version 0.04



  Octcube color quantization

  There are several different octcube/octree based quantizations.
  These can be classified, in the order in which they appear in this
  file, as follows:

  (1) General adaptive octree
  (2) Adaptive octree by population at fixed level
  (3) Adaptive octree using population and with specified number
      of output colors
  (4) Octcube with colormap representation of mixed color/gray
  (5) 256 fixed octcubes covering color space
  (6) Octcubes at fixed level for ncolors <= 256
  (7) Octcubes at fixed level with RGB output
  (8) Quantizing an rgb image using a specified colormap

  (1) Two-pass adaptive octree color quantization
          PIX              *pixOctreeColorQuant()
          PIX              *pixOctreeColorQuantGeneral()

        which calls
          static CQCELL  ***octreeGenerateAndPrune()
          static PIX       *pixOctreeQuantizePixels()

        which calls
          static l_int32    octreeFindColorCell()

      Helper cqcell functions
          static CQCELL  ***cqcellTreeCreate()
          static void       cqcellTreeDestroy()

      Helper index functions
          l_int32           makeRGBToIndexTables()
          void              getOctcubeIndexFromRGB()
          static void       getRGBFromOctcube()
          static l_int32    getOctcubeIndices()
          static l_int32    octcubeGetCount()

  (2) Adaptive octree quantization based on population at a fixed level
          PIX              *pixOctreeQuantByPopulation()
          static l_int32    pixDitherOctindexWithCmap()

  (3) Adaptive octree quantization to 4 and 8 bpp with specified
      number of output colors in colormap
          PIX              *pixOctreeQuantNumColors()

  (4) Mixed color/gray quantization with specified number of colors
          PIX              *pixOctcubeQuantMixedWithGray()

  (5) Fixed partition octcube quantization with 256 cells
          PIX              *pixFixedOctcubeQuant256()

  (6) Fixed partition quantization for images with few colors
          PIX              *pixFewColorsOctcubeQuant1()
          PIX              *pixFewColorsOctcubeQuant2()
          PIX              *pixFewColorsOctcubeQuantMixed()

  (7) Fixed partition octcube quantization at specified level
      with quantized output to RGB
          PIX              *pixFixedOctcubeQuantGenRGB()

  (8) Color quantize RGB image using existing colormap
          PIX              *pixQuantFromCmap()  [high-level wrapper]
          PIX              *pixOctcubeQuantFromCmap()
          PIX              *pixOctcubeQuantFromCmapLUT()

      Generation of octcube histogram
          NUMA             *pixOctcubeHistogram()

      Get filled octcube table from colormap
          l_int32          *pixcmapToOctcubeLUT()

      Strip out unused elements in colormap
          l_int32           pixRemoveUnusedColors()

      Find number of occupied octcubes at the specified level
          l_int32           pixNumberOccupiedOctcubes()

  Note: leptonica also provides color quantization using a modified
        form of median cut.  See colorquant2.c for details.



void getOctcubeIndexFromRGB ( l_int32 rval, l_int32 gval, l_int32 bval, l_uint32 *rtab, l_uint32 *gtab, l_uint32 *btab, l_uint32 *pindex )


      Input:  rval, gval, bval
              rtab, gtab, btab  (generated with makeRGBToIndexTables())
              &index (<return>)
      Return: void

  Note: no error checking!


l_int32 makeRGBToIndexTables ( l_uint32 **prtab, l_uint32 **pgtab, l_uint32 **pbtab, l_int32 cqlevels )


      Input:  &rtab, &gtab, &btab  (<return> tables)
              cqlevels (can be 1, 2, 3, 4, 5 or 6)
      Return: 0 if OK; 1 on error

  Set up tables.  e.g., for cqlevels = 5, we need an integer 0 < i < 2^15:
      rtab = (0  i7  0   0  i6  0   0  i5  0   0   i4  0   0   i3  0   0)
      gtab = (0  0   i7  0   0  i6  0   0  i5  0   0   i4  0   0   i3  0)
      btab = (0  0   0   i7  0  0   i6  0  0   i5  0   0   i4  0   0   i3)

  The tables are then used to map from rbg --> index as follows:
      index = (0  r7  g7  b7  r6  g6  b6  r5  g5  b5  r4  g4  b4  r3  g3  b3)

    e.g., for cqlevels = 4, we map to
      index = (0  0   0   0   r7  g7  b7  r6  g6  b6  r5  g5  b5  r4  g4  b4)

  This may look a bit strange.  The notation 'r7' means the MSBit of
  the r value (which has 8 bits, going down from r7 to r0).
  Keep in mind that r7 is actually the r component bit for level 1 of
  the octtree.  Level 1 is composed of 8 octcubes, represented by
  the bits (r7 g7 b7), which divide the entire color space into
  8 cubes.  At level 2, each of these 8 octcubes is further divided into
  8 cubes, each labeled by the second most significant bits (r6 g6 b6)
  of the rgb color.


