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
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Verifying an algorithm computing Discrete VectorFields 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 Educacion 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
Simplicial complex
Homology groups
C0 = Z2[vertices]C1 = Z2[edges]C2 = Z2[triangles]
H1 = Z2 ⊕ Z2 ⊕ Z2
H0 = Z2 ⊕ Z2
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
Simplicial complex
Homology groups
C0 = Z2[vertices]C1 = Z2[edges]C2 = Z2[triangles]
H1 = Z2 ⊕ Z2 ⊕ Z2
H0 = Z2 ⊕ Z2
ReducedChain 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
Simplicial complex Chain complex
Homology groups
C0 = Z2[vertices]C1 = Z2[edges]C2 = Z2[triangles]
H1 = Z2 ⊕ Z2 ⊕ Z2
H0 = Z2 ⊕ Z2
ReducedChain 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
Simplicial complex Chain complex
Homology groups
C0 = Z2[vertices]C1 = Z2[edges]C2 = Z2[triangles]
H1 = Z2 ⊕ Z2 ⊕ Z2
H0 = Z2 ⊕ Z2
ReducedChain 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
Simplicial complex Chain complex
Homology groups
C0 = Z2[vertices]C1 = Z2[edges]C2 = Z2[triangles]
H1 = Z2 ⊕ Z2 ⊕ Z2
H0 = Z2 ⊕ Z2
ReducedChain 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
Simplicial complex Chain complex
Homology groups
C0 = Z2[vertices]C1 = Z2[edges]C2 = Z2[triangles]
H1 = Z2 ⊕ Z2 ⊕ Z2
H0 = Z2 ⊕ Z2
ReducedChain 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
Requirements:
EfficiencyReliability
Goal
A formally verified efficient method to compute homology fromdigital 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:
EfficiencyReliability
Goal
A formally verified efficient method to compute homology fromdigital 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:
EfficiencyReliability
Goal
A formally verified efficient method to compute homology fromdigital images
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 3/31
Motivation
Goal
Digital Image
Simplicial Complex
Chain Complex
Homology
triangulation
graded structure
computing
properties
reduction
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 4/31
Motivation
Goal
Digital Image
Simplicial Complex
Chain Complex
Homology
triangulation
graded structure
computing
properties
reduction
J. Heras, M. Denes, G. Mata, A. Mortberg, M. Poza, and V. Siles. Towards acertified computation of homology groups. In proceedings 4th InternationalWorkshop on Computational Topology in Image Context. Lecture Notes inComputer 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
Simplicial Complex
Chain Complex
Homology
triangulation
graded structure
computing
properties
reduction
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
Simplicial Complex
Chain Complex
Homology
triangulation
graded structure
computing
properties
reduction
Goal of this work
Formalization in Coq/SSReflect of a procedure to reduce the sizeof 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
1 Mathematical background
2 An abstract method
3 An effective method
4 Application
5 Conclusions and Further work
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 5/31
Mathematical background
Table of Contents
1 Mathematical background
2 An abstract method
3 An effective method
4 Application
5 Conclusions and Further work
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∗ = (Cq , dq)q∈Z where:
For every q ∈ Z, the component Cq is a Z2-module, the chain group of degree q
For every q ∈ Z, the component dq is a module morphism dq : Cq → Cq−1, thedifferential map
For every q ∈ Z, the composition dqdq+1 is null: dqdq+1 = 0
Definition
If C∗ = (Cq , dq)q∈Z is a chain complex:
The image Bq = im dq+1 ⊆ Cq is the (sub)module of q-boundaries
The kernel Zq = ker dq ⊆ Cq is the (sub)module of q-cycles
