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Pgmminkowski User Manual
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pgmminkowski
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Updated: 29 October 2002
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Table Of Contents
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NAME
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pgmminkowski - compute Minkowski integral
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SYNOPSIS
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\fBpgmminkowski\fR \fIpgmfile\fR
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DESCRIPTION
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This program is part of Netpbm.
\fBpgmminkowski\fR computes the 3 Minkowski integrals of a PGM image. 
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The Minkowski integrals mathematically characterize the shapes in the
image and hence are the basis of "morphological image analysis."
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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.
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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.
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For a grayscale image, these numbers are a function of the threshold
of what you want to call black or white.  \fBpgmminkowski\fR 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.
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For more information about Minkowski integrals, see e.g. 
.IP \(bu
 K. Michielsen and
H. De Raedt, "Integral-Geometry Morphological Image Analysis",�
Phys. Rep. 347, 461-538 (2001).
.IP \(bu
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)
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The output is suitable for direct use as a datafile in \fBgnuplot\fR.
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In addition to the three Minkowski integrals, \fBpgmminkowski\fR also
lists the horizontal and vertical edge counts.
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SEE ALSO
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\fBpgmmorphconv\fR
\fBpgm\fR
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AUTHORS
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Luuk van Dijk, 2001.
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Based on work which is Copyright (C) 1989, 1991 by Jef Poskanzer.
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\l'5i'
.SH 2
Table Of Contents
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.IP \(bu
SYNOPSIS
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DESCRIPTION
.IP \(bu
SEE ALSO
.IP \(bu
AUTHORS
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