public software.sextractor

/*

*                               back.c

*

* Functions dealing with the image background.

*

*%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

*

*       This file part of:      SExtractor

*

*       Copyright:              (C) 1993-2010 Emmanuel Bertin -- IAP/CNRS/UPMC

*

*       License:                GNU General Public License

*

*       SExtractor is free software: you can redistribute it and/or modify

*       it under the terms of the GNU General Public License as published by

*       the Free Software Foundation, either version 3 of the License, or

*       (at your option) any later version.

*       SExtractor 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 General Public License for more details.

*       You should have received a copy of the GNU General Public License

*       along with SExtractor. If not, see <http://www.gnu.org/licenses/>.

*

*       Last modified:          11/10/2010

*

*%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%*/

#ifdef HAVE_CONFIG_H

#include        "config.h"

#endif

#include        <math.h>

#include        <stdio.h>

#include        <stdlib.h>

#include        <string.h>

#include        "define.h"

#include        "globals.h"

#include        "prefs.h"

#include        "fits/fitscat.h"

#include        "back.h"

#include        "field.h"

#include        "weight.h"

/******************************** makeback ***********************************/

/*

Background maps are established from the images themselves; thus we need to

make at least one first pass through the data.

*/

void    makeback(picstruct *field, picstruct *wfield)

  {

   backstruct   *backmesh,*wbackmesh, *bm,*wbm;

   PIXTYPE      *buf,*wbuf, *buft,*wbuft;

   OFF_T        fcurpos,wfcurpos, wfcurpos2,fcurpos2, bufshift, jumpsize;

   size_t       bufsize, bufsize2,

                size,meshsize;

   int          i,j,k,m,n, step, nlines,

                w,bw, bh, nx,ny,nb,

                lflag, nr;

   float        *ratio,*ratiop, *weight, *sigma,

                sratio;

/* If the weight-map is not an external one, no stats are needed for it */

  if (wfield && wfield->flags&(INTERP_FIELD|BACKRMS_FIELD))

    wfield= NULL;

  w = field->width;

  bw = field->backw;

  bh = field->backh;

  nx = field->nbackx;

  ny = field->nbacky;

  nb = field->nback;

  NFPRINTF(OUTPUT, "Setting up background maps");

/* Decide if it is worth displaying progress each 16 lines */

  lflag = (field->width*field->backh >= (size_t)65536);

/* Save current positions in files */

  wfcurpos = wfcurpos2 = 0;              /* to avoid gcc -Wall warnings */

  QFTELL(field->file, fcurpos, field->filename);

  if (wfield)

    QFTELL(wfield->file, wfcurpos, wfield->filename);

/* Allocate a correct amount of memory to store pixels */

  bufsize = (OFF_T)w*bh;

  meshsize = (size_t)bufsize;

  nlines = 0;

  if (bufsize > (size_t)BACK_BUFSIZE)

    {

    nlines = BACK_BUFSIZE/w;

    step = (field->backh-1)/nlines+1;

    bufsize = (size_t)(nlines = field->backh/step)*w;

    bufshift = (step/2)*(OFF_T)w;

    jumpsize = (step-1)*(OFF_T)w;

    }

  else

    bufshift = jumpsize = 0;             /* to avoid gcc -Wall warnings */

/* Allocate some memory */

  QMALLOC(backmesh, backstruct, nx);            /* background information */

  QMALLOC(buf, PIXTYPE, bufsize);               /* pixel buffer */

  free(field->back);

  QMALLOC(field->back, float, nb);              /* background map */

  free(field->backline);

  QMALLOC(field->backline, PIXTYPE, w);         /* current background line */

  free(field->sigma);

  QMALLOC(field->sigma, float, nb);             /* sigma map */

  if (wfield)

    {

    QMALLOC(wbackmesh, backstruct, nx);         /* background information */

    QMALLOC(wbuf, PIXTYPE, bufsize);            /* pixel buffer */

    free(wfield->back);

    QMALLOC(wfield->back, float, nb);           /* background map */

    free(wfield->backline);

    QMALLOC(wfield->backline, PIXTYPE, w);      /* current background line */

    free(wfield->sigma);

    QMALLOC(wfield->sigma, float, nb);          /* sigma map */

    wfield->sigfac = 1.0;