PIX * pixFewColorsOctcubeQuant1 ( PIX *pixs, l_int32 level )


      Input:  pixs (32 bpp rgb)
              level (significant bits for each of RGB; valid in [1...6])
      Return: pixd (quantized to octcube) or null on error

      (1) Generates a colormapped image, where the colormap table values
          are the averages of all pixels that are found in the octcube.
      (2) This fails if there are more than 256 colors (i.e., more
          than 256 occupied octcubes).
      (3) Often level 3 (512 octcubes) will succeed because not more
          than half of them are occupied with 1 or more pixels.
      (4) The depth of the result, which is either 2, 4 or 8 bpp,
          is the minimum required to hold the number of colors that
          are found.
      (5) This can be useful for quantizing orthographically generated
          images such as color maps, where there may be more than 256 colors
          because of aliasing or jpeg artifacts on text or lines, but
          there are a relatively small number of solid colors.  Then,
          use with level = 3 can often generate a compact and accurate
          representation of the original RGB image.  For this purpose,
          it is better than pixFewColorsOctcubeQuant2(), because it
          uses the average value of pixels in the octcube rather
          than the first found pixel.  It is also simpler to use,
          because it generates the histogram internally.


PIX * pixFewColorsOctcubeQuant2 ( PIX *pixs, l_int32 level, NUMA *na, l_int32 ncolors, l_int32 *pnerrors )


      Input:  pixs (32 bpp rgb)
              level (of octcube indexing, for histogram: 3, 4, 5, 6)
              na (histogram of pixel occupation in octree leaves at
                  given level)
              ncolors (number of occupied octree leaves at given level)
              &nerrors (<optional return> num of pixels not exactly
                        represented in the colormap)
      Return: pixd (2, 4 or 8 bpp with colormap), or null on error

      (1) Generates a colormapped image, where the colormap table values
          are the averages of all pixels that are found in the octcube.
      (2) This fails if there are more than 256 colors (i.e., more
          than 256 occupied octcubes).
      (3) Often level 3 (512 octcubes) will succeed because not more
          than half of them are occupied with 1 or more pixels.
      (4) For an image with not more than 256 colors, it is unlikely
          that two pixels of different color will fall in the same
          octcube at level = 4.   However it is possible, and this
          function optionally returns @nerrors, the number of pixels
          where, because more than one color is in the same octcube,
          the pixel color is not exactly reproduced in the colormap.
          The colormap for an occupied leaf of the octree contains
          the color of the first pixel encountered in that octcube.
      (5) This differs from pixFewColorsOctcubeQuant1(), which also
          requires not more than 256 occupied leaves, but represents
          the color of each leaf by an average over the pixels in
          that leaf.  This also requires precomputing the histogram
          of occupied octree leaves, which is generated using
      (6) This is used in pixConvertRGBToColormap() for images that
          are determined, by their histogram, to have relatively few
          colors.  This typically happens with orthographically
          produced images (as oppopsed to natural images), where
          it is expected that most of the pixels within a leaf
          octcube have exactly the same color, and quantization to
          that color is lossless.


PIX * pixFewColorsOctcubeQuantMixed ( PIX *pixs, l_int32 level, l_int32 darkthresh, l_int32 lightthresh, l_int32 diffthresh, l_float32 minfract, l_int32 maxspan )


      Input:  pixs (32 bpp rgb)
              level (significant octcube bits for each of RGB;
                     valid in [1...6]; use 0 for default)
              darkthresh (threshold near black; if the lightest component
                          is below this, the pixel is not considered to
                          be gray or color; uses 0 for default)
              lightthresh (threshold near white; if the darkest component
                           is above this, the pixel is not considered to
                           be gray or color; use 0 for default)
              diffthresh (thresh for the max difference between component
                          values; for differences below this, the pixel
                          is considered to be gray; use 0 for default)
                          considered gray; use 0 for default)
              minfract (min fraction of pixels for gray histo bin;
                        use 0.0 for default)
              maxspan (max size of gray histo bin; use 0 for default)
      Return: pixd (8 bpp, quantized to octcube for pixels that are
                    not gray; gray pixels are quantized separately
                    over the full gray range), or null on error

      (1) First runs pixFewColorsOctcubeQuant1().  If this succeeds,
          it separates the color from gray(ish) entries in the cmap,
          and re-quantizes the gray pixels.  The result has some pixels
          in color and others in gray.
      (2) This fails if there are more than 256 colors (i.e., more
          than 256 occupied octcubes in the color quantization).
      (3) Level 3 (512 octcubes) will usually succeed because not more
          than half of them are occupied with 1 or more pixels.
      (4) This uses the criterion from pixColorFraction() for deciding
          if a colormap entry is color; namely, if the color components
          are not too close to either black or white, and the maximum
          difference between component values equals or exceeds a threshold.
      (5) For quantizing the gray pixels, it uses a histogram-based
          method where input parameters determining the buckets are
          the minimum population fraction and the maximum allowed size.
      (6) Recommended input parameters are:
              @level:  3 or 4  (3 is default)
              @darkthresh:  20
              @lightthresh: 244
              @diffthresh: 20
              @minfract: 0.05
              @maxspan: 15
          These numbers are intended to be conservative (somewhat over-
          sensitive) in color detection,  It's usually better to pay
          extra with octcube quantization of a grayscale image than
          to use grayscale quantization on an image that has some
          actual color.  Input 0 on any of these to get the default.
      (7) This can be useful for quantizing orthographically generated
          images such as color maps, where there may be more than 256 colors
          because of aliasing or jpeg artifacts on text or lines, but
          there are a relatively small number of solid colors.  It usually
          gives results that are better than pixOctcubeQuantMixedWithGray(),
          both in size and appearance.  But it is a bit slower.


