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![An Adaptive Metric
Idea: Let’s replace our variable radii balls with
equal radii balls in a different metric.
(t)
: [0, a] :! Rd](https://image.slidesharecdn.com/osu16sampling-160613132440/85/Some-Thoughts-on-Sampling-8-320.jpg)
![An Adaptive Metric
Idea: Let’s replace our variable radii balls with
equal radii balls in a different metric.
(t)
: [0, a] :! Rd
len( ) :=
Z a
0
k 0
(t)kL
dt](https://image.slidesharecdn.com/osu16sampling-160613132440/85/Some-Thoughts-on-Sampling-9-320.jpg)
![An Adaptive Metric
Idea: Let’s replace our variable radii balls with
equal radii balls in a different metric.
(t)
: [0, a] :! Rd
len( ) :=
Z a
0
k 0
(t)kL
dt](https://image.slidesharecdn.com/osu16sampling-160613132440/85/Some-Thoughts-on-Sampling-10-320.jpg)
![An Adaptive Metric
Idea: Let’s replace our variable radii balls with
equal radii balls in a different metric.
(t)
: [0, a] :! Rd
len( ) :=
Z a
0
k 0
(t)kL
dt
=
Z a
0
k 0
(t)k
fL( (t))
dt](https://image.slidesharecdn.com/osu16sampling-160613132440/85/Some-Thoughts-on-Sampling-11-320.jpg)
![An Adaptive Metric
Idea: Let’s replace our variable radii balls with
equal radii balls in a different metric.
(t)
: [0, a] :! Rd
len( ) :=
Z a
0
k 0
(t)kL
dt
=
Z a
0
k 0
(t)k
fL( (t))
dt
=
Z
dz
fL(z)](https://image.slidesharecdn.com/osu16sampling-160613132440/85/Some-Thoughts-on-Sampling-12-320.jpg)
![An Adaptive Metric
Idea: Let’s replace our variable radii balls with
equal radii balls in a different metric.
(t)
: [0, a] :! Rd
len( ) :=
Z a
0
k 0
(t)kL
dt
=
Z a
0
k 0
(t)k
fL( (t))
dt
=
Z
dz
fL(z)
dL
(x, y) := inf
2Path(x,y)
len( )
The metric induced by L.](https://image.slidesharecdn.com/osu16sampling-160613132440/85/Some-Thoughts-on-Sampling-13-320.jpg)
![An Adaptive Metric
Idea: Let’s replace our variable radii balls with
equal radii balls in a different metric.
(t)
: [0, a] :! Rd
len( ) :=
Z a
0
k 0
(t)kL
dt
=
Z a
0
k 0
(t)k
fL( (t))
dt
=
Z
dz
fL(z)
dL
(x, y) := inf
2Path(x,y)
len( )
The metric induced by L.
Coming up: Adaptive samples correspond to uniform samples
in the metric induced by L.](https://image.slidesharecdn.com/osu16sampling-160613132440/85/Some-Thoughts-on-Sampling-14-320.jpg)
![kNN Sampling by Delaunay Refinement
maintain Voronoi/Delaunay incrementally
loop until no more points are added:
if some Voronoi cell V contains k or more points:
insert the farthest corner of V
!
if some Delaunay circumball C contains k or more points:
insert the center of C
!
while any Voronoi cell V has aspect ratio > tau:
insert the farthest corner of V
Based on Sparse Voronoi Refinement [Hudson et al ’06]](https://image.slidesharecdn.com/osu16sampling-160613132440/85/Some-Thoughts-on-Sampling-83-320.jpg)
Uploaded byDon Sheehy
Some Thoughts on Sampling
The document discusses surface reconstruction and adaptive sampling techniques in the context of geometric data analysis. It highlights the importance of homology inference, critical point theory, and properties of distance functions in effective reconstruction and sampling strategies. Additionally, it addresses mesh generation and the quality criteria for decomposing a domain into simple elements.
More Related Content
Some Thoughts on Sampling
- 1. Some thoughts onSampling Don Sheehy University of Connecticut ! joint work with Nick Cavanna and Kirk Gardner
- 2.