    }

  else

    {

    wbackmesh = NULL;

    wbuf = NULL;

    }

/* Loop over the data packets */

  for (j=0; j<ny; j++)

    {

    if (lflag && j)

      NPRINTF(OUTPUT, "\33[1M> Setting up background map at line:%5d\n\33[1A",

              j*bh);

    if (!nlines)

      {

/*---- The image is small enough so that we can make exhaustive stats */

      if (j == ny-1 && field->npix%bufsize)

        bufsize = field->npix%bufsize;

      read_body(field->tab, buf, bufsize);

      if (wfield)

        {

        read_body(wfield->tab, wbuf, bufsize);

        weight_to_var(wfield, wbuf, bufsize);

        }

/*---- Build the histograms */

      backstat(backmesh, wbackmesh, buf, wbuf, bufsize,nx, w, bw,

        wfield?wfield->weight_thresh:0.0);

      bm = backmesh;

      for (m=nx; m--; bm++)

        if (bm->mean <= -BIG)

          bm->histo=NULL;

        else

          QCALLOC(bm->histo, LONG, bm->nlevels);

      if (wfield)

        {

        wbm = wbackmesh;

        for (m=nx; m--; wbm++)

          if (wbm->mean <= -BIG)

            wbm->histo=NULL;

          else

            QCALLOC(wbm->histo, LONG, wbm->nlevels);

        }

      backhisto(backmesh, wbackmesh, buf, wbuf, bufsize,nx, w, bw,

        wfield?wfield->weight_thresh:0.0);

      }

    else

      {

/*---- Image size too big, we have to skip a few data !*/

      QFTELL(field->file, fcurpos2, field->filename);

      if (wfield)

        QFTELL(wfield->file, wfcurpos2, wfield->filename);

      if (j == ny-1 && (n=field->height%field->backh))

        {

        meshsize = n*(size_t)w;

        nlines = BACK_BUFSIZE/w;

        step = (n-1)/nlines+1;

        bufsize = (nlines = n/step)*(size_t)w;

        bufshift = (step/2)*(OFF_T)w;

        jumpsize = (step-1)*(OFF_T)w;

        free(buf);

        QMALLOC(buf, PIXTYPE, bufsize);         /* pixel buffer */

        if (wfield)

          {

          free(wbuf);

          QMALLOC(wbuf, PIXTYPE, bufsize);      /* pixel buffer */

          }

        }

/*---- Read and skip, read and skip, etc... */

      QFSEEK(field->file, bufshift*(OFF_T)field->bytepix, SEEK_CUR,

                field->filename);

      buft = buf;

      for (i=nlines; i--; buft += w)

        {

        read_body(field->tab, buft, w);

        if (i)

          QFSEEK(field->file, jumpsize*(OFF_T)field->bytepix, SEEK_CUR,

                field->filename);

        }

      if (wfield)

        {

/*------ Read and skip, read and skip, etc... now on the weight-map */

        QFSEEK(wfield->file, bufshift*(OFF_T)wfield->bytepix, SEEK_CUR,

                wfield->filename);

        wbuft = wbuf;

        for (i=nlines; i--; wbuft += w)

          {

          read_body(wfield->tab, wbuft, w);

          weight_to_var(wfield, wbuft, w);

          if (i)

            QFSEEK(wfield->file, jumpsize*(OFF_T)wfield->bytepix, SEEK_CUR,

                wfield->filename);

          }

        }

      backstat(backmesh, wbackmesh, buf, wbuf, bufsize, nx, w, bw,

        wfield?wfield->weight_thresh:0.0);

      QFSEEK(field->file, fcurpos2, SEEK_SET, field->filename);

      bm = backmesh;

      for (m=nx; m--; bm++)

        if (bm->mean <= -BIG)

          bm->histo=NULL;

        else

          QCALLOC(bm->histo, LONG, bm->nlevels);

      if (wfield)

        {

        QFSEEK(wfield->file, wfcurpos2, SEEK_SET, wfield->filename);

        wbm = wbackmesh;

        for (m=nx; m--; wbm++)

          if (wbm->mean <= -BIG)

            wbm->histo=NULL;

          else

            QCALLOC(wbm->histo, LONG, wbm->nlevels);

