Normal and curvature estimation using the k-Voronoi covariance measure

Normal and curvature estimation using the k-Voronoi covariance

measure

Louis Cuel

Collaboration : J.O. Lachaud, Q. M´erigot, B. Thibert

Universit´e de Grenoble Universit´e de Savoie Laboratoire Jean Kuntzman LAMA

Geometric inference

Given :

What can we say about the geometry and the topology of K?

#connected components intrinsec dimension

— A finite set of points P ⊆ R^d approximating K.

— An unknown object K (compact set) of R^d

Previous work

1. Voronoi-based algorithms :

−Amenta et Bern, 1999, Surface reconstruction by Voronoi filtering

−Cohen-Steiner et al., 2007,Voronoi-based Variational Reconstruction

−G. M. O., 2009,Robust Voronoi-based Curvature and Feature Estimation

Previous work

1. Voronoi-based algorithms :

−Amenta et Bern, 1999, Surface reconstruction by Voronoi filtering

−Cohen-Steiner et al., 2007,Voronoi-based Variational Reconstruction

−G. M. O., 2009,Robust Voronoi-based Curvature and Feature Estimation

2. Distance to a measure :

−Chazal, Cohen-Steiner, Merigot, 2010,Geometric Inference for probability measure

Previous work

1. Voronoi-based algorithms :

−Amenta et Bern, 1999, Surface reconstruction by Voronoi filtering

−Cohen-Steiner et al., 2007,Voronoi-based Variational Reconstruction

−G. M. O., 2009,Robust Voronoi-based Curvature and Feature Estimation

2. Distance to a measure :

−Chazal, Cohen-Steiner, Merigot, 2010,Geometric Inference for probability measure

our approach = 1 + 2

Normal estimation based on Voronoi

Voronoi cell: Vor^P (q) = { points whose closest point in P is q}

P = {p₁, . . . , p_N } ⊆ R^d

Definition :

Normal estimation based on Voronoi

Voronoi cell: Vor^P (q) = { points whose closest point in P is q}

P = {p₁, . . . , p_N } ⊆ R^d p

pole_P (p)

Definition :

Amenta, Bern, Discrete and Computational Geometry 22 (1999) pole_P (p) := farthest point of p in Vor_P (p)

Poles’ method

Normal estimation based on Voronoi

Voronoi cell: Vor^P (q) = { points whose closest point in P is q}

P = {p₁, . . . , p_N } ⊆ R^d

Definition :

Amenta, Bern, Discrete and Computational Geometry 22 (1999) pole_P (p) := farthest point of p in Vor_P (p)

Poles’ method

Normal estimation based on Voronoi

Voronoi cell: Vor^P (q) = { points whose closest point in P is q}

P = {p₁, . . . , p_N } ⊆ R^d

Idea : integrate to obtain stability.

Definition :

Amenta, Bern, Discrete and Computational Geometry 22 (1999) pole_P (p) := farthest point of p in Vor_P (p)

Poles’ method

Alliez, Cohen-Steiner, Tong, Desbruns, Proc. Symposium Geometry Processing 2007

Covariance Matrix: cov_p(Ω) := R

Ω(x − p)(x − p)^t d x.

Eigenvectors of cov_p(Ω) are the principal axes of Ω (viewed from p).

Voronoi Covariance

Alliez, Cohen-Steiner, Tong, Desbruns, Proc. Symposium Geometry Processing 2007

Covariance Matrix: cov_p(Ω) := R

Ω(x − p)(x − p)^t d x.

Eigenvectors of cov_p(Ω) are the principal axes of Ω (viewed from p).

Algorithm:

• The normal is estimated by the eigenvector corre- sponding to the largest eigenvalue (in red).

• They take : Ω = Vor_P (p_i) ∩ E

• Stability to noise : Sum matrices over a neighborhod.

Voronoi Covariance

Voronoi covariance measure

Definition: Offset of P of radius R : P ^R = ∪_p∈P B(p, R).

Voronoi covariance measure

V(P, R) := PN

i=1 A(p)δ_p

A(p) := cov_p(Vor_P (p_i) ∩ P ^R)

Definition: The Voronoi covariance measure of P of offset radius R is : Definition: Offset of P of radius R : P ^R = ∪_p∈P B(p, R).

V(P, R) := PN

i=1 A(p)δ_p

A(p) := cov_p(Vor_P (p_i) ∩ P ^R)

Definition: The Voronoi covariance measure of P of offset radius R is :

V(P, R) ∗ χ_r(p) := P

p_i∈B(p,r) A(p_i)

Voronoi covariance measure

The VCM is defined for all compacts.

Definition: Offset of P of radius R : P ^R = ∪_p∈P B(p, R).

V(P, R) := PN

i=1 A(p)δ_p

A(p) := cov_p(Vor_P (p_i) ∩ P ^R)

Definition: The Voronoi covariance measure of P of offset radius R is :

V(P, R) ∗ χ_r(p) := P

p_i∈B(p,r) A(p_i)

Voronoi covariance measure

The VCM is defined for all compacts.

