NAME

       pgmminkowski - compute Minkowski integral

SYNOPSIS

       pgmminkowski pgmfile

DESCRIPTION

       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)
              ⟨http://rugth30.phys.rug.nl/pdf/prechaos.pdf⟩

       •      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.

OPTIONS

       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 ⟨index.html#commonoptions⟩ .)

SEE ALSO

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

AUTHORS

       Luuk van Dijk, 2001.

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

DOCUMENT SOURCE

       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