geometry and topology from point cloud data
TRANSCRIPT
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Geometry and Topology from Point Cloud Data
Tamal K. Dey
Department of Computer Science and EngineeringThe Ohio State University
Dey (2011) Geometry and Topology from Point Cloud Data WALCOM 11 1 / 51
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Outline
Problems
Two and Three dimensions:
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Outline
Problems
Two and Three dimensions:
Curve and surface reconstruction
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Outline
Problems
Two and Three dimensions:
Curve and surface reconstruction
High dimensions:
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Outline
Problems
Two and Three dimensions:
Curve and surface reconstruction
High dimensions:
Manifold reconstruction
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Outline
Problems
Two and Three dimensions:
Curve and surface reconstruction
High dimensions:
Manifold reconstructionHomological attributes computation
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Reconstruction
Surface Reconstruction
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Topology Background
Basic Topology
d -ball Bd {x ∈ Rd | ||x || ≤ 1}
d -sphere Sd {x ∈ Rd | ||x || = 1}
Homeomorphism h : T1 → T2
where h is continuous, bijectiveand has continuous inverse
k-manifold: neighborhoods homeomorphic to open k-ball
2-sphere, torus, double torus are 2-manifolds
k-manifold with boundary: interior points, boundary points
Bd is a d-manifold with boundary where bd(Bd) = Sd−1
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Topology Background
Basic Topology
Smooth Manifolds
Triangulation
k-simplex
Simplicial complex K :
(i) t ∈ K if t is a face of t ′ ∈ K
(ii) t1, t2 ∈ K ⇒ t1 ∩ t2 is a face of both
K is a triangulation of a topologicalspace T if T ≈ |K |
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Sampling
Sampling
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Sampling
Medial Axis
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Sampling
Local Feature Size
f (x) is the distanceto medial axis
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Sampling
ε-sample (Amenta-Bern-Eppstein 98)
Each x has a samplewithin εf (x) distance
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Sampling
Voronoi Diagram & Delaunay Triangulation
Definition
Voronoi diagram Vor P : collection of Voronoi cells {Vp} and its facesVp = {x ∈ R
3 | ||x − p|| ≤ ||x − q|| for all q ∈ P}
Definition
Delaunay triangulation Del P : dual of Vor P , a simplicial complex
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Curve Reconstruction
Curve samples and Voronoi
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Curve Reconstruction
Curve Reconstruction Algorithms
Crust algorithm(Amenta-Bern-Eppstein 98)
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Curve Reconstruction
Curve Reconstruction Algorithms
Crust algorithm(Amenta-Bern-Eppstein 98)
Nearest neighbor algorithm(Dey-Kumar 99)
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Curve Reconstruction
Curve Reconstruction Algorithms
Crust algorithm(Amenta-Bern-Eppstein 98)
Nearest neighbor algorithm(Dey-Kumar 99)
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Curve Reconstruction
Curve Reconstruction Algorithms
Crust algorithm(Amenta-Bern-Eppstein 98)
Nearest neighbor algorithm(Dey-Kumar 99)
many variations(DMR99,Gie00,GS00,FR01,AM02..)
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Surface Reconstruction
Difficulties in 3D
Voronoi vertices can comeclose to the surface . . .slivers are nasty
There is no unique ‘correct’surface for reference
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Surface Reconstruction
Restricted Voronoi/Delaunay
Definition
Restricted Voronoi: Vor P |Σ = {fP |Σ = f ∩ Σ | f ∈ Vor P}
Definition
Restricted Delaunay: Del P |Σ = {σ |Vσ ∩ Σ 6= ∅}
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Surface Reconstruction
Topology
Closed Ball property (Edelsbrunner, Shah 94)
If restricted Voronoi cell is a closed ball in each dimension, thenDel P |Σ is homeomorphic to Σ.
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Surface Reconstruction
Topology
Closed Ball property (Edelsbrunner, Shah 94)
If restricted Voronoi cell is a closed ball in each dimension, thenDel P |Σ is homeomorphic to Σ.