        }

/*---- Build (progressively this time) the histograms */

      for(size=meshsize, bufsize2=bufsize; size>0; size -= bufsize2)

        {

        if (bufsize2>size)

          bufsize2 = size;

        read_body(field->tab, buf, bufsize2);

        if (wfield)

          {

          read_body(wfield->tab, wbuf, bufsize2);

          weight_to_var(wfield, wbuf, bufsize2);

          }

        backhisto(backmesh, wbackmesh, buf, wbuf, bufsize2, nx, w, bw,

                wfield?wfield->weight_thresh:0.0);

        }

      }

    /*-- Compute background statistics from the histograms */

    bm = backmesh;

    for (m=0; m<nx; m++, bm++)

      {

      k = m+nx*j;

      backguess(bm, field->back+k, field->sigma+k);

      free(bm->histo);

      }

    if (wfield)

      {

      wbm = wbackmesh;

      for (m=0; m<nx; m++, wbm++)

        {

        k = m+nx*j;

        backguess(wbm, wfield->back+k, wfield->sigma+k);

        free(wbm->histo);

        }

      }

    }

/* Free memory */

  free(buf);

  free(backmesh);

  if (wfield)

    {

    free(wbackmesh);

    free(wbuf);

    }

/* Go back to the original position */

  QFSEEK(field->file, fcurpos, SEEK_SET, field->filename);

  if (wfield)

    QFSEEK(wfield->file, wfcurpos, SEEK_SET, wfield->filename);

/* Median-filter and check suitability of the background map */

  NFPRINTF(OUTPUT, "Filtering background map(s)");

  filterback(field);

  if (wfield)

    filterback(wfield);

/* Compute normalization for variance- or weight-maps*/

  if (wfield && wfield->flags&(VAR_FIELD|WEIGHT_FIELD))

    {

    nr = 0;

    QMALLOC(ratio, float, wfield->nback);

    ratiop = ratio;

    weight = wfield->back;

    sigma = field->sigma;

    for (i=wfield->nback; i--; sigma++)

      if ((sratio=*(weight++)) > 0.0

                && (sratio = *sigma/sqrt(sratio)) > 0.0)

        {

        *(ratiop++) = sratio;

        nr++;

        }

    wfield->sigfac = fqmedian(ratio, nr);

    for (i=0; i<nr && ratio[i]<=0.0; i++);

    if (i<nr)

      wfield->sigfac = fqmedian(ratio+i, nr-i);

    else

      {

      warning("Null or negative global weighting factor:","defaulted to 1");

      wfield->sigfac = 1.0;

      }

    free(ratio);

    }

/* Compute 2nd derivatives along the y-direction */

  NFPRINTF(OUTPUT, "Computing background d-map");

  free(field->dback);

  field->dback = makebackspline(field, field->back);

  NFPRINTF(OUTPUT, "Computing background-noise d-map");

  free(field->dsigma);

  field->dsigma = makebackspline(field, field->sigma);

/* If asked for, force the backmean parameter to the supplied value */

  if (field->back_type == BACK_ABSOLUTE)

    field->backmean = (float)prefs.back_val[(field->flags&DETECT_FIELD)?0:1];

/* Set detection/measurement threshold */

  if (prefs.ndthresh > 1)

    {

     double     dval;

    if (fabs(dval=prefs.dthresh[0] - prefs.dthresh[1])> 70.0)

      error(EXIT_FAILURE,

        "*Error*: I cannot deal with such extreme thresholds!", "");

    field->dthresh = field->pixscale*field->pixscale*pow(10.0, -0.4*dval);

    }

  else if (prefs.thresh_type[0]==THRESH_ABSOLUTE)

    field->dthresh = prefs.dthresh[0];

  else

    field->dthresh = prefs.dthresh[0]*field->backsig;

  if (prefs.nthresh > 1)

    {

     double     dval;

    if (fabs(dval=prefs.thresh[0] - prefs.thresh[1]) > 70.0)

      error(EXIT_FAILURE,

        "*Error*: I cannot deal with such extreme thresholds!", "");

    field->thresh = field->pixscale*field->pixscale*pow(10.0, -0.4*dval);

    }

  else if (prefs.thresh_type[1]==THRESH_ABSOLUTE)

    field->thresh = prefs.thresh[0];

  else

    field->thresh = prefs.thresh[0]*field->backsig;

