice1my; \
    \
    const double twice_1px_times_ym1over2 = twice1px * ym1over2; \
    const double twice_1mx_times_ym1over2 = twice1mx * ym1over2; \
    const double xp1over2_times_ym1over2  = xp1over2 * ym1over2; \
    const double xm1over2_times_ym1over2  = xm1over2 * ym1over2; \
    \
    const double xm1over2_times_yp1over2  = xm1over2 * yp1over2; \
    const double xp1over2_times_yp1over2  = xp1over2 * yp1over2; \
    const double twice_1mx_times_yp1over2 = twice1mx * yp1over2; \
    const double twice_1px_times_yp1over2 = twice1px * yp1over2; \
    \
    const double c00 = \
      four_times_1px_times_1py * xm1over2sq_times_ym1over2sq; \
    const double c00dx = \
      twice_xp1over2_times_1py * xm1over2sq_times_ym1over2sq; \
    const double c00dy = \
      twice_1px_times_yp1over2 * xm1over2sq_times_ym1over2sq; \
    const double c00dxdy = \
       xp1over2_times_yp1over2 * xm1over2sq_times_ym1over2sq; \
    \
    const double c10 = \
      four_times_1mx_times_1py * xp1over2sq_times_ym1over2sq; \
    const double c10dx = \
      twice_xm1over2_times_1py * xp1over2sq_times_ym1over2sq; \
    const double c10dy = \
      twice_1mx_times_yp1over2 * xp1over2sq_times_ym1over2sq; \
    const double c10dxdy = \
       xm1over2_times_yp1over2 * xp1over2sq_times_ym1over2sq; \
    \
    const double c01 = \
      four_times_1px_times_1my * xm1over2sq_times_yp1over2sq; \
    const double c01dx = \
      twice_xp1over2_times_1my * xm1over2sq_times_yp1over2sq; \
    const double c01dy = \
      twice_1px_times_ym1over2 * xm1over2sq_times_yp1over2sq; \
    const double c01dxdy = \
       xp1over2_times_ym1over2 * xm1over2sq_times_yp1over2sq; \
    \
    const double c11 = \
      four_times_1mx_times_1my * xp1over2sq_times_yp1over2sq; \
    const double c11dx = \
      twice_xm1over2_times_1my * xp1over2sq_times_yp1over2sq; \
    const double c11dy = \
      twice_1mx_times_ym1over2 * xp1over2sq_times_yp1over2sq; \
    const double c11dxdy = \
       xm1over2_times_ym1over2 * xp1over2sq_times_yp1over2sq; \
    \
    int band = bands; \
    \
    do \
      { \
        const double double_result =        \
          lbbicubic( c00,                   \
                     c10,                   \
                     c01,                   \
                     c11,                   \
                     c00dx,                 \
                     c10dx,                 \
                     c01dx,                 \
                     c11dx,                 \
                     c00dy,                 \
                     c10dy,                 \
                     c01dy,                 \
                     c11dy,                 \
                     c00dxdy,               \
                     c10dxdy,               \
                     c01dxdy,               \
                     c11dxdy,               \
                     in[ uno_one_shift ],   \
                     in[ uno_two_shift ],   \
                     in[ uno_thr_shift ],   \
                     in[ uno_fou_shift ],   \
                     in[ dos_one_shift ],   \
                     in[             0 ],   \
                     in[ dos_thr_shift ],   \
                     in[ dos_fou_shift ],   \
                     in[ tre_one_shift ],   \
                     in[ tre_two_shift ],   \
                     in[ tre_thr_shift ],   \
                     in[ tre_fou_shift ],   \
                     in[ qua_one_shift ],   \
                     in[ qua_two_shift ],   \
                     in[ qua_thr_shift ],   \
                     in[ qua_fou_shift ] ); \
        \
        const T result = to_ ## conversion<T>( double_result ); \
        in++; \
        *out++ = result; \
      } while (--band); \
  }

