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Pgmminkowski User Manual(0) Pgmminkowski User Manual(0)
pgmminkowski - compute Minkowski integral
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
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 pic-
tures of different crystal lattices under a scanning tunnelling elec-
For more info, see e.g.
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.
Luuk van Dijk, 2001.
Based on work which is Copyright (C) 1989, 1991 by Jef Poskanzer.
netpbm documentation 29 October 2002 Pgmminkowski User Manual(0)
Man(1) output converted with