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# Homology of parametric images by Saveliev

### From Mathematics Is A Science

*Homology of parametric images* by Peter Saveliev

Draft

Homology theory is not broadly used in image processing. One of the reasons is that homology theory studies sets, i.e., binary images. Meanwhile, images typically seen in practical applications, such as gray scale images or color photos, have parameters. We will discuss a simple approach to homology theory of images with parameters and its applications in image processing.

The paper was presented initially as a talk *Homology of color images* at the Workshop on Applications of Algebraic Topology in Sciences and Engineering, Mathematical Sciences Research Institute, September 2006. Note: The formula in slide 19 (an important one!) is incorrect but fixed in the paper.

Soon after the paper was submitted, I realized that a couple of important references are missing:

- J. Andrew Bangham, J. R. Hidalgo, Richard Harvey, Gavin C. Cawley, The Segmentation of Images via Scale-Space Trees, British Machine Vision Conference, 1998.
- P. Monasse and F. Guichard. Fast computation of a contrast invariant image representation. IEEE Transactions on Image Processing, 9(5):860–872, 2000.

The papers take some similar approaches to the problem. The differences are quite important. They don't use homology. This makes their approach inapplicable to images of dimension above 2. Color images are not discussed. Object counting is not addressed. The way image simplification works is very similar though.

The paper isn't as interesting as the reviews it received, still to come.

Broader issues are discussed under Computational topology.

Full text: Homology of parametric images