- 3. Surface Reconstruction Unknown surfaceX. Finite sample P Medial Axis L
- 4. We say Pis an (adaptive) "-sample if for all x 2 X, there is a p 2 P s.t. kx pk fL(x) ". Surface Reconstruction Unknown surface X. Finite sample P Medial Axis L
- 5. We say Pis an (adaptive) "-sample if for all x 2 X, there is a p 2 P s.t. kx pk fL(x) ". Surface Reconstruction Unknown surface X. Finite sample P Medial Axis L Several known algorithms for producing a homeomorphic reconstruction given such a sample.
- 6. We say Pis an (adaptive) "-sample if for all x 2 X, there is a p 2 P s.t. kx pk fL(x) ". Surface Reconstruction Unknown surface X. Finite sample P Medial Axis L Several known algorithms for producing a homeomorphic reconstruction given such a sample. fL(x) := min y2L kx yk local feature size
- 7. An Adaptive Metric Idea:Let’s replace our variable radii balls with equal radii balls in a different metric.
- 8. An Adaptive Metric Idea:Let’s replace our variable radii balls with equal radii balls in a different metric. (t) : [0, a] :! Rd
- 9. An Adaptive Metric Idea:Let’s replace our variable radii balls with equal radii balls in a different metric. (t) : [0, a] :! Rd len( ) := Z a 0 k 0 (t)kL dt
- 10. An Adaptive Metric Idea:Let’s replace our variable radii balls with equal radii balls in a different metric. (t) : [0, a] :! Rd len( ) := Z a 0 k 0 (t)kL dt
- 11. An Adaptive Metric Idea:Let’s replace our variable radii balls with equal radii balls in a different metric. (t) : [0, a] :! Rd len( ) := Z a 0 k 0 (t)kL dt = Z a 0 k 0 (t)k fL( (t)) dt
- 12. An Adaptive Metric Idea:Let’s replace our variable radii balls with equal radii balls in a different metric. (t) : [0, a] :! Rd len( ) := Z a 0 k 0 (t)kL dt = Z a 0 k 0 (t)k fL( (t)) dt = Z dz fL(z)
- 13. An Adaptive Metric Idea:Let’s replace our variable radii balls with equal radii balls in a different metric. (t) : [0, a] :! Rd len( ) := Z a 0 k 0 (t)kL dt = Z a 0 k 0 (t)k fL( (t)) dt = Z dz fL(z) dL (x, y) := inf 2Path(x,y) len( ) The metric induced by L.
- 14. An Adaptive Metric Idea:Let’s replace our variable radii balls with equal radii balls in a different metric. (t) : [0, a] :! Rd len( ) := Z a 0 k 0 (t)kL dt = Z a 0 k 0 (t)k fL( (t)) dt = Z dz fL(z) dL (x, y) := inf 2Path(x,y) len( ) The metric induced by L. Coming up: Adaptive samples correspond to uniform samples in the metric induced by L.
- 15.
- 16.
- 17.
- 18. Niyogi-Smale-Weinberger Homology Inference of Submanifolds Takea union of balls.1
- 19. Niyogi-Smale-Weinberger Homology Inference of Submanifolds Takea union of balls.1 Compute the homology of the resulting Cech complex.2
- 20. Niyogi-Smale-Weinberger Homology Inference of Submanifolds Takea union of balls.1 Compute the homology of the resulting Cech complex.2
- 21. Niyogi-Smale-Weinberger Homology Inference of Submanifolds Takea union of balls.1 Compute the homology of the resulting Cech complex.2 0 Denoise the data.
- 22. Niyogi-Smale-Weinberger Homology Inference of Submanifolds Takea union of balls.1 Compute the homology of the resulting Cech complex.2 0 Denoise the data.
- 23. Niyogi-Smale-Weinberger Homology Inference of Submanifolds Takea union of balls.1 Compute the homology of the resulting Cech complex.2 0 Denoise the data.
- 24. Niyogi-Smale-Weinberger Homology Inference of Submanifolds Takea union of balls.1 Compute the homology of the resulting Cech complex.2 0 Denoise the data.
- 25. Niyogi-Smale-Weinberger Homology Inference of Submanifolds Takea union of balls.1 Compute the homology of the resulting Cech complex.2 0 Denoise the data.
- 26. Niyogi-Smale-Weinberger Homology Inference of Submanifolds Takea union of balls.1 Compute the homology of the resulting Cech complex.2 0 Denoise the data. To prove: The density is bounded from below near M and from above far from M. (Related to kNN density estimator) ! Explicitly construct homotopy equivalence between balls and the submanifold (using the distance).
