A Certified Reduction Strategy for Homological Image Processing

Transcription

1 A Certified Reduction Strategy for Homological Image Processing M. Poza, C. Domínguez, J. Heras, and J. Rubio Department of Mathematics and Computer Science, University of La Rioja 19 September 2014 PROLE 14 Partially supported by Ministerio de Educación y Ciencia, project MTM C02-01, and by European Commission FP7, STREP project ForMath, n J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 1/24

2 Goal Motivation Goal A formally-verified and efficient method to analyse digital images. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 2/24

3 Motivation Goal Goal A formally-verified and efficient method to analyse digital images. A motivating example: counting synapses Synapses are the points of connection between neurons. Relevance: Computational capabilities of the brain. Procedures to modify the synaptic density may be an important asset in the treatment of neurological diseases. An automated, efficient, and reliable method is necessary. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 2/24

4 Counting Synapses Motivation Counting Synapses J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 3/24

5 Counting Synapses Motivation Counting Synapses J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 3/24

6 Counting Synapses Motivation Counting Synapses J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 3/24

7 Motivation Algebraic Topology and Digital Images Algebraic Topology and Digital Images Digital Image Homology groups H 0 = Z Z H 1 = Z Z Z Simplicial Complex C 0 = vertices C 1 = edges C 2 = triangles Chain Complex J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 4/24

8 Motivation Algebraic Topology and Digital Images Algebraic Topology and Digital Images Digital Image Homology groups H 0 = Z Z H 1 = Z Z Z Simplicial Complex C 0 = vertices C 1 = edges C 2 = triangles Chain Complex J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 4/24

9 Motivation Algebraic Topology and Digital Images Algebraic Topology and Digital Images Digital Image Homology groups H 0 = Z Z H 1 = Z Z Z Simplicial Complex C 0 = vertices C 1 = edges C 2 = triangles Chain Complex J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 4/24

10 Motivation Algebraic Topology and Digital Images Algebraic Topology and Digital Images Digital Image Homology groups H 0 = Z Z H 1 = Z Z Z Simplicial Complex C 0 = vertices C 1 = edges C 2 = triangles Chain Complex J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 4/24

11 Motivation Algebraic Topology and Digital Images Algebraic Topology and Digital Images Digital Image Homology groups H 0 = Z Z H 1 = Z Z Z Simplicial Complex C 0 = vertices C 1 = edges C 2 = triangles Chain Complex J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 4/24

12 Motivation Interactive Proof Assistants Interactive Proof Assistants What is an Interactive Proof Assistant? Software tool for the development of formal proofs. Man-Machine collaboration: Human: design the proofs. Machine: fill the gaps. Examples: Isabelle, Hol, ACL2, Coq,... J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 5/24

13 Motivation Interactive Proof Assistants Interactive Proof Assistants What is an Interactive Proof Assistant? Software tool for the development of formal proofs. Man-Machine collaboration: Human: design the proofs. Machine: fill the gaps. Examples: Isabelle, Hol, ACL2, Coq,... Applications: Mathematical proofs: Four Colour Theorem. Feit-Thompson Theorem. Kepler conjecture. Software and Hardware verification: C compiler. AMD5K86 microprocessor. sel4: operating-system kernel. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 5/24

14 Coq/SSReflect Motivation Interactive Proof Assistants Coq: An Interactive Proof Assistant. Based on Calculus of Inductive Constructions. Interesting feature: program extraction from a constructive proof. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 6/24

15 Coq/SSReflect Motivation Interactive Proof Assistants Coq: An Interactive Proof Assistant. Based on Calculus of Inductive Constructions. Interesting feature: program extraction from a constructive proof. SSReflect: Extension of Coq. Developed while formalising the Four Colour Theorem by G. Gonthier. It has been used in the formalisation of the Feit-Thompson Theorem. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 6/24

16 Motivation Problem Problem Goal A formally-verified (using Coq) and efficient method (from Algebraic Topology) to analyse digital images. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 7/24

17 Motivation Problem Problem Goal A formally-verified (using Coq) and efficient method (from Algebraic Topology) to analyse digital images. Problem Coq is not designed for computing, but for proving. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 7/24

18 Motivation Problem Problem Goal A formally-verified (using Coq) and efficient method (from Algebraic Topology) to analyse digital images. Problem Coq is not designed for computing, but for proving. Demo. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 7/24

19 Outline A Reduction Algorithm 1 A Reduction Algorithm 2 The Role of Haskell 3 Conclusions and Further Work J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 8/24

20 Outline A Reduction Algorithm 1 A Reduction Algorithm 2 The Role of Haskell 3 Conclusions and Further Work J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 9/24

