Tolerance Volume Manish Mandad David Cohen-Steiner Pierre Alliez - - PowerPoint PPT Presentation

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Tolerance Volume Manish Mandad David Cohen-Steiner Pierre Alliez - - PowerPoint PPT Presentation

Sep 19, 2023 263 likes •1.12k views

Isotopic Approximation within a Tolerance Volume Manish Mandad David Cohen-Steiner Pierre Alliez Inria Sophia Antipolis - 1 Goals and Motivation - 2 Goals and Motivation Input: Tolerance volume of a surface geometry - 3 Goals and

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SLIDE 1

Isotopic Approximation within a Tolerance Volume

Manish Mandad David Cohen-Steiner Pierre Alliez

  • 1

Inria Sophia Antipolis

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SLIDE 2

Goals and Motivation

  • 2
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SLIDE 3

Goals and Motivation

  • Input:
  • Tolerance volume of

a surface geometry

  • 3
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SLIDE 4

Goals and Motivation

  • Input:
  • Tolerance volume of

a surface geometry

  • Output:
  • Surface triangle mesh
  • Properties:

 Within tolerance volume Motivation: control global approximation error

  • 4
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SLIDE 5

Goals and Motivation

  • Input:
  • Tolerance volume of

a surface geometry

  • Output:
  • Surface triangle mesh
  • Properties:

 Within tolerance volume  Intersection free Motivation: simulation, machining, printing etc.

  • 5
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SLIDE 6

Goals and Motivation

  • Input:
  • Tolerance volume of

a surface geometry

  • Output:
  • Surface triangle mesh
  • Properties:

 Within tolerance volume  Intersection free  Low vertex count Motivation: get an approximated mesh, low polygon count

  • 6
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SLIDE 7

Goals and Motivation A Condition for Isotopic Approximation

[Chazal, Cohen-Steiner 04]

  • 7

Constructive algorithm for this theoretical result

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SLIDE 8

Goals and Motivation

  • 8

Ω

Ω : Topological thickening of S

A Condition for Isotopic Approximation

[Chazal, Cohen-Steiner 04] ∂Ω2 ∂Ω1

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SLIDE 9

Goals and Motivation

  • 9

Ω : Topological thickening of S If

  • S’ is included and separates sides of Ω,
  • S’ is connected and
  • Genus of S’ does not exceed genus of S

Then, S’ and S are isotopic.

S’

A Condition for Isotopic Approximation

[Chazal, Cohen-Steiner 04] Ω

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SLIDE 10

Related Work

  • 10
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SLIDE 11

Related Work

  • 11

Simplification Envelopes Cohen et al. 1996 Multiresolution decimation based on global error Ciampalini et al. 1997 Adaptively Sampled Distance Fields Frisken et al. 2000 Permission Grids: Practical, Error-Bounded Simplification Zelinka – Garland 2002 Intersection Free Simplification Gumhold et al. 2003 GPU-based Tolerance Volumes for Mesh Processing Botsch et al. 2004 Simplification of surface mesh using Hausdorff envelope Borouchaki – Frey 2005

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SLIDE 12

Related Work

Simplification Envelopes [Cohen et al. 96]

  • 12
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SLIDE 13

Related Work

Simplification Envelopes [Cohen et al. 96]

  • 13

― Very good for simple geometry ― Input dependent ― Tolerance hinders simplification

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SLIDE 14

Overview

  • 14

Tolerance Volume Ω Sampling Refinement Simplification Output

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SLIDE 15

Algorithm

  • 15
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SLIDE 16

Algorithm (Initialization)

  • S: Point sample of ∂Ω with labels
  • 16

Initialization

f = -1 f = +1 σ radii balls at S cover ∂Ω

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SLIDE 17

Algorithm (Initialization)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box

+1 +1 +1 +1

f = +1 f = +1 f = -1

  • 17

Initialization

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SLIDE 18

Algorithm (Initialization)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box

+1 +1 +1 +1

f = +1 f = +1 f = -1 fT

  • 18

Initialization

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SLIDE 19

Algorithm (Initialization)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box

+1 +1 +1 +1

Error Error µ(s)=|f µ(s)=|f(s) s)–fT(s (s)| )| f = +1 f = +1 f = -1 fT s

  • 19

Initialization

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SLIDE 20

Algorithm (coarse-to-fine)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box
  • While (S not classified)
  • Insert Steiner point to T

Sample point (s ∈ S) with maximum error

  • 20

Refinement

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SLIDE 21

Algorithm (coarse-to-fine)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box
  • While (S not classified)
  • Insert Steiner point to T

f = +1 f = +1 f = -1

fT = 0 Zero-set

  • 21

Refinement

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SLIDE 22

Algorithm (coarse-to-fine)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box
  • While (S not classified)
  • Insert Steiner point to T
  • 22

