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Pgmminkowski User Manual(1)            General Commands Manual           Pgmminkowski User Manual(1)

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 pre‐
       served 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 recogni‐
       tion.

       Basically, the Minkowski integrals are the area, total perimeter length, and the Euler  char‐
       acteristic of the image, where these metrics apply to the foreground image, not the rectangu‐
       lar  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  con‐
       sider  the  other two functions, the boundary length and the Euler characteristic, as a func‐
       tion of the first one, the surface, you get two functions that are a fingerprint of the  pic‐
       ture.   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 ver‐
       tical 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 .)


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

netpbm documentation                       29 October 2002               Pgmminkowski User Manual(1)