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“Porosity”in 3D digital images of heterogeneous materials: a homological approach Pedro Real Computational Topology and Applied Math Team E.T.S. Ingeniería Informática UNIVERSITY OF SEVILLE (Spain) SADIEL, November 2008 Seville

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Emergent project of topological quantification of nD digital objects (Computational Algebraic Topology) –Project R+D+I del MEC MTM “A New Model of algebraic-topological representation of digital volumes: the AT-model”. –Project R+D+I of Excelence of Junta Andalucia P06- TIC “Topological Analysis of nD digital images”. –Andalusian Reseach Group investigación FQM-296 “Computational Topology and Applied Mathematics”.

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Facing to the problem of “porosity” in heterogeneous digital volumes For us POROSITY means more or less, quantification of an accurate GEOMETRICAL DESCRIPTION of the TOPOLOGICAL COMPLEXITY of the object (and perhaps of its complement)

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Estructura ósea trabecular Trabecular bone architecture: quantification of its topological complexity

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Pedro Real Topología Algebraica Computacional e Imágenes Médicas Cracks in the architecture without apparent loss of material

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Trabecular Bone Anatomy

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Digital Volume A voxel is a unit of graphic information defining a point in three- dimensional space. Voxels are structured in a digital volume in a regular cubic grid

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Digital volumes We consider 3D binary images, i.e., volume images consisting of object and background. Each voxel has 26 neighbours in its immediate neighbourhood: six face neighbours,twelve edge neighbours, and eight vertex neighbours. Object voxels having at least one face neighbour in the background are called border voxels. We use 26-connectedness for the object and 6-connectedness for the background,

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Method of Saha et al, 2003: topological classification of voxels of the thinned volume

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DESCOMPOSITION IN CILINDERS Y RODs STAUBER & Müller, 2006 METHOD: ELEMENTAL CHARACTERIZATION OF TRABECULAR BONE MICRO_CT VOLUME “TOPOLOGICAL” SQUELETONIZATION SEGMENTOS (VOXELES) DILATION DESCOMPOSITION IN CILINDERS AND RODS VOLUMETRIC AND SUBVOLUMETRIC STUDY

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AT-model microMR bone volume Image of a porous binary volumetric object consisting of voxels Cellular version of the volume

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Objective: homological analysis Image ↔ Algebra -ALGEBRAIC-TOPOLOGICAL MODEL (Gonzalez-Diaz and Real, Discrete Applied Math., 2003) -CHAIN HOMOTOPIES FOR OBJECT TOPOLOGICAL REPRESENTATION (Gonzalez-Díaz, Jimenez, Medrano and Real, Discrete Applied Mathematics, 2008)

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El modelo AT (algebro-topológico) Operador de homotopia de cadenas caracterizando topológicamente al complejo simplicial asociado al objeto digital (modelo AT)

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Result of the incremental algorithm of homological calculus in 31 seconds: 23 connected components, 1826 “tunnels” (it is shown the representative cycles) and 0 cavities. The AT-model description is in the right.

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Global Topological Quantification of VOIs of porous media DIGITAL VOLUME (41642 voxels)

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Cellularization of the volume

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AT-model

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4) Visualization: 10 connected components, 4332 “tunels” and 27 cavities

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Process of topological computation in real cases Sequences of 2D binary images obtained by uMR (ETH Zurich) Cellularization of three slices

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Problems Facing to the problem of “porosity” in heterogeneous digital volumes The representative 1-cycle of a homology generator in the AT-model must represent one true “tunnel” If one succeed, the representative 1-cycle of a homology generator in the AT-model must “fits well” the tunnel (a good “geometric homology generator”)

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AT-model: At present, theoretical solution to deformation problems Deforming a homological 1-cycle (in red) to a homologous cycle (in black). More difficult in 3D!! To solve problems of topological tracking in FEA simulations. Objective: to determine fractures or topological changes. More dificult in 3D!! Simulation

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Tracking of “holes” in bone computational models?