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Surface Reconstruction
Topology
Closed Ball property (Edelsbrunner, Shah 94)
If restricted Voronoi cell is a closed ball in each dimension, thenDel P |Σ is homeomorphic to Σ.
Theorem
For a sufficiently small ε if P is anε-sample of Σ, then (P, Σ) satisfiesthe closed ball property, and henceDel P |Σ ≈ Σ.
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Surface Reconstruction
Normals and Voronoi Cells 3D (Amenta-Bern 98)
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Surface Reconstruction
Long Voronoi cells and Poles
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Surface Reconstruction
Normal Approximation
Lemma (Pole Vector)
∠((p+ − p),np) = 2 arcsin ε
1−ε
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Surface Reconstruction
Crust in 3D (Amenta-Bern 98)
Compute Voronoi diagram Vor P
Recompute the Voronoi diagramafter introducing poles
Filter crust triangles from Delaunay
Filter by normals
Extract manifold
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Surface Reconstruction
Cocone
vp = p+ − p is the pole vector
Space spanned by vectors within theVoronoi cell making angle > 3π
8with
vp or −vp
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Surface Reconstruction
Cocone Algorithm
Cocone(P)
1 compute Vor P ;2 T = ∅;3 for each p ∈ P do
4 Tp = CandidateTriangles(Vp);5 T := T ∪ Tp;6 end for
7 M := ExtractManifold(T );8 output M
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Surface Reconstruction
Candidate Triangle Properties
The following properties hold for sufficiently small ε (ε < 0.06)
Candidate triangles include the restricted Delaunay triangles
Their circumradii are small O(ε)f (p)
Their normals make only O(ε) angle with the surface normals atthe vertices
Candidate triangles include restricted Delaunay triangles
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Surface Reconstruction
Manifold Extraction: Prune and Walk
Remove Sharp edges with their triangles
Walk outside or inside the remaining triangles
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Surface Reconstruction
Homeomorphism
Let M be the triangulated surface obtained after the manifoldextraction.
Define h : R3 → Σ where h(q) is the closest point on Σ. h is well
defined except at the medial axis points.
Lemma (Homeomorphism)
The restriction of h to M, h : M → Σ, is a homeomorphism.
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Surface Reconstruction
Cocone Guarantees
Theorem
Any point x ∈ Σ is within O(ε)f (x) distance from a point in theoutput. Conversely, any point of the output surface has a pointx ∈ Σ within O(ε)f (x) distance for ε < 0.06.
Theorem (Amenta-Choi-Dey-Leekha)
The output surface computed by Cocone from an ε − sample ishomeomorphic to the sampled surface for ε < 0.06.
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Input Variations
Boundaries
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Input Variations
Boundaries
Ambiguity in reconstruction
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Input Variations
Boundaries
Non-homeomorphic Restricted Delaunay [DLRW09]
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Input Variations
Boundaries
Non-orientabilty
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Input Variations
Boundaries
Theorem (Dey-Li-Ramos-Wenger 2009)
Let P be a sample of a smooth compact Σ with boundary whered(x , P) ≤ ερ, ρ = infx lfs(x). For sufficiently small ε > 0 and6ερ ≤ α ≤ 6ερ + O(ερ), Peel(P , α) computes a Delaunay meshisotopic to Σ.
