Verifying an Algorithm Computing Discrete Vector Fields for Digital Imaging

In this paper, we present a formalization of an algorithm to construct admissible discrete vector fields in the Coq theorem prover taking advantage of the SSReflect library. Discrete vector fields are a tool which has been welcomed in the homological analysis of digital images since it provides a procedure to reduce the amount of information but preserving the homological properties. In particular, thanks to discrete vector fields, we are able to compute, inside Coq, homological properties of biomedical images which otherwise are out of the reach of this system.

[1]  Assia Mahboubi,et al.  An introduction to small scale reflection in Coq , 2010, J. Formaliz. Reason..

[2]  Ana Romero,et al.  Discrete Vector Fields and Fundamental Algebraic Topology , 2010, ArXiv.

[3]  Francisco-Jesús Martín-Mateos,et al.  Applying ACL2 to the Formalization of Algebraic Topology: Simplicial Polynomials , 2011, ITP.

[4]  R. Forman.  Morse Theory for Cell Complexes , 1998 .

[5]  N. Jacobson,et al.  Basic Algebra II , 1989 .

[6]  César Domínguez,et al.  Effective homology of bicomplexes, formalized in Coq , 2011, Theor. Comput. Sci..

[7]  Anders Mörtberg,et al.  Constructive Algebra in Functional Programming and Type Theory , 2010 .

[8]  Paul Graham.  ANSI Common Lisp , 1995 .

[9]  W. Marsden.  I and J , 2012 .

[10]  Georges Gonthier,et al.  Formal Proof—The Four- Color Theorem , 2008 .

[11]  Peyton Jones,et al.  Haskell 98 language and libraries : the revised report , 2003 .

[12]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[13]  Jesús Aransay,et al.  A Mechanized Proof of the Basic Perturbation Lemma , 2008, Journal of Automated Reasoning.

[14]  Pierre Courtieu,et al.  Efficient Reasoning about Executable Specifications in Coq , 2002, TPHOLs.

[15]  Krzysztof Galkowski,et al.  Conclusions and Further Work , 2007 .

[16]  Ana Romero,et al.  Homotopy groups of suspended classifying spaces: An experimental approach , 2013, Math. Comput..

[17]  Jónathan Heras,et al.  Towards a Certified Computation of Homology Groups for Digital Images , 2012, CTIC.

[18]  Germán Cuesto,et al.  Phosphoinositide-3-Kinase Activation Controls Synaptogenesis and Spinogenesis in Hippocampal Neurons , 2011, The Journal of Neuroscience.

[19]  Jónathan Heras,et al.  Proving with ACL2 the Correctness of Simplicial Sets in the Kenzo System , 2010, LOPSTR.

[20]  Koen Claessen,et al.  QuickCheck: a lightweight tool for random testing of Haskell programs , 2000, ICFP.

[21]  Assia Mahboubi,et al.  Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination , 2012, Log. Methods Comput. Sci..

[22]  Francis Sergeraert,et al.  Constructive algebraic topology , 1999, SIGS.

[23]  I. K. Wood,et al.  Neuroscience: Exploring the brain , 1996 .