Download - Topological Inference via Meshing
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Topological Inference via MeshingBenoit Hudson, Gary Miller, Steve Oudot and Don Sheehy
SoCG 2010
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Mesh Generationand
Persistent Homology
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The Problem
Input: Points in Euclidean space sampled from some unknown object.
Output: Information about the topology of the unknown object.
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Points, offsets, homology, and persistence.
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Points, offsets, homology, and persistence.
Input: P ! Rd
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Points, offsets, homology, and persistence.
P! =!
p!P
ball(p,!)
Input: P ! Rd
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Points, offsets, homology, and persistence.
P! =!
p!P
ball(p,!)
Input: P ! Rd
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Points, offsets, homology, and persistence.
P! =!
p!P
ball(p,!)
Input: P ! Rd
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Points, offsets, homology, and persistence.
P! =!
p!P
ball(p,!)
Input: P ! Rd
Offsets
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Points, offsets, homology, and persistence.
P! =!
p!P
ball(p,!)
Input: P ! Rd
Offsets
Compute the Homology
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Points, offsets, homology, and persistence.
P! =!
p!P
ball(p,!)
Input: P ! Rd
Offsets
Compute the Homology
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Points, offsets, homology, and persistence.
P! =!
p!P
ball(p,!)
Input: P ! Rd
Offsets
Compute the Homology
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Points, offsets, homology, and persistence.
P! =!
p!P
ball(p,!)
Input: P ! Rd
Offsets
Compute the Homology
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Points, offsets, homology, and persistence.
P! =!
p!P
ball(p,!)
Input: P ! Rd
Offsets
Compute the Homology
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Points, offsets, homology, and persistence.
P! =!
p!P
ball(p,!)
Input: P ! Rd
Offsets
Compute the Homology
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Points, offsets, homology, and persistence.
P! =!
p!P
ball(p,!)
Input: P ! Rd
Offsets
Compute the Homology
Persistent
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Persistence Diagrams
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Persistence Diagrams
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Persistence Diagrams
d!B = maxi
|pi ! qi|!
Bottleneck Distance
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Persistence Diagrams
d!B = maxi
|pi ! qi|!
Bottleneck Distance
Approximate
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Persistence Diagrams
d!B = maxi
|pi ! qi|!
Bottleneck Distance
Approximate Birth and Death times differ by a constant factor.
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Persistence Diagrams
d!B = maxi
|pi ! qi|!
Bottleneck Distance
This is just the bottleneck distance of the log-scale diagrams.
Approximate Birth and Death times differ by a constant factor.
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Persistence Diagrams
d!B = maxi
|pi ! qi|!
Bottleneck Distance
This is just the bottleneck distance of the log-scale diagrams.
log a ! log b < !
log a
b< !
a
b< 1 + !
Approximate Birth and Death times differ by a constant factor.
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We need to build a filtered simplicial complex.
Associate a birth time with each simplex in complex K.
At timeα, we have a complex Kα consisting of all simplices born at or before time α.
time
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There are two phases, one is geometric the other is topological.
Geometry Topology(linear algebra)
Build a filtration, i.e. a filtered simplicial
complex.
Compute the persistence diagram
(Run the Persistence Algorithm).
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We’ll focus on this side.
There are two phases, one is geometric the other is topological.
Geometry Topology(linear algebra)
Build a filtration, i.e. a filtered simplicial
complex.
Compute the persistence diagram
(Run the Persistence Algorithm).
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We’ll focus on this side.
There are two phases, one is geometric the other is topological.
Geometry Topology(linear algebra)
Build a filtration, i.e. a filtered simplicial
complex.
Compute the persistence diagram
(Run the Persistence Algorithm).
Running time: O(N3).N is the size of the complex.
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Idea 1: Use the Delaunay Triangulation
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Idea 1: Use the Delaunay Triangulation
Good: It works, (alpha-complex filtration).
