# Man Pages

pgmminkowski(1) - phpMan pgmminkowski(1) - phpMan

Command: man perldoc info search(apropos)

```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 "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 defor-
mations 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  thresh-
old.  It turns out that if you consider the other two functions, the boundary length and the Euler characteris-
tic, 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.

?
K. Michielsen and H. De Raedt,  "Integral-Geometry  Morphological  Image  Analysis",  Phys.  Rep.  347,
461-538 (2001).

?
J.S.  Kole,  K.  Michielsen,  and  H. De Raedt, "Morphological Image Analysis of Quantum Motion in Bil-
liards", Phys. Rev. E 63, 016201-1 - 016201-7 (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.