Towards a Certified Computation of Homology Groups for Digital Images

Abstract

In this paper we report on a project to obtain a verified computation of homology groups of digital images. The methodology is based on programming and executing inside the Coq proof assistant. Though more research is needed to integrate and make efficient more processing tools, we present some examples partially computed in Coq from real biomedical images.

Computing persistent homology within Coq/SSReflect

“…In previous work, see [28,27], we have formalized the notions presented in subsections 2.1 and 2.2. However, for the sake of clarity of the exposition we include the main definitions and results which have been developed previously.…”

“…Homological techniques have been successfully applied in the biomedical context, see [36,35,27]. In this environment it is necessary to have both efficient and reliable software systems; therefore the use of formally verified efficient algorithms seems desirable.…”

Verifying an Algorithm Computing Discrete Vector Fields for Digital Imaging

“…In the context of Algebraic Digital Topology, this issue can be tackled by means of the computation of the homology group H 0 of the monochromatic image. This task can be performed in Coq through the formally verified programs which were presented in [15]. Nevertheless, such programs are not able to handle images like the one of the right side of Figure 1 due to its size (let us remark that Coq is a proof assistant tool and not a computer algebra system).…”

“…Nevertheless, such programs are not able to handle images like the one of the right side of Figure 1 due to its size (let us remark that Coq is a proof assistant tool and not a computer algebra system). In order to overcome this drawback, as we have explained at the end of the previous section, we have integrate our reduction programs with the ones presented in [15]. Using this approach, we can successfully compute the homology of the biomedical images in just 25 seconds, a remarkable time for an execution inside Coq.…”

A Coq Formalization of Finitely Presented Modules

“…A previous work, that one of the authors was involved in, studied ways to compute homology groups of vector spaces [18,17] in Coq. When generalizing this to commutative rings the universal coefficient theorem of homology [15] states that most of the homological information of an R-module over a ring R can be computed by only doing computations with elements in Z.…”