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Finite Element Analysis ( _ FEA) DETERMINATION of BONE RIDIGITY IN A COMPUTATIONAL MODEL

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Composition of integral operators It is immediate to see that the composition of sucessive chain contractions defined by corresponding integral operators is a new chain contraction determined by a chain homotopy operator Φ:C(K)→C(K), satisfying that ΦΦ=0, ΦdΦ=Φ and the property of strongly 2- nilpotence condition. C(K) M (K) f g Φ

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Some methods for getting homology Algebraic-Topological thinning based on the sucesive application of integral operators erasing cells of the border of the objects for getting a topological skeleton Merging internal cells for getting a homological segmentation “saving” the geometry of the border of the object. Using an incremental technique of adding a simplex and updating the homology in each step. In a regular grid (that is the case of digital object) to predetermine the integral operators of each simplex in the grid and to adapt this information to the corresponding subset.

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Integral operator and geometry ↓ ↗ ↖ ← ↖

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Integral Operators as algebraic generalization of gradient vector fields (Morse theory) Let Φ: K →C be the integral operator defined by Φ(u k )= σ, for some simplices a and b. (Z[K],d) ( Z[K \ u k, σ}],],d-dΦd) Π=1-dΦ-Φd incl Φ

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Homological Thinning C(K) H(K) f g Φ

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An example of topological thinning C(K) H(K) f g Φ

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Homological Merging C(K) H(K) f g Φ

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Bibliography [1] REAL P., GONZALEZ-DIAZ, R., Towards Digital Cohomology. Lecture Notes in Computer Science LNCS 2886, Springer-Verlag, [2] GONZALEZ-DIAZ, R., REAL, P., On the cohomology of 3D digital images. Discrete Applied Mathematics 147, [3] GONZALEZ-DIAZ, R., MEDRANO, B., REAL, P., SANCHEZ-PELAEZ, J Understanding volumes from an algebraic topological perspective. Application of Computer Algebra ACA 2005, Julio 31-Agosto 3, Nara (Japón). Abstracts ACA2005, Editor: K. Shirayanagi,p. 30. ISBN: [4] GONZALEZ-DIAZ, R., MEDRANO, B., REAL, P., SANCHEZ-PELAEZ, J., Algebraic- topological analysis of time-sequence of digital images. Lecture Notes in Computer Science LNCS 3718, [5] GONZALEZ-DIAZ, R., MEDRANO, B., REAL, P., SANCHEZ-PELAEZ, J., Técnicas Algebraicas para el Control Topológico en Imágenes Digitales. I Congreso Español de Informática (Sesión: XV Congreso Español de Informática Gráfica), Septiembre, Proceedings del XV CEIG, [6] GONZALEZ-DIAZ, R., MEDRANO, B., REAL, P., SANCHEZ-PELAEZ, J., Algebraic Topological Techniques in 3D and 4D digital Imagery. XVI Coloquio Latinoamericano de Algebra, Colonia (Uruguay). [7] GONZALEZ-DIAZ R., Medrano B., SÁNCHEZ-PELÁEZ J., REAL P, 2006: Simplicial Perturbation Techniques and Effective Homology, Lecture Notes in Computer Science, Volumen:4191, [8] GONZALEZ-DIAZ R., Medrano B., SÁNCHEZ-PELÁEZ J., REAL P, 2006: Reusing integer homology information in binary 3D digital images,: Lecture Notes in Computer Science, Volumen: 4245, [9] REAL P., 2007: Análisis Topológico de imágenes 3D médicas: el ejemplo de Osteoporósis. XII Convención y Expo Internacional INFORMATICA 2007 (Salud). Lugar celebración: La Habana (Cuba)Fecha: 12 al 16 de Febrero de 2007.

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Emergent project of topological quantification of nD digital objects –Project R+D+I del MEC MTM “A New Model of algebraic-topological representation of digital volumes: the AT-model”. –Project R+D+I of Excelence of Junta Andalucia P06- TIC “Topological Analysis of nD digital images”. –Andalusian Reseach Group investigación FQM-296 “Computational Topology and Applied Mathematics”.

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CONCEPTS AND NOTATIONS: Homological Algebra C, a chain complex: C={C q,d q } q≥0 o C q = abelian groups. o d q: C q C q-1 homomorphisms such that d q d q+1 =0 – differential of C Chain Contractions: (f, g, Φ): C C’ C C’ f g Φ f and g are chains maps fg=id C’ Φ d+ d Φ =id C -gf fΦ =0, Φg=0, ΦΦ =0 o C’ has fewer or the same number of generators than C o C and C’ have isomorphic homology groups o The chain homotopy operator Φ is a gradient function for C