PIX * pixFixedOctcubeQuant256 ( PIX *pixs, l_int32 ditherflag )


      Input:  pixs  (32 bpp; 24-bit color)
              ditherflag  (1 for dithering; 0 for no dithering)
      Return: pixd (8 bit with colormap), or null on error

  This simple 1-pass color quantization works by breaking the
  color space into 256 pieces, with 3 bits quantized for each of
  red and green, and 2 bits quantized for blue.  We shortchange
  blue because the eye is least sensitive to blue.  This
  division of the color space is into two levels of octrees,
  followed by a further division by 4 (not 8), where both
  blue octrees have been combined in the third level.

  The color map is generated from the 256 color centers by
  taking the representative color to be the center of the
  cell volume.  This gives a maximum error in the red and
  green values of 16 levels, and a maximum error in the
  blue sample of 32 levels.

  Each pixel in the 24-bit color image is placed in its containing
  cell, given by the relevant MSbits of the red, green and blue
  samples.  An error-diffusion dithering is performed on each
  color sample to give the appearance of good average local color.
  Dithering is required; without it, the contouring and visible
  color errors are very bad.

  I originally implemented this algorithm in two passes,
  where the first pass was used to compute the weighted average
  of each sample in each pre-allocated region of color space.
  The idea was to use these centroids in the dithering algorithm
  of the second pass, to reduce the average error that was
  being dithered.  However, with dithering, there is
  virtually no difference, so there is no reason to make the
  first pass.  Consequently, this 1-pass version just assigns
  the pixels to the centers of the pre-allocated cells.
  We use dithering to spread the difference between the sample
  value and the location of the center of the cell.  For speed
  and simplicity, we use integer dithering and propagate only
  to the right, down, and diagonally down-right, with ratios
  3/8, 3/8 and 1/4, respectively.  The results should be nearly
  as good, and a bit faster, with propagation only to the right
  and down.

  The algorithm is very fast, because there is no search,
  only fast generation of the cell index for each pixel.
  We use a simple mapping from the three 8 bit rgb samples
  to the 8 bit cell index; namely, (r7 r6 r5 g7 g6 g5 b7 b6).
  This is not in an octcube format, but it doesn't matter.
  There are no storage requirements.  We could keep a
  running average of the center of each sample in each
  cluster, rather than using the center of the cell, but
  this is just extra work, esp. with dithering.

  This method gives surprisingly good results with dithering.
  However, without dithering, the loss of color accuracy is
  evident in regions that are very light or that have subtle
  blending of colors.


PIX * pixFixedOctcubeQuantGenRGB ( PIX *pixs, l_int32 level )


      Input:  pixs (32 bpp rgb)
              level (significant bits for each of r,g,b)
      Return: pixd (rgb; quantized to octcube centers), or null on error

      (1) Unlike the other color quantization functions, this one
          generates an rgb image.
      (2) The pixel values are quantized to the center of each octcube
          (at the specified level) containing the pixel.  They are
          not quantized to the average of the pixels in that octcube.


l_int32 pixNumberOccupiedOctcubes ( PIX *pix, l_int32 level, l_int32 mincount, l_float32 minfract, l_int32 *pncolors )


      Input:  pix (32 bpp)
              level (of octcube)
              mincount (minimum num pixels in an octcube to be counted;
                        -1 to not use)
              minfract (minimum fract of pixels in an octcube to be
                        counted; -1 to not use)
              &ncolors (<return> number of occupied octcubes)
      Return: 0 if OK, 1 on error

      (1) Exactly one of (@mincount, @minfract) must be -1, so, e.g.,
          if @mincount == -1, then we use @minfract.
      (2) If all occupied octcubes are to count, set @mincount == 1.
          Setting @minfract == 0.0 is taken to mean the same thing.


NUMA * pixOctcubeHistogram ( PIX *pixs, l_int32 level, l_int32 *pncolors )


      Input:  pixs (32 bpp rgb)
              level (significant bits for each of RGB; valid in [1...6])
              &ncolors (<optional return> number of occupied cubes)
      Return: numa (histogram of color pixels, or null on error)

      (1) Input NULL for &ncolors to prevent computation and return value.


PIX * pixOctcubeQuantFromCmap ( PIX *pixs, PIXCMAP *cmap, l_int32 mindepth, l_int32 level, l_int32 metric )


      Input:  pixs  (32 bpp rgb)
              cmap  (to quantize to; insert copy into dest pix)
              mindepth (minimum depth of pixd: can be 2, 4 or 8 bpp)
              level (of octcube used for finding nearest color in cmap)
      Return: pixd  (2, 4 or 8 bpp, colormapped), or null on error

