NAME

Image::Leptonica::Func::numafunc2

VERSION

version 0.04

numafunc2.c

numafunc2.c

   Morphological (min/max) operations
       NUMA        *numaErode()
       NUMA        *numaDilate()
       NUMA        *numaOpen()
       NUMA        *numaClose()

   Other transforms
       NUMA        *numaTransform()
       l_int32      numaWindowedStats()
       NUMA        *numaWindowedMean()
       NUMA        *numaWindowedMeanSquare()
       l_int32      numaWindowedVariance()
       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      numaDiscretizeRankAndIntensity()
       l_int32      numaGetRankBinValues()

   Splitting a distribution
       l_int32      numaSplitDistribution()

   Comparing two histograms
       l_int32      numaEarthMoverDistance()

   Extrema finding
       NUMA        *numaFindPeaks()
       NUMA        *numaFindExtrema()
       l_int32     *numaCountReversals()

   Threshold crossings and frequency analysis
       l_int32      numaSelectCrossingThreshold()
       NUMA        *numaCrossingsByThreshold()
       NUMA        *numaCrossingsByPeaks()
       NUMA        *numaEvalBestHaarParameters()
       l_int32      numaEvalHaarSum()

 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.

 (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.

FUNCTIONS

numaClose

NUMA * numaClose ( NUMA *nas, l_int32 size )

numaClose()

    Input:  nas
            size (of sel; greater than 0, odd; origin implicitly in center)
    Return: nad (opened), or null on error

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.

numaConvertToInt

NUMA * numaConvertToInt ( NUMA *nas )

numaConvertToInt()

    Input:  na
    Return: na with all values rounded to nearest integer, or
            null on error

numaCountReversals

l_int32 numaCountReversals ( NUMA *nas, l_float32 minreversal, l_int32 *pnr, l_float32 *pnrpl )

numaCountReversals()

    Input:  nas (input values)
            minreversal (relative amount to resolve peaks and valleys)
            &nr (<optional return> number of reversals
            &nrpl (<optional return> reversal density: reversals/length)
    Return: 0 if OK, 1 on error

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.

numaCrossingsByPeaks

NUMA * numaCrossingsByPeaks ( NUMA *nax, NUMA *nay, l_float32 delta )

numaCrossingsByPeaks()

    Input:  nax (<optional> numa of abscissa values)
            nay (numa of ordinate values, corresponding to nax)
            delta (parameter used to identify when a new peak can be found)
    Return: nad (abscissa pts at threshold), or null on error

Notes:
    (1) If nax == NULL, we use startx and delx from nay to compute
        the crossing values in nad.

numaCrossingsByThreshold

NUMA * numaCrossingsByThreshold ( NUMA *nax, NUMA *nay, l_float32 thresh )

numaCrossingsByThreshold()

    Input:  nax (<optional> numa of abscissa values; can be NULL)
            nay (numa of ordinate values, corresponding to nax)
            thresh (threshold value for nay)
    Return: nad (abscissa pts at threshold), or null on error

Notes:
    (1) If nax == NULL, we use startx and delx from nay to compute
        the crossing values in nad.

numaDilate

NUMA * numaDilate ( NUMA *nas, l_int32 size )

numaDilate()

    Input:  nas
            size (of sel; greater than 0, odd; origin implicitly in center)
    Return: nad (dilated), or null on error

Notes:
    (1) The structuring element (sel) is linear, all "hits"
    (2) If size == 1, this returns a copy

numaDiscretizeRankAndIntensity

l_int32 numaDiscretizeRankAndIntensity ( NUMA *na, l_int32 nbins, NUMA **pnarbin, NUMA **pnam, NUMA **pnar, NUMA **pnabb )

numaDiscretizeRankAndIntensity()

    Input:  na (normalized histogram of probability density vs intensity)
            nbins (number of bins at which the rank is divided)
            &pnarbin (<optional return> rank bin value vs intensity)
            &pnam (<optional return> median intensity in a bin vs
                   rank bin value, with @nbins of discretized rank values)
            &pnar (<optional return> rank vs intensity; this is
                   a cumulative norm histogram)
            &pnabb (<optional return> intensity at the right bin boundary
                    vs rank bin)
    Return: 0 if OK, 1 on error

