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Verifying an algorithm computing Discrete Vector Fields for digital imaging ∗ J. Heras, M. Poza, and J. Rubio Department of Mathematics and Computer Science, University of La Rioja Calculemus 2012 ∗ Partially supported by Ministerio de Educaci´ on y Ciencia, project MTM2009-13842-C02-01, and by European Commission FP7, STREP project ForMath, n. 243847 J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 1/31

Motivation Algebraic Topology and Digital Images Digital Image Homology groups H 0 = Z 2 ⊕ Z 2 H 1 = Z 2 ⊕ Z 2 ⊕ Z 2 C 0 = Z 2 [ vertices ] C 1 = Z 2 [ edges ] C 2 = Z 2 [ triangles ] Simplicial complex J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 2/31

Motivation Algebraic Topology and Digital Images Digital Image Homology groups H 0 = Z 2 ⊕ Z 2 H 1 = Z 2 ⊕ Z 2 ⊕ Z 2 Reduced Chain complex C 0 = Z 2 [ vertices ] C 1 = Z 2 [ edges ] C 2 = Z 2 [ triangles ] Simplicial complex J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 2/31

Motivation Algebraic Topology and Digital Images Digital Image Homology groups H 0 = Z 2 ⊕ Z 2 H 1 = Z 2 ⊕ Z 2 ⊕ Z 2 Reduced Chain Complex C 0 = Z 2 [ vertices ] C 1 = Z 2 [ edges ] C 2 = Z 2 [ triangles ] Simplicial complex Chain complex J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 2/31

Motivation Algebraic Topology and Digital Images Digital Image Homology groups H 0 = Z 2 ⊕ Z 2 H 1 = Z 2 ⊕ Z 2 ⊕ Z 2 Reduced Chain Complex C 0 = Z 2 [ vertices ] C 1 = Z 2 [ edges ] C 2 = Z 2 [ triangles ] Simplicial complex Chain complex J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 2/31

Motivation Algebraic Topology and Digital Images Digital Image Homology groups H 0 = Z 2 ⊕ Z 2 H 1 = Z 2 ⊕ Z 2 ⊕ Z 2 Reduced Chain Complex C 0 = Z 2 [ vertices ] C 1 = Z 2 [ edges ] C 2 = Z 2 [ triangles ] Simplicial complex Chain complex J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 2/31

Motivation Algebraic Topology and Digital Images Digital Image Homology groups H 0 = Z 2 ⊕ Z 2 H 1 = Z 2 ⊕ Z 2 ⊕ Z 2 Reduced Chain Complex C 0 = Z 2 [ vertices ] C 1 = Z 2 [ edges ] C 2 = Z 2 [ triangles ] Simplicial complex Chain complex J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 2/31

Motivation Goal Application: Analysis of biomedical images J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 3/31

Motivation Goal Application: Analysis of biomedical images Requirements: Efficiency Reliability J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 3/31

Motivation Goal Application: Analysis of biomedical images Requirements: Efficiency Reliability Goal A formally verified efficient method to compute homology from digital images J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 3/31

Motivation Goal Digital Image triangulation Simplicial Complex graded structure properties Chain Complex reduction computing Homology J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 4/31

Motivation Goal Digital Image triangulation Simplicial Complex graded structure properties Chain Complex reduction computing Homology J. Heras, M. D´ en` es, G. Mata, A. M¨ ortberg, M. Poza, and V. Siles. Towards a certified computation of homology groups. In proceedings 4th International Workshop on Computational Topology in Image Context. Lecture Notes in Computer Science, 7309, pages 49–57, 2012. J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 4/31

Motivation Goal Digital Image triangulation Simplicial Complex graded structure properties Chain Complex reduction computing Homology Bottleneck Compute Homology from Chain Complexes J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 4/31

Motivation Goal Digital Image triangulation Simplicial Complex graded structure properties Chain Complex reduction computing Homology Goal of this work Formalization in Coq/SSReflect of a procedure to reduce the size of Chain Complexes but preserving homology J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 4/31

Table of Contents Mathematical background 1 An abstract method 2 An effective method 3 Application 4 Conclusions and Further work 5 J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 5/31

Mathematical background Table of Contents Mathematical background 1 An abstract method 2 An effective method 3 Application 4 Conclusions and Further work 5 J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 6/31

Mathematical background Chain Complexes Definition A chain complex C ∗ is a pair of sequences C ∗ = ( C q , d q ) q ∈ Z where: For every q ∈ Z , the component C q is a Z 2 -module, the chain group of degree q For every q ∈ Z , the component d q is a module morphism d q : C q → C q − 1 , the differential map For every q ∈ Z , the composition d q d q +1 is null: d q d q +1 = 0 J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 7/31

Mathematical background Chain Complexes Definition A chain complex C ∗ is a pair of sequences C ∗ = ( C q , d q ) q ∈ Z where: For every q ∈ Z , the component C q is a Z 2 -module, the chain group of degree q For every q ∈ Z , the component d q is a module morphism d q : C q → C q − 1 , the differential map For every q ∈ Z , the composition d q d q +1 is null: d q d q +1 = 0 Definition If C ∗ = ( C q , d q ) q ∈ Z is a chain complex: The image B q = im d q +1 ⊆ C q is the (sub)module of q-boundaries The kernel Z q = ker d q ⊆ C q is the (sub)module of q-cycles J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 7/31

Mathematical background Chain Complexes Definition A chain complex C ∗ is a pair of sequences C ∗ = ( C q , d q ) q ∈ Z where: For every q ∈ Z , the component C q is a Z 2 -module, the chain group of degree q For every q ∈ Z , the component d q is a module morphism d q : C q → C q − 1 , the differential map For every q ∈ Z , the composition d q d q +1 is null: d q d q +1 = 0 Definition If C ∗ = ( C q , d q ) q ∈ Z is a chain complex: The image B q = im d q +1 ⊆ C q is the (sub)module of q-boundaries The kernel Z q = ker d q ⊆ C q is the (sub)module of q-cycles Definition Let C ∗ = ( C q , d q ) q ∈ Z be a chain complex. For each degree n ∈ Z , the n-homology module of C ∗ is defined as the quotient module H n ( C ∗ ) = Z n B n J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 7/31

� � Mathematical background Effective Homology Theory Reduction Definition A reduction ρ between two chain complexes C ∗ y D ∗ (denoted by ρ : C ∗ ⇒ ⇒ D ∗ ) is a tern ρ = ( f , g , h ) h f � D ∗ C ∗ g satisfying the following relations: 1) fg = id D ∗ ; 2) d C h + hd C = id C ∗ − gf ; 3) fh = 0 ; hg = 0 ; hh = 0 . Theorem ⇒ D ∗ , then C ∗ ∼ If C ∗ ⇒ = D ∗ ⊕ A ∗ , with A ∗ acyclic, what implies that H n ( C ∗ ) ∼ = H n ( D ∗ ) for all n. J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 8/31

Mathematical background Discrete Morse Theory Discrete Morse Theory A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology, 2010. http://arxiv.org/abs/1005.5685v1. J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 9/31

Mathematical background Discrete Morse Theory Discrete Morse Theory A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology, 2010. http://arxiv.org/abs/1005.5685v1. J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 9/31

Mathematical background Discrete Morse Theory Discrete Morse Theory A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology, 2010. http://arxiv.org/abs/1005.5685v1. J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 9/31

Mathematical background Discrete Morse Theory Discrete Morse Theory A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology, 2010. http://arxiv.org/abs/1005.5685v1. J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 9/31