LBB_CONVERSION( fptypes )
LBB_CONVERSION( withsign )
LBB_CONVERSION( nosign )

#define CALL( T, conversion )          \
  lbb_ ## conversion<T>( out,          \
                         p,            \
                         bands,        \
                         lskip,        \
                         relative_x,   \
                         relative_y );

/*
 * We need C linkage:
 */
extern "C" {
G_DEFINE_TYPE( VipsInterpolateLbb, vips_interpolate_lbb,
	VIPS_TYPE_INTERPOLATE );
}

static void
vips_interpolate_lbb_interpolate( VipsInterpolate* restrict interpolate,
                                  void*            restrict out,
                                  VipsRegion*      restrict in,
                                  double                    absolute_x,
                                  double                    absolute_y )
{
  /* absolute_x and absolute_y are always >= 1.0 (see double-check assert
   * below), so we don't need floor(). 
   *
   * It's 1 not 0 since have a window_offset of 1.
   */
  const int ix = (int) absolute_x;
  const int iy = (int) absolute_y;

  /*
   * Move the pointer to (the first band of) the top/left pixel of the
   * 2x2 group of pixel centers which contains the sampling location
   * in its convex hull:
   */
  const VipsPel* restrict p = VIPS_REGION_ADDR( in, ix, iy );

  const double relative_x = absolute_x - ix;
  const double relative_y = absolute_y - iy;

  /*
   * VIPS versions of Nicolas's pixel addressing values.
   */
  const int lskip = VIPS_REGION_LSKIP( in ) / 
	  VIPS_IMAGE_SIZEOF_ELEMENT( in->im );
  /*
   * Double the bands for complex images to account for the real and
   * imaginary parts being computed independently:
   */
  const int actual_bands = in->im->Bands;
  const int bands =
    vips_band_format_iscomplex( in->im->BandFmt ) ? 
      2 * actual_bands : actual_bands;

  g_assert( ix - 1 >= in->valid.left );
  g_assert( iy - 1 >= in->valid.top );
  g_assert( ix + 2 < VIPS_RECT_RIGHT( &in->valid ) );
  g_assert( iy + 2 < VIPS_RECT_BOTTOM( &in->valid ) );

  /* Confirm that absolute_x and absolute_y are >= 1, see above. 
   */
  g_assert( absolute_x >= 1.0 );
  g_assert( absolute_y >= 1.0 );

  switch( in->im->BandFmt ) {
  case VIPS_FORMAT_UCHAR:
    CALL( unsigned char, nosign );
    break;

  case VIPS_FORMAT_CHAR:
    CALL( signed char, withsign );
    break;

  case VIPS_FORMAT_USHORT:
    CALL( unsigned short, nosign );
    break;

  case VIPS_FORMAT_SHORT:
    CALL( signed short, withsign );
    break;

  case VIPS_FORMAT_UINT:
    CALL( unsigned int, nosign );
    break;

  case VIPS_FORMAT_INT:
    CALL( signed int, withsign );
    break;

  /*
   * Complex images are handled by doubling of bands.
   */
  case VIPS_FORMAT_FLOAT:
  case VIPS_FORMAT_COMPLEX:
    CALL( float, fptypes );
    break;

  case VIPS_FORMAT_DOUBLE:
  case VIPS_FORMAT_DPCOMPLEX:
    CALL( double, fptypes );
    break;

  default:
    g_assert( 0 );
    break;
  }
}

static void
vips_interpolate_lbb_class_init( VipsInterpolateLbbClass *klass )
{
  VipsObjectClass *object_class = VIPS_OBJECT_CLASS( klass );
  VipsInterpolateClass *interpolate_class =
    VIPS_INTERPOLATE_CLASS( klass );

  object_class->nickname    = "lbb";
  object_class->description = _( "reduced halo bicubic" );

  interpolate_class->interpolate   = vips_interpolate_lbb_interpolate;
  interpolate_class->window_size   = 4;
}

static void
vips_interpolate_lbb_init( VipsInterpolateLbb *lbb )
{
}