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Computational Homological Algebra Reciprocaly, a chain homotopy operator Φ: C q C q+1, satisfying the conditions Φ Φ=0 and Φ d Φ = Φ gives rise to a chain contraction (f,g, Φ): C Im(π), where π = 1- d Φ – Φ d, f=π and g=inclusion. (Lambe and Barnes, 1991) In the case in which Φ also satisfies the condition d Φ d = d, then the associated chain contraction (f,g, Φ) connects C to its homology H(C). Given a chain contraction: (f, g, Φ): C C’, it is clear that Φ is a chain homotopy operator satisfying ΦΦ =0 and ΦdΦ = Φ. Φ (C,d) ( Im(π),d-dΦd) π incl Φ

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Integral operators and simplicity by pairs Let C=Z[ c ] be a chain complex and Φ: C →C be the integral operator defined by Φ(a)=b, for some chains a,b. → (C,d) ( Z[ C \{a b}],d-dΦd ) Π=1-dΦ-Φd incl Φ * (Z[K],d) ( Z[K \{a b}],d-dΦd) π incl Φ Let C=Z[ K ] be the chain complex cannonically associated to a cell or simplicial complex K and Φ: K →C be the integral operator defined by Φ(a)=b, for some simplices a and b.

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Integral operators for homology over a field (Type 1) Let K be a simplicial complex and σ a simplex of K of dimension t. If d(σ)=∑λ i u i, such that λ k =1, for some k, take Φ:K t-1 → K t defined by Φ(u k )= σ, and zero in the rest. Then Φ satisfy ΦΦ=0 and ΦdΦ=Φ and it generates the following chain contraction: (Z[K],d) ( Z[K \{u k, σ}],d-dΦd) f g Φ (f, g, Φ)=(1-d Φ - Φ d, incl, Φ) : (Z[K],d) ↔ Z[K \{u k, σ}],d’) f(u k )= -∑ i≠k λ i u i ; f(σ)=0 ; f(σ’)=σ’- ασ for σ’ \ d(σ’)= αu k +…; f=1 for the rest This is equivalent to card({σ’}) column operations and one row operation in the matrix of the differential in dimension t, three line elimination operations in dimenstion t and t-1 and card({σ’}) row operations in dimension t+1 (basis changing). If u k is face of only one simplex σ, there is no need to do a basis changing

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Integral operators for integer homology (Type 2) Let C a finite chain complex and σ be an element of C of degree t, basis of C. If d(σ)=∑λ i u i is such that λ k ≠1, for all I and g.c.d(λ j, λ k )=1[and 1= α λ j + β λ k ] for some pair of indices j and k, let us take Φ:C t-1 → C t defined by Φ(u k )= ασ,Φ(u k )=βσ, and zero in the rest. Then Φ satisfy ΦΦ=0 and ΦdΦ=Φ and it generates the following chain contraction: (C,d) ( Z[ C \{u j, σ}], d-dΦd) f g Φ (f, g, Φ)=(1-d Φ - Φ d, incl, Φ) : (C,d) ↔ Z[ C \{u j, σ}], d’) f(u j )= (1-α) u j -∑ i≠j λ i u i ; f(u k )= (1-β) u k -∑ i≠k λ i u i ; f(σ)=0; f(σ’)=σ’- (γα+θβ) σ for σ’ \ d(σ’)= γu k +θu j …; f=1 for the rest

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Integral operators for integer homology (Type 3) Let C a finite chain complex and σ 1 and σ 2 be two elements of degree t of C, basis of C. If d(σ r )=∑λ i,r u i (r=1,2) is such that g.c.d(λ k1, λ k2 )=1 for some k [Bezout identity 1= α λ k1 + β λ k2 ], and let us take Φ: C t-1 → C t defined by Φ(u k )= ασ 1 +βσ 2, and zero in the rest. Then Φ satisfy ΦΦ=0 and ΦdΦ=Φ and it generates the following chain contraction: (C,d) ( Z[ C \{u k, σ 1 }], d-dΦd) f g Φ (f, g, Φ)=(1-d Φ - Φ d, incl, Φ) : (C,d) ↔ (Z[ C \{u k, σ 1 }], d’) f(u j )= (1-α) u j -∑ i≠j λ i u i ; f(u k )= (1-β) u k -∑ i≠k λ i u i ; f(σ)=0 f(σ)=0 ; f(σ’)=σ’- (γα+θβ) σ for σ’ \ d(σ’)= γu k +θu j …; f=1 for the rest

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