Definition: Offset of P of radius R : P ^R = ∪_p∈P B(p, R).

Theorem: Let P, K be two compacts and p ∈ R^d.

kV(P, R) ∗ χ_r(p) − V(K, R) ∗ χ_r(p)k = O(d_H(K, P) ¹² )

Distance to a measure

Definition: The distance to a mea- sure of parameter k (or k-distance) :

d²_k,P (p) = _k¹ P

p_i∈N N_k(p) ||p − p_i||² p

P

Distance to a measure

Definition: The distance to a mea- sure of parameter k (or k-distance) :

d²_k,P (p) = _k¹ P

p_i∈N N_k(p) ||p − p_i||² p

P

Theorem : Let P, K be two compact sets and R^d. kd_P,k − d_K,kk ≤ ^√1

k W₂(µ_P , µ_K)

k-Voronoi covariance measure

VCM k-VCM

k-Voronoi covariance measure

R

Vor_P(p)∩P ^R(x − π(x))(x − π(x))t d x.

R

Vor^P_k (p)∩P˜^R(x − πk(x))(x − πk(x))t d x.

VCM k-VCM

A(p):= Ak(p) :=

Vk(P, R) ∗ χr(p) := P

pi∈B(p,r) Ak(pi) V(P, R) ∗ χr(p) := P

pi∈B(p,r) A(pi)

k-Voronoi covariance measure

R

Vor_P(p)∩P ^R(x − π(x))(x − π(x))t d x.

R

Vor^P_k (p)∩P˜^R(x − πk(x))(x − πk(x))t d x.

VCM k-VCM

A(p):= Ak(p) :=

Vk(P, R) ∗ χr(p) := P

pi∈B(p,r) Ak(pi) V(P, R) ∗ χr(p) := P

pi∈B(p,r) A(pi)

kV(P, R) ∗ χ_r(p) − V(K, R) ∗ χ_r(p)k Theorem: Let P, K be two compact sets

kV_k(P, R) ∗ χ_r(p) − V(K, R) ∗ χ_r(p)k and p ∈ R^d.

Theorem: Let P, K be two compact sets and p ∈ R^d.

Idea of the proof

• Control the symetric difference of the two integration domains :

h

Vor^P_k (p) ∩ d⁻¹_P,k([0, R])i

4

Vor^K(p) ∩ K^R

= O(kd_P,k − d_Kk_∞)

R

Vor^P_k (p)∩d⁻¹_P,k([0,R])(x − πP

k (x))(x − πP

k (x))t d x −

R

Vor^K(p)∩K^R(x − πK(x))(x − πK(x))t d x

Idea of the proof

• Control the symetric difference of the two integration domains :

h

Vor^P_k (p) ∩ d⁻¹_P,k([0, R])i

4

Vor^K(p) ∩ K^R

= O(kd_P,k − d_Kk_∞)

• Control of A_P (x) − A_K(x) :

• Theorem : f and g locally convex on E s.t.

diam(Of(E) ∪ Og(E)) is bounded, then

1

R

Vor^P_k (p)∩d⁻¹_P,k([0,R])(x − πP

k (x))(x − πP

k (x))t d x −

R

Vor^K(p)∩K^R(x − πK(x))(x − πK(x))t d x

kA_P (x) − A_K(x)k ≤ (kR² + 2R)kπ_k^P (x) − π^K(x)k

Generalization : δ -VCM

Definition: Let δ : R³ −→ R be a 1-lipschitz function.

δ is a distance-like function if ψ_δ(x) = ||x||² − δ²(x) is convexe.

Exemple: d_K and d_k,K are distance-like.

Generalization : δ -VCM

Definition: Let δ : R³ −→ R be a 1-lipschitz function.

δ is a distance-like function if ψ_δ(x) = ||x||² − δ²(x) is convexe.

Exemple: d_K and d_k,K are distance-like.

Definition : δ-VCM :

R

Vor^P_δ (p)∩P˜^R(x −

O

O

Aδ(p) :=

Vδ(P, R) ∗ χr(p) := P

pi∈B(p,r) Aδ(pi)

kV_δ(P, R) ∗ χ_r(p) − V(K, R) ∗ χ_r(p)k = O(kδ − d_Kk

1

∞2 ) Theorem: Let P, K be two compact sets and p ∈ R^d.

11-1

curvature direction estimation

Bimba 100k points

12-1

Normal estimation : Ellipsoid

ε = 0.4 ε = 0

Sensibility to parameters.

ε = 0.2

Normal Estimation : Ellipsoid

Normale deviation in function of noise.

Hausdorff noise + outliers Gaussian noise Comparison with the Jet fitting and the VCM.

14-1

Sharp feature estimation

Input

fandisk 50k points

Output

Conclusion

• k-VCM is a tool to estimate normal and curvature direction on a point cloud.

• Resilient to Hausdorff noise and outliers.

Summary:

Conclusion

• k-VCM is a tool to estimate normal and curvature direction on a point cloud.

• Resilient to Hausdorff noise and outliers.

Summary:

Next work:

• Applying k-VCM to pixels/voxels sets.

• parameter choice k, R et r.