1Pgmminkowski User Manual(0)                        Pgmminkowski User Manual(0)
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NAME

6       pgmminkowski - compute Minkowski integral
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SYNOPSIS

10       pgmminkowski pgmfile
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DESCRIPTION

14       This program is part of Netpbm(1).
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16       pgmminkowski computes the 3 Minkowski integrals of a PGM image.
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18       The  Minkowski  integrals mathematically characterize the shapes in the
19       image and hence are the basis of "morphological image analysis."
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21       Hadwiger's theorem has it that these integrals  are  the  only  motion-
22       invariant,  additive  and  conditionally continuous functions of a two-
23       dimensional image, which means that they are  preserved  under  certain
24       kinds of deformations of the image.  On top of that, they are very easy
25       and quickly calculated.  This makes them of interest for certain  kinds
26       of pattern recognition.
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28       Basically,  the  Minkowski  integrals  are  the  area,  total perimeter
29       length, and the Euler characteristic of the image, where these  metrics
30       apply  to  the  foreground image, not the rectangular PGM image itself.
31       The foreground image consists of all the pixels in the image  that  are
32       white.   For  a  grayscale  image, there is some threshold of intensity
33       applied to categorize pixels into black and white,  and  the  Minkowski
34       integrals  are  calculated  as  a function of this threshold value. The
35       total surface area refers to the number of white pixels in the PGM  and
36       the  perimeter  is the sum of perimeters of each closed white region in
37       the PGM.
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39       For a grayscale image, these numbers are a function of the threshold of
40       what  you want to call black or white.  pgmminkowski reports these num‐
41       bers as a function of the threshold for all possible threshold  values.
42       Since  the  total  surface  area can increase only as a function of the
43       threshold, it is a reparameterization of the threshold.  It  turns  out
44       that  if  you consider the other two functions, the boundary length and
45       the Euler characteristic, as a function of the first one, the  surface,
46       you get two functions that are a fingerprint of the picture.  This fin‐
47       gerprint is e.g. sufficient to recognize the  difference  between  pic‐
48       tures  of  different crystal lattices under a scanning tunnelling elec‐
49       tron microscope.
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51       For more information about Minkowski integrals, see e.g.
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54       ·
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56               K. Michielsen and H. De Raedt, "Integral-Geometry Morphological
57              Image    Analysis",    Phys.    Rep.    347,   461-538   (2001).
58              ⟨http://rugth30.phys.rug.nl/compphys0/2001.htm⟩
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61       ·
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63               J.S. Kole, K. Michielsen, and H. De Raedt, "Morphological Image
64              Analysis  of  Quantum  Motion  in  Billiards",  Phys. Rev. E 63,
65              016201-1             -              016201-7              (2001)
66              ⟨http://rugth30.phys.rug.nl/pdf/prechaos.pdf⟩
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69       The output is suitable for direct use as a datafile in gnuplot.
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71       In  addition  to the three Minkowski integrals, pgmminkowski also lists
72       the horizontal and vertical edge counts.
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SEE ALSO

78       pgmmorphconv(1) pbmminkowski(1) pgm(1)
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AUTHORS

82       Luuk van Dijk, 2001.
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84       Based on work which is Copyright (C) 1989, 1991 by Jef Poskanzer.
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DOCUMENT SOURCE

87       This manual page was generated by the Netpbm tool 'makeman'  from  HTML
88       source.  The master documentation is at
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90              http://netpbm.sourceforge.net/doc/pgmminkowski.html
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92netpbm documentation            29 October 2002    Pgmminkowski User Manual(0)