Python2/PyMuPDF: mupdf-source/thirdparty/leptonica/src/numafunc2.c comparison

comparison mupdf-source/thirdparty/leptonica/src/numafunc2.c @ 2:b50eed0cc0ef upstream

ADD: MuPDF v1.26.7: the MuPDF source as downloaded by a default build of PyMuPDF 1.26.4. The directory name has changed: no version number in the expanded directory now.

author	Franz Glasner <fzglas.hg@dom66.de>
date	Mon, 15 Sep 2025 11:43:07 +0200
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-:1d09e1dec1d9
+:b50eed0cc0ef
+/*====================================================================*
+-  Copyright (C) 2001 Leptonica.  All rights reserved.
+-
+-  Redistribution and use in source and binary forms, with or without
+-  modification, are permitted provided that the following conditions
+-  are met:
+-  1. Redistributions of source code must retain the above copyright
+-     notice, this list of conditions and the following disclaimer.
+-  2. Redistributions in binary form must reproduce the above
+-     copyright notice, this list of conditions and the following
+-     disclaimer in the documentation and/or other materials
+-     provided with the distribution.
+-
+-  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+-  ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+-  LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
+-  A PARTICULAR PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL ANY
+-  CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
+-  EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
+-  PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
+-  PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
+-  OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+-  NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+-  SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+*====================================================================*/
+/*!
+* \file  numafunc2.c
+* <pre>
+*
+*      --------------------------------------
+*      This file has these Numa utilities:
+*         - morphological operations
+*         - arithmetic transforms
+*         - windowed statistical operations
+*         - histogram extraction
+*         - histogram comparison
+*         - extrema finding
+*         - frequency and crossing analysis
+*      --------------------------------------
+*      Morphological (min/max) operations
+*          NUMA        *numaErode()
+*          NUMA        *numaDilate()
+*          NUMA        *numaOpen()
+*          NUMA        *numaClose()
+*
+*      Other transforms
+*          NUMA        *numaTransform()
+*          l_int32      numaSimpleStats()
+*          l_int32      numaWindowedStats()
+*          NUMA        *numaWindowedMean()
+*          NUMA        *numaWindowedMeanSquare()
+*          l_int32      numaWindowedVariance()
+*          NUMA        *numaWindowedMedian()
+*          NUMA        *numaConvertToInt()
+*
+*      Histogram generation and statistics
+*          NUMA        *numaMakeHistogram()
+*          NUMA        *numaMakeHistogramAuto()
+*          NUMA        *numaMakeHistogramClipped()
+*          NUMA        *numaRebinHistogram()
+*          NUMA        *numaNormalizeHistogram()
+*          l_int32      numaGetStatsUsingHistogram()
+*          l_int32      numaGetHistogramStats()
+*          l_int32      numaGetHistogramStatsOnInterval()
+*          l_int32      numaMakeRankFromHistogram()
+*          l_int32      numaHistogramGetRankFromVal()
+*          l_int32      numaHistogramGetValFromRank()
+*          l_int32      numaDiscretizeSortedInBins()
+*          l_int32      numaDiscretizeHistoInBins()
+*          l_int32      numaGetRankBinValues()
+*          NUMA        *numaGetUniformBinSizes()
+*
+*      Splitting a distribution
+*          l_int32      numaSplitDistribution()
+*
+*      Comparing histograms
+*          l_int32      grayHistogramsToEMD()
+*          l_int32      numaEarthMoverDistance()
+*          l_int32      grayInterHistogramStats()
+*
+*      Extrema finding
+*          NUMA        *numaFindPeaks()
+*          NUMA        *numaFindExtrema()
+*          NUMA        *numaFindLocForThreshold()
+*          l_int32     *numaCountReversals()
+*
+*      Threshold crossings and frequency analysis
+*          l_int32      numaSelectCrossingThreshold()
+*          NUMA        *numaCrossingsByThreshold()
+*          NUMA        *numaCrossingsByPeaks()
+*          NUMA        *numaEvalBestHaarParameters()
+*          l_int32      numaEvalHaarSum()
+*
+*      Generating numbers in a range under constraints
+*          NUMA        *genConstrainedNumaInRange()
+*
+*    Things to remember when using the Numa:
+*
+*    (1) The numa is a struct, not an array.  Always use accessors
+*        (see numabasic.c), never the fields directly.
+*
+*    (2) The number array holds l_float32 values.  It can also
+*        be used to store l_int32 values.  See numabasic.c for
+*        details on using the accessors.  Integers larger than
+*        about 10M will lose accuracy due on retrieval due to round-off.
+*        For large integers, use the dna (array of l_float64) instead.
+*
+*    (3) Occasionally, in the comments we denote the i-th element of a
+*        numa by na[i].  This is conceptual only -- the numa is not an array!
+*
+*    Some general comments on histograms:
+*
+*    (1) Histograms are the generic statistical representation of
+*        the data about some attribute.  Typically they're not
+*        normalized -- they simply give the number of occurrences
+*        within each range of values of the attribute.  This range
+*        of values is referred to as a 'bucket'.  For example,
+*        the histogram could specify how many connected components
+*        are found for each value of their width; in that case,
+*        the bucket size is 1.
+*
+*    (2) In leptonica, all buckets have the same size.  Histograms
+*        are therefore specified by a numa of occurrences, along
+*        with two other numbers: the 'value' associated with the
+*        occupants of the first bucket and the size (i.e., 'width')
+*        of each bucket.  These two numbers then allow us to calculate
+*        the value associated with the occupants of each bucket.
+*        These numbers are fields in the numa, initialized to
+*        a startx value of 0.0 and a binsize of 1.0.  Accessors for
+*        these fields are functions numa*Parameters().  All histograms
+*        must have these two numbers properly set.
+* </pre>
+*/
+#ifdef HAVE_CONFIG_H
+#include <config_auto.h>
+#endif  /* HAVE_CONFIG_H */
+#include <math.h>
+#include "allheaders.h"
+/* bin sizes in numaMakeHistogram() */
+static const l_int32 BinSizeArray[] = {2, 5, 10, 20, 50, 100, 200, 500, 1000,\
+2000, 5000, 10000, 20000, 50000, 100000, 200000,\
+500000, 1000000, 2000000, 5000000, 10000000,\
+200000000, 50000000, 100000000};
+static const l_int32 NBinSizes = 24;
+#ifndef  NO_CONSOLE_IO
+#define  DEBUG_HISTO        0
+#define  DEBUG_CROSSINGS    0
+#define  DEBUG_FREQUENCY    0
+#endif  /* ~NO_CONSOLE_IO */
+/*----------------------------------------------------------------------*
+*                     Morphological operations                         *
+*----------------------------------------------------------------------*/
+/*!
+* \brief   numaErode()
+*
+* \param[in]    nas
+* \param[in]    size   of sel; greater than 0, odd.  The origin
+*                      is implicitly in the center.
+* \return  nad eroded, or NULL on error
+*
+* <pre>
+* Notes:
+*      (1) The structuring element (sel) is linear, all "hits"
+*      (2) If size == 1, this returns a copy
+*      (3) General comment.  The morphological operations are equivalent
+*          to those that would be performed on a 1-dimensional fpix.
+*          However, because we have not implemented morphological
+*          operations on fpix, we do this here.  Because it is only
+*          1 dimensional, there is no reason to use the more
+*          complicated van Herk/Gil-Werman algorithm, and we do it
+*          by brute force.
+* </pre>
+*/
+NUMA *
+numaErode(NUMA    *nas,
+l_int32  size)
+{
+l_int32     i, j, n, hsize, len;
+l_float32   minval;
+l_float32  *fa, *fas, *fad;
+NUMA       *nad;
+if (!nas)
+return (NUMA *)ERROR_PTR("nas not defined", __func__, NULL);
+if (size <= 0)
+return (NUMA *)ERROR_PTR("size must be > 0", __func__, NULL);
+if ((size & 1) == 0 ) {
+L_WARNING("sel size must be odd; increasing by 1\n", __func__);
+size++;
+}
+if (size == 1)
+return numaCopy(nas);
+/* Make a source fa (fas) that has an added (size / 2) boundary
+* on left and right, contains a copy of nas in the interior region
+* (between 'size' and 'size + n', and has large values
+* inserted in the boundary (because it is an erosion). */
+n = numaGetCount(nas);
+hsize = size / 2;
+len = n + 2 * hsize;
+if ((fas = (l_float32 *)LEPT_CALLOC(len, sizeof(l_float32))) == NULL)
+return (NUMA *)ERROR_PTR("fas not made", __func__, NULL);
+for (i = 0; i < hsize; i++)
+fas[i] = 1.0e37f;
+for (i = hsize + n; i < len; i++)
+fas[i] = 1.0e37f;
+fa = numaGetFArray(nas, L_NOCOPY);
+for (i = 0; i < n; i++)
+fas[hsize + i] = fa[i];
+nad = numaMakeConstant(0, n);
+numaCopyParameters(nad, nas);
+fad = numaGetFArray(nad, L_NOCOPY);
+for (i = 0; i < n; i++) {
+minval = 1.0e37f;  /* start big */
+for (j = 0; j < size; j++)
+minval = L_MIN(minval, fas[i + j]);
+fad[i] = minval;
+}
+LEPT_FREE(fas);
+return nad;
+}
+/*!
+* \brief   numaDilate()
+*
+* \param[in]    nas
+* \param[in]    size   of sel; greater than 0, odd.  The origin
+*                      is implicitly in the center.
+* \return  nad dilated, or NULL on error
+*
+* <pre>
+* Notes:
+*      (1) The structuring element (sel) is linear, all "hits"
+*      (2) If size == 1, this returns a copy
+* </pre>
+*/
+NUMA *
+numaDilate(NUMA    *nas,
+l_int32  size)
+{
+l_int32     i, j, n, hsize, len;
+l_float32   maxval;
+l_float32  *fa, *fas, *fad;
+NUMA       *nad;
+if (!nas)
+return (NUMA *)ERROR_PTR("nas not defined", __func__, NULL);
+if (size <= 0)
+return (NUMA *)ERROR_PTR("size must be > 0", __func__, NULL);
+if ((size & 1) == 0 ) {
+L_WARNING("sel size must be odd; increasing by 1\n", __func__);
+size++;
+}
+if (size == 1)
+return numaCopy(nas);
+/* Make a source fa (fas) that has an added (size / 2) boundary
+* on left and right, contains a copy of nas in the interior region
+* (between 'size' and 'size + n', and has small values
+* inserted in the boundary (because it is a dilation). */
+n = numaGetCount(nas);
+hsize = size / 2;
+len = n + 2 * hsize;
+if ((fas = (l_float32 *)LEPT_CALLOC(len, sizeof(l_float32))) == NULL)
+return (NUMA *)ERROR_PTR("fas not made", __func__, NULL);
+for (i = 0; i < hsize; i++)
+fas[i] = -1.0e37f;
+for (i = hsize + n; i < len; i++)
+fas[i] = -1.0e37f;
+fa = numaGetFArray(nas, L_NOCOPY);
+for (i = 0; i < n; i++)
+fas[hsize + i] = fa[i];
+nad = numaMakeConstant(0, n);
+numaCopyParameters(nad, nas);
+fad = numaGetFArray(nad, L_NOCOPY);
+for (i = 0; i < n; i++) {
+maxval = -1.0e37f;  /* start small */
+for (j = 0; j < size; j++)
+maxval = L_MAX(maxval, fas[i + j]);
+fad[i] = maxval;
+}
+LEPT_FREE(fas);
+return nad;
+}
+/*!
+* \brief   numaOpen()
+*
+* \param[in]    nas
+* \param[in]    size   of sel; greater than 0, odd.  The origin
+*                      is implicitly in the center.
+* \return  nad opened, or NULL on error
+*
+* <pre>
+* Notes:
+*      (1) The structuring element (sel) is linear, all "hits"
+*      (2) If size == 1, this returns a copy
+* </pre>
+*/
+NUMA *
+numaOpen(NUMA    *nas,
+l_int32  size)
+{
+NUMA  *nat, *nad;
+if (!nas)
+return (NUMA *)ERROR_PTR("nas not defined", __func__, NULL);
+if (size <= 0)
+return (NUMA *)ERROR_PTR("size must be > 0", __func__, NULL);
+if ((size & 1) == 0 ) {
+L_WARNING("sel size must be odd; increasing by 1\n", __func__);
+size++;
+}
+if (size == 1)
+return numaCopy(nas);
+nat = numaErode(nas, size);
+nad = numaDilate(nat, size);
+numaDestroy(&nat);
+return nad;
+}
+/*!
+* \brief   numaClose()
+*
+* \param[in]    nas
+* \param[in]    size   of sel; greater than 0, odd.  The origin
+*                      is implicitly in the center.
+* \return  nad  closed, or NULL on error
+*
+* <pre>
+* Notes:
+*      (1) The structuring element (sel) is linear, all "hits"
+*      (2) If size == 1, this returns a copy
+*      (3) We add a border before doing this operation, for the same
+*          reason that we add a border to a pix before doing a safe closing.
+*          Without the border, a small component near the border gets
+*          clipped at the border on dilation, and can be entirely removed
+*          by the following erosion, violating the basic extensivity
+*          property of closing.
+* </pre>
+*/
+NUMA *
+numaClose(NUMA    *nas,
+l_int32  size)
+{
+NUMA  *nab, *nat1, *nat2, *nad;
+if (!nas)
+return (NUMA *)ERROR_PTR("nas not defined", __func__, NULL);
+if (size <= 0)
+return (NUMA *)ERROR_PTR("size must be > 0", __func__, NULL);
+if ((size & 1) == 0 ) {
+L_WARNING("sel size must be odd; increasing by 1\n", __func__);
+size++;
+}
+if (size == 1)
+return numaCopy(nas);
+nab = numaAddBorder(nas, size, size, 0);  /* to preserve extensivity */
+nat1 = numaDilate(nab, size);
+nat2 = numaErode(nat1, size);
+nad = numaRemoveBorder(nat2, size, size);
+numaDestroy(&nab);
+numaDestroy(&nat1);
+numaDestroy(&nat2);
+return nad;
+}
+/*----------------------------------------------------------------------*
+*                            Other transforms                          *
+*----------------------------------------------------------------------*/
+/*!