Refinement

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SLIDE 23

Algorithm (coarse-to-fine)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box
  • While (S not classified)
  • Insert Steiner point to T
  • 23

Refinement

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SLIDE 24

Algorithm (coarse-to-fine)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box
  • While (S not classified)
  • Insert Steiner point to T
  • 24

Refinement

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SLIDE 25

Algorithm (coarse-to-fine)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box
  • While (S not classified)
  • Insert Steiner point to T
  • 25

Refinement

Boundary of simplicial tolerance (∂Ɣ)

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SLIDE 26

Algorithm (coarse-to-fine)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box
  • While (S not classified)
  • Insert Steiner point to T

Extension to 3D? Guarantees?

  • 26

Refinement

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SLIDE 27

Refinement (conditions)

  • 27

Zero set may cross ∂Ω due to flat tetrahedra

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SLIDE 28

Refinement (conditions)

Ensure interpolated function is Lipschitz

  • 28

Zero set may cross ∂Ω due to flat tetrahedra

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SLIDE 29

Refinement (conditions)

Ensure interpolated function is Lipschitz

  • μ(s) < 1 – α,

given some α ∈ [0, 1] μ(s) < 1

Classify samples with a margin

  • 28

Zero set may cross ∂Ω due to flat tetrahedra

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SLIDE 30

Refinement (conditions)

  • μ(s) < 1 – α,

given some α ∈ [0, 1]

  • h > 2σ/α
  • 29

Ensure interpolated function is Lipschitz

Zero set may cross ∂Ω due to flat tetrahedra

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SLIDE 31

Refinement (conditions)

What else do we need to fix ?

  • 30
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SLIDE 32

Refinement (conditions)

  • Misoriented elements
  • 31
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SLIDE 33

Refinement (conditions)

  • Misoriented elements
  • Interpolated function should classify well local geometry

fT defined on ΔABC should classify well samples of S nearest (orange) to a shrunk triangle (green)

  • 32
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SLIDE 34

Refinement (conditions)

S not classified : μ(s)>1

{

  • μ(s)

) ≥ 1–α given ven some me α∈[0;1 [0;1]

  • h

h ≤ 2σ/α

  • f does

f does not cla

  • t classifies

ifies local ge cal geometry etry

}

  • 33
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SLIDE 35

Algorithm (fine-to-coarse)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box
  • While (S not classified)
  • Insert Steiner point to T
  • Collapse edges of ∂Ɣ
  • 34

Simplification

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SLIDE 36

Simplification (conditions)

  • 36
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SLIDE 37

Simplification (conditions)

  • Validity of Triangulation
  • Combinatorial Topology [Dey et al. 98]
  • Valid embedding – Visibility kernel of 1-ring [Lee-Preparata 79]
  • 37

Edge PQ to be collapsed Link condition Valid embedding kernel

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SLIDE 38

Simplification (conditions)

  • Validity of Triangulation
  • Combinatorial Topology [Dey et al. 98]
  • Valid embedding – Visibility kernel of 1-ring [Lee-Preparata 79]
  • 38

Valid embedding kernel Edge PQ to be collapsed

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SLIDE 39

Simplification (conditions)

  • Validity of Triangulation
  • Combinatorial Topology [Dey et al. 98]
  • Valid embedding – Visibility kernel of 1-ring [Lee-Preparata 79]
  • Preserve classification of S – Non convex problem
  • 39

Valid embedding kernel Region that preserve classification of S Edge PQ to be collapsed

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SLIDE 40

Simplification (conditions)

  • Validity of Triangulation
  • Combinatorial Topology [Dey et al. 98]
  • Valid embedding – Visibility kernel of 1-ring [Lee-Preparata 79]
  • Preserve classification of S – Non convex problem
  • Optimal location : Minimizes sum of squared distances

between target vertex and 2-ring planes

  • 40
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SLIDE 41

Simplification (conditions)

  • Validity of Triangulation
  • Combinatorial Topology [Dey et al. 98]
  • Valid embedding – Visibility kernel of 1-ring [Lee-Preparata 79]
  • Preserve classification of S – Non convex problem
  • Optimal location : Minimizes sum of squared distances

between target vertex and 2-ring planes

  • Faithful normals (same as in refinement)
  • 42
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SLIDE 42

Algorithm (fine-to-coarse)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box
  • While (S not classified)
  • Insert Steiner point to T
  • Collapse edges of ∂Ɣ
  • 40