#ifdef  QUALITY_CHECK

  printf("%-10g %-10g %-10g\n", field->backmean, field->backsig,

        (field->flags & DETECT_FIELD)? field->dthresh : field->thresh);

#endif

  if (field->dthresh<=0.0 || field->thresh<=0.0)

    error(EXIT_FAILURE,

        "*Error*: I cannot deal with zero or negative thresholds!", "");

  if (prefs.detect_type == PHOTO

        && field->backmean+3*field->backsig > 50*field->ngamma)

    error(EXIT_FAILURE,

        "*Error*: The density range of this image is too large for ",

        "PHOTO mode");

  return;

  }

/******************************** backstat **********************************/

/*

Compute robust statistical estimators in a row of meshes.

*/

void    backstat(backstruct *backmesh, backstruct *wbackmesh,

                PIXTYPE *buf, PIXTYPE *wbuf, size_t bufsize,

                        int n, int w, int bw, PIXTYPE wthresh)

  {

   backstruct   *bm, *wbm;

   double       pix,wpix, sig, mean,wmean, sigma,wsigma, step;

   PIXTYPE      *buft,*wbuft,

                lcut,wlcut, hcut,whcut;

   int          m,h,x,y, npix,wnpix, offset, lastbite;

  h = bufsize/w;

  bm = backmesh;

  wbm = wbackmesh;

  offset = w - bw;

  step = sqrt(2/PI)*QUANTIF_NSIGMA/QUANTIF_AMIN;

  wmean = wsigma = wlcut = whcut = 0.0; /* to avoid gcc -Wall warnings */

  for (m = n; m--; bm++,buf+=bw)

    {

    if (!m && (lastbite=w%bw))

      {

      bw = lastbite;

      offset = w-bw;

      }

    mean = sigma = 0.0;

    buft=buf;

/*-- We separate the weighted case at this level to avoid penalty in CPU */

    npix = 0;

    if (wbackmesh)

      {

      wmean = wsigma = 0.0;

      wbuft = wbuf;

      for (y=h; y--; buft+=offset,wbuft+=offset)

        for (x=bw; x--;)

          {

          pix = *(buft++);

          if ((wpix = *(wbuft++)) < wthresh && pix > -BIG)

            {

            wmean += wpix;

            wsigma += wpix*wpix;

            mean += pix;

            sigma += pix*pix;

            npix++;

            }

          }

      }

    else

      for (y=h; y--; buft+=offset)

        for (x=bw; x--;)

          if ((pix = *(buft++)) > -BIG)

            {

            mean += pix;

            sigma += pix*pix;

            npix++;

            }

/*-- If not enough valid pixels, discard this mesh */

    if ((float)npix < (float)(bw*h*BACK_MINGOODFRAC))

      {

      bm->mean = bm->sigma = -BIG;

      if (wbackmesh)

        {

        wbm->mean = wbm->sigma = -BIG;

        wbm++;

        wbuf += bw;

        }

      continue;

      }

    if (wbackmesh)

      {

      wmean /= (double)npix;

      wsigma = (sig = wsigma/npix - wmean*wmean)>0.0? sqrt(sig):0.0;

      wlcut = wbm->lcut = (PIXTYPE)(wmean - 2.0*wsigma);

      whcut = wbm->hcut = (PIXTYPE)(wmean + 2.0*wsigma);

      }

    mean /= (double)npix;

    sigma = (sig = sigma/npix - mean*mean)>0.0? sqrt(sig):0.0;

    lcut = bm->lcut = (PIXTYPE)(mean - 2.0*sigma);

    hcut = bm->hcut = (PIXTYPE)(mean + 2.0*sigma);

    mean = sigma = 0.0;

    npix = wnpix = 0;

    buft = buf;

    if (wbackmesh)

      {

      wmean = wsigma = 0.0;

      wbuft=wbuf;

      for (y=h; y--; buft+=offset, wbuft+=offset)

        for (x=bw; x--;)

          {

          pix = *(buft++);

          if ((wpix = *(wbuft++))<wthresh && pix<=hcut && pix>=lcut)

            {

            mean += pix;

            sigma += pix*pix;

            npix++;

            if (wpix<=whcut && wpix>=wlcut)