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Idea 1: Use the Delaunay Triangulation
Good: It works, (alpha-complex filtration).
Bad: It can have size nO(d).
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Idea 2: Connect up everything close.
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Idea 2: Connect up everything close.
Čech Filtration: Add a k-simplex for every k+1 points that have a smallest enclosing ball of radius at mostα.
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Idea 2: Connect up everything close.
Čech Filtration: Add a k-simplex for every k+1 points that have a smallest enclosing ball of radius at mostα.
Rips Filtration: Add a k-simplex for every k+1 points that have all pairwise distances at mostα.
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Idea 2: Connect up everything close.
Čech Filtration: Add a k-simplex for every k+1 points that have a smallest enclosing ball of radius at mostα.
Rips Filtration: Add a k-simplex for every k+1 points that have all pairwise distances at mostα.
Still nd, but we can quit early.
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Our Idea: Build a quality mesh.
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Our Idea: Build a quality mesh.
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Our Idea: Build a quality mesh.
We can build meshes of size 2O(d )n.2
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Meshing Counter-intuition
Delaunay Refinement can take less time and space than
Delaunay Triangulation.
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Meshing Counter-intuition
Delaunay Refinement can take less time and space than
Delaunay Triangulation.
Theorem [Hudson, Miller, Phillips, ’06]:A quality mesh of a point set canbe constructed in O(n log !) time,where ! is the spread.
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Meshing Counter-intuition
Delaunay Refinement can take less time and space than
Delaunay Triangulation.
Theorem [Hudson, Miller, Phillips, ’06]:
Theorem [Miller, Phillips, Sheehy, ’08]:
A quality mesh of a point set canbe constructed in O(n log !) time,where ! is the spread.
A quality mesh of a well-paced
point set has size O(n).
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The α-mesh filtration
1. Build a mesh M.
2. Assign birth times to vertices based on distance to P (special case points very close to P).
3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices.
4. Feed this filtered complex to the persistence algorithm.
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The α-mesh filtration
1. Build a mesh M.
2. Assign birth times to vertices based on distance to P (special case points very close to P).
3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices.
4. Feed this filtered complex to the persistence algorithm.
![Page 43: Topological Inference via Meshing](https://reader033.vdocuments.us/reader033/viewer/2022052619/555095fcb4c90595208b45e3/html5/thumbnails/43.jpg)
The α-mesh filtration
1. Build a mesh M.
2. Assign birth times to vertices based on distance to P (special case points very close to P).
3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices.
4. Feed this filtered complex to the persistence algorithm.
![Page 44: Topological Inference via Meshing](https://reader033.vdocuments.us/reader033/viewer/2022052619/555095fcb4c90595208b45e3/html5/thumbnails/44.jpg)
The α-mesh filtration
1. Build a mesh M.
2. Assign birth times to vertices based on distance to P (special case points very close to P).
3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices.
4. Feed this filtered complex to the persistence algorithm.
![Page 45: Topological Inference via Meshing](https://reader033.vdocuments.us/reader033/viewer/2022052619/555095fcb4c90595208b45e3/html5/thumbnails/45.jpg)
The α-mesh filtration
1. Build a mesh M.
2. Assign birth times to vertices based on distance to P (special case points very close to P).
3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices.
4. Feed this filtered complex to the persistence algorithm.
![Page 46: Topological Inference via Meshing](https://reader033.vdocuments.us/reader033/viewer/2022052619/555095fcb4c90595208b45e3/html5/thumbnails/46.jpg)
The α-mesh filtration
1. Build a mesh M.
2. Assign birth times to vertices based on distance to P (special case points very close to P).
3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices.
4. Feed this filtered complex to the persistence algorithm.
![Page 47: Topological Inference via Meshing](https://reader033.vdocuments.us/reader033/viewer/2022052619/555095fcb4c90595208b45e3/html5/thumbnails/47.jpg)
The α-mesh filtration
1. Build a mesh M.
2. Assign birth times to vertices based on distance to P (special case points very close to P).
3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices.
