pgmminkowski - compute Minkowski integral

       Luuk van Dijk, 2001.

       Based on work which is Copyright (C) 1989, 1991 by Jef Poskanzer.

       This program is part of Netpbm(1).

       pgmminkowski computes the 3 Minkowski integrals of a PGM image.

       The  Minkowski  integrals  mathematically characterize the shapes in the image and hence are the basis of
       "morphological image analysis."

       Hadwiger's theorem has it that these integrals are the only motion-invariant, additive and  conditionally
       continuous  functions of a two-dimensional image, which means that they are preserved under certain kinds
       of deformations of the image.  On top of that, they are very easy and  quickly  calculated.   This  makes
       them of interest for certain kinds of pattern recognition.

       Basically,  the Minkowski integrals are the area, total perimeter length, and the Euler characteristic of
       the image, where these metrics apply to the foreground image, not the rectangular PGM image itself.   The
       foreground image consists of all the pixels in the image that are white.  For a grayscale image, there is
       some  threshold  of  intensity  applied  to  categorize  pixels  into  black and white, and the Minkowski
       integrals are calculated as a function of this threshold value. The total  surface  area  refers  to  the
       number  of white pixels in the PGM and the perimeter is the sum of perimeters of each closed white region
       in the PGM.

       For a grayscale image, these numbers are a function of the threshold of what you want to  call  black  or
       white.   pgmminkowski  reports  these  numbers  as a function of the threshold for all possible threshold
       values.  Since the total surface area can increase  only  as  a  function  of  the  threshold,  it  is  a
       reparameterization  of  the  threshold.   It  turns out that if you consider the other two functions, the
       boundary length and the Euler characteristic, as a function of the first one, the surface,  you  get  two
       functions  that  are  a fingerprint of the picture.  This fingerprint is e.g. sufficient to recognize the
       difference  between  pictures  of  different  crystal  lattices  under  a  scanning  tunnelling  electron
       microscope.

       For more information about Minkowski integrals, see e.g.

       •       J.S.  Kole,  K.  Michielsen,  and H. De Raedt, "Morphological Image Analysis of Quantum Motion in
              Billiards", Phys. Rev. E 63, 016201-1 - 016201-7 (2001)

       •      K. Michielsen and H. De Raedt, "Integral-Geometry Morphological Image Analysis", Phys.  Rep.  347,
              461-538 (2001).

       The output is suitable for direct use as a datafile in gnuplot.

       In  addition  to  the three Minkowski integrals, pgmminkowski also lists the horizontal and vertical edge
       counts.

       This  manual  page was generated by the Netpbm tool 'makeman' from HTML source.  The master documentation
       is at

              http://netpbm.sourceforge.net/doc/pgmminkowski.html

netpbm documentation                             29 October 2002                     PgmminkowskiUserManual(1)

       pgmminkowski - compute Minkowski integral

       There are no command line options defined specifically for pgmminkowski, but it  recognizes  the  options
       common to all programs based on libnetpbm (See  Common Options .)

pgmmorphconv(1) pbmminkowski(1) pgm(1)

pgmminkowskipgmfile