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Input Variations
Noisy Data: Ram Head
Hausdorff distance dH(P , Σ) is εf (p)
Theoretical guarantees [Dey-Goswami04, Amenta et al.05]
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Input Variations
Nonsmoothness
Guarantee of homeomorphism is open
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High Dimensions
High Dimensional PCD
Curse of dimensionality (intrinsic vs. extrinsic)
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High Dimensions
High Dimensional PCD
Curse of dimensionality (intrinsic vs. extrinsic)
Reconstruction of submanifolds brings ambiguity
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High Dimensions
High Dimensional PCD
Curse of dimensionality (intrinsic vs. extrinsic)
Reconstruction of submanifolds brings ambiguity
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High Dimensions
High Dimensional PCD
Curse of dimensionality (intrinsic vs. extrinsic)
Reconstruction of submanifolds brings ambiguity
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High Dimensions
High Dimensional PCD
Curse of dimensionality (intrinsic vs. extrinsic)
Reconstruction of submanifolds brings ambiguity
Use (ε, δ)-sampling
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High Dimensions
High Dimensional PCD
Curse of dimensionality (intrinsic vs. extrinsic)
Reconstruction of submanifolds brings ambiguity
Use (ε, δ)-sampling
Restricted Delaunay does not capture topology
Slivers are arbitrarily oriented [CDR05] ⇒ DelP|Σ 6≈ Σ nomatter how dense P is.
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High Dimensions
High Dimensional PCD
Curse of dimensionality (intrinsic vs. extrinsic)
Reconstruction of submanifolds brings ambiguity
Use (ε, δ)-sampling
Restricted Delaunay does not capture topology
Slivers are arbitrarily oriented [CDR05] ⇒ DelP|Σ 6≈ Σ nomatter how dense P is.
Delaunay triangulation becomes harder
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High Dimensions
Reconstruction
Theorem (Cheng-Dey-Ramos 2005)
Given an (ε, δ)-sample P of a smooth manifold Σ ⊂ Rd for
appropriate ε, δ > 0, there is a weight assignment of P so thatDel P |Σ ≈ Σ which can be computed efficiently.
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High Dimensions
Reconstruction
Theorem (Cheng-Dey-Ramos 2005)
Given an (ε, δ)-sample P of a smooth manifold Σ ⊂ Rd for
appropriate ε, δ > 0, there is a weight assignment of P so thatDel P |Σ ≈ Σ which can be computed efficiently.
Theorem (Chazal-Lieutier 2006)
Given an ε-noisy sample P of manifold Σ ⊂ Rd , there exists
rp ≤ ρ(Σ) for each p ∈ P so that the union of balls B(p, rp) ishomotopy equivalent to Σ.
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High Dimensions
Reconstructing Compacts
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High Dimensions
Reconstructing Compacts
lfs vanishes, introduce µ-reach and define (ε, µ)-samples.
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High Dimensions
Reconstructing Compacts
lfs vanishes, introduce µ-reach and define (ε, µ)-samples.
Theorem (Chazal-Cohen-S.-Lieutier 2006)
Given an (ε, µ)-sample P of a compact K ⊂ Rd for appropriate
ε, µ > 0, there is an α so that union of balls B(p, α) is homotopyequivalent to K η for arbitrarily small η.
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Homology
Homology from PCD
Point cloud
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Homology
Homology from PCD
Point cloud Loops
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Homology
PCD→complex→homology
Point cloud
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Homology
PCD→complex→homology
Point cloud Rips complex
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Homology
PCD→complex→homology
Point cloud Rips complex Loops
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Homology Definitions
Boundary
Definition
A p-boundary ∂p+1c of a (p + 1)-chain c is defined as the sum ofboundaries of its simplices
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Homology Definitions
Boundary
Definition
A p-boundary ∂p+1c of a (p + 1)-chain c is defined as the sum ofboundaries of its simplices
a
b
c
d
e
Simplicial complex
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Homology Definitions
Boundary
Definition
A p-boundary ∂p+1c of a (p + 1)-chain c is defined as the sum ofboundaries of its simplices