Definition
Let C∗ = (Cq , dq)q∈Z be a chain complex. For each degree n ∈ Z, the n-homologymodule of C∗ is defined as the quotient module
Hn(C∗) =Zn
Bn
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∗ = (Cq , dq)q∈Z where:
For every q ∈ Z, the component Cq is a Z2-module, the chain group of degree q
For every q ∈ Z, the component dq is a module morphism dq : Cq → Cq−1, thedifferential map
For every q ∈ Z, the composition dqdq+1 is null: dqdq+1 = 0
Definition
If C∗ = (Cq , dq)q∈Z is a chain complex:
The image Bq = im dq+1 ⊆ Cq is the (sub)module of q-boundaries
The kernel Zq = ker dq ⊆ Cq is the (sub)module of q-cycles
Definition
Let C∗ = (Cq , dq)q∈Z be a chain complex. For each degree n ∈ Z, the n-homologymodule of C∗ is defined as the quotient module
Hn(C∗) =Zn
Bn
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∗ = (Cq , dq)q∈Z where:
For every q ∈ Z, the component Cq is a Z2-module, the chain group of degree q
For every q ∈ Z, the component dq is a module morphism dq : Cq → Cq−1, thedifferential map
For every q ∈ Z, the composition dqdq+1 is null: dqdq+1 = 0
Definition
If C∗ = (Cq , dq)q∈Z is a chain complex:
The image Bq = im dq+1 ⊆ Cq is the (sub)module of q-boundaries
The kernel Zq = ker dq ⊆ Cq is the (sub)module of q-cycles
Definition
Let C∗ = (Cq , dq)q∈Z be a chain complex. For each degree n ∈ Z, the n-homologymodule of C∗ is defined as the quotient module
Hn(C∗) =Zn
Bn
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 atern ρ = (f , g , h)
C∗
h
�� f++D∗
g
kk
satisfying the following relations:
1) fg = idD∗ ;
2) dC h + hdC = idC∗ − gf ;
3) fh = 0; hg = 0; hh = 0.
Theorem
If C∗⇒⇒D∗, then C∗ ∼= D∗ ⊕ A∗, with A∗ acyclic, what implies thatHn(C∗) ∼= Hn(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 AlgebraicTopology, 2010. http://arxiv.org/abs/1005.5685v1.
Given a chain complex C∗ and a dvf , V over C∗
C∗⇒⇒C c∗
generators of C c∗ are critical cells of C∗
0← Z162
d1←− Z322
d2←− Z162 ← 0
↓
0← Z2d1←− Z2
d2←− 0← 0
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 AlgebraicTopology, 2010. http://arxiv.org/abs/1005.5685v1.
Given a chain complex C∗ and a dvf , V over C∗
C∗⇒⇒C c∗
generators of C c∗ are critical cells of C∗
0← Z162
d1←− Z322
d2←− Z162 ← 0
↓
0← Z2d1←− Z2
d2←− 0← 0
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 AlgebraicTopology, 2010. http://arxiv.org/abs/1005.5685v1.
Given a chain complex C∗ and a dvf , V over C∗
C∗⇒⇒C c∗
generators of C c∗ are critical cells of C∗
0← Z162
d1←− Z322
d2←− Z162 ← 0
↓
0← Z2d1←− Z2
d2←− 0← 0
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 AlgebraicTopology, 2010. http://arxiv.org/abs/1005.5685v1.
Given a chain complex C∗ and a dvf , V over C∗
C∗⇒⇒C c∗
generators of C c∗ are critical cells of C∗
0← Z162
d1←− Z322
d2←− Z162 ← 0
↓
0← Z2d1←− Z2
d2←− 0← 0
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 AlgebraicTopology, 2010. http://arxiv.org/abs/1005.5685v1.
Given a chain complex C∗ and a dvf , V over C∗
C∗⇒⇒C c∗
generators of C c∗ are critical cells of C∗
0← Z162
d1←− Z322
d2←− Z162 ← 0
↓
0← Z2d1←− Z2
d2←− 0← 0
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 AlgebraicTopology, 2010. http://arxiv.org/abs/1005.5685v1.
Given a chain complex C∗ and a dvf , V over C∗
C∗⇒⇒C c∗
generators of C c∗ are critical cells of C∗
0← Z162
d1←− Z322
d2←− Z162 ← 0
↓
0← Z2d1←− Z2
d2←− 0← 0
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 AlgebraicTopology, 2010. http://arxiv.org/abs/1005.5685v1.
Given a chain complex C∗ and a dvf , V over C∗
C∗⇒⇒C c∗
generators of C c∗ are critical cells of C∗
0← Z162
d1←− Z322
d2←− Z162 ← 0
↓
0← Z2d1←− Z2
d2←− 0← 0
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 AlgebraicTopology, 2010. http://arxiv.org/abs/1005.5685v1.
