pgmminkowski

   Updated: 29 October 2002
   Table Of Contents

NAME

   pgmminkowski - compute Minkowski integral

SYNOPSIS

   pgmminkowski pgmfile

DESCRIPTION

   This	  program  is part of Netpbm. 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  func-
tions of a
   two-dimensional image, which means that they are preserved un-
der 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 perime-
ter length, and
   the Euler characteristic of the image, where these metrics ap-
ply 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 catego-
rize pixels
   into	  black and white, and the Minkowski integrals are calcu-
lated 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 charac-
teristic, 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. suffi-
cient to recognize
   the	difference between pictures of different crystal lattices
under a
   scanning tunnelling electron microscope.

   For	more info, see e.g. K. Michielsen and H. De Raedt, "Inte-
gral-Geometry
   Morphological  Image	 Analysis",   Phys.  Rep.  347,	  461-538
(2001).

   The	output	is  suitable for direct use as a datafile in gnu-
plot.

   In addition to the three Minkowski integrals, pgmminkowski al-
so lists the
   horizontal and vertical edge counts.

SEE ALSO

   pgmmorph pgm

AUTHORS

   Luuk van Dijk, 2001.

   Based  on  work  which  is  Copyright  (C)  1989,  1991 by Jef
Poskanzer.
     _________________________________________________________________



Table Of Contents

     * NAME
     * SYNOPSIS
     * DESCRIPTION
     * SEE ALSO
     * AUTHORS

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