+* \brief   numaTransform()
+*
+* \param[in]    nas
+* \param[in]    shift    add this to each number
+* \param[in]    scale    multiply each number by this
+* \return  nad with all values shifted and scaled, or NULL on error
+*
+* <pre>
+* Notes:
+*      (1) Each number is shifted before scaling.
+* </pre>
+*/
+NUMA *
+numaTransform(NUMA      *nas,
+l_float32  shift,
+l_float32  scale)
+{
+l_int32    i, n;
+l_float32  val;
+NUMA      *nad;
+if (!nas)
+return (NUMA *)ERROR_PTR("nas not defined", __func__, NULL);
+n = numaGetCount(nas);
+if ((nad = numaCreate(n)) == NULL)
+return (NUMA *)ERROR_PTR("nad not made", __func__, NULL);
+numaCopyParameters(nad, nas);
+for (i = 0; i < n; i++) {
+numaGetFValue(nas, i, &val);
+val = scale * (val + shift);
+numaAddNumber(nad, val);
+}
+return nad;
+}
+/*!
+* \brief   numaSimpleStats()
+*
+* \param[in]    na       input numa
+* \param[in]    first    first element to use
+* \param[in]    last     last element to use; -1 to go to the end
+* \param[out]   pmean    [optional] mean value
+* \param[out]   pvar     [optional] variance
+* \param[out]   prvar    [optional] rms deviation from the mean
+* \return  0 if OK, 1 on error
+*/
+l_ok
+numaSimpleStats(NUMA       *na,
+l_int32     first,
+l_int32     last,
+l_float32  *pmean,
+l_float32  *pvar,
+l_float32  *prvar)
+{
+l_int32    i, n, ni;
+l_float32  sum, sumsq, val, mean, var;
+if (pmean) *pmean = 0.0;
+if (pvar) *pvar = 0.0;
+if (prvar) *prvar = 0.0;
+if (!pmean && !pvar && !prvar)
+return ERROR_INT("nothing requested", __func__, 1);
+if (!na)
+return ERROR_INT("na not defined", __func__, 1);
+if ((n = numaGetCount(na)) == 0)
+return ERROR_INT("na is empty", __func__, 1);
+first = L_MAX(0, first);
+if (last < 0) last = n - 1;
+if (first >= n)
+return ERROR_INT("invalid first", __func__, 1);
+if (last >= n) {
+L_WARNING("last = %d is beyond max index = %d; adjusting\n",
+__func__, last, n - 1);
+last = n - 1;
+}
+if (first > last)
+return ERROR_INT("first > last\n", __func__, 1);
+ni = last - first + 1;
+sum = sumsq = 0.0;
+for (i = first; i <= last; i++) {
+numaGetFValue(na, i, &val);
+sum += val;
+sumsq += val * val;
+}
+mean = sum / ni;
+if (pmean)
+*pmean = mean;
+if (pvar || prvar) {
+var = sumsq / ni - mean * mean;
+if (pvar) *pvar = var;
+if (prvar) *prvar = sqrtf(var);
+}
+return 0;
+}
+/*!
+* \brief   numaWindowedStats()
+*
+* \param[in]    nas     input numa
+* \param[in]    wc      half width of the window
+* \param[out]   pnam    [optional] mean value in window
+* \param[out]   pnams   [optional] mean square value in window
+* \param[out]   pnav    [optional] variance in window
+* \param[out]   pnarv   [optional] rms deviation from the mean
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*      (1) This is a high-level convenience function for calculating
+*          any or all of these derived arrays.
+*      (2) These statistical measures over the values in the
+*          rectangular window are:
+*            ~ average value: [x]  (nam)
+*            ~ average squared value: [x*x] (nams)
+*            ~ variance: [(x - [x])*(x - [x])] = [x*x] - [x]*[x]  (nav)
+*            ~ square-root of variance: (narv)
+*          where the brackets [ .. ] indicate that the average value is
+*          to be taken over the window.
+*      (3) Note that the variance is just the mean square difference from
+*          the mean value; and the square root of the variance is the
+*          root mean square difference from the mean, sometimes also
+*          called the 'standard deviation'.
+*      (4) Internally, use mirrored borders to handle values near the
+*          end of each array.
+* </pre>
+*/
+l_ok
+numaWindowedStats(NUMA    *nas,
+l_int32  wc,
+NUMA   **pnam,
+NUMA   **pnams,
+NUMA   **pnav,
+NUMA   **pnarv)
+{
+NUMA  *nam, *nams;
+if (!nas)
+return ERROR_INT("nas not defined", __func__, 1);
+if (2 * wc + 1 > numaGetCount(nas))
+L_WARNING("filter wider than input array!\n", __func__);
+if (!pnav && !pnarv) {
+if (pnam) *pnam = numaWindowedMean(nas, wc);
+if (pnams) *pnams = numaWindowedMeanSquare(nas, wc);
+return 0;
+}
+nam = numaWindowedMean(nas, wc);
+nams = numaWindowedMeanSquare(nas, wc);
+numaWindowedVariance(nam, nams, pnav, pnarv);
+if (pnam)
+*pnam = nam;
+else
+numaDestroy(&nam);
+if (pnams)
+*pnams = nams;
+else
+numaDestroy(&nams);
+return 0;
+}
+/*!
+* \brief   numaWindowedMean()
+*
+* \param[in]    nas
+* \param[in]    wc    half width of the convolution window
+* \return  nad after low-pass filtering, or NULL on error
+*
+* <pre>
+* Notes:
+*      (1) This is a convolution.  The window has width = 2 * %wc + 1.
+*      (2) We add a mirrored border of size %wc to each end of the array.
+* </pre>
+*/
+NUMA *
+numaWindowedMean(NUMA    *nas,
+l_int32  wc)
+{
+l_int32     i, n, n1, width;
+l_float32   sum, norm;
+l_float32  *fa1, *fad, *suma;
+NUMA       *na1, *nad;
+if (!nas)
+return (NUMA *)ERROR_PTR("nas not defined", __func__, NULL);
+n = numaGetCount(nas);
+width = 2 * wc + 1;  /* filter width */
+if (width > n)
+L_WARNING("filter wider than input array!\n", __func__);
+na1 = numaAddSpecifiedBorder(nas, wc, wc, L_MIRRORED_BORDER);
+n1 = n + 2 * wc;
+fa1 = numaGetFArray(na1, L_NOCOPY);
+nad = numaMakeConstant(0, n);
+fad = numaGetFArray(nad, L_NOCOPY);
+/* Make sum array; note the indexing */
+if ((suma = (l_float32 *)LEPT_CALLOC(n1 + 1, sizeof(l_float32))) == NULL) {
+numaDestroy(&na1);
+numaDestroy(&nad);
+return (NUMA *)ERROR_PTR("suma not made", __func__, NULL);
+}
+sum = 0.0;
+suma[0] = 0.0;
+for (i = 0; i < n1; i++) {
+sum += fa1[i];
+suma[i + 1] = sum;
+}
+norm = 1.f / (2 * wc + 1);
+for (i = 0; i < n; i++)
+fad[i] = norm * (suma[width + i] - suma[i]);
+LEPT_FREE(suma);
+numaDestroy(&na1);
+return nad;
+}
+/*!
+* \brief   numaWindowedMeanSquare()
+*
+* \param[in]    nas
+* \param[in]    wc    half width of the window
+* \return  nad containing windowed mean square values, or NULL on error
+*
+* <pre>
+* Notes:
+*      (1) The window has width = 2 * %wc + 1.
+*      (2) We add a mirrored border of size %wc to each end of the array.
+* </pre>
+*/
+NUMA *
+numaWindowedMeanSquare(NUMA    *nas,
+l_int32  wc)
+{
+l_int32     i, n, n1, width;
+l_float32   sum, norm;
+l_float32  *fa1, *fad, *suma;
+NUMA       *na1, *nad;
+if (!nas)
+return (NUMA *)ERROR_PTR("nas not defined", __func__, NULL);
+n = numaGetCount(nas);
+width = 2 * wc + 1;  /* filter width */
+if (width > n)
+L_WARNING("filter wider than input array!\n", __func__);
+na1 = numaAddSpecifiedBorder(nas, wc, wc, L_MIRRORED_BORDER);
+n1 = n + 2 * wc;
+fa1 = numaGetFArray(na1, L_NOCOPY);
+nad = numaMakeConstant(0, n);
+fad = numaGetFArray(nad, L_NOCOPY);
+/* Make sum array; note the indexing */
+if ((suma = (l_float32 *)LEPT_CALLOC(n1 + 1, sizeof(l_float32))) == NULL) {
+numaDestroy(&na1);
+numaDestroy(&nad);
+return (NUMA *)ERROR_PTR("suma not made", __func__, NULL);
+}
+sum = 0.0;
+suma[0] = 0.0;
+for (i = 0; i < n1; i++) {
+sum += fa1[i] * fa1[i];
+suma[i + 1] = sum;
+}
+norm = 1.f / (2 * wc + 1);
+for (i = 0; i < n; i++)
+fad[i] = norm * (suma[width + i] - suma[i]);
+LEPT_FREE(suma);
+numaDestroy(&na1);
+return nad;
+}
+/*!
+* \brief   numaWindowedVariance()
+*
+* \param[in]    nam    windowed mean values
+* \param[in]    nams   windowed mean square values
+* \param[out]   pnav   [optional] numa of variance -- the ms deviation
+*                      from the mean
+* \param[out]   pnarv  [optional] numa of rms deviation from the mean
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*      (1) The numas of windowed mean and mean square are precomputed,
+*          using numaWindowedMean() and numaWindowedMeanSquare().
+*      (2) Either or both of the variance and square-root of variance
+*          are returned, where the variance is the average over the
+*          window of the mean square difference of the pixel value
+*          from the mean:
+*                [(x - [x])*(x - [x])] = [x*x] - [x]*[x]
+* </pre>
+*/
+l_ok
+numaWindowedVariance(NUMA   *nam,
+NUMA   *nams,
+NUMA  **pnav,
+NUMA  **pnarv)
+{
+l_int32     i, nm, nms;
+l_float32   var;
+l_float32  *fam, *fams, *fav = NULL, *farv = NULL;
+NUMA       *nav, *narv;  /* variance and square root of variance */
+if (pnav) *pnav = NULL;
+if (pnarv) *pnarv = NULL;
+if (!pnav && !pnarv)
+return ERROR_INT("neither &nav nor &narv are defined", __func__, 1);
+if (!nam)
+return ERROR_INT("nam not defined", __func__, 1);
+if (!nams)
+return ERROR_INT("nams not defined", __func__, 1);
+nm = numaGetCount(nam);
+nms = numaGetCount(nams);
+if (nm != nms)
+return ERROR_INT("sizes of nam and nams differ", __func__, 1);
+if (pnav) {
+nav = numaMakeConstant(0, nm);
+*pnav = nav;
+fav = numaGetFArray(nav, L_NOCOPY);
+}
+if (pnarv) {
+narv = numaMakeConstant(0, nm);
+*pnarv = narv;
+farv = numaGetFArray(narv, L_NOCOPY);
+}
+fam = numaGetFArray(nam, L_NOCOPY);
+fams = numaGetFArray(nams, L_NOCOPY);
+for (i = 0; i < nm; i++) {
+var = fams[i] - fam[i] * fam[i];
+if (pnav)
+fav[i] = var;
+if (pnarv)
+farv[i] = sqrtf(var);
+}
+return 0;
+}
+/*!
+* \brief   numaWindowedMedian()
+*
+* \param[in]    nas
+* \param[in]    halfwin   half width of window over which the median is found
+* \return  nad after windowed median filtering, or NULL on error
+*
+* <pre>
+* Notes:
+*      (1) The requested window has width = 2 * %halfwin + 1.
+*      (2) If the input nas has less then 3 elements, return a copy.
+*      (3) If the filter is too small (%halfwin <= 0), return a copy.
+*      (4) If the filter is too large, it is reduced in size.
+*      (5) We add a mirrored border of size %halfwin to each end of
+*          the array to simplify the calculation by avoiding end-effects.
+* </pre>
+*/
+NUMA *
+numaWindowedMedian(NUMA    *nas,
+l_int32  halfwin)
+{
+l_int32    i, n;
+l_float32  medval;
+NUMA      *na1, *na2, *nad;
+if (!nas)
+return (NUMA *)ERROR_PTR("nas not defined", __func__, NULL);
+if ((n = numaGetCount(nas)) < 3)
+return numaCopy(nas);
+if (halfwin <= 0) {
+L_ERROR("filter too small; returning a copy\n", __func__);
+return numaCopy(nas);
+}
+if (halfwin > (n - 1) / 2) {
+halfwin = (n - 1) / 2;
+L_INFO("reducing filter to halfwin = %d\n", __func__, halfwin);
+}
+/* Add a border to both ends */
+na1 = numaAddSpecifiedBorder(nas, halfwin, halfwin, L_MIRRORED_BORDER);
+/* Get the median value at the center of each window, corresponding
+* to locations in the input nas. */
+nad = numaCreate(n);
+for (i = 0; i < n; i++) {
+na2 = numaClipToInterval(na1, i, i + 2 * halfwin);
+numaGetMedian(na2, &medval);
+numaAddNumber(nad, medval);
+numaDestroy(&na2);
+}
+numaDestroy(&na1);
+return nad;
+}
+/*!
+* \brief   numaConvertToInt()
+*
+* \param[in]    nas   source numa
+* \return  na with all values rounded to nearest integer, or
+*              NULL on error
+*/
+NUMA *
+numaConvertToInt(NUMA  *nas)
+{
+l_int32  i, n, ival;
+NUMA    *nad;
+if (!nas)
+return (NUMA *)ERROR_PTR("nas not defined", __func__, NULL);
+n = numaGetCount(nas);
+if ((nad = numaCreate(n)) == NULL)
+return (NUMA *)ERROR_PTR("nad not made", __func__, NULL);
+numaCopyParameters(nad, nas);
+for (i = 0; i < n; i++) {
+numaGetIValue(nas, i, &ival);
+numaAddNumber(nad, ival);
+}
+return nad;
+}
+/*----------------------------------------------------------------------*
+*                 Histogram generation and statistics                  *
+*----------------------------------------------------------------------*/
+/*!