Notes:
    (1) We are inverting the rank(intensity) function to get
        the intensity(rank) function at @nbins equally spaced
        values of rank between 0.0 and 1.0.  We save integer values
        for the intensity.
    (2) We are using the word "intensity" to describe the type of
        array values, but any array of non-negative numbers will work.
    (3) The output arrays give the following mappings, where the
        input is a normalized histogram of array values:
           array values     -->  rank bin number  (narbin)
           rank bin number  -->  median array value in bin (nam)
           array values     -->  cumulative norm = rank  (nar)
           rank bin number  -->  array value at right bin edge (nabb)

numaEarthMoverDistance

l_int32 numaEarthMoverDistance ( NUMA *na1, NUMA *na2, l_float32 *pdist )

 numaEarthMoverDistance()

     Input:  na1, na2 (two numas of the same size, typically histograms)
             &dist (<return> EM distance)
     Return: 0 if OK, 1 on error

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.
        Further normalization, by the number of buckets (minus 1),
        gives the 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.

numaErode

NUMA * numaErode ( NUMA *nas, l_int32 size )

numaErode()

    Input:  nas
            size (of sel; greater than 0, odd; origin implicitly in center)
    Return: nad (eroded), or null on error

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.

numaEvalBestHaarParameters

l_int32 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 )

numaEvalBestHaarParameters()

    Input:  nas (numa of non-negative signal values)
            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.)
            nwidth (number of widths to consider)
            nshift (number of shifts to consider for each width)
            minwidth (smallest width to consider)
            maxwidth (largest width to consider)
            &bestwidth (<return> width giving largest score)
            &bestshift (<return> shift giving largest score)
            &bestscore (<optional return> convolution with
                        "Haar"-like comb)
    Return: 0 if OK, 1 on error

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.

numaEvalHaarSum

l_int32 numaEvalHaarSum ( NUMA *nas, l_float32 width, l_float32 shift, l_float32 relweight, l_float32 *pscore )

numaEvalHaarSum()

    Input:  nas (numa of non-negative signal values)
            width (distance between +1 and -1 in convolution comb)
            shift (phase of the comb: location of first +1)
            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.)
            &score (<return> convolution with "Haar"-like comb)
    Return: 0 if OK, 1 on error

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.

numaFindExtrema

NUMA * numaFindExtrema ( NUMA *nas, l_float32 delta )

numaFindExtrema()

    Input:  nas (input values)
            delta (relative amount to resolve peaks and valleys)
    Return: nad (locations of extrema), or null on error

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 looking for a peak.
        We are keeping the largest value we've seen since the
        last valley, and are looking for a value that is delta
        BELOW our current peak.  When we find such a value,
        we label the peak, use the current value to label the
        valley, and then do the same operation in reverse (looking
        for a valley).

numaFindPeaks

NUMA * numaFindPeaks ( NUMA *nas, l_int32 nmax, l_float32 fract1, l_float32 fract2 )

 numaFindPeaks()

     Input:  source na
             max number of peaks to be found
             fract1  (min fraction of peak value)
             fract2  (min slope)
     Return: peak na, or null on error.

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

numaGetHistogramStats

l_int32 numaGetHistogramStats ( NUMA *nahisto, l_float32 startx, l_float32 deltax, l_float32 *pxmean, l_float32 *pxmedian, l_float32 *pxmode, l_float32 *pxvariance )

numaGetHistogramStats()

    Input:  nahisto (histogram: y(x(i)), i = 0 ... nbins - 1)
            startx (x value of first bin: x(0))
            deltax (x increment between bins; the bin size; x(1) - x(0))
            &xmean (<optional return> mean value of histogram)
            &xmedian (<optional return> median value of histogram)
            &xmode (<optional return> mode value of histogram:
                   xmode = x(imode), where y(xmode) >= y(x(i)) for
                   all i != imode)
            &xvariance (<optional return> variance of x)
    Return: 0 if OK, 1 on error

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

numaGetHistogramStatsOnInterval

l_int32 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 )

numaGetHistogramStatsOnInterval()

    Input:  nahisto (histogram: y(x(i)), i = 0 ... nbins - 1)
            startx (x value of first bin: x(0))
            deltax (x increment between bins; the bin size; x(1) - x(0))
            ifirst (first bin to use for collecting stats)
            ilast (last bin for collecting stats; use 0 to go to the end)
            &xmean (<optional return> mean value of histogram)
            &xmedian (<optional return> median value of histogram)
            &xmode (<optional return> mode value of histogram:
                   xmode = x(imode), where y(xmode) >= y(x(i)) for
                   all i != imode)
            &xvariance (<optional return> variance of x)
    Return: 0 if OK, 1 on error