a
b
c
d
e
2-chain bcd + bde (under Z2)
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Homology Definitions
Boundary
Definition
A p-boundary ∂p+1c of a (p + 1)-chain c is defined as the sum ofboundaries of its simplices
a
b
c
d
e
1-boundary bc+cd+db+bd+de+eb = bc+cd+de+eb = ∂2(bcd+bde)
(under Z2)
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Homology Definitions
Cycles
Definition
A p-cycle is a p-chain that has an empty boundary
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Homology Definitions
Cycles
Definition
A p-cycle is a p-chain that has an empty boundary
a
b
c
d
e
Simplicial complex
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Homology Definitions
Cycles
Definition
A p-cycle is a p-chain that has an empty boundary
a
b
c
d
e
1-cycle ab + bc + cd + de + ea (under Z2)
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Homology Definitions
Cycles
Definition
A p-cycle is a p-chain that has an empty boundary
a
b
c
d
e
1-cycle ab + bc + cd + de + ea (under Z2)
Each p-boundary is a p-cycle: ∂p ◦ ∂p+1 = 0
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Homology Definitions
Homology
Definition
The p-dimensional homology group is defined asHp(K) = Zp(K)/Bp(K)
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Homology Definitions
Homology
Definition
The p-dimensional homology group is defined asHp(K) = Zp(K)/Bp(K)
Definition
Two p-chains c and c ′ are homologous if c = c ′ + ∂p+1d for somechain d
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Homology Definitions
Homology
Definition
The p-dimensional homology group is defined asHp(K) = Zp(K)/Bp(K)
Definition
Two p-chains c and c ′ are homologous if c = c ′ + ∂p+1d for somechain d
(a) (b) (c)
(a) trivial (null-homologous) cycle; (b), (c) nontrivial homologous cyclesDey (2011) Geometry and Topology from Point Cloud Data WALCOM 11 36 / 51
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Homology Definitions
Complexes
Let P ⊂ Rd be a point set
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Homology Definitions
Complexes
Let P ⊂ Rd be a point set
B(p, r) denotes an open d -ball centered at p with radius r
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Homology Definitions
Complexes
Let P ⊂ Rd be a point set
B(p, r) denotes an open d -ball centered at p with radius r
Definition
The Cech complex Cr (P) is a simplicial complex where a simplexσ ∈ Cr (P) iff Vert(σ) ⊆ P and ∩p∈Vert(σ)B(p, r/2) 6= 0
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Homology Definitions
Complexes
Let P ⊂ Rd be a point set
B(p, r) denotes an open d -ball centered at p with radius r
Definition
The Cech complex Cr (P) is a simplicial complex where a simplexσ ∈ Cr (P) iff Vert(σ) ⊆ P and ∩p∈Vert(σ)B(p, r/2) 6= 0
Definition
The Rips complex Rr (P) is a simplicial complex where a simplexσ ∈ Rr (P) iff Vert(σ) are within pairwise Euclidean distance of r
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Homology Definitions
Complexes
Let P ⊂ Rd be a point set
B(p, r) denotes an open d -ball centered at p with radius r
Definition
The Cech complex Cr (P) is a simplicial complex where a simplexσ ∈ Cr (P) iff Vert(σ) ⊆ P and ∩p∈Vert(σ)B(p, r/2) 6= 0
Definition
The Rips complex Rr (P) is a simplicial complex where a simplexσ ∈ Rr (P) iff Vert(σ) are within pairwise Euclidean distance of r
Proposition
For any finite set P ⊂ Rd and any r ≥ 0, Cr (P) ⊆ Rr (P) ⊆ C2r (P)
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Homology Definitions
Point set P
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Homology Definitions
Balls B(p, r/2) for p ∈ P
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Homology Definitions
Cech complex Cr(P)
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Homology Definitions
Rips complex Rr(P)
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Homology Rank
Homology rank from PCD
Results of Chazal and Oudot (Main idea):
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Homology Rank
Homology rank from PCD
Results of Chazal and Oudot (Main idea):
Consider inclusion of Rips complexes i : Rr (P) → R4r (P).
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Homology Rank
Homology rank from PCD
Results of Chazal and Oudot (Main idea):
Consider inclusion of Rips complexes i : Rr (P) → R4r (P).
