1Pgmminkowski User Manual(0) Pgmminkowski User Manual(0)
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6 pgmminkowski - compute Minkowski integral
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10 pgmminkowski pgmfile
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14 This program is part of Netpbm(1).
15 pgmminkowski computes the 3 Minkowski integrals of a PGM image.
17 The Minkowski integrals mathematically characterize the shapes in the
19 image and hence are the basis of "morphological image analysis."
20 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.
27 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.
38 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.
50 For more information about Minkowski integrals, see e.g.
52 ·
55 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⟩
59 ·
62 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⟩
67 The output is suitable for direct use as a datafile in gnuplot.
70 In addition to the three Minkowski integrals, pgmminkowski also lists
72 the horizontal and vertical edge counts.
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78 There are no command line options defined specifically for pgm‐
79 minkowski, but it recognizes the options common to all programs based
80 on libnetpbm (See
81 Common Options ⟨index.html#commonoptions⟩ .)
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85 pgmmorphconv(1) pbmminkowski(1) pgm(1)
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89 Luuk van Dijk, 2001.
90 Based on work which is Copyright (C) 1989, 1991 by Jef Poskanzer.
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94 This manual page was generated by the Netpbm tool 'makeman' from HTML
95 source. The master documentation is at
96 http://netpbm.sourceforge.net/doc/pgmminkowski.html
98netpbm documentation 29 October 2002 Pgmminkowski User Manual(0)