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Under GNU General Public License
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Pgmminkowski User Manual(0) Pgmminkowski User Manual(0)
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-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 numbers 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 fingerprint is e.g. sufficient to recognize the difference between
pictures of different crystal lattices under a scanning tunnelling electron microscope.
For more information about Minkowski integrals, see e.g.
o
K. Michielsen and H. De Raedt, "Integral-Geometry Morphological Image Analysis",
Phys. Rep. 347, 461-538 (2001). <http://rugth30.phys.rug.nl/compphys0/2001.htm>
o
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)
<http://rugth30.phys.rug.nl/pdf/prechaos.pdf>
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.
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(0)
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On Apache
Under GNU General Public License
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