21 General Idea A Reduction Algorithm Digital Image triangulation Simplicial Complex graded structure properties Chain Complex construction Matrices computing reduction Homology J. Heras, M. Dénès, G. Mata, A. Mörtberg, 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, J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 10/24

22 A Reduction Algorithm General Idea Digital Image triangulation Simplicial Complex graded structure properties Chain Complex construction Matrices computing reduction Homology Concrete Goal Formalise an algorithm that reduces the information but keeps the homological properties. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 10/24

23 Discrete Morse Theory A Reduction Algorithm A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology, J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 11/24

24 Discrete Morse Theory A Reduction Algorithm A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology, J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 11/24

25 Discrete Morse Theory A Reduction Algorithm A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology, J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 11/24

26 Discrete Morse Theory A Reduction Algorithm A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology, J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 11/24

27 Discrete Morse Theory A Reduction Algorithm A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology, J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 11/24

28 Discrete Morse Theory A Reduction Algorithm A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology, J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 11/24

29 Discrete Morse Theory A Reduction Algorithm A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology, J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 11/24

30 A Reduction Algorithm Discrete Morse Theory A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology, From a digital image, we construct 2 incidence matrices: Vertex-Edge matrix. Edge-Triangle matrix. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 11/24

31 A Reduction Algorithm Discrete Morse Theory A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology, From a digital image, we construct 2 incidence matrices: Vertex-Edge matrix. Edge-Triangle matrix. Compute an admissible discrete vector field dvf. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 11/24

32 A Reduction Algorithm Discrete Morse Theory A. Romero and F. Sergeraert. Discrete Vector Fields and Fundamental Algebraic Topology, From a digital image, we construct 2 incidence matrices: Vertex-Edge matrix. Edge-Triangle matrix. Compute an admissible discrete vector field dvf. Use dvf to reduce the matrices: Vertex-Edge matrix: Edge-Triangle matrix: J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 11/24

33 A Reduction Algorithm Formalised Results Two main results have been formalised: The Coq s implementation of Romero-Sergeraert s is correct. Given incidence matrices and a discrete vector field for them, we can reduce the size of the matrices preserving their homological properties. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 12/24

34 A Reduction Algorithm Formalised Results Two main results have been formalised: The Coq s implementation of Romero-Sergeraert s is correct. Given incidence matrices and a discrete vector field for them, we can reduce the size of the matrices preserving their homological properties. Coq development: 45,000 lines of code, 2,500 theorems, and 1,700 definitions. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 12/24

35 A Reduction Algorithm Formalised Results Two main results have been formalised: The Coq s implementation of Romero-Sergeraert s is correct. Given incidence matrices and a discrete vector field for them, we can reduce the size of the matrices preserving their homological properties. Coq development: 45,000 lines of code, 2,500 theorems, and 1,700 definitions. Coq development about homology of digital images: 3,800 lines of code, 260 theorems, and 200 definitions. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 12/24

36 Outline The Role of Haskell 1 A Reduction Algorithm 2 The Role of Haskell Fast Proof Prototyping Improving the Performance of the Computations 3 Conclusions and Further Work J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 13/24

37 Outline The Role of Haskell Fast Proof Prototyping 1 A Reduction Algorithm 2 The Role of Haskell Fast Proof Prototyping Improving the Performance of the Computations 3 Conclusions and Further Work J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 14/24

38 The Role of Haskell Fast Proof Prototyping Fast Proof Prototyping 3rd Workshop on Mechanizing Metatheory (2008) The use of proof assistants in programming language research is still not commonplace: the available tools are confusingly diverse, difficult to learn, inadequately documented, and lacking in specific library facilities required for work in programming languages. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 15/24

39 The Role of Haskell Fast Proof Prototyping Fast Proof Prototyping 3rd Workshop on Mechanizing Metatheory (2008) The use of proof assistants in programming language research is still not commonplace: the available tools are confusingly diverse, difficult to learn, inadequately documented, and lacking in specific library facilities required for work in programming languages. Proof assistants have improved in the last few years, but they are not there yet. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 15/24

40 The Role of Haskell Fast Proof Prototyping From computation to verification through testing Haskell as programming language. QuickCheck to test the programs. Coq/SSReflect to verify the correctness of the programs. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 16/24

41 The Role of Haskell Fast Proof Prototyping From computation to verification through testing Haskell: Algorithm (gen adm dvf ) Input: A matrix M. Output: An admissible discrete vector field for M. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 17/24

42 The Role of Haskell Fast Proof Prototyping From computation to verification through testing Haskell: Algorithm (gen adm dvf ) Input: A matrix M. Output: An admissible discrete vector field for M. QuickCheck: A specification of the properties which our program must verify. Testing them: Towards verification. Detect bugs in early stages. > quickcheck M -> admissible (gen_adm_dvf M) OK, passed 100 tests J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 17/24

