Noise-Adaptive Shape Reconstruction from Raw Point Sets 60 Simon - - PowerPoint PPT Presentation

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July 5, 2013 Simon Giraudot Titane Inria Sophia Antipolis - M´ editerran´ ee EUROGRAPHICS Symposium on Geometry Processing

Noise-Adaptive Shape Reconstruction from Raw Point Sets

Simon Giraudot, David Cohen-Steiner & Pierre Alliez

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Focus

Robust shape reconstruction

Resilience to raw, defect-laden point sets

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Focus

Robust shape reconstruction

Resilience to raw, defect-laden point sets

◮ Variable noise (sensor noise, acquisition noise, registration

noise)

◮ Outliers (structured, unstructured, background noise) ◮ Missing data

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Previous Work

Noise Outliers Variable noise Laplace

• perator

[Kazhdan et al., 2006]

Data clustering

[Song, 2010]

Automatic MLS scale

[Wang et al., 2009]

Scale space

[Digne, 2010]

Spectral methods

[Kolluri et al., 2004]

Adjusting statistical estimator

[Unnikrish- nan et al., 2010]

Delaunay triangu- lation properties

[Dey & Goswami, 2006]

Robust distances

[Chazal et al., 2011]

Adaptive bandwidth

[Mellado et al., 2012]

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Previous Work

Growing Least Squares for the Continuous Analysis of Manifolds in Scale-Space

1 2 3 1 2 3

[N. Mellado, G. Guennebaud, P. Barla, P. Reuter, C. Schlick, 2012] ◮ Analysis of geometric variation ν

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Previous Work

Growing Least Squares for the Continuous Analysis of Manifolds in Scale-Space

1 2 3 1 2 3

[N. Mellado, G. Guennebaud, P. Barla, P. Reuter, C. Schlick, 2012] ◮ Analysis of geometric variation ν ◮ Threshold for ∂ν ∂t

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Problem Statement

Input: raw point set P sampling submanifold of known dimension k, possibly defect-laden with:

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Problem Statement

Input: raw point set P sampling submanifold of known dimension k, possibly defect-laden with:

◮ Variable noise ◮ Outliers ◮ Missing data

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Problem Statement

Input: raw point set P sampling submanifold of known dimension k, possibly defect-laden with:

◮ Variable noise ◮ Outliers ◮ Missing data

Output: smooth closed shape defined by implicit function

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Overview

Point set

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Overview

1 — Unsigned noise-adaptive distance function

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Overview

2 — Sign guess

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Overview

3 — Signed implicit function

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1

Adaptive Distance Function

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Non-Adaptive Distance Function

Unsigned distance function to a measure [Chazal et al., 2011]

d2

µ,m : Rn → R, q → 1

m

• B(q,rµ,m(q))

q − y2dµ(y)

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Non-Adaptive Distance Function

Unsigned distance function to a measure [Chazal et al., 2011]

d2

µ,m : Rn → R, q → 1

m

• B(q,rµ,m(q))

q − y2dµ(y)

q

B(q,rµ,m(q))

m

||q-y||2

µ

rµ,m(q)

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Non-Adaptive Distance Function

Unsigned distance function to a measure [Chazal et al., 2011]

d2

µ,m : Rn → R, q → 1

m

• B(q,rµ,m(q))

q − y2dµ(y)

q

B(q,rµ,m(q))

m

||q-y||2

µ

rµ,m(q)

Note: scale parameter m

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Non-Adaptive Distance Function

Unsigned distance function to a measure [Chazal et al., 2011]

d2

µ,m : Rn → R, q → 1

m

• B(q,rµ,m(q))

q − y2dµ(y)

q

B(q,rµ,m(q))

m

||q-y||2

µ

rµ,m(q)

Note: scale parameter m

◮ User-specified

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Non-Adaptive Distance Function

Unsigned distance function to a measure [Chazal et al., 2011]

d2

µ,m : Rn → R, q → 1

m

• B(q,rµ,m(q))

q − y2dµ(y)

q

B(q,rµ,m(q))

m

||q-y||2

µ

rµ,m(q)

