This program is part of

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.

numbers as a function of the threshold for all possible threshold val-

ues. 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 sur-

face, 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 tun-

nelling electron microscope.

For more information about Minkowski integrals, see e.g.

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The output is suitable for direct use as a datafile in

In addition to the three Minkowski integrals,

the horizontal and vertical edge counts.

Luuk van Dijk, 2001.

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

netpbm documentation 29 October 2002 Pgmminkowski User Manual(0)