43 The Role of Haskell Fast Proof Prototyping From computation to verification through testing Haskell: Algorithm (gen adm dvf ) Input: A matrix M. Output: An admissible discrete vector field for M. QuickCheck: A specification of the properties which our program must verify. Testing them: Towards verification. Detect bugs in early stages. > quickcheck M -> admissible (gen_adm_dvf M) OK, passed 100 tests Coq/SSReflect: SSReflect Theorem: Theorem gen_adm_dvf_is_admissible (M : seq (seq Z2)) : admissible (gen_adm_dvf M). J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 17/24

44 Outline The Role of Haskell Improving the Performance of the Computations 1 A Reduction Algorithm 2 The Role of Haskell Fast Proof Prototyping Improving the Performance of the Computations 3 Conclusions and Further Work J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 18/24

45 The Role of Haskell Improving the Performance of the Computations How much is the performance improved? 500 randomly generated matrices: Initial size of the matrices: Time: 12 seconds. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 19/24

46 The Role of Haskell Improving the Performance of the Computations How much is the performance improved? 500 randomly generated matrices: Initial size of the matrices: Time: 12 seconds. After reduction: Size of matrices: Time: milliseconds. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 19/24

47 The Role of Haskell Improving the Performance of the Computations How much is the performance improved? 500 randomly generated matrices: Initial size of the matrices: Time: 12 seconds. After reduction: Size of matrices: Time: milliseconds. Problem For bigger matrices, the reduction process takes a lot of time. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 19/24

48 Solution The Role of Haskell Improving the Performance of the Computations Use Haskell as an Oracle: J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 20/24

49 Solution The Role of Haskell Improving the Performance of the Computations Use Haskell as an Oracle: 1 Detect computational bottlenecks. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 20/24

50 The Role of Haskell Improving the Performance of the Computations Solution Use Haskell as an Oracle: 1 Detect computational bottlenecks. 2 When an expensive computation is required, call Haskell. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 20/24

51 The Role of Haskell Improving the Performance of the Computations Solution Use Haskell as an Oracle: 1 Detect computational bottlenecks. 2 When an expensive computation is required, call Haskell. 3 Check in Coq that the result is correct. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 20/24

52 The Role of Haskell Improving the Performance of the Computations Solution Use Haskell as an Oracle: 1 Detect computational bottlenecks. 2 When an expensive computation is required, call Haskell. 3 Check in Coq that the result is correct. 4 Continue the computation in Coq. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 20/24

53 The Role of Haskell Improving the Performance of the Computations Solution Use Haskell as an Oracle: 1 Detect computational bottlenecks. 2 When an expensive computation is required, call Haskell. 3 Check in Coq that the result is correct. 4 Continue the computation in Coq. Example (Inverse of a matrix): Problem related to handling big datastructures. Given a matrix M, compute (in Haskell) its inverse matrix M 1. Prove in Coq that M M 1 = 1 (by compute). Continue the computation in Coq. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 20/24

54 Outline Conclusions and Further Work 1 A Reduction Algorithm 2 The Role of Haskell 3 Conclusions and Further Work J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 21/24

55 Conclusions and Further Work Conclusions and Further Work Conclusions: Formalisation of reduction algorithm for digital images. Methodology to overcome efficiency problems. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 22/24

56 Conclusions and Further Work Conclusions and Further Work Conclusions: Formalisation of reduction algorithm for digital images. Methodology to overcome efficiency problems. Further work: Obtain better performance: Improve algorithms. Implement more efficient data structures. Improve the running environments in proof assistants. Extract certified code from Coq. Apply methodology and techniques to: Persistent homology and Zigzag persistence. Application: automate the location of neuronal structure from a stack of images. J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 22/24

57 Conclusions and Further Work Bibliography about this work M. Poza (2013): Certifying homological algorithms to study biomedical images. Ph.D. thesis, University of La Rioja. M. Poza, C. Domínguez, J. Heras, and J. Rubio (2014): A certified reduction strategy for homological image processing. ACM Transactions on Computational Logic 15(3). M. Poza, C. Domínguez, J. Heras, and J. Rubio (2014): A certified reduction strategy for homological image processing. Arxiv: M. Poza, C. Domínguez, J. Heras, and J. Rubio (2014): J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 23/24

58 Thank you for your attention Questions? A Certified Reduction Strategy for Homological Image Processing M. Poza, C. Domínguez, J. Heras, and J. Rubio Department of Mathematics and Computer Science, University of La Rioja 19 September 2014 PROLE 14 J. Heras (U. of La Rioja) A Certified Reduction Strategy for Homological Image Processing 24/24