+* \brief   numaMakeHistogram()
+*
+* \param[in]    na
+* \param[in]    maxbins    max number of histogram bins
+* \param[out]   pbinsize   [optional] size of histogram bins
+* \param[out]   pbinstart  [optional] start val of minimum bin;
+*                          input NULL to force start at 0
+* \return  na consisiting of histogram of integerized values,
+*              or NULL on error.
+*
+* <pre>
+* Notes:
+*      (1) This simple interface is designed for integer data.
+*          The bins are of integer width and start on integer boundaries,
+*          so the results on float data will not have high precision.
+*      (2) Specify the max number of input bins.   Then %binsize,
+*          the size of bins necessary to accommodate the input data,
+*          is returned.  It is optionally returned and one of the sequence:
+*                {1, 2, 5, 10, 20, 50, ...}.
+*      (3) If &binstart is given, all values are accommodated,
+*          and the min value of the starting bin is returned.
+*          Otherwise, all negative values are discarded and
+*          the histogram bins start at 0.
+* </pre>
+*/
+NUMA *
+numaMakeHistogram(NUMA     *na,
+l_int32   maxbins,
+l_int32  *pbinsize,
+l_int32  *pbinstart)
+{
+l_int32    i, n, ival, hval;
+l_int32    iminval, imaxval, range, binsize, nbins, ibin;
+l_float32  val, ratio;
+NUMA      *nai, *nahist;
+if (pbinsize) *pbinsize = 0;
+if (pbinstart) *pbinstart = 0;
+if (!na)
+return (NUMA *)ERROR_PTR("na not defined", __func__, NULL);
+if (maxbins < 1)
+return (NUMA *)ERROR_PTR("maxbins < 1", __func__, NULL);
+/* Determine input range */
+numaGetMin(na, &val, NULL);
+iminval = (l_int32)(val + 0.5);
+numaGetMax(na, &val, NULL);
+imaxval = (l_int32)(val + 0.5);
+if (pbinstart == NULL) {  /* clip negative vals; start from 0 */
+iminval = 0;
+if (imaxval < 0)
+return (NUMA *)ERROR_PTR("all values < 0", __func__, NULL);
+}
+/* Determine binsize */
+range = imaxval - iminval + 1;
+if (range > maxbins - 1) {
+ratio = (l_float32)range / (l_float32)maxbins;
+binsize = 0;
+for (i = 0; i < NBinSizes; i++) {
+if (ratio < BinSizeArray[i]) {
+binsize = BinSizeArray[i];
+break;
+}
+}
+if (binsize == 0)
+return (NUMA *)ERROR_PTR("numbers too large", __func__, NULL);
+} else {
+binsize = 1;
+}
+if (pbinsize) *pbinsize = binsize;
+nbins = 1 + range / binsize;  /* +1 seems to be sufficient */
+/* Redetermine iminval */
+if (pbinstart && binsize > 1) {
+if (iminval >= 0)
+iminval = binsize * (iminval / binsize);
+else
+iminval = binsize * ((iminval - binsize + 1) / binsize);
+}
+if (pbinstart) *pbinstart = iminval;
+#if  DEBUG_HISTO
+lept_stderr(" imaxval = %d, range = %d, nbins = %d\n",
+imaxval, range, nbins);
+#endif  /* DEBUG_HISTO */
+/* Use integerized data for input */
+if ((nai = numaConvertToInt(na)) == NULL)
+return (NUMA *)ERROR_PTR("nai not made", __func__, NULL);
+n = numaGetCount(nai);
+/* Make histogram, converting value in input array
+* into a bin number for this histogram array. */
+if ((nahist = numaCreate(nbins)) == NULL) {
+numaDestroy(&nai);
+return (NUMA *)ERROR_PTR("nahist not made", __func__, NULL);
+}
+numaSetCount(nahist, nbins);
+numaSetParameters(nahist, iminval, binsize);
+for (i = 0; i < n; i++) {
+numaGetIValue(nai, i, &ival);
+ibin = (ival - iminval) / binsize;
+if (ibin >= 0 && ibin < nbins) {
+numaGetIValue(nahist, ibin, &hval);
+numaSetValue(nahist, ibin, hval + 1.0f);
+}
+}
+numaDestroy(&nai);
+return nahist;
+}
+/*!
+* \brief   numaMakeHistogramAuto()
+*
+* \param[in]    na       numa of floats; these may be integers
+* \param[in]    maxbins  max number of histogram bins; >= 1
+* \return  na consisiting of histogram of quantized float values,
+*              or NULL on error.
+*
+* <pre>
+* Notes:
+*      (1) This simple interface is designed for accurate binning
+*          of both integer and float data.
+*      (2) If the array data is integers, and the range of integers
+*          is smaller than %maxbins, they are binned as they fall,
+*          with binsize = 1.
+*      (3) If the range of data, (maxval - minval), is larger than
+*          %maxbins, or if the data is floats, they are binned into
+*          exactly %maxbins bins.
+*      (4) Unlike numaMakeHistogram(), these bins in general have
+*          non-integer location and width, even for integer data.
+* </pre>
+*/
+NUMA *
+numaMakeHistogramAuto(NUMA    *na,
+l_int32  maxbins)
+{
+l_int32    i, n, imin, imax, irange, ibin, ival, allints;
+l_float32  minval, maxval, range, binsize, fval;
+NUMA      *nah;
+if (!na)
+return (NUMA *)ERROR_PTR("na not defined", __func__, NULL);
+maxbins = L_MAX(1, maxbins);
+/* Determine input range */
+numaGetMin(na, &minval, NULL);
+numaGetMax(na, &maxval, NULL);
+/* Determine if values are all integers */
+n = numaGetCount(na);
+numaHasOnlyIntegers(na, &allints);
+/* Do simple integer binning if possible */
+if (allints && (maxval - minval < maxbins)) {
+imin = (l_int32)minval;
+imax = (l_int32)maxval;
+irange = imax - imin + 1;
+nah = numaCreate(irange);
+numaSetCount(nah, irange);  /* init */
+numaSetParameters(nah, minval, 1.0);
+for (i = 0; i < n; i++) {
+numaGetIValue(na, i, &ival);
+ibin = ival - imin;
+numaGetIValue(nah, ibin, &ival);
+numaSetValue(nah, ibin, ival + 1.0f);
+}
+return nah;
+}
+/* Do float binning, even if the data is integers. */
+range = maxval - minval;
+binsize = range / (l_float32)maxbins;
+if (range == 0.0) {
+nah = numaCreate(1);
+numaSetParameters(nah, minval, binsize);
+numaAddNumber(nah, n);
+return nah;
+}
+nah = numaCreate(maxbins);
+numaSetCount(nah, maxbins);
+numaSetParameters(nah, minval, binsize);
+for (i = 0; i < n; i++) {
+numaGetFValue(na, i, &fval);
+ibin = (l_int32)((fval - minval) / binsize);
+ibin = L_MIN(ibin, maxbins - 1);  /* "edge" case; stay in bounds */
+numaGetIValue(nah, ibin, &ival);
+numaSetValue(nah, ibin, ival + 1.0f);
+}
+return nah;
+}
+/*!
+* \brief   numaMakeHistogramClipped()
+*
+* \param[in]    na
+* \param[in]    binsize    typically 1.0
+* \param[in]    maxsize    of histogram ordinate
+* \return  na histogram of bins of size %binsize, starting with
+*                  the na[0] (x = 0.0 and going up to a maximum of
+*                  x = %maxsize, by increments of %binsize), or NULL on error
+*
+* <pre>
+* Notes:
+*      (1) This simple function generates a histogram of values
+*          from na, discarding all values < 0.0 or greater than
+*          min(%maxsize, maxval), where maxval is the maximum value in na.
+*          The histogram data is put in bins of size delx = %binsize,
+*          starting at x = 0.0.  We use as many bins as are
+*          needed to hold the data.
+* </pre>
+*/
+NUMA *
+numaMakeHistogramClipped(NUMA      *na,
+l_float32  binsize,
+l_float32  maxsize)
+{
+l_int32    i, n, nbins, ival, ibin;
+l_float32  val, maxval;
+NUMA      *nad;
+if (!na)
+return (NUMA *)ERROR_PTR("na not defined", __func__, NULL);
+if (binsize <= 0.0)
+return (NUMA *)ERROR_PTR("binsize must be > 0.0", __func__, NULL);
+if (binsize > maxsize)
+binsize = maxsize;  /* just one bin */
+numaGetMax(na, &maxval, NULL);
+n = numaGetCount(na);
+maxsize = L_MIN(maxsize, maxval);
+nbins = (l_int32)(maxsize / binsize) + 1;
+/*    lept_stderr("maxsize = %7.3f, nbins = %d\n", maxsize, nbins); */
+if ((nad = numaCreate(nbins)) == NULL)
+return (NUMA *)ERROR_PTR("nad not made", __func__, NULL);
+numaSetParameters(nad, 0.0, binsize);
+numaSetCount(nad, nbins);  /* interpret zeroes in bins as data */
+for (i = 0; i < n; i++) {
+numaGetFValue(na, i, &val);
+ibin = (l_int32)(val / binsize);
+if (ibin >= 0 && ibin < nbins) {
+numaGetIValue(nad, ibin, &ival);
+numaSetValue(nad, ibin, ival + 1.0f);
+}
+}
+return nad;
+}
+/*!
+* \brief   numaRebinHistogram()
+*
+* \param[in]    nas      input histogram
+* \param[in]    newsize  number of old bins contained in each new bin
+* \return  nad more coarsely re-binned histogram, or NULL on error
+*/
+NUMA *
+numaRebinHistogram(NUMA    *nas,
+l_int32  newsize)
+{
+l_int32    i, j, ns, nd, index, count, val;
+l_float32  start, oldsize;
+NUMA      *nad;
+if (!nas)
+return (NUMA *)ERROR_PTR("nas not defined", __func__, NULL);
+if (newsize <= 1)
+return (NUMA *)ERROR_PTR("newsize must be > 1", __func__, NULL);
+if ((ns = numaGetCount(nas)) == 0)
+return (NUMA *)ERROR_PTR("no bins in nas", __func__, NULL);
+nd = (ns + newsize - 1) / newsize;
+if ((nad = numaCreate(nd)) == NULL)
+return (NUMA *)ERROR_PTR("nad not made", __func__, NULL);
+numaGetParameters(nad, &start, &oldsize);
+numaSetParameters(nad, start, oldsize * newsize);
+for (i = 0; i < nd; i++) {  /* new bins */
+count = 0;
+index = i * newsize;
+for (j = 0; j < newsize; j++) {
+if (index < ns) {
+numaGetIValue(nas, index, &val);
+count += val;
+index++;
+}
+}
+numaAddNumber(nad, count);
+}
+return nad;
+}
+/*!
+* \brief   numaNormalizeHistogram()
+*
+* \param[in]    nas   input histogram
+* \param[in]    tsum  target sum of all numbers in dest histogram; e.g., use
+*                     %tsum= 1.0 if this represents a probability distribution
+* \return  nad normalized histogram, or NULL on error
+*/
+NUMA *
+numaNormalizeHistogram(NUMA      *nas,
+l_float32  tsum)
+{
+l_int32    i, ns;
+l_float32  sum, factor, fval;
+NUMA      *nad;
+if (!nas)
+return (NUMA *)ERROR_PTR("nas not defined", __func__, NULL);
+if (tsum <= 0.0)
+return (NUMA *)ERROR_PTR("tsum must be > 0.0", __func__, NULL);
+if ((ns = numaGetCount(nas)) == 0)
+return (NUMA *)ERROR_PTR("no bins in nas", __func__, NULL);
+numaGetSum(nas, &sum);
+factor = tsum / sum;
+if ((nad = numaCreate(ns)) == NULL)
+return (NUMA *)ERROR_PTR("nad not made", __func__, NULL);
+numaCopyParameters(nad, nas);
+for (i = 0; i < ns; i++) {
+numaGetFValue(nas, i, &fval);
+fval *= factor;
+numaAddNumber(nad, fval);
+}
+return nad;
+}
+/*!
+* \brief   numaGetStatsUsingHistogram()
+*
+* \param[in]    na        an arbitrary set of numbers; not ordered and not
+*                         a histogram
+* \param[in]    maxbins   the maximum number of bins to be allowed in
+*                         the histogram; use an integer larger than the
+*                         largest number in %na for consecutive integer bins
+* \param[out]   pmin      [optional] min value of set
+* \param[out]   pmax      [optional] max value of set
+* \param[out]   pmean     [optional] mean value of set
+* \param[out]   pvariance [optional] variance
+* \param[out]   pmedian   [optional] median value of set
+* \param[in]    rank      in [0.0 ... 1.0]; median has a rank 0.5;
+*                         ignored if &rval == NULL
+* \param[out]   prval     [optional] value in na corresponding to %rank
+* \param[out]   phisto    [optional] Numa histogram; use NULL to prevent
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*      (1) This is a simple interface for gathering statistics
+*          from a numa, where a histogram is used 'under the covers'
+*          to avoid sorting if a rank value is requested.  In that case,
+*          by using a histogram we are trading speed for accuracy, because
+*          the values in %na are quantized to the center of a set of bins.
+*      (2) If the median, other rank value, or histogram are not requested,
+*          the calculation is all performed on the input Numa.
+*      (3) The variance is the average of the square of the
+*          difference from the mean.  The median is the value in na
+*          with rank 0.5.
+*      (4) There are two situations where this gives rank results with
+*          accuracy comparable to computing stastics directly on the input
+*          data, without binning into a histogram:
+*           (a) the data is integers and the range of data is less than
+*               %maxbins, and
+*           (b) the data is floats and the range is small compared to
+*               %maxbins, so that the binsize is much less than 1.
+*      (5) If a histogram is used and the numbers in the Numa extend
+*          over a large range, you can limit the required storage by
+*          specifying the maximum number of bins in the histogram.
+*          Use %maxbins == 0 to force the bin size to be 1.
+*      (6) This optionally returns the median and one arbitrary rank value.