              {

              wmean += wpix;

              wsigma += wpix*wpix;

              wnpix++;

              }

            }

          }

      }

    else

      for (y=h; y--; buft+=offset)

        for (x=bw; x--;)

          {

          pix = *(buft++);

          if (pix<=hcut && pix>=lcut)

            {

            mean += pix;

            sigma += pix*pix;

            npix++;

            }

          }

    bm->npix = npix;

    mean /= (double)npix;

    sig = sigma/npix - mean*mean;

    sigma = sig>0.0 ? sqrt(sig):0.0;

    bm->mean = mean;

    bm->sigma = sigma;

    if ((bm->nlevels = (int)(step*npix+1)) > QUANTIF_NMAXLEVELS)

      bm->nlevels = QUANTIF_NMAXLEVELS;

    bm->qscale = sigma>0.0? 2*QUANTIF_NSIGMA*sigma/bm->nlevels : 1.0;

    bm->qzero = mean - QUANTIF_NSIGMA*sigma;

    if (wbackmesh)

      {

      wbm->npix = wnpix;

      wmean /= (double)wnpix;

      sig = wsigma/wnpix - wmean*wmean;

      wsigma = sig>0.0 ? sqrt(sig):0.0;

      wbm->mean = wmean;

      wbm->sigma = wsigma;

      if ((wbm->nlevels = (int)(step*wnpix+1)) > QUANTIF_NMAXLEVELS)

        wbm->nlevels = QUANTIF_NMAXLEVELS;

      wbm->qscale = wsigma>0.0? 2*QUANTIF_NSIGMA*wsigma/wbm->nlevels : 1.0;

      wbm->qzero = wmean - QUANTIF_NSIGMA*wsigma;

      wbm++;

      wbuf += bw;

      }

    }

  return;

  }

/******************************** backhisto *********************************/

/*

Compute robust statistical estimators in a row of meshes.

*/

void    backhisto(backstruct *backmesh, backstruct *wbackmesh,

                PIXTYPE *buf, PIXTYPE *wbuf, size_t bufsize,

                        int n, int w, int bw, PIXTYPE wthresh)

  {

   backstruct   *bm,*wbm;

   PIXTYPE      *buft,*wbuft;

   float        qscale,wqscale, cste,wcste, wpix;

   LONG         *histo,*whisto;

   int          h,m,x,y, nlevels,wnlevels, lastbite, offset, bin;

  h = bufsize/w;

  bm = backmesh;

  wbm = wbackmesh;

  offset = w - bw;

  for (m=0; m++<n; bm++ , buf+=bw)

    {

    if (m==n && (lastbite=w%bw))

      {

      bw = lastbite;

      offset = w-bw;

      }

/*-- Skip bad meshes */

    if (bm->mean <= -BIG)

      {

      if (wbackmesh)

        {

        wbm++;

        wbuf += bw;

        }

      continue;

      }

    nlevels = bm->nlevels;

    histo = bm->histo;

    qscale = bm->qscale;

    cste = 0.499999 - bm->qzero/qscale;

    buft = buf;

    if (wbackmesh)

      {

      wnlevels = wbm->nlevels;

      whisto = wbm->histo;

      wqscale = wbm->qscale;

      wcste = 0.499999 - wbm->qzero/wqscale;

      wbuft = wbuf;

      for (y=h; y--; buft+=offset, wbuft+=offset)

        for (x=bw; x--;)

          {

          bin = (int)(*(buft++)/qscale + cste);

          if ((wpix = *(wbuft++))<wthresh && bin<nlevels && bin>=0)

            {

            (*(histo+bin))++;

            bin = (int)(wpix/wqscale + wcste);

            if (bin>=0 && bin<wnlevels)

              (*(whisto+bin))++;

            }

          }

      wbm++;

      wbuf += bw;

      }

    else

      for (y=h; y--; buft += offset)

        for (x=bw; x--;)

          {

          bin = (int)(*(buft++)/qscale + cste);

          if (bin>=0 && bin<nlevels)

            (*(histo+bin))++;

          }

    }

  return;

  }

/******************************* backguess **********************************/

/*

Estimate the background from a histogram;

*/

float   backguess(backstruct *bkg, float *mean, float *sigma)