      (1) In typical use, we are doing an operation, such as
          interpolative scaling, on a colormapped pix, where it is
          necessary to remove the colormap before the operation.
          We then want to re-quantize the RGB result using the same
      (2) The level is used to divide the color space into octcubes.
          Each input pixel is, in effect, placed at the center of an
          octcube at the given level, and it is mapped into the
          exact color (given in the colormap) that is the closest
          to that location.  We need to know that distance, for each color
          in the colormap.  The higher the level of the octtree, the smaller
          the octcubes in the color space, and hence the more accurately
          we can determine the closest color in the colormap; however,
          the size of the LUT, which is the total number of octcubes,
          increases by a factor of 8 for each increase of 1 level.
          The time required to acquire a level 4 mapping table, which has
          about 4K entries, is less than 1 msec, so that is the
          recommended minimum size to be used.  At that size, the
          octcubes have their centers 16 units apart in each (r,g,b)
          direction.  If two colors are in the same octcube, the one
          closest to the center will always be chosen.  The maximum
          error for any component occurs when the correct color is
          at a cube corner and there is an incorrect color just inside
          the cube next to the opposite corner, giving an error of
          14 units (out of 256) for each component.   Using a level 5
          mapping table reduces the maximum error to 6 units.
      (3) Typically you should use the Euclidean metric, because the
          resulting voronoi cells (which are generated using the actual
          colormap values as seeds) are convex for Euclidean distance
          but not for Manhattan distance.  In terms of the octcubes,
          convexity of the voronoi cells means that if the 8 corners
          of any cube (of which the octcubes are special cases)
          are all within a cell, then every point in the cube will
          lie within the cell.
      (4) The depth of the output pixd is equal to the maximum of
          (a) @mindepth and (b) the minimum (2, 4 or 8 bpp) necessary
          to hold the indices in the colormap.
      (5) We build a mapping table from octcube to colormap index so
          that this function can run in a time (otherwise) independent
          of the number of colors in the colormap.  This avoids a
          brute-force search for the closest colormap color to each
          pixel in the image.
      (6) This is similar to the function pixAssignToNearestColor()
          used for color segmentation.
      (7) Except for very small images or when using level > 4,
          it takes very little time to generate the tables,
          compared to the generation of the colormapped dest pix,
          so one would not typically use the low-level version.


PIX * pixOctcubeQuantFromCmapLUT ( PIX *pixs, PIXCMAP *cmap, l_int32 mindepth, l_int32 *cmaptab, l_uint32 *rtab, l_uint32 *gtab, l_uint32 *btab )


      Input:  pixs  (32 bpp rgb)
              cmap  (to quantize to; insert copy into dest pix)
              mindepth (minimum depth of pixd: can be 2, 4 or 8 bpp)
              cmaptab  (table mapping from octindex to colormap index)
              rtab, gtab, btab (tables mapping from RGB to octindex)
      Return: pixd  (2, 4 or 8 bpp, colormapped), or null on error

      (1) See the notes in the higher-level function
          pixOctcubeQuantFromCmap().  The octcube level for
          the generated octree is specified there, along with
          the distance metric for determining the closest
          color in the colormap to each octcube.
      (2) If the colormap, level and metric information have already
          been used to construct the set of mapping tables,
          this low-level function can be used directly (i.e.,
          independently of pixOctcubeQuantFromCmap()) to build
          a colormapped pix that uses the specified colormap.


PIX * pixOctcubeQuantMixedWithGray ( PIX *pixs, l_int32 depth, l_int32 graylevels, l_int32 delta )


      Input:  pixs (32 bpp rgb)
              depth (of output pix)
              graylevels (grayscale)
              delta (threshold for deciding if a pix is color or grayscale)
      Return: pixd (quantized to octcube and gray levels) or null on error

      (1) Generates a colormapped image, where the colormap table values
          have two components: octcube values representing pixels with
          color content, and grayscale values for the rest.
      (2) The threshold (delta) is the maximum allowable difference of
          the max abs value of | r - g |, | r - b | and | g - b |.
      (3) The octcube values are the averages of all pixels that are
          found in the octcube, and that are far enough from gray to
          be considered color.  This can roughly be visualized as all
          the points in the rgb color cube that are not within a "cylinder"
          of diameter approximately 'delta' along the main diagonal.
      (4) We want to guarantee full coverage of the rgb color space; thus,
          if the output depth is 4, the octlevel is 1 (2 x 2 x 2 = 8 cubes)
          and if the output depth is 8, the octlevel is 2 (4 x 4 x 4
          = 64 cubes).
      (5) Consequently, we have the following constraint on the number
          of allowed gray levels: for 4 bpp, 8; for 8 bpp, 192.


PIX * pixOctreeColorQuant ( PIX *pixs, l_int32 colors, l_int32 ditherflag )


      Input:  pixs  (32 bpp; 24-bit color)
              colors  (in colormap; some number in range [128 ... 256];
                      the actual number of colors used will be smaller)
              ditherflag  (1 to dither, 0 otherwise)
      Return: pixd (8 bpp with colormap), or null on error

  I found one description in the literature of octree color
  quantization, using progressive truncation of the octree,
  by M. Gervautz and W. Purgathofer in Graphics Gems, pp.
  287-293, ed. A. Glassner, Academic Press, 1990.
  Rather than setting up a fixed partitioning of the color
  space ab initio, as we do here, they allow the octree to be
  progressively truncated as new pixels are added.  They
  need to set up some data structures that are traversed
  with the addition of each 24 bit pixel, in order to decide
  either (1) in which cluster (sub-branch of the octree) to put
  the pixel, or (2) whether to truncate the octree further
  to place the pixel in an existing cluster, or (3) which
  two existing clusters should be merged so that the pixel
  can be left to start a truncated leaf of the octree.  Such dynamic
  truncation is considerably more complicated, and Gervautz et
  al. did not explain how they did it in anywhere near the
  detail required to check their implementation.