- 27. weak feature sizeand mu-reach
- 28. weak feature sizeand mu-reach Key Technique: The Critical Point Theory of Distance Functions ! Applies to compact subsets of Riemannian Manifolds
- 29. weak feature sizeand mu-reach Key Technique: The Critical Point Theory of Distance Functions ! Applies to compact subsets of Riemannian Manifolds Weak Feature Size: Let L be the critical points of the distance to X. ! mu-reach: Let L be the points where the gradient of the distance to X is at most mu.
- 30. weak feature sizeand mu-reach Key Technique: The Critical Point Theory of Distance Functions ! Applies to compact subsets of Riemannian Manifolds Weak Feature Size: Let L be the critical points of the distance to X. ! mu-reach: Let L be the points where the gradient of the distance to X is at most mu. See Chazal, Lieutier ’05,’06,’09 and Chazal, Cohen-Steiner-Lieutier ’09.
- 31. Where does themetric come from?
- 32. Where does themetric come from? fL(x) := min y2L kx yk fX(x) := min y2X kx yk Distance to a set
- 33. Where does themetric come from? fL(x) := min y2L kx yk fX(x) := min y2X kx yk Distance to a set fL X(y) := min x2X dL (x, y) = min x2X inf 2Path(x,y) Z dz fL(z) Distance to a set in the induced metric
- 34. Where does themetric come from? fL(x) := min y2L kx yk fX(x) := min y2X kx yk Distance to a set fL X(y) := min x2X dL (x, y) = min x2X inf 2Path(x,y) Z dz fL(z) Distance to a set in the induced metric cfL X(y) := min x2X kx yk fL(x) Approximate distance to a set, using adaptivity in the Euclidean metric
- 35. Where does themetric come from? fL(x) := min y2L kx yk fX(x) := min y2X kx yk Distance to a set fL X(y) := min x2X dL (x, y) = min x2X inf 2Path(x,y) Z dz fL(z) Distance to a set in the induced metric cfL X(y) := min x2X kx yk fL(x) Approximate distance to a set, using adaptivity in the Euclidean metric Offsets Filtrations AL X (r) := {x 2 Rd | fL X(x) r}
- 36. Where does themetric come from? fL(x) := min y2L kx yk fX(x) := min y2X kx yk Distance to a set fL X(y) := min x2X dL (x, y) = min x2X inf 2Path(x,y) Z dz fL(z) Distance to a set in the induced metric cfL X(y) := min x2X kx yk fL(x) Approximate distance to a set, using adaptivity in the Euclidean metric Offsets Filtrations AL X (r) := {x 2 Rd | fL X(x) r} BL X(r) := {x 2 Rd | cfL X (x) r}
- 37. Where does themetric come from? fL(x) := min y2L kx yk fX(x) := min y2X kx yk Distance to a set fL X(y) := min x2X dL (x, y) = min x2X inf 2Path(x,y) Z dz fL(z) Distance to a set in the induced metric cfL X(y) := min x2X kx yk fL(x) Approximate distance to a set, using adaptivity in the Euclidean metric Offsets Filtrations AL X (r) := {x 2 Rd | fL X(x) r} BL X(r) := {x 2 Rd | cfL X (x) r} = [ x2X ball(x, r fL(x))
- 38. Uniform Samples ofAdaptive Metrics
- 39. Uniform Samples ofAdaptive Metrics dL (x, Y ) := min y2Y dL (x, y)
- 40. Uniform Samples ofAdaptive Metrics dL (x, Y ) := min y2Y dL (x, y) dL H(X, Y ) := max{max x2X dL (x, Y ), max y2Y dL (y, X)}
- 41. Uniform Samples ofAdaptive Metrics dL (x, Y ) := min y2Y dL (x, y) dL H(X, Y ) := max{max x2X dL (x, Y ), max y2Y dL (y, X)} Uniform Sample:dL H(P, X) "
- 42. Uniform Samples ofAdaptive Metrics dL (x, Y ) := min y2Y dL (x, y) dL H(X, Y ) := max{max x2X dL (x, Y ), max y2Y dL (y, X)} Uniform Sample:dL H(P, X) " Theorem Let L and X be compact sets, let P ⇢ X be a sample, and let " 2 [0, 1) be a constant. If P is an "-sample of X with respect to the distance to L, then dL H(X, P) " 1 " . Also, if dL H(X, P) " < 1 2 , then P is an " 1 " -sample of X with respect to the distance to L.