Note: scale parameter m

◮ User-specified ◮ Depends on point set properties

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Non-Adaptive Distance Function

Unsigned distance function to a measure [Chazal et al., 2011]

d2

µ,m : Rn → R, q → 1

m

• B(q,rµ,m(q))

q − y2dµ(y)

q

B(q,rµ,m(q))

m

||q-y||2

µ

rµ,m(q)

Note: scale parameter m

◮ User-specified ◮ Depends on point set properties ◮ Global: not noise-adaptive

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Non-Adaptive Distance Function

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8

K = 6 K = 70

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Case of an Ambiant Noise

Uniform measure in d-dimensional space

µ

q m

α

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Case of an Ambiant Noise

Uniform measure in d-dimensional space

µ

q m

d2

µ,m(q) = c · m

2 d

α

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Case of an Ambiant Noise

Uniform measure in d-dimensional space

µ

q m

d2

µ,m(q) = c · m

2 d

dµ,m(q) ∝ m

1 d for q fixed

α

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Case of an Ambiant Noise

Uniform measure in d-dimensional space

µ

q m

d2

µ,m(q) = c · m

2 d

dµ,m(q) ∝ m

1 d for q fixed

dµ,m(q) mα

decreasing for α > 1

d

dµ,m(q) m

α

m

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Case of a Submanifold

Uniform measure on k-submanifold

m h q

µ

α

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Case of a Submanifold

Uniform measure on k-submanifold

m h q

µ

d2

µ,m(q) = c · m

2 k + h2

α

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Case of a Submanifold

Uniform measure on k-submanifold

m h q

µ

d2

µ,m(q) = c · m

2 k + h2

dµ,m(q) ∝ m

1 k for q fixed

α

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Case of a Submanifold

Uniform measure on k-submanifold

m h q

µ

d2

µ,m(q) = c · m

2 k + h2

dµ,m(q) ∝ m

1 k for q fixed

dµ,m(q) mα

increasing for α < 1

k

dµ,m(q) m

α

m

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Discrete Noisy Case

Scale m = 10 nearest neighbors

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Discrete Noisy Case

Scale m = 10 nearest neighbors

◮ Apparent dimension = 2

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Discrete Noisy Case

Scale m = 10 nearest neighbors

◮ Apparent dimension = 2 ◮ Ambiant noise in 2D

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Discrete Noisy Case

Scale m = 10 nearest neighbors

◮ Apparent dimension = 2 ◮ Ambiant noise in 2D ◮ dµ,m(q) mα

decreasing for α > 1

2 dµ,m(q) m

α

m

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Discrete Noisy Case

Scale m = 10 nearest neighbors

◮ Apparent dimension = 2 ◮ Ambiant noise in 2D ◮ dµ,m(q) mα

decreasing for α > 1

2

Scale m = 30 nearest neighbors

dµ,m(q) m

α

m

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Discrete Noisy Case

Scale m = 10 nearest neighbors

◮ Apparent dimension = 2 ◮ Ambiant noise in 2D ◮ dµ,m(q) mα

decreasing for α > 1

2

Scale m = 30 nearest neighbors

◮ Apparent dimension = 1 dµ,m(q) m

α

m

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Discrete Noisy Case

Scale m = 10 nearest neighbors

◮ Apparent dimension = 2 ◮ Ambiant noise in 2D ◮ dµ,m(q) mα

decreasing for α > 1

2

Scale m = 30 nearest neighbors

◮ Apparent dimension = 1 ◮ 1-submanifold in 2D dµ,m(q) m

α

m

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Discrete Noisy Case

Scale m = 10 nearest neighbors

◮ Apparent dimension = 2 ◮ Ambiant noise in 2D ◮ dµ,m(q) mα

decreasing for α > 1

2

Scale m = 30 nearest neighbors

◮ Apparent dimension = 1 ◮ 1-submanifold in 2D ◮ dµ,m(q) mα

increasing for α < 1

dµ,m(q) m

α

m

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Discrete Noisy Case

Scale m = 10 nearest neighbors

◮ Apparent dimension = 2 ◮ Ambiant noise in 2D ◮ dµ,m(q) mα

decreasing for α > 1

2

Scale m = 30 nearest neighbors

◮ Apparent dimension = 1 ◮ 1-submanifold in 2D ◮ dµ,m(q) mα

increasing for α < 1

dµ,m(q) m

α

m δμ(q)

