Pgmminkowski User Manual(1) General Commands Manual Pgmminkowski User Manual(1)
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-in-
variant, additive and conditionally continuous functions of a two-dimen-
sional 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 fore-
ground 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 pictures
of different crystal lattices under a scanning tunnelling electron mi-
croscope.
For more information about Minkowski integrals, see e.g.
• ]8;;http://rugth30.phys.rug.nl/pdf/prechaos.pdf\ 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)]8;;\
• 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.
OPTIONS
There are no command line options defined specifically for pgmminkowski,
but it recognizes the options common to all programs based on libnetpbm
(See ]8;;index.html#commonoptions\ Common Options]8;;\ .)
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.
DOCUMENT SOURCE
This manual page was generated by the Netpbm tool 'makeman' from HTML
source. The master documentation is at
http://netpbm.sourceforge.net/doc/pgmminkowski.html
netpbm documentation 29 October 2002 Pgmminkowski User Manual(1)
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