+*          If you need several rank values, return the histogram and use
+*               numaHistogramGetValFromRank(nah, rank, &rval)
+*          multiple times.
+* </pre>
+*/
+l_ok
+numaGetStatsUsingHistogram(NUMA       *na,
+l_int32     maxbins,
+l_float32  *pmin,
+l_float32  *pmax,
+l_float32  *pmean,
+l_float32  *pvariance,
+l_float32  *pmedian,
+l_float32   rank,
+l_float32  *prval,
+NUMA      **phisto)
+{
+l_int32    i, n;
+l_float32  minval, maxval, fval, mean, sum;
+NUMA      *nah;
+if (pmin) *pmin = 0.0;
+if (pmax) *pmax = 0.0;
+if (pmean) *pmean = 0.0;
+if (pvariance) *pvariance = 0.0;
+if (pmedian) *pmedian = 0.0;
+if (prval) *prval = 0.0;
+if (phisto) *phisto = NULL;
+if (!na)
+return ERROR_INT("na not defined", __func__, 1);
+if ((n = numaGetCount(na)) == 0)
+return ERROR_INT("numa is empty", __func__, 1);
+numaGetMin(na, &minval, NULL);
+numaGetMax(na, &maxval, NULL);
+if (pmin) *pmin = minval;
+if (pmax) *pmax = maxval;
+if (pmean || pvariance) {
+sum = 0.0;
+for (i = 0; i < n; i++) {
+numaGetFValue(na, i, &fval);
+sum += fval;
+}
+mean = sum / (l_float32)n;
+if (pmean) *pmean = mean;
+}
+if (pvariance) {
+sum = 0.0;
+for (i = 0; i < n; i++) {
+numaGetFValue(na, i, &fval);
+sum += fval * fval;
+}
+*pvariance = sum / (l_float32)n - mean * mean;
+}
+if (!pmedian && !prval && !phisto)
+return 0;
+nah = numaMakeHistogramAuto(na, maxbins);
+if (pmedian)
+numaHistogramGetValFromRank(nah, 0.5, pmedian);
+if (prval)
+numaHistogramGetValFromRank(nah, rank, prval);
+if (phisto)
+*phisto = nah;
+else
+numaDestroy(&nah);
+return 0;
+}
+/*!
+* \brief   numaGetHistogramStats()
+*
+* \param[in]    nahisto     histogram: y(x(i)), i = 0 ... nbins - 1
+* \param[in]    startx      x value of first bin: x(0)
+* \param[in]    deltax      x increment between bins; the bin size; x(1) - x(0)
+* \param[out]   pxmean      [optional] mean value of histogram
+* \param[out]   pxmedian    [optional] median value of histogram
+* \param[out]   pxmode      [optional] mode value of histogram:
+*                           xmode = x(imode), where y(xmode) >= y(x(i)) for
+*                           all i != imode
+* \param[out]   pxvariance  [optional] variance of x
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*      (1) If the histogram represents the relation y(x), the
+*          computed values that are returned are the x values.
+*          These are NOT the bucket indices i; they are related to the
+*          bucket indices by
+*                x(i) = startx + i * deltax
+* </pre>
+*/
+l_ok
+numaGetHistogramStats(NUMA       *nahisto,
+l_float32   startx,
+l_float32   deltax,
+l_float32  *pxmean,
+l_float32  *pxmedian,
+l_float32  *pxmode,
+l_float32  *pxvariance)
+{
+if (pxmean) *pxmean = 0.0;
+if (pxmedian) *pxmedian = 0.0;
+if (pxmode) *pxmode = 0.0;
+if (pxvariance) *pxvariance = 0.0;
+if (!nahisto)
+return ERROR_INT("nahisto not defined", __func__, 1);
+return numaGetHistogramStatsOnInterval(nahisto, startx, deltax, 0, -1,
+pxmean, pxmedian, pxmode,
+pxvariance);
+}
+/*!
+* \brief   numaGetHistogramStatsOnInterval()
+*
+* \param[in]    nahisto    histogram: y(x(i)), i = 0 ... nbins - 1
+* \param[in]    startx     x value of first bin: x(0)
+* \param[in]    deltax     x increment between bins; the bin size; x(1) - x(0)
+* \param[in]    ifirst     first bin to use for collecting stats
+* \param[in]    ilast      last bin for collecting stats; -1 to go to the end
+* \param[out]   pxmean     [optional] mean value of histogram
+* \param[out]   pxmedian   [optional] median value of histogram
+* \param[out]   pxmode     [optional] mode value of histogram:
+*                          xmode = x(imode), where y(xmode) >= y(x(i)) for
+*                          all i != imode
+* \param[out]   pxvariance [optional] variance of x
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*      (1) If the histogram represents the relation y(x), the
+*          computed values that are returned are the x values.
+*          These are NOT the bucket indices i; they are related to the
+*          bucket indices by
+*                x(i) = startx + i * deltax
+* </pre>
+*/
+l_ok
+numaGetHistogramStatsOnInterval(NUMA       *nahisto,
+l_float32   startx,
+l_float32   deltax,
+l_int32     ifirst,
+l_int32     ilast,
+l_float32  *pxmean,
+l_float32  *pxmedian,
+l_float32  *pxmode,
+l_float32  *pxvariance)
+{
+l_int32    i, n, imax;
+l_float32  sum, sumval, halfsum, moment, var, x, y, ymax;
+if (pxmean) *pxmean = 0.0;
+if (pxmedian) *pxmedian = 0.0;
+if (pxmode) *pxmode = 0.0;
+if (pxvariance) *pxvariance = 0.0;
+if (!nahisto)
+return ERROR_INT("nahisto not defined", __func__, 1);
+if (!pxmean && !pxmedian && !pxmode && !pxvariance)
+return ERROR_INT("nothing to compute", __func__, 1);
+n = numaGetCount(nahisto);
+ifirst = L_MAX(0, ifirst);
+if (ilast < 0) ilast = n - 1;
+if (ifirst >= n)
+return ERROR_INT("invalid ifirst", __func__, 1);
+if (ilast >= n) {
+L_WARNING("ilast = %d is beyond max index = %d; adjusting\n",
+__func__, ilast, n - 1);
+ilast = n - 1;
+}
+if (ifirst > ilast)
+return ERROR_INT("ifirst > ilast", __func__, 1);
+for (sum = 0.0, moment = 0.0, var = 0.0, i = ifirst; i <= ilast ; i++) {
+x = startx + i * deltax;
+numaGetFValue(nahisto, i, &y);
+sum += y;
+moment += x * y;
+var += x * x * y;
+}
+if (sum == 0.0) {
+L_INFO("sum is 0\n", __func__);
+return 0;
+}
+if (pxmean)
+*pxmean = moment / sum;
+if (pxvariance)
+*pxvariance = var / sum - moment * moment / (sum * sum);
+if (pxmedian) {
+halfsum = sum / 2.0f;
+for (sumval = 0.0, i = ifirst; i <= ilast; i++) {
+numaGetFValue(nahisto, i, &y);
+sumval += y;
+if (sumval >= halfsum) {
+*pxmedian = startx + i * deltax;
+break;
+}
+}
+}
+if (pxmode) {
+imax = -1;
+ymax = -1.0e10;
+for (i = ifirst; i <= ilast; i++) {
+numaGetFValue(nahisto, i, &y);
+if (y > ymax) {
+ymax = y;
+imax = i;
+}
+}
+*pxmode = startx + imax * deltax;
+}
+return 0;
+}
+/*!
+* \brief   numaMakeRankFromHistogram()
+*
+* \param[in]    startx   xval corresponding to first element in nay
+* \param[in]    deltax   x increment between array elements in nay
+* \param[in]    nasy     input histogram, assumed equally spaced
+* \param[in]    npts     number of points to evaluate rank function
+* \param[out]   pnax     [optional] array of x values in range
+* \param[out]   pnay     rank array of specified npts
+* \return  0 if OK, 1 on error
+*/
+l_ok
+numaMakeRankFromHistogram(l_float32  startx,
+l_float32  deltax,
+NUMA      *nasy,
+l_int32    npts,
+NUMA     **pnax,
+NUMA     **pnay)
+{
+l_int32    i, n;
+l_float32  sum, fval;
+NUMA      *nan, *nar;
+if (pnax) *pnax = NULL;
+if (!pnay)
+return ERROR_INT("&nay not defined", __func__, 1);
+*pnay = NULL;
+if (!nasy)
+return ERROR_INT("nasy not defined", __func__, 1);
+if ((n = numaGetCount(nasy)) == 0)
+return ERROR_INT("no bins in nas", __func__, 1);
+/* Normalize and generate the rank array corresponding to
+* the binned histogram. */
+nan = numaNormalizeHistogram(nasy, 1.0);
+nar = numaCreate(n + 1);  /* rank numa corresponding to nan */
+sum = 0.0;
+numaAddNumber(nar, sum);  /* first element is 0.0 */
+for (i = 0; i < n; i++) {
+numaGetFValue(nan, i, &fval);
+sum += fval;
+numaAddNumber(nar, sum);
+}
+/* Compute rank array on full range with specified
+* number of points and correspondence to x-values. */
+numaInterpolateEqxInterval(startx, deltax, nar, L_LINEAR_INTERP,
+startx, startx + n * deltax, npts,
+pnax, pnay);
+numaDestroy(&nan);
+numaDestroy(&nar);
+return 0;
+}
+/*!
+* \brief   numaHistogramGetRankFromVal()
+*
+* \param[in]    na     histogram
+* \param[in]    rval   value of input sample for which we want the rank
+* \param[out]   prank  fraction of total samples below rval
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*      (1) If we think of the histogram as a function y(x), normalized
+*          to 1, for a given input value of x, this computes the
+*          rank of x, which is the integral of y(x) from the start
+*          value of x to the input value.
+*      (2) This function only makes sense when applied to a Numa that
+*          is a histogram.  The values in the histogram can be ints and
+*          floats, and are computed as floats.  The rank is returned
+*          as a float between 0.0 and 1.0.
+*      (3) The numa parameters startx and binsize are used to
+*          compute x from the Numa index i.
+* </pre>
+*/
+l_ok
+numaHistogramGetRankFromVal(NUMA       *na,
+l_float32   rval,
+l_float32  *prank)
+{
+l_int32    i, ibinval, n;
+l_float32  startval, binsize, binval, maxval, fractval, total, sum, val;
+if (!prank)
+return ERROR_INT("prank not defined", __func__, 1);
+*prank = 0.0;
+if (!na)
+return ERROR_INT("na not defined", __func__, 1);
+numaGetParameters(na, &startval, &binsize);
+n = numaGetCount(na);
+if (rval < startval)
+return 0;
+maxval = startval + n * binsize;
+if (rval > maxval) {
+*prank = 1.0;
+return 0;
+}
+binval = (rval - startval) / binsize;
+ibinval = (l_int32)binval;
+if (ibinval >= n) {
+*prank = 1.0;
+return 0;
+}
+fractval = binval - (l_float32)ibinval;
+sum = 0.0;
+for (i = 0; i < ibinval; i++) {
+numaGetFValue(na, i, &val);
+sum += val;
+}
+numaGetFValue(na, ibinval, &val);
+sum += fractval * val;
+numaGetSum(na, &total);
+*prank = sum / total;
+/*    lept_stderr("binval = %7.3f, rank = %7.3f\n", binval, *prank); */
+return 0;
+}
+/*!
+* \brief   numaHistogramGetValFromRank()
+*
+* \param[in]    na     histogram
+* \param[in]    rank   fraction of total samples
+* \param[out]   prval  approx. to the bin value
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*      (1) If we think of the histogram as a function y(x), this returns
+*          the value x such that the integral of y(x) from the start
+*          value to x gives the fraction 'rank' of the integral
+*          of y(x) over all bins.
+*      (2) This function only makes sense when applied to a Numa that
+*          is a histogram.  The values in the histogram can be ints and
+*          floats, and are computed as floats.  The val is returned
+*          as a float, even though the buckets are of integer width.
+*      (3) The numa parameters startx and binsize are used to
+*          compute x from the Numa index i.
+* </pre>
+*/
+l_ok
+numaHistogramGetValFromRank(NUMA       *na,
+l_float32   rank,
+l_float32  *prval)
+{
+l_int32    i, n;
+l_float32  startval, binsize, rankcount, total, sum, fract, val;
+if (!prval)
+return ERROR_INT("prval not defined", __func__, 1);
+*prval = 0.0;
+if (!na)
+return ERROR_INT("na not defined", __func__, 1);
+if (rank < 0.0) {
+L_WARNING("rank < 0; setting to 0.0\n", __func__);
+rank = 0.0;
+}
+if (rank > 1.0) {
+L_WARNING("rank > 1.0; setting to 1.0\n", __func__);
+rank = 1.0;
+}
+n = numaGetCount(na);
+numaGetParameters(na, &startval, &binsize);
+numaGetSum(na, &total);
+rankcount = rank * total;  /* count that corresponds to rank */
+sum = 0.0;
+val = 0.0;
+for (i = 0; i < n; i++) {
+numaGetFValue(na, i, &val);
+if (sum + val >= rankcount)
+break;
+sum += val;
+}
+if (val <= 0.0)  /* can == 0 if rank == 0.0 */
+fract = 0.0;
+else  /* sum + fract * val = rankcount */
+fract = (rankcount - sum) / val;
+/* The use of the fraction of a bin allows a simple calculation
+* for the histogram value at the given rank. */
+*prval = startval + binsize * ((l_float32)i + fract);
+/*    lept_stderr("rank = %7.3f, val = %7.3f\n", rank, *prval); */
+return 0;
+}
+/*!
+* \brief   numaDiscretizeSortedInBins()
+*
+* \param[in]    na          sorted
+* \param[in]    nbins       number of equal population bins (> 1)
+* \param[out]   pnabinval   average "gray" values in each bin
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*      (1) The input %na is sorted in increasing value.
+*      (2) The output array has the following mapping:
+*             bin number  -->  average array value in bin (nabinval)
+*      (3) With %nbins == 100, nabinval is the average gray value in
+*          each of the 100 equally populated bins.  It is the function
+*                gray[100 * rank].
+*          Thus it is the inverse of
+*                rank[gray]
+*      (4) Contast with numaDiscretizeHistoInBins(), where the input %na
+*          is a histogram.
