Reconstruction de formes en grandes dimensions Dominique Attali Co-authors: André Lieutier, David Salinas Conférence Mathématiques et Grandes Dimensions de la théorie aux développements industriels 10 décembre 2012 Lyon
Shape Approximation Simplicial complex n points Betti numbers Reconstruction Processing Volume Medial axis Signatures Input Output . . . 2
in 2D in R 2 Medial axis n points Simplicial complex Reconstruction Processing Input Output 3
in 2D in R 2 Medial axis n points Simplicial complex Reconstruction Processing (1995 – 2005) Building Heuristics (Crust, Power crust, Co-cone, Wrap, . . . ) Delaunay complex ✴ In R 2 , has size O ( n ) Empty circle property Delaunay of 10M points in 2D ≈ 10 s 4
in 3D in R 3 Medial axis n points Simplicial complex Reconstruction Processing (1995 – 2005) Building (Crust, Power crust, Co-cone, Wrap, . . . ) Delaunay complex ✴ In R 3 , has size O ( n 2 ) ✴ In practice, has size O ( n ) Empty sphere property Delaunay of 10M points in 3D ≈ 80 s 5
Shape in dD R d in R d n points Simplicial complex Betti numbers Reconstruction Processing Volume Medial axis Signatures . . . Building Delaunay complex curse of dimensionality ✴ In R d , has size O ( n � d/ 2 � ) ✴ The bound is tight (and achieved for points that sample curves). 6
Shape in dD R d in R d n points Simplicial complex Betti numbers Reconstruction Processing Volume Medial axis Signatures . . . Delaunay complex Building / / / / / / / / / / / / / How to reconstruct without Delaunay? 7
Shape in dD Guaranties on the result? R d in R d n points Simplicial complex Betti numbers Reconstruction Processing Volume Medial axis Signatures . . . Delaunay complex Building / / / / / / / / / / / / / How to reconstruct without Delaunay? 8
How to reconstruct without building the whole Delaunay complex? witnesses Landmarks weak Delaunay triangulation [V. de Silva 2008] tangent plane tangential Delaunay complexes [J. D. Boissonnat & A. Ghosh 2010] Rips complexes our approach with André Lieutier and David Salinas 9
Rips complexes α b a c Rips( P, α ) = { σ ⊂ P | Diameter( σ ) ≤ 2 α } Rips( P, α ) ⊃ Cech( P, α ) ✹ proximity graph connects every pair of points within G α 2 α ✹ [Flag G = largest complex whose 1-skeleton is G ] Rips( P, α ) = Flag G α ✹ compressed form of storage through the 1-skeleton ✹ easy to compute 10
Overview Is it possible to find sampling conditions which guarantee? Shape ➋ ≈ ➊ � R d Point cloud in Rips complex Triangulation Reconstruction Simplification Can be high-dimensional! 11
Simplification by iteratively applying elementary operations c Contraction b Edge contraction ab �→ c a Identifies vertices a and b to vertex c Preserves homotopy type if Lk( ab ) = Lk( a ) ∩ Lk( b ) The result may not be a flag complex anymore . . . ⇒ data structure = (1-skeleton, blocker set) = σ blocker of K iff dim σ ≥ 2 , ∀ τ � σ , τ ∈ K and σ �∈ K x x Collapse of a simplex σ Collapse ab b a Removes σ and its cofaces y y Preserves homotopy type if Lk( σ ) is a cone The result is a flag complex if σ a vertex or an edge 12
Example Physical system Point cloud in R 128 2 Polygonal curve Rips complex Correct intrinsic dimension Correct homotopy type Is high-dimensional! 13
Reconstruction theorems A Sampling conditions: ⇒ � = d H ( A, P ) < λ feature size( A ) Reconstruction( P, α ) P Input Output 14
Reconstruction theorems A Sampling conditions: ⇒ � = d H ( A, P ) < λ feature size( A ) Reconstruction( P, α ) P Input Output 14
Nerve � C , where C finite collection of sets If sets in C are convex Nerve Lemma. � � Nerve C = { η ⊂ C | η � = ∅} 15
Cech complex P α = � B ( p, α ) α -o ff set of P p ∈ P α p Nerve Lemma. � Cech( P, α ) = Nerve { B ( p, α ) | p ∈ P } Can be high-dimensional! & expensive to compute 16
Cech complex A P α � ? Nerve Lemma. � Cech( P, α ) P Reconstruction Input Output 17
Shapes and Reach 18
Shapes and Reach 18
Shapes and Reach 19
Shapes and Reach A m Medial Axis of A MedialAxis( A ) = { m ∈ R d | m has at least two closest points in A } Reach A = d ( A, MedialAxis( A )) 20