  The simple method in pixFixedOctcubeQuant256() is very
  fast, and with dithering the results are good, but you
  can do better if the color clusters are selected adaptively
  from the image.  We want a method that makes much better
  use of color samples in regions of color space with high
  pixel density, while also fairly representing small numbers
  of color pixels in low density regions.  Such adaptation
  requires two passes through the image: the first for generating
  the pruned tree of color cubes and the second for computing the index
  into the color table for each pixel.

  A relatively simple adaptive method is pixOctreeQuantByPopulation().
  That function first determines if the image has very few colors,
  and, if so, quantizes to those colors.  If there are more than
  256 colors, it generates a histogram of octcube leaf occupancy
  at level 4, chooses the 192 most populated such leaves as
  the first 192 colors, and sets the remaining 64 colors to the
  residual average pixel values in each of the 64 level 2 octcubes.
  This is a bit faster than pixOctreeColorQuant(), and does very
  well without dithering, but for most images with dithering it
  is clearly inferior.

  We now describe pixOctreeColorQuant().  The first pass is done
  on a subsampled image, because we do not need to use all the
  pixels in the image to generate the tree.  Subsampling
  down to 0.25 (1/16 of the pixels) makes the program run
  about 1.3 times faster.

  Instead of dividing the color space into 256 equal-sized
  regions, we initially divide it into 2^12 or 2^15 or 2^18
  equal-sized octcubes.  Suppose we choose to use 2^18 octcubes.
  This gives us 6 octree levels.  We then prune back,
  starting from level 6.  For every cube at level 6, there
  are 8 cubes at level 5.  Call the operation of putting a
  cube aside as a color table entry (CTE) a "saving."
  We use a (in general) level-dependent threshold, and save
  those level 6 cubes that are above threshold.
  The rest are combined into the containing level 5 cube.
  If between 1 and 7 level 6 cubes within a level 5
  cube have been saved by thresholding, then the remaining
  level 6 cubes in that level 5 cube are automatically
  saved as well, without applying a threshold.  This greatly
  simplifies both the description of the CTEs and the later
  classification of each pixel as belonging to a CTE.
  This procedure is iterated through every cube, starting at
  level 5, and then 4, 3, and 2, successively.  The result is that
  each CTE contains the entirety of a set of from 1 to 7 cubes
  from a given level that all belong to a single cube at the
  level above.   We classify the CTEs in terms of the
  condition in which they are made as either being "threshold"
  or "residual."  They are "threshold" CTEs if no subcubes
  are CTEs (that is, they contain every pixel within the cube)
  and the number of pixels exceeds the threshold for making
  a CTE.  They are "residual" CTEs if at least one but not more
  than 7 of the subcubes have already been determined to be CTEs;
  this happens automatically -- no threshold is applied.
  If all 8 subcubes are determined to be CTEs, the cube is
  marked as having all pixels accounted for ('bleaf' = 1) but
  is not saved as a CTE.

  We stop the pruning at level 2, at which there are 64
  sub-cubes.  Any pixels not already claimed in a CTE are
  put in these cubes.

  As the cubes are saved as color samples in the color table,
  the number of remaining pixels P and the number of
  remaining colors in the color table N are recomputed,
  along with the average number of pixels P/N (ppc) to go in
  each of the remaining colors.  This running average number is
  used to set the threshold at the current level.

  Because we are going to very small cubes at levels 6 or 5,
  and will dither the colors for errors, it is not necessary
  to compute the color center of each cluster; we can simply
  use the center of the cube.  This gives us a minimax error
  condition: the maximum error is half the width of the
  level 2 cubes -- 32 color values out of 256 -- for each color
  sample.  In practice, most of the pixels will be very much
  closer to the center of their cells.  And with dithering,
  the average pixel color in a small region will be closer still.
  Thus with the octree quantizer, we are able to capture
  regions of high color pdf (probability density function) in small
  but accurate CTEs, and to have only a small number of pixels
  that end up a significant distance (with a guaranteed maximum)
  from their true color.

  How should the threshold factor vary?  Threshold factors
  are required for levels 2, 3, 4 and 5 in the pruning stage.
  The threshold for level 5 is actually applied to cubes at
  level 6, etc.  From various experiments, it appears that
  the results do not vary appreciably for threshold values near 1.0.
  If you want more colors in smaller cubes, the threshold
  factors can be set lower than 1.0 for cubes at levels 4 and 5.
  However, if the factor is set much lower than 1.0 for
  levels 2 and 3, we can easily run out of colors.
  We put aside 64 colors in the calculation of the threshold
  values, because we must have 64 color centers at level 2,
  that will have very few pixels in most of them.
  If we reduce the factor for level 5 to 0.4, this will
  generate many level 6 CTEs, and consequently
  many residual cells will be formed up from those leaves,
  resulting in the possibility of running out of colors.
  Remember, the residual CTEs are mandatory, and are formed
  without using the threshold, regardless of the number of
  pixels that are absorbed.

  The implementation logically has four parts:

       (1) accumulation into small, fixed cells
       (2) pruning back into selected CTE cubes
       (3) organizing the CTEs for fast search to find
           the CTE to which any image pixel belongs
       (4) doing a second scan to code the image pixels by CTE

  Step (1) is straightforward; we use 2^15 cells.

  We've already discussed how the pruning step (2) will be performed.