+* </pre>
+*/
+l_ok
+numaDiscretizeSortedInBins(NUMA    *na,
+l_int32  nbins,
+NUMA   **pnabinval)
+{
+NUMA      *nabinval;  /* average gray value in the bins */
+NUMA      *naeach;
+l_int32    i, ntot, bincount, binindex, binsize;
+l_float32  sum, val, ave;
+if (!pnabinval)
+return ERROR_INT("&nabinval not defined", __func__, 1);
+*pnabinval = NULL;
+if (!na)
+return ERROR_INT("na not defined", __func__, 1);
+if (nbins < 2)
+return ERROR_INT("nbins must be > 1", __func__, 1);
+/* Get the number of items in each bin */
+ntot = numaGetCount(na);
+if ((naeach = numaGetUniformBinSizes(ntot, nbins)) == NULL)
+return ERROR_INT("naeach not made", __func__, 1);
+/* Get the average value in each bin */
+sum = 0.0;
+bincount = 0;
+binindex = 0;
+numaGetIValue(naeach, 0, &binsize);
+nabinval = numaCreate(nbins);
+for (i = 0; i < ntot; i++) {
+numaGetFValue(na, i, &val);
+bincount++;
+sum += val;
+if (bincount == binsize) {  /* add bin entry */
+ave = sum / binsize;
+numaAddNumber(nabinval, ave);
+sum = 0.0;
+bincount = 0;
+binindex++;
+if (binindex == nbins) break;
+numaGetIValue(naeach, binindex, &binsize);
+}
+}
+*pnabinval = nabinval;
+numaDestroy(&naeach);
+return 0;
+}
+/*!
+* \brief   numaDiscretizeHistoInBins()
+*
+* \param[in]    na          histogram
+* \param[in]    nbins       number of equal population bins (> 1)
+* \param[out]   pnabinval   average "gray" values in each bin
+* \param[out]   pnarank     [optional] rank value of input histogram;
+*                           this is a cumulative norm histogram.
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*      (1) With %nbins == 100, nabinval is the average gray value in
+*          each of the 100 equally populated bins.  It is the function
+*                gray[100 * rank].
+*          Thus it is the inverse of
+*                rank[gray]
+*          which is optionally returned in narank.
+*      (2) The "gray value" is the index into the input histogram.
+*      (3) The two output arrays give the following mappings, where the
+*          input is an un-normalized histogram of array values:
+*             bin number  -->  average array value in bin (nabinval)
+*             array values     -->  cumulative normalized histogram (narank)
+* </pre>
+*/
+l_ok
+numaDiscretizeHistoInBins(NUMA    *na,
+l_int32  nbins,
+NUMA   **pnabinval,
+NUMA   **pnarank)
+{
+NUMA      *nabinval;  /* average gray value in the bins */
+NUMA      *naeach, *nan;
+l_int32    i, j, nxvals, occup, count, bincount, binindex, binsize;
+l_float32  sum, ave, ntot;
+if (pnarank) *pnarank = NULL;
+if (!pnabinval)
+return ERROR_INT("&nabinval not defined", __func__, 1);
+*pnabinval = NULL;
+if (!na)
+return ERROR_INT("na not defined", __func__, 1);
+if (nbins < 2)
+return ERROR_INT("nbins must be > 1", __func__, 1);
+nxvals = numaGetCount(na);
+numaGetSum(na, &ntot);
+occup = ntot / nxvals;
+if (occup < 1) L_INFO("average occupancy %d < 1\n", __func__, occup);
+/* Get the number of items in each bin */
+if ((naeach = numaGetUniformBinSizes(ntot, nbins)) == NULL)
+return ERROR_INT("naeach not made", __func__, 1);
+/* Get the average value in each bin */
+sum = 0.0;
+bincount = 0;
+binindex = 0;
+numaGetIValue(naeach, 0, &binsize);
+nabinval = numaCreate(nbins);
+for (i = 0; i < nxvals; i++) {
+numaGetIValue(na, i, &count);
+for (j = 0; j < count; j++) {
+bincount++;
+sum += i;
+if (bincount == binsize) {  /* add bin entry */
+ave = sum / binsize;
+numaAddNumber(nabinval, ave);
+sum = 0.0;
+bincount = 0;
+binindex++;
+if (binindex == nbins) break;
+numaGetIValue(naeach, binindex, &binsize);
+}
+}
+if (binindex == nbins) break;
+}
+*pnabinval = nabinval;
+if (binindex != nbins)
+L_ERROR("binindex = %d != nbins = %d\n", __func__, binindex, nbins);
+/* Get cumulative normalized histogram (rank[gray value]).
+* This is the partial sum operating on the normalized histogram. */
+if (pnarank) {
+nan = numaNormalizeHistogram(na, 1.0);
+*pnarank = numaGetPartialSums(nan);
+numaDestroy(&nan);
+}
+numaDestroy(&naeach);
+return 0;
+}
+/*!
+* \brief   numaGetRankBinValues()
+*
+* \param[in]    na       an array of values
+* \param[in]    nbins    number of bins at which the rank is divided
+* \param[out]   pnam     mean intensity in a bin vs rank bin value,
+*                        with %nbins of discretized rank values
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*      (1) Simple interface for getting a binned rank representation
+*          of an input array of values.  This returns:
+*             rank bin number -->  average array value in each rank bin (nam)
+*      (2) Uses bins either a sorted array or a histogram, depending on
+*          the values in the array and the size of the array.
+* </pre>
+*/
+l_ok
+numaGetRankBinValues(NUMA    *na,
+l_int32  nbins,
+NUMA   **pnam)
+{
+NUMA      *na1;
+l_int32    maxbins, type;
+l_float32  maxval, delx;
+if (!pnam)
+return ERROR_INT("&pnam not defined", __func__, 1);
+*pnam = NULL;
+if (!na)
+return ERROR_INT("na not defined", __func__, 1);
+if (numaGetCount(na) == 0)
+return ERROR_INT("na is empty", __func__, 1);
+if (nbins < 2)
+return ERROR_INT("nbins must be > 1", __func__, 1);
+/* Choose between sorted array and a histogram.
+* If the input array is has a small number of numbers with
+* a large maximum, we will sort it.  At the other extreme, if
+* the array has many numbers with a small maximum, such as the
+* values of pixels in an 8 bpp grayscale image, generate a histogram.
+* If type comes back as L_BIN_SORT, use a histogram. */
+type = numaChooseSortType(na);
+if (type == L_SHELL_SORT) {  /* sort the array */
+L_INFO("sort the array: input size = %d\n", __func__, numaGetCount(na));
+na1 = numaSort(NULL, na, L_SORT_INCREASING);
+numaDiscretizeSortedInBins(na1, nbins, pnam);
+numaDestroy(&na1);
+return 0;
+}
+/* Make the histogram.  Assuming there are no negative values
+* in the array, if the max value in the array does not exceed
+* about 100000, the bin size for generating the histogram will
+* be 1; maxbins refers to the number of entries in the histogram. */
+L_INFO("use a histogram: input size = %d\n", __func__, numaGetCount(na));
+numaGetMax(na, &maxval, NULL);
+maxbins = L_MIN(100002, (l_int32)maxval + 2);
+na1 = numaMakeHistogram(na, maxbins, NULL, NULL);
+/* Warn if there is a scale change.  This shouldn't happen
+* unless the max value is above 100000.  */
+numaGetParameters(na1, NULL, &delx);
+if (delx > 1.0)
+L_WARNING("scale change: delx = %6.2f\n", __func__, delx);
+/* Rank bin the results */
+numaDiscretizeHistoInBins(na1, nbins, pnam, NULL);
+numaDestroy(&na1);
+return 0;
+}
+/*!
+* \brief   numaGetUniformBinSizes()
+*
+* \param[in]    ntotal   number of values to be split up
+* \param[in]    nbins    number of bins
+* \return  naeach   number of values to go in each bin, or NULL on error
+*
+* <pre>
+* Notes:
+*      (1) The numbers in the bins can differ by 1.  The sum of
+*          bin numbers in %naeach is %ntotal.
+* </pre>
+*/
+NUMA *
+numaGetUniformBinSizes(l_int32  ntotal,
+l_int32  nbins)
+{
+l_int32  i, start, end;
+NUMA    *naeach;
+if (ntotal <= 0)
+return (NUMA *)ERROR_PTR("ntotal <= 0", __func__, NULL);
+if (nbins <= 0)
+return (NUMA *)ERROR_PTR("nbins <= 0", __func__, NULL);
+if ((naeach = numaCreate(nbins)) == NULL)
+return (NUMA *)ERROR_PTR("naeach not made", __func__, NULL);
+if (ntotal < nbins) {  /* put 1 in each of %ntotal bins */
+for (i = 0; i < ntotal; i++)
+numaAddNumber(naeach, 1);
+return naeach;
+}
+start = 0;
+for (i = 0; i < nbins; i++) {
+end = ntotal * (i + 1) / nbins;
+numaAddNumber(naeach, end - start);
+start = end;
+}
+return naeach;
+}
+/*----------------------------------------------------------------------*
+*                      Splitting a distribution                        *
+*----------------------------------------------------------------------*/
+/*!
+* \brief   numaSplitDistribution()
+*
+* \param[in]    na           histogram
+* \param[in]    scorefract   fraction of the max score, used to determine
+*                            range over which the histogram min is searched
+* \param[out]   psplitindex  [optional] index for splitting
+* \param[out]   pave1        [optional] average of lower distribution
+* \param[out]   pave2        [optional] average of upper distribution
+* \param[out]   pnum1        [optional] population of lower distribution
+* \param[out]   pnum2        [optional] population of upper distribution
+* \param[out]   pnascore     [optional] for debugging; otherwise use NULL
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*      (1) This function is intended to be used on a distribution of
+*          values that represent two sets, such as a histogram of
+*          pixel values for an image with a fg and bg, and the goal
+*          is to determine the averages of the two sets and the
+*          best splitting point.
+*      (2) The Otsu method finds a split point that divides the distribution
+*          into two parts by maximizing a score function that is the
+*          product of two terms:
+*            (a) the square of the difference of centroids, (ave1 - ave2)^2
+*            (b) fract1 * (1 - fract1)
+*          where fract1 is the fraction in the lower distribution.
+*      (3) This works well for images where the fg and bg are
+*          each relatively homogeneous and well-separated in color.
+*          However, if the actual fg and bg sets are very different
+*          in size, and the bg is highly varied, as can occur in some
+*          scanned document images, this will bias the split point
+*          into the larger "bump" (i.e., toward the point where the
+*          (b) term reaches its maximum of 0.25 at fract1 = 0.5.
+*          To avoid this, we define a range of values near the
+*          maximum of the score function, and choose the value within
+*          this range such that the histogram itself has a minimum value.
+*          The range is determined by scorefract: we include all abscissa
+*          values to the left and right of the value that maximizes the
+*          score, such that the score stays above (1 - scorefract) * maxscore.
+*          The intuition behind this modification is to try to find
+*          a split point that both has a high variance score and is
+*          at or near a minimum in the histogram, so that the histogram
+*          slope is small at the split point.
+*      (4) We normalize the score so that if the two distributions
+*          were of equal size and at opposite ends of the numa, the
+*          score would be 1.0.
+* </pre>
+*/
+l_ok
+numaSplitDistribution(NUMA       *na,
+l_float32   scorefract,
+l_int32    *psplitindex,
+l_float32  *pave1,
+l_float32  *pave2,
+l_float32  *pnum1,
+l_float32  *pnum2,
+NUMA      **pnascore)
+{
+l_int32    i, n, bestsplit, minrange, maxrange, maxindex;
+l_float32  ave1, ave2, ave1prev, ave2prev;
+l_float32  num1, num2, num1prev, num2prev;
+l_float32  val, minval, sum, fract1;
+l_float32  norm, score, minscore, maxscore;
+NUMA      *nascore, *naave1, *naave2, *nanum1, *nanum2;
+if (psplitindex) *psplitindex = 0;
+if (pave1) *pave1 = 0.0;
+if (pave2) *pave2 = 0.0;
+if (pnum1) *pnum1 = 0.0;
+if (pnum2) *pnum2 = 0.0;
+if (pnascore) *pnascore = NULL;
+if (!na)
+return ERROR_INT("na not defined", __func__, 1);
+n = numaGetCount(na);
+if (n <= 1)
+return ERROR_INT("n = 1 in histogram", __func__, 1);
+numaGetSum(na, &sum);
+if (sum <= 0.0)
+return ERROR_INT("sum <= 0.0", __func__, 1);
+norm = 4.0f / ((l_float32)(n - 1) * (n - 1));
+ave1prev = 0.0;
+numaGetHistogramStats(na, 0.0, 1.0, &ave2prev, NULL, NULL, NULL);
+num1prev = 0.0;
+num2prev = sum;
+maxindex = n / 2;  /* initialize with something */
+/* Split the histogram with [0 ... i] in the lower part
+* and [i+1 ... n-1] in upper part.  First, compute an otsu
+* score for each possible splitting.  */
+if ((nascore = numaCreate(n)) == NULL)
+return ERROR_INT("nascore not made", __func__, 1);
+naave1 = (pave1) ? numaCreate(n) : NULL;
+naave2 = (pave2) ? numaCreate(n) : NULL;
+nanum1 = (pnum1) ? numaCreate(n) : NULL;
+nanum2 = (pnum2) ? numaCreate(n) : NULL;
+maxscore = 0.0;
+for (i = 0; i < n; i++) {
+numaGetFValue(na, i, &val);
+num1 = num1prev + val;
+if (num1 == 0)
+ave1 = ave1prev;
+else
+ave1 = (num1prev * ave1prev + i * val) / num1;
+num2 = num2prev - val;
+if (num2 == 0)
+ave2 = ave2prev;
+else
+ave2 = (num2prev * ave2prev - i * val) / num2;
+fract1 = num1 / sum;
+score = norm * (fract1 * (1 - fract1)) * (ave2 - ave1) * (ave2 - ave1);
+numaAddNumber(nascore, score);
+if (pave1) numaAddNumber(naave1, ave1);
+if (pave2) numaAddNumber(naave2, ave2);
+if (pnum1) numaAddNumber(nanum1, num1);
+if (pnum2) numaAddNumber(nanum2, num2);
+if (score > maxscore) {
+maxscore = score;
+maxindex = i;
+}
+num1prev = num1;
+num2prev = num2;
+ave1prev = ave1;
+ave2prev = ave2;
+}
+/* Next, for all contiguous scores within a specified fraction
+* of the max, choose the split point as the value with the
+* minimum in the histogram. */
+minscore = (1.f - scorefract) * maxscore;
+for (i = maxindex - 1; i >= 0; i--) {
+numaGetFValue(nascore, i, &val);
+if (val < minscore)
+break;
+}
+minrange = i + 1;
+for (i = maxindex + 1; i < n; i++) {
+numaGetFValue(nascore, i, &val);
+if (val < minscore)
+break;
+}
+maxrange = i - 1;
+numaGetFValue(na, minrange, &minval);
+bestsplit = minrange;
+for (i = minrange + 1; i <= maxrange; i++) {
+numaGetFValue(na, i, &val);
+if (val < minval) {
+minval = val;
+bestsplit = i;
+}
+}
+/* Add one to the bestsplit value to get the threshold value,
+* because when we take a threshold, as in pixThresholdToBinary(),
+* we always choose the set with values below the threshold. */
+bestsplit = L_MIN(255, bestsplit + 1);
+if (psplitindex) *psplitindex = bestsplit;
+if (pave1) numaGetFValue(naave1, bestsplit, pave1);
+if (pave2) numaGetFValue(naave2, bestsplit, pave2);
+if (pnum1) numaGetFValue(nanum1, bestsplit, pnum1);
+if (pnum2) numaGetFValue(nanum2, bestsplit, pnum2);
+if (pnascore) {  /* debug mode */
+lept_stderr("minrange = %d, maxrange = %d\n", minrange, maxrange);
+lept_stderr("minval = %10.0f\n", minval);
+gplotSimple1(nascore, GPLOT_PNG, "/tmp/lept/nascore",
+"Score for split distribution");
+*pnascore = nascore;
+} else {
+numaDestroy(&nascore);
+}
+if (pave1) numaDestroy(&naave1);
+if (pave2) numaDestroy(&naave2);
+if (pnum1) numaDestroy(&nanum1);
+if (pnum2) numaDestroy(&nanum2);
+return 0;
+}
+/*----------------------------------------------------------------------*
+*                         Comparing histograms                         *
+*----------------------------------------------------------------------*/
+/*!