Cech complex A P α [Niyogi Smale Weinberger 2004] deformation retracts to if √ d H ( A, P ) ≤ ε < (3 − 8) Reach A √ α = (2 + 2) ε Nerve Lemma. � Cech( P, α ) P Reconstruction Input Output 21
Short proof R = Reach A � β = R − ( R − ε ) 2 − α 2 √ 2) ε } β < α − ε } = ε < (3 − 8) R α < R − ε P α deformation retracts to A β ⇒ ⇒ = √ α = (2 + P α x β A β y p p � A a α prove that � a − p � � ≤ β = ⇒ y lies between x and p � 22
Short proof R = Reach A � β = R − ( R − ε ) 2 − α 2 R = Reach A P α z x β A β y ε A ε p p � A a α � p − p � � ≤ α } � z − p � 2 − � p − p � � 2 ≤ β � � z − p � ≥ R − ε � a − p � � = R − 23
Rips complexes α b a c Rips( P, α ) = { σ ⊂ P | Diameter( σ ) ≤ 2 α } Rips( P, α ) ⊃ Cech( P, α ) ✹ proximity graph connects every pair of points within G α 2 α ✹ [Flag G = largest complex whose 1-skeleton is G ] Rips( P, α ) = Flag G α ✹ compressed form of storage through the 1-skeleton ✹ easy to compute 24
Rips complexes with L ∞ When distances are measured using L ∞ b a α c Rips( P, α ) = { σ ⊂ P | Diameter( σ ) ≤ 2 α } = Rips( P, α ) Cech( P, α ) ✹ proximity graph connects every pair of points within G α 2 α ✹ [Flag G = largest complex whose 1-skeleton is G ] Rips( P, α ) = Flag G α ✹ compressed form of storage through the 1-skeleton ✹ easy to compute 25
Rips complexes with L ∞ P + α B ∞ (0 , 1) A � ? Nerve Lemma. � = Rips( P, α ) Cech( P, α ) P Reconstruction Input Output easy to compute 26
Minkowski sum P + αC A � ? compact where C = convex set 27
Minkowski sum P + αC A � ? compact where C = convex set 28
Minkowski sum P + αC A � ? compact where C = convex set 29
Minkowski sum inclusion homotopy equivalence P + α C A if α 4 ε P ⊂ A ε and A ⊂ P + ε C Reach A small enough and and ε = 1 − ξ where C compact convex set that satisfies: (i) B (0 , 1) ⊂ C ⊂ δ B (0 , 1) for some δ ≥ 1; (“distortion” to unit ball) C is ( θ, κ )-round for θ = arccos( − 1 (ii) d ) and κ > 0; (“curvature”) (iii) C is ξ -eccentric for ξ < 1. (“distance” between � q ∈ Q ( q + C ) and Hull( Q )) excludes excludes n 1 b a C c 1 c 2 m a + C b + C n 2 30
Minkowski sum inclusion homotopy equivalence P + α C A if α 4 ε P ⊂ A ε and A ⊂ P + ε C Reach A small enough and and ε = 1 − ξ ➊ d -balls satisfy (i) (ii) and (iii) for δ = 1, κ = 1 and ξ = 0. ➋ √ d -cubes satisfy (i) (ii) and (iii) for δ = d 1 � � cos π 4 + cos π if d = 2, √ 12 2 2 1 if d = 3, = κ √ 6 1 if d ≥ 4, √ ( d − 2) d ξ = 1 − 2 d 31
Minkowski sum inclusion homotopy equivalence P + α C A if ε P ⊂ A ε and A ⊂ P + ε C α Reach A < λ and and ε = η Admissible values of ε and α are solutions of a system of equations that depends upon ( δ, κ , ξ ). C d λ η √ √ d -ball with [NSW04] ∀ d 3 - 8 ≈ 0 . 17 2 + 2 ≈ 3 . 41 d -ball with this proof ∀ d 0.077 3.96 2 0.04 4.04 3 0.01 6.14 4 0.004 8.18 d -cube 5 0.002 10.2 10 0.0002 20.23 [ Rips( P, α ) with � ∞ ] 100 0.0000002 200.23 32
What now? ε ✺ The largest ratio Reach A that we get for Rips( P, α ) with � ∞ : ✺ Decreases quickly with d ✺ Is it tight? ✺ ➟ � ∞ � 2 33
Rips complexes with L 2 A P α Cech( P, α ) Rips( P, α ) P ⊂ Input Output easy to compute 34
Rips complexes with L 2 A P α Nerve Lemma � Cech( P, α ) Rips( P, α ) P ⊂ Input Output easy to compute 34
Rips complexes with L 2 A P α deformation retracts to [NSW 04] Nerve Lemma � Cech( P, α ) Rips( P, α ) P ⊂ Input Output easy to compute 34
Rips complexes with L 2 A P α deformation retracts to [NSW 04] Nerve Lemma � Cech( P, α ) Rips( P, α ) P ⊂ deformation retracts to ? Input Output easy to compute 34
Rips complexes with L 2 A P α deformation retracts to [NSW 04] Nerve Lemma � Cech( P, α ) Rips( P, α ) P ⊂ deformation retracts to ? ≈ sphere � circle Rips and Cech complexes generally don’t share the same topology, but ... 35
Roadmap P ϑ d α P α � � Cech( P, ϑ d α ) Cech( P, α ) Rips( P, α ) ⊂ ⊂ ≈ sphere � circle ≈ 5-ball � 2 d for ϑ d = d +1 36
Roadmap P ϑ d α P α Find a condition under which the topology of ➊ { Cech( P, t ) } α ≤ t ≤ ϑ d α is “stable” � � sequence of collapses ? Cech( P, ϑ d α ) Cech( P, α ) Rips( P, α ) ⊂ ⊂ � 2 d for ϑ d = d +1 37
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