- 43.
- 44. Interleaving Filtrations A pairof filtrations (F, G) is (h1, h2)-interleaved in (s, t) if F(r) ✓ G(h1(r)) whenever r, h1(r) 2 (s, t) and G(r) ✓ F(h2(r)) whenever r, h2(r) 2 (s, t). (h1, h2 must be nondecreasing in (s, t)).
- 45. Interleaving Filtrations A pairof filtrations (F, G) is (h1, h2)-interleaved in (s, t) if F(r) ✓ G(h1(r)) whenever r, h1(r) 2 (s, t) and G(r) ✓ F(h2(r)) whenever r, h2(r) 2 (s, t). (h1, h2 must be nondecreasing in (s, t)). If (F, G) is (h1, h2)-interleaved in (s1, t1), and (G, H) is (h3, h4)-interleaved in (s2, t2), then (F, H) is (h3 h1, h2 h4)-interleaved in (s3, t3), where s3 = max{s1, s2} and t3 = min{t1, t2}. Lemma (Composing Interleavings)
- 46. Replacing X witha sample
- 47. Replacing X witha sample Lemma 1 If dL H( bX, X) ", then (AL X , AL bX ) is (h1, h1)-interleaved in (0, 1), where h1(r) = r + ".
- 48. Replacing X witha sample Lemma 1 Follows from triangle inequality and definition of Hausdorff distance. If dL H( bX, X) ", then (AL X , AL bX ) is (h1, h1)-interleaved in (0, 1), where h1(r) = r + ".
- 49. Using the EuclideanMetric
- 50. Using the EuclideanMetric Lemma 2 The pair (AL bX , BL bX ) are (h2, h2)-interleaved in (0, 1), where h2(r) = r 1 r .
- 51. Using the EuclideanMetric Lemma 2 The pair (AL bX , BL bX ) are (h2, h2)-interleaved in (0, 1), where h2(r) = r 1 r . x Key step is to prove the interleaving for a single point.
- 52.
- 53. Approximating L Lemma 3 Ifd bX H (L, bL) for some < 1, then (BL bX , B bL bX ) is (h3, h3)- interleaved in (0, 1), where h3(r) = r 1 .
- 54. Approximating L Lemma 3 Ifd bX H (L, bL) for some < 1, then (BL bX , B bL bX ) is (h3, h3)- interleaved in (0, 1), where h3(r) = r 1 . Sampling condition for L is dual to that for X. ! This shows that an “adaptive sample” of L suffices to give the desired interleaving.
- 55.
- 56. The Big Interleaving Theorem LetL, bL ⇢ Rd and X, bX ⇢ Rd (L [ bL) be compact sets. If d bX H (L, bL) < 1 and dL H( bX, X) " < 1, then (AL X , B bL bX ) are (h4, h5)-interleaved in (0, 1), where h4(r) = r+" (1 r ")(1 ) and h5(r) = r 1 r + ".
- 57. The Big Interleaving Takeaway:Sufficient conditions on sampling X and L to guarantee an interleaving. Theorem Let L, bL ⇢ Rd and X, bX ⇢ Rd (L [ bL) be compact sets. If d bX H (L, bL) < 1 and dL H( bX, X) " < 1, then (AL X , B bL bX ) are (h4, h5)-interleaved in (0, 1), where h4(r) = r+" (1 r ")(1 ) and h5(r) = r 1 r + ".
- 58.
- 59. kNN Sampling Let Pbe a finite set of points in a bounding region B.
- 60. kNN Sampling fP,k(x) :=distance to kth nearest point of P Let P be a finite set of points in a bounding region B.
- 61. kNN Sampling fP,k(x) :=distance to kth nearest point of P Let P be a finite set of points in a bounding region B. Goal: Compute a set M such that M is ⌧-well-spaced, and for all x 2 B ↵fP,k(x) fM,2(x) fP,k(x) for some constants ↵ and .