+* \brief   grayHistogramsToEMD()
+*
+* \param[in]    naa1, naa2    two numaa, each with one or more 256-element
+*                             histograms
+* \param[out]   pnad          nad of EM distances for each histogram
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*     (1) The two numaas must be the same size and have corresponding
+*         256-element histograms.  Pairs do not need to be normalized
+*         to the same sum.
+*     (2) This is typically used on two sets of histograms from
+*         corresponding tiles of two images.  The similarity of two
+*         images can be found with the scoring function used in
+*         pixCompareGrayByHisto():
+*             score S = 1.0 - k * D, where
+*                 k is a constant, say in the range 5-10
+*                 D = EMD
+*             for each tile; for multiple tiles, take the Min(S) over
+*             the set of tiles to be the final score.
+* </pre>
+*/
+l_ok
+grayHistogramsToEMD(NUMAA  *naa1,
+NUMAA  *naa2,
+NUMA  **pnad)
+{
+l_int32     i, n, nt;
+l_float32   dist;
+NUMA       *na1, *na2, *nad;
+if (!pnad)
+return ERROR_INT("&nad not defined", __func__, 1);
+*pnad = NULL;
+if (!naa1 || !naa2)
+return ERROR_INT("na1 and na2 not both defined", __func__, 1);
+n = numaaGetCount(naa1);
+if (n != numaaGetCount(naa2))
+return ERROR_INT("naa1 and naa2 numa counts differ", __func__, 1);
+nt = numaaGetNumberCount(naa1);
+if (nt != numaaGetNumberCount(naa2))
+return ERROR_INT("naa1 and naa2 number counts differ", __func__, 1);
+if (256 * n != nt)  /* good enough check */
+return ERROR_INT("na sizes must be 256", __func__, 1);
+nad = numaCreate(n);
+*pnad = nad;
+for (i = 0; i < n; i++) {
+na1 = numaaGetNuma(naa1, i, L_CLONE);
+na2 = numaaGetNuma(naa2, i, L_CLONE);
+numaEarthMoverDistance(na1, na2, &dist);
+numaAddNumber(nad, dist / 255.f);  /* normalize to [0.0 - 1.0] */
+numaDestroy(&na1);
+numaDestroy(&na2);
+}
+return 0;
+}
+/*!
+* \brief   numaEarthMoverDistance()
+*
+* \param[in]    na1, na2    two numas of the same size, typically histograms
+* \param[out]   pdist       earthmover distance
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*     (1) The two numas must have the same size.  They do not need to be
+*         normalized to the same sum before applying the function.
+*     (2) For a 1D discrete function, the implementation of the EMD
+*         is trivial.  Just keep filling or emptying buckets in one numa
+*         to match the amount in the other, moving sequentially along
+*         both arrays.
+*     (3) We divide the sum of the absolute value of everything moved
+*         (by 1 unit at a time) by the sum of the numa (amount of "earth")
+*         to get the average distance that the "earth" was moved.
+*         This is the value returned here.
+*     (4) The caller can do a further normalization, by the number of
+*         buckets (minus 1), to get the EM distance as a fraction of
+*         the maximum possible distance, which is n-1.  This fraction
+*         is 1.0 for the situation where all the 'earth' in the first
+*         array is at one end, and all in the second array is at the
+*         other end.
+* </pre>
+*/
+l_ok
+numaEarthMoverDistance(NUMA       *na1,
+NUMA       *na2,
+l_float32  *pdist)
+{
+l_int32     n, norm, i;
+l_float32   sum1, sum2, diff, total;
+l_float32  *array1, *array3;
+NUMA       *na3;
+if (!pdist)
+return ERROR_INT("&dist not defined", __func__, 1);
+*pdist = 0.0;
+if (!na1 || !na2)
+return ERROR_INT("na1 and na2 not both defined", __func__, 1);
+n = numaGetCount(na1);
+if (n != numaGetCount(na2))
+return ERROR_INT("na1 and na2 have different size", __func__, 1);
+/* Generate na3; normalize to na1 if necessary */
+numaGetSum(na1, &sum1);
+numaGetSum(na2, &sum2);
+norm = (L_ABS(sum1 - sum2) < 0.00001 * L_ABS(sum1)) ? 1 : 0;
+if (!norm)
+na3 = numaTransform(na2, 0, sum1 / sum2);
+else
+na3 = numaCopy(na2);
+array1 = numaGetFArray(na1, L_NOCOPY);
+array3 = numaGetFArray(na3, L_NOCOPY);
+/* Move earth in n3 from array elements, to match n1 */
+total = 0;
+for (i = 1; i < n; i++) {
+diff = array1[i - 1] - array3[i - 1];
+array3[i] -= diff;
+total += L_ABS(diff);
+}
+*pdist = total / sum1;
+numaDestroy(&na3);
+return 0;
+}
+/*!
+* \brief   grayInterHistogramStats()
+*
+* \param[in]    naa      numaa with two or more 256-element histograms
+* \param[in]    wc       half-width of the smoothing window
+* \param[out]   pnam     [optional] mean values
+* \param[out]   pnams    [optional] mean square values
+* \param[out]   pnav     [optional] variances
+* \param[out]   pnarv    [optional] rms deviations from the mean
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*     (1) The %naa has two or more 256-element numa histograms, which
+*         are to be compared value-wise at each of the 256 gray levels.
+*         The result are stats (mean, mean square, variance, root variance)
+*         aggregated across the set of histograms, and each is output
+*         as a 256 entry numa.  Think of these histograms as a matrix,
+*         where each histogram is one row of the array.  The stats are
+*         then aggregated column-wise, between the histograms.
+*     (2) These stats are:
+*            ~ average value: <v>  (nam)
+*            ~ average squared value: <v*v> (nams)
+*            ~ variance: <(v - <v>)*(v - <v>)> = <v*v> - <v>*<v>  (nav)
+*            ~ square-root of variance: (narv)
+*         where the brackets < .. > indicate that the average value is
+*         to be taken over each column of the array.
+*     (3) The input histograms are optionally smoothed before these
+*         statistical operations.
+*     (4) The input histograms are normalized to a sum of 10000.  By
+*         doing this, the resulting numbers are independent of the
+*         number of samples used in building the individual histograms.
+*     (5) A typical application is on a set of histograms from tiles
+*         of an image, to distinguish between text/tables and photo
+*         regions.  If the tiles are much larger than the text line
+*         spacing, text/table regions typically have smaller variance
+*         across tiles than photo regions.  For this application, it
+*         may be useful to ignore values near white, which are large for
+*         text and would magnify the variance due to variations in
+*         illumination.  However, because the variance of a drawing or
+*         a light photo can be similar to that of grayscale text, this
+*         function is only a discriminator between darker photos/drawings
+*         and light photos/text/line-graphics.
+* </pre>
+*/
+l_ok
+grayInterHistogramStats(NUMAA   *naa,
+l_int32  wc,
+NUMA   **pnam,
+NUMA   **pnams,
+NUMA   **pnav,
+NUMA   **pnarv)
+{
+l_int32      i, j, n, nn;
+l_float32  **arrays;
+l_float32    mean, var, rvar;
+NUMA        *na1, *na2, *na3, *na4;
+if (pnam) *pnam = NULL;
+if (pnams) *pnams = NULL;
+if (pnav) *pnav = NULL;
+if (pnarv) *pnarv = NULL;
+if (!pnam && !pnams && !pnav && !pnarv)
+return ERROR_INT("nothing requested", __func__, 1);
+if (!naa)
+return ERROR_INT("naa not defined", __func__, 1);
+n = numaaGetCount(naa);
+for (i = 0; i < n; i++) {
+nn = numaaGetNumaCount(naa, i);
+if (nn != 256) {
+L_ERROR("%d numbers in numa[%d]\n", __func__, nn, i);
+return 1;
+}
+}
+if (pnam) *pnam = numaCreate(256);
+if (pnams) *pnams = numaCreate(256);
+if (pnav) *pnav = numaCreate(256);
+if (pnarv) *pnarv = numaCreate(256);
+/* First, use mean smoothing, normalize each histogram,
+* and save all results in a 2D matrix. */
+arrays = (l_float32 **)LEPT_CALLOC(n, sizeof(l_float32 *));
+for (i = 0; i < n; i++) {
+na1 = numaaGetNuma(naa, i, L_CLONE);
+na2 = numaWindowedMean(na1, wc);
+na3 = numaNormalizeHistogram(na2, 10000.);
+arrays[i] = numaGetFArray(na3, L_COPY);
+numaDestroy(&na1);
+numaDestroy(&na2);
+numaDestroy(&na3);
+}
+/* Get stats between histograms */
+for (j = 0; j < 256; j++) {
+na4 = numaCreate(n);
+for (i = 0; i < n; i++) {
+numaAddNumber(na4, arrays[i][j]);
+}
+numaSimpleStats(na4, 0, -1, &mean, &var, &rvar);
+if (pnam) numaAddNumber(*pnam, mean);
+if (pnams) numaAddNumber(*pnams, mean * mean);
+if (pnav) numaAddNumber(*pnav, var);
+if (pnarv) numaAddNumber(*pnarv, rvar);
+numaDestroy(&na4);
+}
+for (i = 0; i < n; i++)
+LEPT_FREE(arrays[i]);
+LEPT_FREE(arrays);
+return 0;
+}
+/*----------------------------------------------------------------------*
+*                             Extrema finding                          *
+*----------------------------------------------------------------------*/
+/*!
+* \brief   numaFindPeaks()
+*
+* \param[in]    nas      source numa
+* \param[in]    nmax     max number of peaks to be found
+* \param[in]    fract1   min fraction of peak value
+* \param[in]    fract2   min slope
+* \return  peak na, or NULL on error.
+*
+* <pre>
+* Notes:
+*     (1) The returned na consists of sets of four numbers representing
+*         the peak, in the following order:
+*            left edge; peak center; right edge; normalized peak area
+* </pre>
+*/
+NUMA *
+numaFindPeaks(NUMA      *nas,
+l_int32    nmax,
+l_float32  fract1,
+l_float32  fract2)
+{
+l_int32    i, k, n, maxloc, lloc, rloc;
+l_float32  fmaxval, sum, total, newtotal, val, lastval;
+l_float32  peakfract;
+NUMA      *na, *napeak;
+if (!nas)
+return (NUMA *)ERROR_PTR("nas not defined", __func__, NULL);
+n = numaGetCount(nas);
+numaGetSum(nas, &total);
+/* We munge this copy */
+if ((na = numaCopy(nas)) == NULL)
+return (NUMA *)ERROR_PTR("na not made", __func__, NULL);
+if ((napeak = numaCreate(4 * nmax)) == NULL) {
+numaDestroy(&na);
+return (NUMA *)ERROR_PTR("napeak not made", __func__, NULL);
+}
+for (k = 0; k < nmax; k++) {
+numaGetSum(na, &newtotal);
+if (newtotal == 0.0)   /* sanity check */
+break;
+numaGetMax(na, &fmaxval, &maxloc);
+sum = fmaxval;
+lastval = fmaxval;
+lloc = 0;
+for (i = maxloc - 1; i >= 0; --i) {
+numaGetFValue(na, i, &val);
+if (val == 0.0) {
+lloc = i + 1;
+break;
+}
+if (val > fract1 * fmaxval) {
+sum += val;
+lastval = val;
+continue;
+}
+if (lastval - val > fract2 * lastval) {
+sum += val;
+lastval = val;
+continue;
+}
+lloc = i;
+break;
+}
+lastval = fmaxval;
+rloc = n - 1;
+for (i = maxloc + 1; i < n; ++i) {
+numaGetFValue(na, i, &val);
+if (val == 0.0) {
+rloc = i - 1;
+break;
+}
+if (val > fract1 * fmaxval) {
+sum += val;
+lastval = val;
+continue;
+}
+if (lastval - val > fract2 * lastval) {
+sum += val;
+lastval = val;
+continue;
+}
+rloc = i;
+break;
+}
+peakfract = sum / total;
+numaAddNumber(napeak, lloc);
+numaAddNumber(napeak, maxloc);
+numaAddNumber(napeak, rloc);
+numaAddNumber(napeak, peakfract);
+for (i = lloc; i <= rloc; i++)
+numaSetValue(na, i, 0.0);
+}
+numaDestroy(&na);
+return napeak;
+}
+/*!