  Steps (3) and (4) are related, in that the organization
  used by step (3) determines how the search actually
  takes place for each pixel in step (4).

  There are many ways to do step (3).  Let's explore a few.

  (a) The simplest is to order the cubes from highest occupancy
      to lowest, and traverse the list looking for the deepest
      match.  To make this more efficient, so that we know when
      to stop looking, any cube that has separate CTE subcubes
      would be marked as such, so that we know when we hit a
      true leaf.

  (b) Alternatively, we can order the cubes by highest
      occupancy separately each level, and work upward,
      starting at level 5, so that when we find a match we
      know that it will be correct.

  (c) Another approach would be to order the cubes by
      "address" and use a hash table to find the cube
      corresponding to a pixel color.  I don't know how to
      do this with a variable length address, as each CTE
      will have 3*n bits, where n is the level.

  (d) Another approach entirely is to put the CTE cubes into
      a tree, in such a way that starting from the root, and
      using 3 bits of address at a time, the correct branch of
      each octree can be taken until a leaf is found.  Because
      a given cube can be both a leaf and also have branches
      going to sub-cubes, the search stops only when no
      marked subcubes have addresses that match the given pixel.

      In the tree method, we can start with a dense infrastructure,
      and place the leaves corresponding to the N colors
      in the tree, or we can grow from the root only those
      branches that end directly on leaves.

  What we do here is to take approach (d), and implement the tree
  "virtually", as a set of arrays, one array for each level
  of the tree.   Initially we start at level 5, an array with
  2^15 cubes, each with 8 subcubes.  We then build nodes at
  levels closer to the root; at level 4 there are 2^12 nodes
  each with 8 subcubes; etc.  Using these arrays has
  several advantages:

     -  We don't need to keep track of links between cubes
        and subcubes, because we can use the canonical
        addressing on the cell arrays directly to determine
        which nodes are parent cubes and which are sub-cubes.

     -  We can prune directly on this tree

     -  We can navigate the pruned tree quickly to classify
        each pixel in the image.

  Canonical addressing guarantees that the i-th node at level k
  has 8 subnodes given by the 8*i ... 8*i+7 nodes at level k+1.

  The pruning step works as follows.  We go from the lowest
  level up.  At each level, the threshold is found from the
  product of a factor near 1.0 and the ratio of unmarked pixels
  to remaining colors (minus the 64).  We march through
  the space, sequentially considering a cube and its 8 subcubes.
  We first check those subcubes that are not already
  marked as CTE to see if any are above threshold, and if so,
  generate a CTE and mark them as such.
  We then determine if any of the subcubes have been marked.
  If so, and there are subcubes that are not marked,
  we generate a CTE for the cube from the remaining unmarked
  subcubes; this is mandatory and does not depend on how many
  pixels are in the set of subcubes.  If none of the subcubes
  are marked, we aggregate their pixels into the cube
  containing them, but do not mark it as a CTE; that
  will be determined when iterating through the next level up.

  When all the pixels in a cube are accounted for in one or more
  colors, we set the boolean 'bleaf' to true.  This is the
  flag used to mark the cubes in the pruning step.  If a cube
  is marked, and all 8 subcubes are marked, then it is not
  itself given a CTE because all pixels have already been
  accounted for.

  Note that the pruning of the tree and labelling of the CTEs
  (step 2) accomplishes step 3 implicitly, because the marked
  and pruned tree is ready for use in labelling each pixel
  in step 4.  We now, for every pixel in the image, traverse
  the tree from the root, looking for the lowest cube that is a leaf.
  At each level we have a cube and subcube.  If we reach a subcube
  leaf that is marked 0, we know that the color is stored in the
  cube above, and we've found the CTE.  Otherwise, the subcube
  leaf is marked 1.  If we're at the last level, we've reached
  the final leaf and must use it.  Otherwise, continue the
  process at the next level down.

  For robustness, efficiency and high quality output, we do the following:

  (1) Measure the color content of the image.  If there is very little
      color, quantize in grayscale.
  (2) For efficiency, build the octree with a subsampled image if the
      image is larger than some threshold size.
  (3) Reserve an extra set of colors to prevent running out of colors
      when pruning the octree; specifically, during the assignment
      of those level 2 cells (out of the 64) that have unassigned
      pixels.  The problem of running out is more likely to happen
      with small images, because the estimation we use for the
      number of pixels available is not accurate.
  (4) In the unlikely event that we run out of colors, the dithered
      image can be very poor.  As this would only happen with very
      small images, and dithering is not particularly noticeable with
      such images, turn it off.


PIX * pixOctreeColorQuantGeneral ( PIX *pixs, l_int32 colors, l_int32 ditherflag, l_float32 validthresh, l_float32 colorthresh )


      Input:  pixs  (32 bpp; 24-bit color)
              colors  (in colormap; some number in range [128 ... 240];
                      the actual number of colors used will be smaller)
              ditherflag  (1 to dither, 0 otherwise)
              validthresh (minimum fraction of pixels neither near white
                           nor black, required for color quantization;
                           typically ~0.01, but smaller for images that have
                           color but are nearly all white)
              colorthresh (minimum fraction of pixels with color that are
                           not near white or black, that are required
                           for color quantization; typ. ~0.01, but smaller
                           for images that have color along with a
                           significant fraction of gray)
      Return: pixd (8 bit with colormap), or null on error