Simplification

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SLIDE 43

Algorithm (fine-to-coarse)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box
  • While (S not classified)
  • Insert Steiner point to T
  • Collapse edges of ∂Ɣ
  • 41

Simplification

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SLIDE 44

Algorithm (fine-to-coarse)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box
  • While (S not classified)
  • Insert Steiner point to T
  • Collapse edges of ∂Ɣ
  • 42

Simplification

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SLIDE 45

Algorithm (fine-to-coarse)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box
  • While (S not classified)
  • Insert Steiner point to T
  • Collapse edges of ∂Ɣ
  • Insert Z in T
  • 43

Simplification

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SLIDE 46

Algorithm (fine-to-coarse)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box
  • While (S not classified)
  • Insert Steiner point to T
  • Collapse edges of ∂Ɣ
  • Insert Z in T
  • Collapse edges of Z
  • 44

Simplification

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SLIDE 47

Algorithm (fine-to-coarse)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box
  • While (S not classified)
  • Insert Steiner point to T
  • Collapse edges of ∂Ɣ
  • Insert Z in T
  • Collapse edges of Z
  • 45

Simplification

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SLIDE 48

Algorithm (fine-to-coarse)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box
  • While (S not classified)
  • Insert Steiner point to T
  • Collapse edges of ∂Ɣ
  • Insert Z in T
  • Collapse edges of Z
  • 46

Simplification

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SLIDE 49

Algorithm (fine-to-coarse)

  • S: Point sample of ∂Ω with labels
  • Delaunay Triangulation T of loose bounding-box
  • While (S not classified)
  • Insert Steiner point to T
  • Collapse edges of ∂Ɣ
  • Insert Z in T
  • Collapse edges of Z
  • Final output
  • 47

Output

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SLIDE 50

Recap

  • 48
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SLIDE 51

Recap

  • 48
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SLIDE 52

Recap

  • 48
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SLIDE 53

Recap

  • 48
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SLIDE 54

Proof

  • 55
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SLIDE 55

Proof

  • Termination
  • Topology
  • 56

Only assumes separability condition on Ω

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SLIDE 56

Results

  • 62
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SLIDE 57

Blade

  • 63

Input Ω Refinement Simplification

  • n ∂Ɣ

Simplification

  • n Z

Simplification

  • n all edges
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SLIDE 58

Varying Tolerance (fertility)

  • 64
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SLIDE 59

Varying Tolerance (vase lion)

  • 65
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SLIDE 60

Progress of Algorithm

  • 66

tolerance

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SLIDE 61

Robustness to input dataset and noise

  • 67
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SLIDE 62

Robustness to input dataset and noise

  • 68
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SLIDE 63

Robustness to input dataset and noise

  • 69
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SLIDE 64

Robustness to input dataset and noise

  • 70

[Chazal et al. 11]

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SLIDE 65

Comparisons

  • 71
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SLIDE 66

Comparison (Simplification Envelopes)

  • 72

Input Simplification Envelopes H = 60% Our Algorithm H = 10%

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SLIDE 67

Comparisons – Hausdorff distance (output to input)

  • 73
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SLIDE 68

Comparisons – Hausdorff distance (output to input)

  • 74
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SLIDE 69

Comparisons – Hausdorff distance (input to output)

  • 75
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SLIDE 70

Comparisons – Hausdorff distance (input to output)

  • 76
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SLIDE 71

Comparisons – Hausdorff distance (input to output)

  • 77
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SLIDE 72

Comparisons – Hausdorff distance (input to output)

  • 78
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SLIDE 73

Comparisons – Hausdorff distance (output to input)

  • 79

dots: self intersection

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SLIDE 74

Comparisons – Hausdorff distance (input to output)

  • 80
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SLIDE 75

Anisotropy

  • 81
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SLIDE 76

Anisotropy

  • 82
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SLIDE 77

Extensions

  • 83
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SLIDE 78

Non-closed Surfaces

  • 84

rδ ≥2δ During Simplification : Hausdorff distance between Z and boundary is enforced to be δ.

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SLIDE 79

Non-closed Surfaces

  • 85

Input preserving holes repairing holes

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SLIDE 80

Non-manifold

  • 86
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SLIDE 81

Limitations

  • Tolerance volume dependence
  • Compute and memory intensive
  • Slow for small tolerance
  • 87
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SLIDE 82

Conclusions

  • Algorithm for isotopic approximation
  • Always intersection free and within tolerance
  • Low mesh complexity
  • 88
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SLIDE 83

Further Work

  • Out-of-core
  • Progressive algorithm
  • 89
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SLIDE 84

Questions Thank you.

  • 90