4. Feed this filtered complex to the persistence algorithm.
![Page 48: Topological Inference via Meshing](https://reader033.vdocuments.us/reader033/viewer/2022052619/555095fcb4c90595208b45e3/html5/thumbnails/48.jpg)
Approximation via interleaving.
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Definition:
Approximation via interleaving.
Two filtrations, {P!} and {Q!} are!-interleaved if P!!" ! Q! ! P!+"
for all ".
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Definition:
Approximation via interleaving.
Theorem [Chazal et al, ’09]:
Two filtrations, {P!} and {Q!} are!-interleaved if P!!" ! Q! ! P!+"
for all ".
If {P!} and {Q!} are !-interleaved thentheir persistence diagrams are !-close inthe bottleneck distance.
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The Voronoi filtration interleaves with the offset filtration.
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The Voronoi filtration interleaves with the offset filtration.
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The Voronoi filtration interleaves with the offset filtration.
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The Voronoi filtration interleaves with the offset filtration.
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The Voronoi filtration interleaves with the offset filtration.
Theorem:
For all ! > rP , V!/"M ! P!
! V!"M ,
where rP is minimum distance betweenany pair of points in P .
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The Voronoi filtration interleaves with the offset filtration.
Finer refinement yields a tighter interleaving.
Theorem:
For all ! > rP , V!/"M ! P!
! V!"M ,
where rP is minimum distance betweenany pair of points in P .
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The Voronoi filtration interleaves with the offset filtration.
Finer refinement yields a tighter interleaving.
Theorem:
For all ! > rP , V!/"M ! P!
! V!"M ,
where rP is minimum distance betweenany pair of points in P .
Special case for small scales.
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Geometric Approximation
Topologically Equivalent
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Geometric Approximation
Topologically Equivalent
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The Results
1. Build a mesh M.
2. Assign birth times to vertices based on distance to P (special case points very close to P).**
3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices.
4. Feed this filtered complex to the persistence algorithm.
Approximation ratio Complex Size
Previous Work
Simple mesh
filtration
Over-refine the mesh
Linear-Size Meshing
1 nO(d)
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The Results
1. Build a mesh M.
2. Assign birth times to vertices based on distance to P (special case points very close to P).**
3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices.
4. Feed this filtered complex to the persistence algorithm.
Approximation ratio Complex Size
Previous Work
Simple mesh
filtration
Over-refine the mesh
Linear-Size Meshing
!
1 nO(d)
2O(d2)n log !
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The Results
1. Build a mesh M.
2. Assign birth times to vertices based on distance to P (special case points very close to P).**
3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices.
4. Feed this filtered complex to the persistence algorithm.
Approximation ratio Complex Size
Previous Work
Simple mesh
filtration
Over-refine the mesh
Linear-Size Meshing
!
1 nO(d)
2O(d2)n log !=~3
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The Results
1. Build a mesh M.
2. Assign birth times to vertices based on distance to P (special case points very close to P).**
3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices.
4. Feed this filtered complex to the persistence algorithm.
Approximation ratio Complex Size
Previous Work
Simple mesh
filtration
Over-refine the mesh
Linear-Size Meshing
Over-refine it.
!
1 + !
1 nO(d)
2O(d2)n log !
!!O(d2)
n log !
=~3
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The Results
1. Build a mesh M.
2. Assign birth times to vertices based on distance to P (special case points very close to P).**
3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices.
4. Feed this filtered complex to the persistence algorithm.
Approximation ratio Complex Size
Previous Work
Simple mesh
filtration
Over-refine the mesh
Linear-Size Meshing
Over-refine it.Use linear-size meshing.
!
1 + !
1 + ! + 3"
1 nO(d)
2O(d2)n log !
!!O(d2)
n log !
(!")!O(d2)n
=~3
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Thank you.