      (1) The parameters @validthresh and @colorthresh are used to
          determine if color quantization should be used on an image,
          or whether, instead, it should be quantized in grayscale.
          If the image has very few non-white and non-black pixels, or
          if those pixels that are non-white and non-black are all
          very close to either white or black, it is usually better
          to treat the color as accidental and to quantize the image
          to gray only.  These parameters are useful if you know
          something a priori about the image.  Perhaps you know that
          there is only a very small fraction of color pixels, but they're
          important to preserve; then you want to use a smaller value for
          these parameters.  To disable conversion to gray and force
          color quantization, use @validthresh = 0.0 and @colorthresh = 0.0.
      (2) See pixOctreeColorQuant() for algorithmic and implementation
          details.  This function has a more general interface.
      (3) See pixColorFraction() for computing the fraction of pixels
          that are neither white nor black, and the fraction of those
          pixels that have little color.  From the documentation there:
             If pixfract is very small, there are few pixels that are
             neither black nor white.  If colorfract is very small,
             the pixels that are neither black nor white have very
             little color content.  The product 'pixfract * colorfract'
             gives the fraction of pixels with significant color content.
          We test against the product @validthresh * @colorthresh
          to find color in images that have either very few
          intermediate gray pixels or that have many such gray pixels.


PIX * pixOctreeQuantByPopulation ( PIX *pixs, l_int32 level, l_int32 ditherflag )


      Input:  pixs (32 bpp rgb)
              level (significant bits for each of RGB; valid for {3,4},
                     Use 0 for default (level 4; recommended)
              ditherflag  (1 to dither, 0 otherwise)
      Return: pixd (quantized to octcubes) or null on error

      (1) This color quantization method works very well without
          dithering, using octcubes at two different levels:
            (a) the input @level, which is either 3 or 4
            (b) level 2 (64 octcubes to cover the entire color space)
      (2) For best results, using @level = 4 is recommended.
          Why do we provide an option for using level 3?  Because
          there are 512 octcubes at level 3, and for many images
          not more than 256 are filled.  As a result, on some images
          a very accurate quantized representation is possible using
          @level = 3.
      (3) This first breaks up the color space into octcubes at the
          input @level, and computes, for each octcube, the average
          value of the pixels that are in it.
      (4) Then there are two possible situations:
            (a) If there are not more than 256 populated octcubes,
                it returns a cmapped pix with those values assigned.
            (b) Otherwise, it selects 192 octcubes containing the largest
                number of pixels and quantizes pixels within those octcubes
                to their average.  Then, to handle the residual pixels
                that are not in those 192 octcubes, it generates a
                level 2 octree consisting of 64 octcubes, and within
                each octcube it quantizes the residual pixels to their
                average within each of those level 2 octcubes.
      (5) Unpopulated level 2 octcubes are represented in the colormap
          by their centers.  This, of course, has no effect unless
          dithering is used for the output image.
      (6) The depth of pixd is the minumum required to suppport the
          number of colors found at @level; namely, 2, 4 or 8.
      (7) This function works particularly well on images such as maps,
          where there are a relatively small number of well-populated
          colors, but due to antialiasing and compression artifacts
          there may be a large number of different colors.  This will
          pull out and represent accurately the highly populated colors,
          while still making a reasonable approximation for the others.
      (8) The highest level of octcubes allowed is 4.  Use of higher
          levels typically results in having a small fraction of
          pixels in the most populated 192 octcubes.  As a result,
          most of the pixels are represented at level 2, which is
          not sufficiently accurate.
      (9) Dithering shows artifacts on some images.  If you plan to
          dither, pixOctreeColorQuant() and pixFixedOctcubeQuant256()
          usually give better results.


PIX * pixOctreeQuantNumColors ( PIX *pixs, l_int32 maxcolors, l_int32 subsample )


      Input:  pixs (32 bpp rgb)
              maxcolors (8 to 256; the actual number of colors used
                         may be less than this)
              subsample (factor for computing color distribution;
                         use 0 for default)
      Return: pixd (4 or 8 bpp, colormapped), or null on error

  pixOctreeColorQuant() is very flexible in terms of the relative
  depth of different cubes of the octree.   By contrast, this function,
  pixOctreeQuantNumColors() is also adaptive, but it supports octcube
  leaves at only two depths: a smaller depth that guarantees
  full coverage of the color space and octcubes at one level
  deeper for more accurate colors.  Its main virutes are simplicity
  and speed, which are both derived from the natural indexing of
  the octcubes from the RGB values.

  Before describing pixOctreeQuantNumColors(), consider an even simpler
  approach for 4 bpp with either 8 or 16 colors.  With 8 colors,
  you simply go to level 1 octcubes and use the average color
  found in each cube.  For 16 colors, you find which of the three
  colors has the largest variance at the second level, and use two
  indices for that color.  The result is quite poor, because (1) some
  of the cubes are nearly empty and (2) you don't get much color
  differentiation for the extra 8 colors.  Trust me, this method may
  be simple, but it isn't worth anything.

  In pixOctreeQuantNumColors(), we generate colormapped images at
  either 4 bpp or 8 bpp.  For 4 bpp, we have a minimum of 8 colors
  for the level 1 octcubes, plus up to 8 additional colors that
  are determined from the level 2 popularity.  If the number of colors
  is between 8 and 16, the output is a 4 bpp image.  If the number of
  colors is greater than 16, the output is a 8 bpp image.