+* \brief   numaFindExtrema()
+*
+* \param[in]    nas     input values
+* \param[in]    delta   relative amount to resolve peaks and valleys
+* \param[out]   pnav    [optional] values of extrema
+* \return  nad (locations of extrema, or NULL on error
+*
+* <pre>
+* Notes:
+*      (1) This returns a sequence of extrema (peaks and valleys).
+*      (2) The algorithm is analogous to that for determining
+*          mountain peaks.  Suppose we have a local peak, with
+*          bumps on the side.  Under what conditions can we consider
+*          those 'bumps' to be actual peaks?  The answer: if the
+*          bump is separated from the peak by a saddle that is at
+*          least 500 feet below the bump.
+*      (3) Operationally, suppose we are trying to identify a peak.
+*          We have a previous valley, and also the largest value that
+*          we have seen since that valley.  We can identify this as
+*          a peak if we find a value that is delta BELOW it.  When
+*          we find such a value, label the peak, use the current
+*          value to label the starting point for the search for
+*          a valley, and do the same operation in reverse.  Namely,
+*          keep track of the lowest point seen, and look for a value
+*          that is delta ABOVE it.  Once found, the lowest point is
+*          labeled the valley, and continue, looking for the next peak.
+* </pre>
+*/
+NUMA *
+numaFindExtrema(NUMA      *nas,
+l_float32  delta,
+NUMA     **pnav)
+{
+l_int32    i, n, found, loc, direction;
+l_float32  startval, val, maxval, minval;
+NUMA      *nav, *nad;
+if (pnav) *pnav = NULL;
+if (!nas)
+return (NUMA *)ERROR_PTR("nas not defined", __func__, NULL);
+if (delta < 0.0)
+return (NUMA *)ERROR_PTR("delta < 0", __func__, NULL);
+n = numaGetCount(nas);
+nad = numaCreate(0);
+nav =  NULL;
+if (pnav) {
+nav = numaCreate(0);
+*pnav = nav;
+}
+/* We don't know if we'll find a peak or valley first,
+* but use the first element of nas as the reference point.
+* Break when we deviate by 'delta' from the first point. */
+numaGetFValue(nas, 0, &startval);
+found = FALSE;
+for (i = 1; i < n; i++) {
+numaGetFValue(nas, i, &val);
+if (L_ABS(val - startval) >= delta) {
+found = TRUE;
+break;
+}
+}
+if (!found)
+return nad;  /* it's empty */
+/* Are we looking for a peak or a valley? */
+if (val > startval) {  /* peak */
+direction = 1;
+maxval = val;
+} else {
+direction = -1;
+minval = val;
+}
+loc = i;
+/* Sweep through the rest of the array, recording alternating
+* peak/valley extrema. */
+for (i = i + 1; i < n; i++) {
+numaGetFValue(nas, i, &val);
+if (direction == 1 && val > maxval ) {  /* new local max */
+maxval = val;
+loc = i;
+} else if (direction == -1 && val < minval ) {  /* new local min */
+minval = val;
+loc = i;
+} else if (direction == 1 && (maxval - val >= delta)) {
+numaAddNumber(nad, loc);  /* save the current max location */
+if (nav) numaAddNumber(nav, maxval);
+direction = -1;  /* reverse: start looking for a min */
+minval = val;
+loc = i;  /* current min location */
+} else if (direction == -1 && (val - minval >= delta)) {
+numaAddNumber(nad, loc);  /* save the current min location */
+if (nav) numaAddNumber(nav, minval);
+direction = 1;  /* reverse: start looking for a max */
+maxval = val;
+loc = i;  /* current max location */
+}
+}
+/* Save the final extremum */
+/*    numaAddNumber(nad, loc); */
+return nad;
+}
+/*!
+* \brief   numaFindLocForThreshold()
+*
+* \param[in]    nas       input histogram
+* \param[in]    skip      look-ahead distance to avoid false mininma;
+*                         use 0 for default
+* \param[out]   pthresh   threshold value
+* \param[out]   pfract    [optional] fraction below or at threshold
+* \return  0 if OK, 1 on error or if no threshold can be found
+*
+* <pre>
+* Notes:
+*      (1) This finds a good place to set a threshold for a histogram
+*          of values that has two peaks.  The peaks can differ greatly
+*          in area underneath them.  The number of buckets in the
+*          histogram is expected to be 256 (e.g, from an 8 bpp gray image).
+*      (2) The input histogram should have been smoothed with a window
+*          to avoid false peak and valley detection due to noise.  For
+*          example, see pixThresholdByHisto().
+*      (3) A skip value can be input to determine the look-ahead distance
+*          to ignore a false peak on the rise or descent from the first peak.
+*          Input 0 to use the default value (it assumes a histo size of 256).
+*      (4) Optionally, the fractional area under the first peak can
+*          be returned.
+* </pre>
+*/
+l_ok
+numaFindLocForThreshold(NUMA       *na,
+l_int32     skip,
+l_int32    *pthresh,
+l_float32  *pfract)
+{
+l_int32     i, n, start, index, minloc, found;
+l_float32   val, pval, jval, minval, maxval, sum, partsum;
+l_float32  *fa;
+if (pfract) *pfract = 0.0;
+if (!pthresh)
+return ERROR_INT("&thresh not defined", __func__, 1);
+*pthresh = 0;
+if (!na)
+return ERROR_INT("na not defined", __func__, 1);
+if (skip <= 0) skip = 20;
+/* Test for constant value */
+numaGetMin(na, &minval, NULL);
+numaGetMax(na, &maxval, NULL);
+if (minval == maxval)
+return ERROR_INT("all array values are the same", __func__, 1);
+/* Look for the top of the first peak */
+n = numaGetCount(na);
+if (n < 256)
+L_WARNING("array size %d < 256\n", __func__, n);
+fa = numaGetFArray(na, L_NOCOPY);
+pval = fa[0];
+for (i = 1; i < n; i++) {
+val = fa[i];
+index = L_MIN(i + skip, n - 1);
+jval = fa[index];
+if (val < pval && jval < pval)  /* near the top if not there */
+break;
+pval = val;
+}
+if (i > n - 5)  /* just an increasing function */
+return ERROR_INT("top of first peak not found", __func__, 1);
+/* Look for the low point in the valley */
+found = FALSE;
+start = i;
+pval = fa[start];
+for (i = start + 1; i < n; i++) {
+val = fa[i];
+if (val <= pval) {  /* appears to be going down */
+pval = val;
+} else {  /* appears to be going up */
+index = L_MIN(i + skip, n - 1);
+jval = fa[index];  /* junp ahead by 'skip' */
+if (val > jval) {  /* still going down; jump ahead */
+pval = jval;
+i = index;
+} else {  /* really going up; passed the min */
+found = TRUE;
+break;
+}
+}
+}
+if (!found)
+return ERROR_INT("no minimum found", __func__, 1);
+/* Find the location of the minimum in the interval */
+minloc = index;  /* likely passed the min; look backward */
+minval = fa[index];
+for (i = index - 1; i > index - skip; i--) {
+if (fa[i] < minval) {
+minval = fa[i];
+minloc = i;
+}
+}
+/* Is the minimum very near the end of the array? */
+if (minloc > n - 10)
+return ERROR_INT("minimum at end of array; invalid", __func__, 1);
+*pthresh = minloc;
+/* Find the fraction under the first peak */
+if (pfract) {
+numaGetSumOnInterval(na, 0, minloc, &partsum);
+numaGetSum(na, &sum);
+if (sum > 0.0)
+*pfract = partsum / sum;
+}
+return 0;
+}
+/*!
+* \brief   numaCountReversals()
+*
+* \param[in]    nas          input values
+* \param[in]    minreversal  relative amount to resolve peaks and valleys
+* \param[out]   pnr          [optional] number of reversals
+* \param[out]   prd          [optional] reversal density: reversals/length
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*      (1) The input numa is can be generated from pixExtractAlongLine().
+*          If so, the x parameters can be used to find the reversal
+*          frequency along a line.
+*      (2) If the input numa was generated from a 1 bpp pix, the
+*          values will be 0 and 1.  Use %minreversal == 1 to get
+*          the number of pixel flips.  If the only values are 0 and 1,
+*          but %minreversal > 1, set the reversal count to 0 and
+*          issue a warning.
+* </pre>
+*/
+l_ok
+numaCountReversals(NUMA       *nas,
+l_float32   minreversal,
+l_int32    *pnr,
+l_float32  *prd)
+{
+l_int32    i, n, nr, ival, binvals;
+l_int32   *ia;
+l_float32  fval, delx, len;
+NUMA      *nat;
+if (pnr) *pnr = 0;
+if (prd) *prd = 0.0;
+if (!pnr && !prd)
+return ERROR_INT("neither &nr nor &rd are defined", __func__, 1);
+if (!nas)
+return ERROR_INT("nas not defined", __func__, 1);
+if ((n = numaGetCount(nas)) == 0) {
+L_INFO("nas is empty\n", __func__);
+return 0;
+}
+if (minreversal < 0.0)
+return ERROR_INT("minreversal < 0", __func__, 1);
+/* Decide if the only values are 0 and 1 */
+binvals = TRUE;
+for (i = 0; i < n; i++) {
+numaGetFValue(nas, i, &fval);
+if (fval != 0.0 && fval != 1.0) {
+binvals = FALSE;
+break;
+}
+}
+nr = 0;
+if (binvals) {
+if (minreversal > 1.0) {
+L_WARNING("binary values but minreversal > 1\n", __func__);
+} else {
+ia = numaGetIArray(nas);
+ival = ia[0];
+for (i = 1; i < n; i++) {
+if (ia[i] != ival) {
+nr++;
+ival = ia[i];
+}
+}
+LEPT_FREE(ia);
+}
+} else {
+nat = numaFindExtrema(nas, minreversal, NULL);
+nr = numaGetCount(nat);
+numaDestroy(&nat);
+}
+if (pnr) *pnr = nr;
+if (prd) {
+numaGetParameters(nas, NULL, &delx);
+len = delx * n;
+*prd = (l_float32)nr / len;
+}
+return 0;
+}
+/*----------------------------------------------------------------------*
+*                Threshold crossings and frequency analysis            *
+*----------------------------------------------------------------------*/
+/*!
+* \brief   numaSelectCrossingThreshold()
+*
+* \param[in]    nax          [optional] numa of abscissa values; can be NULL
+* \param[in]    nay          signal
+* \param[in]    estthresh    estimated pixel threshold for crossing:
+*                            e.g., for images, white <--> black; typ. ~120
+* \param[out]   pbestthresh  robust estimate of threshold to use
+* \return  0 if OK, 1 on error or warning
+*
+* <pre>
+* Notes:
+*     (1) When a valid threshold is used, the number of crossings is
+*         a maximum, because none are missed.  If no threshold intersects
+*         all the crossings, the crossings must be determined with
+*         numaCrossingsByPeaks().
+*     (2) %estthresh is an input estimate of the threshold that should
+*         be used.  We compute the crossings with 41 thresholds
+*         (20 below and 20 above).  There is a range in which the
+*         number of crossings is a maximum.  Return a threshold
+*         in the center of this stable plateau of crossings.
+*         This can then be used with numaCrossingsByThreshold()
+*         to get a good estimate of crossing locations.
+*     (3) If the count of nay is less than 2, a warning is issued.
+* </pre>
+*/
+l_ok
+numaSelectCrossingThreshold(NUMA       *nax,
+NUMA       *nay,
+l_float32   estthresh,
+l_float32  *pbestthresh)
+{
+l_int32    i, inrun, istart, iend, maxstart, maxend, runlen, maxrunlen;
+l_int32    val, maxval, nmax, count;
+l_float32  thresh, fmaxval, fmodeval;
+NUMA      *nat, *nac;
+if (!pbestthresh)
+return ERROR_INT("&bestthresh not defined", __func__, 1);
+*pbestthresh = 0.0;
+if (!nay)
+return ERROR_INT("nay not defined", __func__, 1);
+if (numaGetCount(nay) < 2) {
+L_WARNING("nay count < 2; no threshold crossing\n", __func__);
+return 1;
+}
+/* Compute the number of crossings for different thresholds */
+nat = numaCreate(41);
+for (i = 0; i < 41; i++) {
+thresh = estthresh - 80.0f + 4.0f * i;
+nac = numaCrossingsByThreshold(nax, nay, thresh);
+numaAddNumber(nat, numaGetCount(nac));
+numaDestroy(&nac);
+}
+/* Find the center of the plateau of max crossings, which
+* extends from thresh[istart] to thresh[iend]. */
+numaGetMax(nat, &fmaxval, NULL);
+maxval = (l_int32)fmaxval;
+nmax = 0;
+for (i = 0; i < 41; i++) {
+numaGetIValue(nat, i, &val);
+if (val == maxval)
+nmax++;
+}
+if (nmax < 3) {  /* likely accidental max; try the mode */
+numaGetMode(nat, &fmodeval, &count);
+if (count > nmax && fmodeval > 0.5 * fmaxval)
+maxval = (l_int32)fmodeval;  /* use the mode */
+}
+inrun = FALSE;
+iend = 40;
+maxrunlen = 0, maxstart = 0, maxend = 0;
+for (i = 0; i < 41; i++) {
+numaGetIValue(nat, i, &val);
+if (val == maxval) {
+if (!inrun) {
+istart = i;
+inrun = TRUE;
+}
+continue;
+}
+if (inrun && (val != maxval)) {
+iend = i - 1;
+runlen = iend - istart + 1;
+inrun = FALSE;
+if (runlen > maxrunlen) {
+maxstart = istart;
+maxend = iend;
+maxrunlen = runlen;
+}
+}
+}
+if (inrun) {
+runlen = i - istart;
+if (runlen > maxrunlen) {
+maxstart = istart;
+maxend = i - 1;
+maxrunlen = runlen;
+}
+}
+*pbestthresh = estthresh - 80.0f + 2.0f * (l_float32)(maxstart + maxend);
+#if  DEBUG_CROSSINGS
+lept_stderr("\nCrossings attain a maximum at %d thresholds, between:\n"
+"  thresh[%d] = %5.1f and thresh[%d] = %5.1f\n",
+nmax, maxstart, estthresh - 80.0 + 4.0 * maxstart,
+maxend, estthresh - 80.0 + 4.0 * maxend);
+lept_stderr("The best choice: %5.1f\n", *pbestthresh);
+lept_stderr("Number of crossings at the 41 thresholds:");
+numaWriteStderr(nat);
+#endif  /* DEBUG_CROSSINGS */
+numaDestroy(&nat);
+return 0;
+}
+/*!