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

numaGetRankBinValues

l_int32 numaGetRankBinValues ( NUMA *na, l_int32 nbins, NUMA **pnarbin, NUMA **pnam )

numaGetRankBinValues()

    Input:  na (just an array of values)
            nbins (number of bins at which the rank is divided)
            &pnarbin (<optional return> rank bin value vs array value)
            &pnam (<optional return> median intensity in a bin vs
                   rank bin value, with @nbins of discretized rank values)
    Return: 0 if OK, 1 on error

Notes:
    (1) Simple interface for getting a binned rank representation
        of an input array of values.  This returns two mappings:
           array value     -->  rank bin number  (narbin)
           rank bin number -->  median array value in each rank bin (nam)

numaGetStatsUsingHistogram

l_int32 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 )

numaGetStatsUsingHistogram()

    Input:  na (an arbitrary set of numbers; not ordered and not
                a histogram)
            maxbins (the maximum number of bins to be allowed in
                     the histogram; use 0 for consecutive integer bins)
            &min (<optional return> min value of set)
            &max (<optional return> max value of set)
            &mean (<optional return> mean value of set)
            &variance (<optional return> variance)
            &median (<optional return> median value of set)
            rank (in [0.0 ... 1.0]; median has a rank 0.5; ignored
                  if &rval == NULL)
            &rval (<optional return> value in na corresponding to @rank)
            &histo (<optional return> Numa histogram; use NULL to prevent)
    Return: 0 if OK, 1 on error

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.

numaHistogramGetRankFromVal

l_int32 numaHistogramGetRankFromVal ( NUMA *na, l_float32 rval, l_float32 *prank )

numaHistogramGetRankFromVal()

    Input:  na (histogram)
            rval (value of input sample for which we want the rank)
            &rank (<return> fraction of total samples below rval)
    Return: 0 if OK, 1 on error

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.

numaHistogramGetValFromRank

l_int32 numaHistogramGetValFromRank ( NUMA *na, l_float32 rank, l_float32 *prval )

numaHistogramGetValFromRank()

    Input:  na (histogram)
            rank (fraction of total samples)
            &rval (<return> approx. to the bin value)
    Return: 0 if OK, 1 on error

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.

numaMakeHistogram

NUMA * numaMakeHistogram ( NUMA *na, l_int32 maxbins, l_int32 *pbinsize, l_int32 *pbinstart )

numaMakeHistogram()

    Input:  na
            maxbins (max number of histogram bins)
            &binsize  (<return> size of histogram bins)
            &binstart (<optional return> start val of minimum bin;
                       input NULL to force start at 0)
    Return: na consisiting of histogram of integerized values,
            or null on error.

Note:
    (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 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.

numaMakeHistogramAuto

NUMA * numaMakeHistogramAuto ( NUMA *na, l_int32 maxbins )

numaMakeHistogramAuto()

    Input:  na (numa of floats; these may be integers)
            maxbins (max number of histogram bins; >= 1)
    Return: na consisiting of histogram of quantized float values,
            or null on error.

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.

numaMakeHistogramClipped

NUMA * numaMakeHistogramClipped ( NUMA *na, l_float32 binsize, l_float32 maxsize )

numaMakeHistogramClipped()

    Input:  na
            binsize (typically 1.0)
            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

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.

numaMakeRankFromHistogram

l_int32 numaMakeRankFromHistogram ( l_float32 startx, l_float32 deltax, NUMA *nasy, l_int32 npts, NUMA **pnax, NUMA **pnay )

numaMakeRankFromHistogram()

    Input:  startx (xval corresponding to first element in nay)
            deltax (x increment between array elements in nay)
            nasy (input histogram, assumed equally spaced)
            npts (number of points to evaluate rank function)
            &nax (<optional return> array of x values in range)
            &nay (<return> rank array of specified npts)
    Return: 0 if OK, 1 on error

numaNormalizeHistogram

NUMA * numaNormalizeHistogram ( NUMA *nas, l_float32 tsum )

numaNormalizeHistogram()