Given a chain complex C∗ and a dvf , V over C∗
C∗⇒⇒C c∗
generators of C c∗ are critical cells of C∗
0← Z162
d1←− Z322
d2←− Z162 ← 0
↓
0← Z2d1←− Z2
d2←− 0← 0
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 AlgebraicTopology, 2010. http://arxiv.org/abs/1005.5685v1.
Given a chain complex C∗ and a dvf , V over C∗
C∗⇒⇒C c∗
generators of C c∗ are critical cells of C∗
0← Z162
d1←− Z322
d2←− Z162 ← 0
↓
0← Z2d1←− Z2
d2←− 0← 0
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 Vector Fields
Definition
Let C∗ = (Cp , dp)p∈Z a free chain complex with distinguished Z2-basis βp ⊂ Cp . A
discrete vector field V on C∗ is a collection of pairs V = {(σi ; τi )}i∈I satisfying the
conditions:
Every σi is some element of βp , in which case τi ∈ βp+1. The degree p dependson i and in general is not constant.
Every component σi is a regular face of the corresponding τi .
Each generator (cell) of C∗ appears at most once in V .
Definition
A V -path of degree p and length m is a sequence π = ((σik , τik ))0≤k<m satisfying:
Every pair (σik , τik ) is a component of V and τik is a p-cell.
For every 0 < k < m, the component σik is a face of τik−1, non necessarily
regular, but different from σik−1.
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 10/31
Mathematical background Discrete Morse Theory
Discrete Vector Fields
Definition
Let C∗ = (Cp , dp)p∈Z a free chain complex with distinguished Z2-basis βp ⊂ Cp . A
discrete vector field V on C∗ is a collection of pairs V = {(σi ; τi )}i∈I satisfying the
conditions:
Every σi is some element of βp , in which case τi ∈ βp+1. The degree p dependson i and in general is not constant.
Every component σi is a regular face of the corresponding τi .
Each generator (cell) of C∗ appears at most once in V .
Definition
A V -path of degree p and length m is a sequence π = ((σik , τik ))0≤k<m satisfying:
Every pair (σik , τik ) is a component of V and τik is a p-cell.
For every 0 < k < m, the component σik is a face of τik−1, non necessarily
regular, but different from σik−1.
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 10/31
Mathematical background Discrete Morse Theory
Discrete Vector Fields
Definition
A discrete vector field V is admissible if for every p ∈ Z, a function λp : βp → N isprovided satisfying the following property: every V -path starting from σ ∈ βp has alength bounded by λp(σ).
Definition
A cell σ which does not appear in a discrete vector field V is called a critical cell.
Theorem
Let C∗ = (Cp , dp)p∈Z be a free chain complex and V = {(σi ; τi )}i∈I be an admissiblediscrete vector field on C∗. Then the vector field V defines a canonical reductionρ = (f , g , h) : (Cp , dp)⇒⇒ (C c
p , d′p) where C c
p = Z2[βcp ] is the free Z2-module generated
by the critical p-cells.
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 11/31
Mathematical background Discrete Morse Theory
Discrete Vector Fields
Definition
A discrete vector field V is admissible if for every p ∈ Z, a function λp : βp → N isprovided satisfying the following property: every V -path starting from σ ∈ βp has alength bounded by λp(σ).
Definition
A cell σ which does not appear in a discrete vector field V is called a critical cell.
Theorem
Let C∗ = (Cp , dp)p∈Z be a free chain complex and V = {(σi ; τi )}i∈I be an admissiblediscrete vector field on C∗. Then the vector field V defines a canonical reductionρ = (f , g , h) : (Cp , dp)⇒⇒ (C c
p , d′p) where C c
p = Z2[βcp ] is the free Z2-module generated
by the critical p-cells.
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 11/31
Mathematical background Discrete Morse Theory
Discrete Vector Fields
Definition
A discrete vector field V is admissible if for every p ∈ Z, a function λp : βp → N isprovided satisfying the following property: every V -path starting from σ ∈ βp has alength bounded by λp(σ).
Definition
A cell σ which does not appear in a discrete vector field V is called a critical cell.
Theorem
Let C∗ = (Cp , dp)p∈Z be a free chain complex and V = {(σi ; τi )}i∈I be an admissiblediscrete vector field on C∗. Then the vector field V defines a canonical reductionρ = (f , g , h) : (Cp , dp)⇒⇒ (C c
p , d′p) where C c
p = Z2[βcp ] is the free Z2-module generated
by the critical p-cells.