Induced homomorphism at homology level:
i∗ : Hk(Rr (P)) → Hk(R
4r (P))
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Homology Rank
Homology rank from PCD
Results of Chazal and Oudot (Main idea):
Consider inclusion of Rips complexes i : Rr (P) → R4r (P).
Induced homomorphism at homology level:
i∗ : Hk(Rr (P)) → Hk(R
4r (P))
Rr (P) R4r (P)
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Homology Rank
Homology rank from PCD
Results of Chazal and Oudot (Main idea):
Consider inclusion of Rips complexes i : Rr (P) → R4r (P).
Induced homomorphism at homology level:
i∗ : Hk(Rr (P)) → Hk(R
4r (P))
Rr (P) R4r (P)
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Homology Rank
Homology rank from PCD
Results of Chazal and Oudot (Main idea):
Consider inclusion of Rips complexes i : Rr (P) → R4r (P).
Induced homomorphism at homology level:
i∗ : Hk(Rr (P)) → Hk(R
4r (P))
Theorem (Chazal-Oudot 2008)
Rank of the image of i∗ equals the rank of Hk(M) if P is densesample of M and r is chosen appropriately.
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Homology Rank
Algorithm for homology rank
1 Compute Rr (P).
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Homology Rank
Algorithm for homology rank
1 Compute Rr (P).
2 Insert simplices of R4r (P) that are not in Rr (P) and computethe rank of the homology classes that survive.
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Homology Rank
Algorithm for homology rank
1 Compute Rr (P).
2 Insert simplices of R4r (P) that are not in Rr (P) and computethe rank of the homology classes that survive.
Step 2: Persistent homology can be computed by the persistencealgorithm [Edelsbrunner-Letscher-Zomorodian 2000].
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Homology basis
OHBP: Optimal Homology Basis Problem
Compute an optimal set of cycles forming a basis
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Homology basis
OHBP: Optimal Homology Basis Problem
Compute an optimal set of cycles forming a basis
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Homology basis
OHBP: Optimal Homology Basis Problem
Compute an optimal set of cycles forming a basis
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Homology basis
OHBP: Optimal Homology Basis Problem
Compute an optimal set of cycles forming a basis
First solution for surfaces: Erickson-Whittlesey [SODA05]
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Homology basis
OHBP: Optimal Homology Basis Problem
Compute an optimal set of cycles forming a basis
First solution for surfaces: Erickson-Whittlesey [SODA05]
General problem NP-hard: Chen-Freedman [SODA10]
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Homology basis
OHBP: Optimal Homology Basis Problem
Compute an optimal set of cycles forming a basis
First solution for surfaces: Erickson-Whittlesey [SODA05]
General problem NP-hard: Chen-Freedman [SODA10]
H1 basis for simplicial complexes: Dey-Sun-Wang [SoCG10]
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Homology basis
Basis
Let Hj(T ) denote the j-dimensional homology group of T underZ2
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Homology basis
Basis
Let Hj(T ) denote the j-dimensional homology group of T underZ2
The elements of H1(T ) are equivalence classes [g ] of1-dimensional cycles g , also called loops
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Homology basis
Basis
Let Hj(T ) denote the j-dimensional homology group of T underZ2
The elements of H1(T ) are equivalence classes [g ] of1-dimensional cycles g , also called loops
Definition
A minimal set {[g1], ..., [gk ]} generating H1(T ) is called its basisHere k = rank H1(T )
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Homology basis
Shortest Basis
We associate a weight w(g) ≥ 0 with each loop g in T
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Homology basis
Shortest Basis
We associate a weight w(g) ≥ 0 with each loop g in T
The length of a set of loops G = {g1, . . . , gk} is given by
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Homology basis
Shortest Basis
We associate a weight w(g) ≥ 0 with each loop g in T
The length of a set of loops G = {g1, . . . , gk} is given by
Len(G) =k
∑
i=1
w(gi)
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Homology basis
Shortest Basis
We associate a weight w(g) ≥ 0 with each loop g in T