    Input:  nas (input histogram)
            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

numaOpen

NUMA * numaOpen ( NUMA *nas, l_int32 size )

numaOpen()

    Input:  nas
            size (of sel; greater than 0, odd; origin implicitly in center)
    Return: nad (opened), or null on error

Notes:
    (1) The structuring element (sel) is linear, all "hits"
    (2) If size == 1, this returns a copy

numaRebinHistogram

NUMA * numaRebinHistogram ( NUMA *nas, l_int32 newsize )

numaRebinHistogram()

    Input:  nas (input histogram)
            newsize (number of old bins contained in each new bin)
    Return: nad (more coarsely re-binned histogram), or null on error

numaSelectCrossingThreshold

l_int32 numaSelectCrossingThreshold ( NUMA *nax, NUMA *nay, l_float32 estthresh, l_float32 *pbestthresh )

numaSelectCrossingThreshold()

    Input:  nax (<optional> numa of abscissa values; can be NULL)
            nay (signal)
            estthresh (estimated pixel threshold for crossing: e.g., for
                       images, white <--> black; typ. ~120)
            &bestthresh (<return> robust estimate of threshold to use)
    Return: 0 if OK, 1 on error

Note:
   (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.

numaSplitDistribution

l_int32 numaSplitDistribution ( NUMA *na, l_float32 scorefract, l_int32 *psplitindex, l_float32 *pave1, l_float32 *pave2, l_float32 *pnum1, l_float32 *pnum2, NUMA **pnascore )

numaSplitDistribution()

    Input:  na (histogram)
            scorefract (fraction of the max score, used to determine
                        the range over which the histogram min is searched)
            &splitindex (<optional return> index for splitting)
            &ave1 (<optional return> average of lower distribution)
            &ave2 (<optional return> average of upper distribution)
            &num1 (<optional return> population of lower distribution)
            &num2 (<optional return> population of upper distribution)
            &nascore (<optional return> for debugging; otherwise use null)
    Return: 0 if OK, 1 on error

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.

numaTransform

NUMA * numaTransform ( NUMA *nas, l_float32 shift, l_float32 scale )

numaTransform()

    Input:  nas
            shift (add this to each number)
            scale (multiply each number by this)
    Return: nad (with all values shifted and scaled, or null on error)

Notes:
    (1) Each number is shifted before scaling.
    (2) The operation sequence is opposite to that for Box and Pta:
        scale first, then shift.

numaWindowedMean

NUMA * numaWindowedMean ( NUMA *nas, l_int32 wc )

numaWindowedMean()

    Input:  nas
            wc (half width of the convolution window)
    Return: nad (after low-pass filtering), or null on error

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.

numaWindowedMeanSquare

NUMA * numaWindowedMeanSquare ( NUMA *nas, l_int32 wc )

numaWindowedMeanSquare()

    Input:  nas
            wc (half width of the window)
    Return: nad (containing windowed mean square values), or null on error

Notes:
    (1) The window has width = 2 * @wc + 1.
    (2) We add a mirrored border of size @wc to each end of the array.

numaWindowedStats

l_int32 numaWindowedStats ( NUMA *nas, l_int32 wc, NUMA **pnam, NUMA **pnams, NUMA **pnav, NUMA **pnarv )

numaWindowedStats()

    Input:  nas (input numa)
            wc (half width of the window)
            &nam (<optional return> mean value in window)
            &nams (<optional return> mean square value in window)
            &pnav (<optional return> variance in window)
            &pnarv (<optional return> rms deviation from the mean)
    Return: 0 if OK, 1 on error

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.

numaWindowedVariance

l_int32 numaWindowedVariance ( NUMA *nam, NUMA *nams, NUMA **pnav, NUMA **pnarv )

numaWindowedVariance()

    Input:  nam (windowed mean values)
            nams (windowed mean square values)
            &pnav (<optional return> numa of variance -- the ms deviation
                   from the mean)
            &pnarv (<optional return> numa of rms deviation from the mean)
    Return: 0 if OK, 1 on error

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>

AUTHOR

Zakariyya Mughal <zmughal@cpan.org>

COPYRIGHT AND LICENSE

This software is copyright (c) 2014 by Zakariyya Mughal.

This is free software; you can redistribute it and/or modify it under the same terms as the Perl 5 programming language system itself.