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Homology of images

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This is how the topology of image is captured by Pixcavator. For the math background see Homology in dimension 1.

The image.
A and B are 0-cycles, C and C’ are 1-cycles.

The homology groups of this image are vector spaces generated by these cycles, as follows. Consider first the image to the right. Its cycles are depicted below. Then the homology groups are the following.

H0 = R2, the basis vectors are A and B.

H1 = R2, the basis vectors are C and C’.

Thus, the dimension of the 0th or 1st homology group is equal to the 0th or 1st Betti number respectively. We do not describe yet how these vector spaces are constructed. The reason why we want to discuss them now and not just the Betti numbers is twofold. First, we want to emphasize that the cycles are not only counted but also captured. Second, as we shall see shortly, this approach allows us to perform algebraic operations with cycles.

We use Rk for homology groups because we want to stay within elementary linear algebra and avoid using groups. Because of the nature of digital images, this will suffice. Using vector spaces over R is also a matter of convenience. In Pixcavator, the computations are carried out in binary arithmetic, so that instead of R2, the homology group would be B2, where B = {0,1} = Z2.

For examples of cycles in real-life images, see Contours.


See also Homology software and Image analysis software.