  We use a priority queue, implemented with a heap, to select the
  requisite number of most populated octcubes at the deepest level
  (level 2 for 64 or fewer colors; level 3 for more than 64 colors).
  These are combined with one color for each octcube one level above,
  which is used to span the color space of octcubes that were not
  included at the deeper level.

  If the deepest level is 2, we combine the popular level 2 octcubes
  (out of a total of 64) with the 8 level 1 octcubes.  If the deepest
  level is 3, we combine the popular level 3 octcubes (out of a
  total 512) with the 64 level 2 octcubes that span the color space.
  In the latter case, we require a minimum of 64 colors for the level 2
  octcubes, plus up to 192 additional colors determined from level 3

  The parameter 'maxlevel' is the deepest octcube level that is used.
  The implementation also uses two LUTs, which are employed in
  two successive traversals of the dest image.  The first maps
  from the src octindex at 'maxlevel' to the color table index,
  which is the value that is stored in the 4 or 8 bpp dest pixel.
  The second LUT maps from that colormap value in the dest to a
  new colormap value for a minimum sized colormap, stored back in
  the dest.  It is used to remove any color map entries that
  correspond to color space regions that have no pixels in the
  source image.  These regions can be either from the higher level
  (e.g., level 1 for 4 bpp), or from octcubes at 'maxlevel' that
  are unoccupied.  This remapping results in the minimum number
  of colors used according to the constraints induced by the
  input 'maxcolors'.  We also compute the average R, G and B color
  values in each region of the color space represented by a
  colormap entry, and store them in the colormap.

  The maximum number of colors is input, which determines the
  following properties of the dest image and octcube regions used:

     Number of colors      dest image depth      maxlevel
     ----------------      ----------------      --------
       8 to 16                  4 bpp               2
       17 to 64                 8 bpp               2
       65 to 256                8 bpp               3

  It may turn out that the number of extra colors, beyond the
  minimum (8 and 64 for maxlevel 2 and 3, respectively), is larger
  than the actual number of occupied cubes at these levels
  In that case, all the pixels are contained in this
  subset of cubes at maxlevel, and no colormap colors are needed
  to represent the remainder pixels one level above.  Thus, for
  example, in use one often finds that the pixels in an image
  occupy less than 192 octcubes at level 3, so they can be represented
  by a colormap for octcubes at level 3 only.


PIX * pixQuantFromCmap ( PIX *pixs, PIXCMAP *cmap, l_int32 mindepth, l_int32 level, l_int32 metric )


      Input:  pixs  (8 bpp grayscale without cmap, or 32 bpp rgb)
              cmap  (to quantize to; insert copy into dest pix)
              mindepth (minimum depth of pixd: can be 2, 4 or 8 bpp)
              level (of octcube used for finding nearest color in cmap)
      Return: pixd  (2, 4 or 8 bpp, colormapped), or null on error

      (1) This is a top-level wrapper for quantizing either grayscale
          or rgb images to a specified colormap.
      (2) The actual output depth is constrained by @mindepth and
          by the number of colors in @cmap.
      (3) For grayscale, @level and @metric are ignored.
      (4) If the cmap has color and pixs is grayscale, the color is
          removed from the cmap before quantizing pixs.


l_int32 pixRemoveUnusedColors ( PIX *pixs )


      Input:  pixs  (colormapped)
      Return: 0 if OK, 1 on error

      (1) This is an in-place operation.
      (2) If the image doesn't have a colormap, returns without error.
      (3) Unusued colors are removed from the colormap, and the
          image pixels are re-numbered.


l_int32 * pixcmapToOctcubeLUT ( PIXCMAP *cmap, l_int32 level, l_int32 metric )


      Input:  cmap
              level (significant bits for each of RGB; valid in [1...6])
      Return: tab[2**(3 * level)]

      (1) This function is used to quickly find the colormap color
          that is closest to any rgb color.  It is used to assign
          rgb colors to an existing colormap.  It can be very expensive
          to search through the entire colormap for the closest color
          to each pixel.  Instead, we first set up this table, which is
          populated by the colormap index nearest to each octcube
          color.  Then we go through the image; for each pixel,
          do two table lookups: first to generate the octcube index
          from rgb and second to use this table to read out the
          colormap index.
      (2) Do a slight modification for white and black.  For level = 4,
          each octcube size is 16.  The center of the whitest octcube
          is at (248, 248, 248), which is closer to 242 than 255.
          Consequently, any gray color between 242 and 254 will
          be selected, even if white (255, 255, 255) exists.  This is
          typically not optimal, because the original color was
          likely white.  Therefore, if white exists in the colormap,
          use it for any rgb color that falls into the most white octcube.
          Do the similar thing for black.
      (3) Here are the actual function calls for quantizing to a
          specified colormap:
            - first make the tables that map from rgb --> octcube index
            - then for each pixel:
                * use the tables to get the octcube index
                * use this table to get the nearest color in the colormap
                     cmap_index = tab[index]
      (4) Distance can be either manhattan or euclidean.
      (5) In typical use, level = 4 gives reasonable results, and
          level = 5 is slightly better.  When this function is used
          for color segmentation, there are typically a small number
          of colors and the number of levels can be small (e.g., level = 3).


Zakariyya Mughal <>


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.