pgmminkowski - compute Minkowski integral
pgmminkowski pgmfile
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.
- •
-
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)
- •
- 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.
There are no command line options defined specifically for
pgmminkowski,
but it recognizes the options common to all programs based on libnetpbm (See
Common Options .)
pgmmorphconv(1) pbmminkowski(1) pgm(1)
Luuk van Dijk, 2001.
Based on work which is Copyright (C) 1989, 1991 by Jef Poskanzer.
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