Pgmminkowski User Manual(0)			   Pgmminkowski User Manual(0)



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 num-
       bers 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 fin-
       gerprint is e.g. sufficient to recognize the  difference	 between  pic-
       tures  of  different crystal lattices under a scanning tunnelling elec-
       tron microscope.

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


       o

	       K. Michielsen and H. De Raedt, "Integral-Geometry Morphological
	      Image    Analysis",    Phys.    Rep.    347,   461-538   (2001).
	      <http://rugth30.phys.rug.nl/compphys0/2001.htm>


       o

	       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>


       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.




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.



netpbm documentation		29 October 2002	   Pgmminkowski User Manual(0)