# FreeBSD Manual Pages

```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-in-
variant,	 additive  and conditionally continuous	functions of a two-di-
mensional 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 ap-
plied to	categorize pixels into black and white,	and the	Minkowski  in-
tegrals 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  pic-
tures  of  different crystal lattices under a scanning tunnelling elec-
tron 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.

pgmmorphconv(1) pbmminkowski(1) pgm(5)

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)
```

NAME | SYNOPSIS | DESCRIPTION | SEE ALSO | AUTHORS | DOCUMENT SOURCE

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