Pgmminkowski User Manual(0)         Pgmminkowski User Manual(0)



Table Of Contents


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 "morpho-
       logical image analysis."

       Hadwiger's theorem has it that these integrals  are  the
       only  motion-invariant,  additive and conditionally con-
       tinuous 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 fore-
       ground 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 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 thresh-
       old.   It  turns  out that if you consider the other two
       functions, the boundary length and the Euler  character-
       istic,  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  lat-
       tices under a scanning tunnelling electron microscope.

       For more info, see e.g.
        K.  Michielsen and H. De Raedt, "Integral-Geometry Mor-
       phological 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,  pgm-
       minkowski  also  lists  the horizontal and vertical edge
       counts.




SEE ALSO
       pgmmorph(1) pgm(1)


AUTHORS
       Luuk van Dijk, 2001.

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



netpbm documentation    29 October 2Pgmminkowski User Manual(0)