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 11/31
Mathematical background Discrete Morse Theory
Example: an admissible discrete vector field
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 12/31
Mathematical background Discrete Morse Theory
Example: an admissible discrete vector field
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 12/31
Mathematical background Discrete Morse Theory
Example: an admissible discrete vector field
Dvf x
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 12/31
Mathematical background Discrete Morse Theory
Example: an admissible discrete vector field
Dvf x Dvf X
Admissible x
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 12/31
Mathematical background Discrete Morse Theory
Example: an admissible discrete vector field
Dvf x Dvf X
Admissible x
Dvf X
Admissible X
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 12/31
Mathematical background Discrete Morse Theory
Vector fields and integer matrices
Differential maps of a Chain Complex can be represented as matrices
. . .← Zm2
M←− Zn2 ← . . .
Definition
An admissible vector field V for M is nothing but a set of integer pairs {(ai , bi )}satisfying these conditions:
1 1 ≤ ai ≤ m and 1 ≤ bi ≤ n
2 The entry M[ai , bi ] of the matrix is 1
3 The indices ai (resp. bi ) are pairwise different
4 Non existence of loops
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 13/31
Mathematical background Discrete Morse Theory
Vector fields and integer matrices
Differential maps of a Chain Complex can be represented as matrices
. . .← Zm2
M←− Zn2 ← . . .
Definition
An admissible vector field V for M is nothing but a set of integer pairs {(ai , bi )}satisfying these conditions:
1 1 ≤ ai ≤ m and 1 ≤ bi ≤ n
2 The entry M[ai , bi ] of the matrix is 1
3 The indices ai (resp. bi ) are pairwise different
4 Non existence of loops
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 13/31
Mathematical background Discrete Morse Theory
Vector fields and integer matrices
Differential maps of a Chain Complex can be represented as matrices
. . .← Zm2
M←− Zn2 ← . . .
Definition
An admissible vector field V for M is nothing but a set of integer pairs {(ai , bi )}satisfying these conditions:
1 1 ≤ ai ≤ m and 1 ≤ bi ≤ n
2 The entry M[ai , bi ] of the matrix is 1
3 The indices ai (resp. bi ) are pairwise different
4 Non existence of loops
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 13/31
Mathematical background Discrete Morse Theory
Vector fields and integer matrices
Differential maps of a Chain Complex can be represented as matrices
. . .← Zm2
M←− Zn2 ← . . .
Definition
An admissible vector field V for M is nothing but a set of integer pairs {(ai , bi )}satisfying these conditions:
1 1 ≤ ai ≤ m and 1 ≤ bi ≤ n
2 The entry M[ai , bi ] of the matrix is 1
3 The indices ai (resp. bi ) are pairwise different
4 Non existence of loops
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 13/31
An abstract method
Table of Contents
1 Mathematical background
2 An abstract method
3 An effective method
4 Application
5 Conclusions and Further work
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 14/31
An abstract method
Coq/SSReflect
Coq:
An Interactive Proof AssistantBased on Calculus of Inductive ConstructionsInteresting feature: program extraction from a constructiveproof
SSReflect:
Extension of CoqDeveloped while formalizing the Four Color Theorem by G.GonthierCurrently, it is used in the formalization of Feit-ThompsonTheorem
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 15/31
An abstract method
Coq/SSReflect
Coq:
An Interactive Proof AssistantBased on Calculus of Inductive ConstructionsInteresting feature: program extraction from a constructiveproof
SSReflect:
Extension of CoqDeveloped while formalizing the Four Color Theorem by G.GonthierCurrently, it is used in the formalization of Feit-ThompsonTheorem
J. Heras, M. Poza, and J. Rubio Verifying an algorithm computing Discrete Vector Fields for digital imaging 15/31
An abstract method
Admissible discrete vector fields in SSReflect
Definition
An admissible discrete vector field V for M is nothing but a set of integer pairs{(ai , bi )} satisfying these conditions:
1 1 ≤ ai ≤ m and 1 ≤ bi ≤ n
2 The entry M[ai , bi ] of the matrix is 1
3 The indices ai (resp. bi ) are pairwise different