A Certified Reduction Strategy for Homological Image Processing

Abstract

The analysis of digital images using homological procedures is an outstanding topic in the area of Computational Algebraic Topology. In this paper, we describe a certified reduction strategy to deal with digital images, but preserving their homological properties. We stress both the advantages of our approach (mainly, the formalisation of the mathematics allowing us to verify the correctness of algorithms) and some limitations (related to the performance of the running systems inside proof assistants). The drawbacks are overcome using techniques that provide an integration of computation and deduction. Our driving application is a problem in bioinformatics, where the accuracy and reliability of computations are specially requested. arXiv:1306.0806v1 [cs.LO] 4 Jun 2013 so-called Basic Perturbation Lemma (or BPL, in short). The proof of this theorem has been already implemented in the Isabelle/HOL proof assistant [1]. The BPL formalisation presented in this paper is much shorter and compact than that of [1]. There are two reasons for this improvement of the formal proof. The former is that in this work we have followed a new and shorter proof of the BPL (due again to Romero and Sergeraert [34]). The latter is that we have built our formal proof on the powerful SSReflect library of Coq [20] (on the contrary, much of the infrastructure required was defined from scratch in [1]).Apart from the efficiency in the writing of proofs, using SSReflect also has other consequences. Since SSReflect is designed to deal only with finite structures, the proof of the BPL presented here only applies over finitely generated groups (the proof formalised in [1] has not this limitation). Furthermore, dealing with finite structures, and inside the constructive logic of Coq, eases the executability of the proofs, and thus the generation of certified programs (the same tasks in Isabelle/HOL pose more difficulties; see [2]). However, it is worth mention that this limitation does not mean any special hindrance in our work, because digital images are always finite structures.In order to prove the correctness of the generated programs, we must establish, and keep, a link among the initial biomedical picture, and the final smaller data structure where the homological calculations are carried out. This implies a big amount of processing, and does not allow us to execute all the steps inside Coq (the full path has been travelled, but only in toy examples). Then we have appealed to a programming language, Haskell [27] in our case, to integrate computation and deduction.Haskell appears in two different steps of our methodology. In the early stages of development, Haskell prototypes of the algorithms are systematically tested by using the QuickCheck tool [11]. This allows us to discharge many small and common errors, which could hinder the proving process in Coq. In the final computational step, Haskell is used as an oracle for Coq. The most hard parts of the calculation (in our case, an important bottleneck is computing inverse matrices) are delega...

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