The length of a set of loops G = {g1, . . . , gk} is given by
Len(G) =k
∑
i=1
w(gi)
Definition
A shortest basis of H1(T ) is a set of k loops with minimal length thatgenerates H1(T )
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Homology basis
Optimal basis for simplicial complex
Theorem (Dey-Sun-Wang 2010)
Let K be a finite simplicial complex with non-negative weights onedges. A shortest basis for H1(K) can be computed in O(n4) timewhere n = |K|
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Homology basis
Approximation from Point Cloud
Let P ⊂ Rd be a point set sampled from a smooth closed
manifold M ⊂ Rd embedded isometrically
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Homology basis
Approximation from Point Cloud
Let P ⊂ Rd be a point set sampled from a smooth closed
manifold M ⊂ Rd embedded isometrically
We want to approximate a shortest basis of H1(M) from P
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Homology basis
Approximation from Point Cloud
Let P ⊂ Rd be a point set sampled from a smooth closed
manifold M ⊂ Rd embedded isometrically
We want to approximate a shortest basis of H1(M) from P
Compute a complex K from P
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Homology basis
Approximation from Point Cloud
Let P ⊂ Rd be a point set sampled from a smooth closed
manifold M ⊂ Rd embedded isometrically
We want to approximate a shortest basis of H1(M) from P
Compute a complex K from P
Compute a shortest basis of H1(K)
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Homology basis
Approximation from Point Cloud
Let P ⊂ Rd be a point set sampled from a smooth closed
manifold M ⊂ Rd embedded isometrically
We want to approximate a shortest basis of H1(M) from P
Compute a complex K from P
Compute a shortest basis of H1(K)
Argue that if P is dense, a subset of computed loopsapproximate a shortest basis of H1(M) within constant factors
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Homology basis
Approximation Theorem
Theorem (Dey-Sun-Wang 2010)
Let M ⊂ Rd be a smooth, closed manifold with l as the length of a
shortest basis of H1(M) and k = rank H1(M).Given a set P ⊂ M of n points which is an ε-sample of M and
4ε ≤ r ≤ min{12
√
35ρ(M), ρc(M)}, one can compute a set of loops
G in O(nn2ent) time where
1
1 + 4r2
3ρ2(M)
l ≤ Len(G) ≤ (1 +4ε
r)l.
Here ne , nt are the number of edges and triangles in R2r (P)
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Conclusions
Conclusions
Reconstructions :
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Conclusions
Conclusions
Reconstructions :
non-smooth surfaces remain open
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Conclusions
Conclusions
Reconstructions :
non-smooth surfaces remain openhigh dimensions still not satisfactory
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Conclusions
Conclusions
Reconstructions :
non-smooth surfaces remain openhigh dimensions still not satisfactory
Homology :
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Conclusions
Conclusions
Reconstructions :
non-smooth surfaces remain openhigh dimensions still not satisfactory
Homology :
Size of the complexes
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Conclusions
Conclusions
Reconstructions :
non-smooth surfaces remain openhigh dimensions still not satisfactory
Homology :
Size of the complexesmore efficient algorithms
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Conclusions
Conclusions
Reconstructions :
non-smooth surfaces remain openhigh dimensions still not satisfactory
Homology :
Size of the complexesmore efficient algorithms
Didn’t talk about :
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Conclusions
Conclusions
Reconstructions :
non-smooth surfaces remain openhigh dimensions still not satisfactory
Homology :
Size of the complexesmore efficient algorithms
Didn’t talk about :
functions on spaces
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Conclusions
Conclusions
Reconstructions :
non-smooth surfaces remain openhigh dimensions still not satisfactory
Homology :
Size of the complexesmore efficient algorithms
Didn’t talk about :
functions on spacespersistence, Reeb graphs, Morse-Smale complexes, Laplacespectra...etc.
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Thank
Thank You
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