#define EPS     (1e-4)  /* a small number */

  {

   LONG         *histo, *hilow, *hihigh, *histot;

   unsigned long lowsum, highsum, sum;

   double       ftemp, mea, sig, sig1, med, dpix;

   int          i, n, lcut,hcut, nlevelsm1, pix;

/* Leave here if the mesh is already classified as `bad' */

  if (bkg->mean<=-BIG)

    {

    *mean = *sigma = -BIG;

    return -BIG;

    }

  histo = bkg->histo;

  hcut = nlevelsm1 = bkg->nlevels-1;

  lcut = 0;

  sig = 10.0*nlevelsm1;

  sig1 = 1.0;

  mea = med = bkg->mean;

  for (n=100; n-- && (sig>=0.1) && (fabs(sig/sig1-1.0)>EPS);)

    {

    sig1 = sig;

    sum = mea = sig = 0.0;

    lowsum = highsum = 0;

    histot = hilow = histo+lcut;

    hihigh = histo+hcut;

    for (i=lcut; i<=hcut; i++)

      {

      if (lowsum<highsum)

        lowsum += *(hilow++);

      else

        highsum +=  *(hihigh--);

      sum += (pix = *(histot++));

      mea += (dpix = (double)pix*i);

      sig += dpix*i;

      }

    med = hihigh>=histo?

        ((hihigh-histo)+0.5+((double)highsum-lowsum)/(2.0*(*hilow>*hihigh?

                                                *hilow:*hihigh)))

       : 0.0;

    if (sum)

      {

      mea /= (double)sum;

      sig = sig/sum - mea*mea;

      }

    sig = sig>0.0?sqrt(sig):0.0;

    lcut = (ftemp=med-3.0*sig)>0.0 ?(int)(ftemp>0.0?ftemp+0.5:ftemp-0.5):0;

    hcut = (ftemp=med+3.0*sig)<nlevelsm1 ?(int)(ftemp>0.0?ftemp+0.5:ftemp-0.5)

                                                                : nlevelsm1;

    }

  *mean = fabs(sig)>0.0? (fabs(bkg->sigma/(sig*bkg->qscale)-1) < 0.0 ?

                            bkg->qzero+mea*bkg->qscale

                            :(fabs((mea-med)/sig)< 0.3 ?

                              bkg->qzero+(2.5*med-1.5*mea)*bkg->qscale

                             :bkg->qzero+med*bkg->qscale))

                       :bkg->qzero+mea*bkg->qscale;

  *sigma = sig*bkg->qscale;

  return *mean;

  }

/******************************* filterback *********************************/

/*

Median filtering of the background map to remove the contribution from bright

sources.

*/

void    filterback(picstruct *field)

  {

   float        *back,*sigma, *back2,*sigma2, *bmask,*smask, *sigmat,

                d2,d2min, fthresh, med, val,sval;

   int          i,j,px,py, np, nx,ny, npx,npx2, npy,npy2, dpx,dpy, x,y, nmin;

  fthresh = prefs.backfthresh;

  nx = field->nbackx;

  ny = field->nbacky;

  np = field->nback;

  npx = field->nbackfx/2;

  npy = field->nbackfy/2;

  npy *= nx;

  QMALLOC(bmask, float, (2*npx+1)*(2*npy+1));

  QMALLOC(smask, float, (2*npx+1)*(2*npy+1));

  QMALLOC(back2, float, np);

  QMALLOC(sigma2, float, np);

  back = field->back;

  sigma = field->sigma;

  val = sval = 0.0;                     /* to avoid gcc -Wall warnings */

/* Look for `bad' meshes and interpolate them if necessary */

  for (i=0,py=0; py<ny; py++)

    for (px=0; px<nx; px++,i++)

      if ((back2[i]=back[i])<=-BIG)

        {

/*------ Seek the closest valid mesh */

        d2min = BIG;

        nmin = 0.0;

        for (j=0,y=0; y<ny; y++)

          for (x=0; x<nx; x++,j++)

            if (back[j]>-BIG)

              {

              d2 = (float)(x-px)*(x-px)+(y-py)*(y-py);

              if (d2<d2min)

                {

                val = back[j];

                sval = sigma[j];

                nmin = 1;

                d2min = d2;

                }

              else if (d2==d2min)