+* \brief   numaCrossingsByThreshold()
+*
+* \param[in]    nax     [optional] numa of abscissa values; can be NULL
+* \param[in]    nay     numa of ordinate values, corresponding to nax
+* \param[in]    thresh  threshold value for nay
+* \return  nad abscissa pts at threshold, or NULL on error
+*
+* <pre>
+* Notes:
+*      (1) If nax == NULL, we use startx and delx from nay to compute
+*          the crossing values in nad.
+* </pre>
+*/
+NUMA *
+numaCrossingsByThreshold(NUMA      *nax,
+NUMA      *nay,
+l_float32  thresh)
+{
+l_int32    i, n;
+l_float32  startx, delx;
+l_float32  xval1, xval2, yval1, yval2, delta1, delta2, crossval, fract;
+NUMA      *nad;
+if (!nay)
+return (NUMA *)ERROR_PTR("nay not defined", __func__, NULL);
+n = numaGetCount(nay);
+if (nax && (numaGetCount(nax) != n))
+return (NUMA *)ERROR_PTR("nax and nay sizes differ", __func__, NULL);
+nad = numaCreate(0);
+if (n < 2) return nad;
+numaGetFValue(nay, 0, &yval1);
+numaGetParameters(nay, &startx, &delx);
+if (nax)
+numaGetFValue(nax, 0, &xval1);
+else
+xval1 = startx;
+for (i = 1; i < n; i++) {
+numaGetFValue(nay, i, &yval2);
+if (nax)
+numaGetFValue(nax, i, &xval2);
+else
+xval2 = startx + i * delx;
+delta1 = yval1 - thresh;
+delta2 = yval2 - thresh;
+if (delta1 == 0.0) {
+numaAddNumber(nad, xval1);
+} else if (delta2 == 0.0) {
+numaAddNumber(nad, xval2);
+} else if (delta1 * delta2 < 0.0) {  /* crossing */
+fract = L_ABS(delta1) / L_ABS(yval1 - yval2);
+crossval = xval1 + fract * (xval2 - xval1);
+numaAddNumber(nad, crossval);
+}
+xval1 = xval2;
+yval1 = yval2;
+}
+return nad;
+}
+/*!
+* \brief   numaCrossingsByPeaks()
+*
+* \param[in]    nax     [optional] numa of abscissa values
+* \param[in]    nay     numa of ordinate values, corresponding to nax
+* \param[in]    delta   parameter used to identify when a new peak can be found
+* \return  nad abscissa pts at threshold, or NULL on error
+*
+* <pre>
+* Notes:
+*      (1) If nax == NULL, we use startx and delx from nay to compute
+*          the crossing values in nad.
+* </pre>
+*/
+NUMA *
+numaCrossingsByPeaks(NUMA      *nax,
+NUMA      *nay,
+l_float32  delta)
+{
+l_int32    i, j, n, np, previndex, curindex;
+l_float32  startx, delx;
+l_float32  xval1, xval2, yval1, yval2, delta1, delta2;
+l_float32  prevval, curval, thresh, crossval, fract;
+NUMA      *nap, *nad;
+if (!nay)
+return (NUMA *)ERROR_PTR("nay not defined", __func__, NULL);
+n = numaGetCount(nay);
+if (nax && (numaGetCount(nax) != n))
+return (NUMA *)ERROR_PTR("nax and nay sizes differ", __func__, NULL);
+/* Find the extrema.  Also add last point in nay to get
+* the last transition (from the last peak to the end).
+* The number of crossings is 1 more than the number of extrema. */
+nap = numaFindExtrema(nay, delta, NULL);
+numaAddNumber(nap, n - 1);
+np = numaGetCount(nap);
+L_INFO("Number of crossings: %d\n", __func__, np);
+/* Do all computation in index units of nax or the delx of nay */
+nad = numaCreate(np);  /* output crossing locations, in nax units */
+previndex = 0;  /* prime the search with 1st point */
+numaGetFValue(nay, 0, &prevval);  /* prime the search with 1st point */
+numaGetParameters(nay, &startx, &delx);
+for (i = 0; i < np; i++) {
+numaGetIValue(nap, i, &curindex);
+numaGetFValue(nay, curindex, &curval);
+thresh = (prevval + curval) / 2.0f;
+if (nax)
+numaGetFValue(nax, previndex, &xval1);
+else
+xval1 = startx + previndex * delx;
+numaGetFValue(nay, previndex, &yval1);
+for (j = previndex + 1; j <= curindex; j++) {
+if (nax)
+numaGetFValue(nax, j, &xval2);
+else
+xval2 = startx + j * delx;
+numaGetFValue(nay, j, &yval2);
+delta1 = yval1 - thresh;
+delta2 = yval2 - thresh;
+if (delta1 == 0.0) {
+numaAddNumber(nad, xval1);
+break;
+} else if (delta2 == 0.0) {
+numaAddNumber(nad, xval2);
+break;
+} else if (delta1 * delta2 < 0.0) {  /* crossing */
+fract = L_ABS(delta1) / L_ABS(yval1 - yval2);
+crossval = xval1 + fract * (xval2 - xval1);
+numaAddNumber(nad, crossval);
+break;
+}
+xval1 = xval2;
+yval1 = yval2;
+}
+previndex = curindex;
+prevval = curval;
+}
+numaDestroy(&nap);
+return nad;
+}
+/*!
+* \brief   numaEvalBestHaarParameters()
+*
+* \param[in]    nas         numa of non-negative signal values
+* \param[in]    relweight   relative weight of (-1 comb) / (+1 comb)
+*                           contributions to the 'convolution'.  In effect,
+*                           the convolution kernel is a comb consisting of
+*                           alternating +1 and -weight.
+* \param[in]    nwidth      number of widths to consider
+* \param[in]    nshift      number of shifts to consider for each width
+* \param[in]    minwidth    smallest width to consider
+* \param[in]    maxwidth    largest width to consider
+* \param[out]   pbestwidth  width giving largest score
+* \param[out]   pbestshift  shift giving largest score
+* \param[out]   pbestscore  [optional] convolution with "Haar"-like comb
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*      (1) This does a linear sweep of widths, evaluating at %nshift
+*          shifts for each width, computing the score from a convolution
+*          with a long comb, and finding the (width, shift) pair that
+*          gives the maximum score.  The best width is the "half-wavelength"
+*          of the signal.
+*      (2) The convolving function is a comb of alternating values
+*          +1 and -1 * relweight, separated by the width and phased by
+*          the shift.  This is similar to a Haar transform, except
+*          there the convolution is performed with a square wave.
+*      (3) The function is useful for finding the line spacing
+*          and strength of line signal from pixel sum projections.
+*      (4) The score is normalized to the size of nas divided by
+*          the number of half-widths.  For image applications, the input is
+*          typically an array of pixel projections, so one should
+*          normalize by dividing the score by the image width in the
+*          pixel projection direction.
+* </pre>
+*/
+l_ok
+numaEvalBestHaarParameters(NUMA       *nas,
+l_float32   relweight,
+l_int32     nwidth,
+l_int32     nshift,
+l_float32   minwidth,
+l_float32   maxwidth,
+l_float32  *pbestwidth,
+l_float32  *pbestshift,
+l_float32  *pbestscore)
+{
+l_int32    i, j;
+l_float32  delwidth, delshift, width, shift, score;
+l_float32  bestwidth, bestshift, bestscore;
+if (pbestscore) *pbestscore = 0.0;
+if (pbestwidth) *pbestwidth = 0.0;
+if (pbestshift) *pbestshift = 0.0;
+if (!pbestwidth || !pbestshift)
+return ERROR_INT("&bestwidth and &bestshift not defined", __func__, 1);
+if (!nas)
+return ERROR_INT("nas not defined", __func__, 1);
+bestscore = bestwidth = bestshift = 0.0;
+delwidth = (maxwidth - minwidth) / (nwidth - 1.0f);
+for (i = 0; i < nwidth; i++) {
+width = minwidth + delwidth * i;
+delshift = width / (l_float32)(nshift);
+for (j = 0; j < nshift; j++) {
+shift = j * delshift;
+numaEvalHaarSum(nas, width, shift, relweight, &score);
+if (score > bestscore) {
+bestscore = score;
+bestwidth = width;
+bestshift = shift;
+#if  DEBUG_FREQUENCY
+lept_stderr("width = %7.3f, shift = %7.3f, score = %7.3f\n",
+width, shift, score);
+#endif  /* DEBUG_FREQUENCY */
+}
+}
+}
+*pbestwidth = bestwidth;
+*pbestshift = bestshift;
+if (pbestscore)
+*pbestscore = bestscore;
+return 0;
+}
+/*!
+* \brief   numaEvalHaarSum()
+*
+* \param[in]    nas         numa of non-negative signal values
+* \param[in]    width       distance between +1 and -1 in convolution comb
+* \param[in]    shift       phase of the comb: location of first +1
+* \param[in]    relweight   relative weight of (-1 comb) / (+1 comb)
+*                           contributions to the 'convolution'.  In effect,
+*                           the convolution kernel is a comb consisting of
+*                           alternating +1 and -weight.
+* \param[out]   pscore      convolution with "Haar"-like comb
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*      (1) This does a convolution with a comb of alternating values
+*          +1 and -relweight, separated by the width and phased by the shift.
+*          This is similar to a Haar transform, except that for Haar,
+*            (1) the convolution kernel is symmetric about 0, so the
+*                relweight is 1.0, and
+*            (2) the convolution is performed with a square wave.
+*      (2) The score is normalized to the size of nas divided by
+*          twice the "width".  For image applications, the input is
+*          typically an array of pixel projections, so one should
+*          normalize by dividing the score by the image width in the
+*          pixel projection direction.
+*      (3) To get a Haar-like result, use relweight = 1.0.  For detecting
+*          signals where you expect every other sample to be close to
+*          zero, as with barcodes or filtered text lines, you can
+*          use relweight > 1.0.
+* </pre>
+*/
+l_ok
+numaEvalHaarSum(NUMA       *nas,
+l_float32   width,
+l_float32   shift,
+l_float32   relweight,
+l_float32  *pscore)
+{
+l_int32    i, n, nsamp, index;
+l_float32  score, weight, val;
+if (!pscore)
+return ERROR_INT("&score not defined", __func__, 1);
+*pscore = 0.0;
+if (!nas)
+return ERROR_INT("nas not defined", __func__, 1);
+if ((n = numaGetCount(nas)) < 2 * width)
+return ERROR_INT("nas size too small", __func__, 1);
+score = 0.0;
+nsamp = (l_int32)((n - shift) / width);
+for (i = 0; i < nsamp; i++) {
+index = (l_int32)(shift + i * width);
+weight = (i % 2) ? 1.0f : -1.0f * relweight;
+numaGetFValue(nas, index, &val);
+score += weight * val;
+}
+*pscore = 2.0f * width * score / (l_float32)n;
+return 0;
+}
+/*----------------------------------------------------------------------*
+*            Generating numbers in a range under constraints           *
+*----------------------------------------------------------------------*/
+/*!
+* \brief   genConstrainedNumaInRange()
+*
+* \param[in]    first     first number to choose; >= 0
+* \param[in]    last      biggest possible number to reach; >= first
+* \param[in]    nmax      maximum number of numbers to select; > 0
+* \param[in]    use_pairs 1 = select pairs of adjacent numbers;
+*                         0 = select individual numbers
+* \return  0 if OK, 1 on error
+*
+* <pre>
+* Notes:
+*     (1) Selection is made uniformly in the range.  This can be used
+*         to select pages distributed as uniformly as possible
+*         through a book, where you are constrained to:
+*          ~ choose between [first, ... biggest],
+*          ~ choose no more than nmax numbers, and
+*         and you have the option of requiring pairs of adjacent numbers.
+* </pre>
+*/
+NUMA *
+genConstrainedNumaInRange(l_int32  first,
+l_int32  last,
+l_int32  nmax,
+l_int32  use_pairs)
+{
+l_int32    i, nsets, val;
+l_float32  delta;
+NUMA      *na;
+first = L_MAX(0, first);
+if (last < first)
+return (NUMA *)ERROR_PTR("last < first!", __func__, NULL);
+if (nmax < 1)
+return (NUMA *)ERROR_PTR("nmax < 1!", __func__, NULL);
+nsets = L_MIN(nmax, last - first + 1);
+if (use_pairs == 1)
+nsets = nsets / 2;
+if (nsets == 0)
+return (NUMA *)ERROR_PTR("nsets == 0", __func__, NULL);
+/* Select delta so that selection covers the full range if possible */
+if (nsets == 1) {
+delta = 0.0;
+} else {
+if (use_pairs == 0)
+delta = (l_float32)(last - first) / (nsets - 1);
+else
+delta = (l_float32)(last - first - 1) / (nsets - 1);
+}
+na = numaCreate(nsets);
+for (i = 0; i < nsets; i++) {
+val = (l_int32)(first + i * delta + 0.5);
+numaAddNumber(na, val);
+if (use_pairs == 1)
+numaAddNumber(na, val + 1);
+}
+return na;
+}

Mercurial > hgrepos > Python2 > PyMuPDF

comparison mupdf-source/thirdparty/leptonica/src/numafunc2.c @ 2:b50eed0cc0ef upstream