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


       pgmminkowski - compute Minkowski integral


       pgmminkowski pgmfile


       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 info, see e.g.
        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.


       pgmmorphconv(1) pgm(1)


       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)

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