local scale Simon Giraudot - Noise-Adaptive Shape Reconstruction July 5, 2013- 15

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Noise-Adaptive Distance function

Assumption

Inferred shape is a submanifold of known dimension

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Noise-Adaptive Distance function

Assumption

Inferred shape is a submanifold of known dimension For a k-submanifold in d-dimensional space:

δµ = inf

m>0

dµ,m mα ,

with α ∈ [ 1

d ; 1 k ]

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Noise-Adaptive Distance function

δµ = inf

m>0

dµ,m mα

Infimum:

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Noise-Adaptive Distance function

δµ = inf

m>0

dµ,m mα

Infimum:

• 1. m as small as possible → no oversmoothing

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Noise-Adaptive Distance function

δµ = inf

m>0

dµ,m mα

Infimum:

• 1. m as small as possible → no oversmoothing
• 2. m large enough → point subset appears as k-submanifold

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Noise-Adaptive Distance function

δµ = inf

m>0

dµ,m mα

Infimum:

• 1. m as small as possible → no oversmoothing
• 2. m large enough → point subset appears as k-submanifold

Setting α (α ∈ [ 1

d ; 1 k ])

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Noise-Adaptive Distance function

δµ = inf

m>0

dµ,m mα

Infimum:

• 1. m as small as possible → no oversmoothing
• 2. m large enough → point subset appears as k-submanifold

Setting α (α ∈ [ 1

d ; 1 k ]) ◮ Curve in 2D: α = 3 4 to satisfy α ∈ [1 2; 1]

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Noise-Adaptive Distance function

δµ = inf

m>0

dµ,m mα

Infimum:

• 1. m as small as possible → no oversmoothing
• 2. m large enough → point subset appears as k-submanifold

Setting α (α ∈ [ 1

d ; 1 k ]) ◮ Curve in 2D: α = 3 4 to satisfy α ∈ [1 2; 1] ◮ Surface in 3D: α = 5 12 to satisfy α ∈ [1 3; 1 2]

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Noise-Adaptive Distance Function

Wasserstein robustness & semiconcavity

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Noise-Adaptive Distance Function

Wasserstein robustness & semiconcavity

◮ ∀m, dµ,m is 1/√m-robust and 1-semiconcave

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Noise-Adaptive Distance Function

Wasserstein robustness & semiconcavity

◮ ∀m, dµ,m is 1/√m-robust and 1-semiconcave ◮ Limiting infimum over values of m above m0: properties

preserved for δµ

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Noise-Adaptive Distance Function

Wasserstein robustness & semiconcavity

◮ ∀m, dµ,m is 1/√m-robust and 1-semiconcave ◮ Limiting infimum over values of m above m0: properties

preserved for δµ → correct topological inference (Geometric Inference for Probability Measures [F. Chazal, D. Cohen-Steiner, Q. M´

erigot, 2011])

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Noise-Adaptive Distance Function

Wasserstein robustness & semiconcavity

◮ ∀m, dµ,m is 1/√m-robust and 1-semiconcave ◮ Limiting infimum over values of m above m0: properties

preserved for δµ → correct topological inference (Geometric Inference for Probability Measures [F. Chazal, D. Cohen-Steiner, Q. M´

erigot, 2011])

In practice

Lower bound m0: 6 nearest neighbors.

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Noise-Adaptive Distance Function

Non-adaptive distance function (reminder)

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8

K = 6 K = 70

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Noise-Adaptive Distance Function

Adaptive distance function

δ

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