- NAME
- VERSION
- numafunc2.c
- FUNCTIONS
- numaClose
- numaConvertToInt
- numaCountReversals
- numaCrossingsByPeaks
- numaCrossingsByThreshold
- numaDilate
- numaDiscretizeRankAndIntensity
- numaEarthMoverDistance
- numaErode
- numaEvalBestHaarParameters
- numaEvalHaarSum
- numaFindExtrema
- numaFindPeaks
- numaGetHistogramStats
- numaGetHistogramStatsOnInterval
- numaGetRankBinValues
- numaGetStatsUsingHistogram
- numaHistogramGetRankFromVal
- numaHistogramGetValFromRank
- numaMakeHistogram
- numaMakeHistogramAuto
- numaMakeHistogramClipped
- numaMakeRankFromHistogram
- numaNormalizeHistogram
- numaOpen
- numaRebinHistogram
- numaSelectCrossingThreshold
- numaSplitDistribution
- numaTransform
- numaWindowedMean
- numaWindowedMeanSquare
- numaWindowedStats
- numaWindowedVariance

- AUTHOR
- COPYRIGHT AND LICENSE

# 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.