- 62. kNN Sampling fP,k(x) :=distance to kth nearest point of P Let P be a finite set of points in a bounding region B. Goal: Compute a set M such that M is ⌧-well-spaced, and for all x 2 B ↵fP,k(x) fM,2(x) fP,k(x) for some constants ↵ and .
- 63.
- 64. Mesh Generation Decompose adomain into simple elements.
- 65. Mesh Generation Decompose adomain into simple elements.
- 66. Mesh Generation Mesh Quality Radius/Edge< const Decompose a domain into simple elements.
- 67. Mesh Generation X X✓ MeshQuality Radius/Edge < const Decompose a domain into simple elements.
- 68. Mesh Generation X X✓ MeshQuality Radius/Edge < const Conforming to Input Decompose a domain into simple elements.
- 69. Mesh Generation X X✓ MeshQuality Radius/Edge < const Conforming to Input Decompose a domain into simple elements.
- 70. Mesh Generation X X✓ MeshQuality Radius/Edge < const Conforming to Input Decompose a domain into simple elements.
- 71. Mesh Generation X X✓ MeshQuality Radius/Edge < const Conforming to Input Decompose a domain into simple elements.
- 72. Mesh Generation X X✓ MeshQuality Radius/Edge < const Conforming to Input Decompose a domain into simple elements. Voronoi Diagram
- 73. Mesh Generation X X✓ MeshQuality Radius/Edge < const OutRadius/InRadius < const Conforming to Input Decompose a domain into simple elements. Voronoi Diagram
- 74. Mesh Generation X X✓ ✓X MeshQuality Radius/Edge < const OutRadius/InRadius < const Conforming to Input Decompose a domain into simple elements. Voronoi Diagram
- 75. Mesh Generation X X✓ ✓X MeshQuality Radius/Edge < const OutRadius/InRadius < const Conforming to Input Decompose a domain into simple elements. Voronoi Diagram
- 76. Mesh Generation X X✓ ✓X MeshQuality Radius/Edge < const OutRadius/InRadius < const Conforming to Input Decompose a domain into simple elements. Voronoi Diagram
- 77. Meshing Point Sets Input:P ⊂ Rd Output: M ⊃ P with a “nice” Voronoi diagram n = |P|, m = |M|
- 78. Meshing Point Sets Input:P ⊂ Rd Output: M ⊃ P with a “nice” Voronoi diagram n = |P|, m = |M|
- 79. Meshing Point Sets Input:P ⊂ Rd Output: M ⊃ P with a “nice” Voronoi diagram n = |P|, m = |M|
- 80. Meshing Point Sets Input:P ⊂ Rd Output: M ⊃ P with a “nice” Voronoi diagram n = |P|, m = |M|
- 81. Meshing Point Sets AspectRatio (quality): Cell Sizing: Constant Local Complexity: Optimality and Running time: v Rv rv Rv rv ≤ τ |M| = Θ(|Optimal|) Running time: O(n log n + |M|) The degree of the 1-skeleton is 2O(d) . Rv = ⇥(fP,2(v))
- 82. kNN Sampling byDelaunay Refinement maintain Voronoi/Delaunay incrementally loop until no more points are added: if some Voronoi cell V contains k or more points: insert the farthest corner of V ! if some Delaunay circumball C contains k or more points: insert the center of C ! while any Voronoi cell V has aspect ratio > tau: insert the farthest corner of V
- 83. kNN Sampling byDelaunay Refinement maintain Voronoi/Delaunay incrementally loop until no more points are added: if some Voronoi cell V contains k or more points: insert the farthest corner of V ! if some Delaunay circumball C contains k or more points: insert the center of C ! while any Voronoi cell V has aspect ratio > tau: insert the farthest corner of V Based on Sparse Voronoi Refinement [Hudson et al ’06]
- 84. kNN Sampling byDelaunay Refinement Key to the analysis: If the aspect ratio is bounded, then any sufficiently small ball that is not contained entirely in a Voronoi cell is contained entirely in a Delaunay circumball.
- 85. Summary There is aRiemannian metric induced by the distance to a compact set that relates uniform and adaptive samples. Interleaving! ! kNN sampling can be done efficiently with a simple variant of Delaunay refinement.
- 86. Summary There is aRiemannian metric induced by the distance to a compact set that relates uniform and adaptive samples. Interleaving! ! kNN sampling can be done efficiently with a simple variant of Delaunay refinement. Thanks.