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-in‐
22       variant,  additive  and conditionally continuous functions of a two-di‐
23       mensional 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 ap‐
33       plied to categorize pixels into black and white, and the Minkowski  in‐
34       tegrals are calculated as a function of this threshold value. The total
35       surface area refers to the number of white pixels in the  PGM  and  the
36       perimeter  is  the sum of perimeters of each closed white region in the
37       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               J.S. Kole, K. Michielsen, and H. De Raedt, "Morphological Image
57              Analysis  of  Quantum  Motion  in  Billiards",  Phys. Rev. E 63,
58              016201-1             -              016201-7              (2001)
59              ⟨http://rugth30.phys.rug.nl/pdf/prechaos.pdf⟩
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62       •      K.  Michielsen and H. De Raedt, "Integral-Geometry Morphological
63              Image Analysis", Phys. Rep. 347, 461-538 (2001).
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67       The output is suitable for direct use as a datafile in gnuplot.
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69       In addition to the three Minkowski integrals, pgmminkowski  also  lists
70       the horizontal and vertical edge counts.
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OPTIONS

76       There  are  no  command  line  options  defined  specifically  for pgm‐
77       minkowski, but it recognizes the options common to all  programs  based
78       on libnetpbm (See
79        Common Options ⟨index.html#commonoptions⟩ .)
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SEE ALSO

83       pgmmorphconv(1) pbmminkowski(1) pgm(1)
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AUTHORS

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

92       This  manual  page was generated by the Netpbm tool 'makeman' from HTML
93       source.  The master documentation is at
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95              http://netpbm.sourceforge.net/doc/pgmminkowski.html
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97netpbm documentation            29 October 2002    Pgmminkowski User Manual(0)