Topics in geometric inference Lecture I: Voronoi covariance measure

Topics in geometric inference

Lecture I: Voronoi covariance measure

Quentin M´erigot

Workshop on Tensor Valuations in Stochastic Geometry and Imaging

CNRS / Universit´e Paris-Dauphine

21–26 September 2014, Sandbjerg Estate, Sønderborg, Denmark Joint works with F. Chazal, D. Cohen-Steiner, L. Cuel,

L. Guibas, J.O Lachaud, M. Ovsjanikov and B. Thibert

Geometric inference

Given:

What amount of the topology and geometry of K can we recover from P ?

# connected components intrinsic dimension

curvature location of sharp features

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

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

3. Distance to a measure and generalized VCM

1. Tube formulas, curvature measures and their stability

2. Voronoi covariance measure

4. Computations

1. Tube formulas and curvature measures

Distance function: d_P : x ∈ R^d 7→ min_p∈P kx − pk

Distance function and offsets

Distance function: d_P : x ∈ R^d 7→ min_p∈P kx − pk

Distance function and offsets

P

x

x⁰

d_P (x)

d_P (x⁰)

r = 1 r = 2 r = 5

Distance function: d_P : x ∈ R^d 7→ min_p∈P kx − pk Offset: P ^r := ∪_p∈P B(p, r) = d⁻¹_P ([0, r])

Distance function and offsets

P ¹ P ² P ⁵

r = 1 r = 2 r = 5

Distance function: d_P : x ∈ R^d 7→ min_p∈P kx − pk Offset: P ^r := ∪_p∈P B(p, r) = d⁻¹_P ([0, r])

Distance function and offsets

P ¹ P ² P ⁵

Hausdorff distance: d_H(P, K) := k d_P − d_K k_∞.

Tube formulas and curvature

K r K^r

α_i

`_j

Theorem (Steiner-Minkowski): For every compact convex subset K of R^d, the function r 7→ H^d(K^r) is a degree d polynomial.

Tube formulas and curvature

K r K^r

α_i

`_j

Theorem (Steiner-Minkowski): For every compact convex subset K of R^d, the function r 7→ H^d(K^r) is a degree d polynomial.

Example: for a polygon P ⊆ R²:

H²(P ^r) = H²(P ) + rH¹(∂P ) + r²π

Tube formulas and curvature

K r K^r

α_i

`_j

Theorem (Steiner-Minkowski): For every compact convex subset K of R^d, the function r 7→ H^d(K^r) is a degree d polynomial.

Example: for a polygon P ⊆ R²:

H²(P ^r) = H²(P ) + rH¹(∂P ) + r²π

Theorem (Weyl): If K ⊆ R^d is a domain with smooth boundary M, then r 7→ vol^d(K^r) is a degree d polynomial on [0, R] for some R > 0.

Tube formulas and curvature

K r K^r

α_i

`_j

Theorem (Steiner-Minkowski): For every compact convex subset K of R^d, the function r 7→ H^d(K^r) is a degree d polynomial.

Example: for a polygon P ⊆ R²:

H²(P ^r) = H²(P ) + rH¹(∂P ) + r²π

Example: if K is bounded by a smooth surface S in R³, H³(K^r) = H³(K) + rΦ²_K + r²Φ¹_K + r³Φ⁰_K

Theorem (Weyl): If K ⊆ R^d is a domain with smooth boundary M, then r 7→ vol^d(K^r) is a degree d polynomial on [0, R] for some R > 0.

Tube formulas and curvature

K r K^r

α_i

`_j

Theorem (Steiner-Minkowski): For every compact convex subset K of R^d, the function r 7→ H^d(K^r) is a degree d polynomial.

Example: for a polygon P ⊆ R²:

H²(P ^r) = H²(P ) + rH¹(∂P ) + r²π

Example: if K is bounded by a smooth surface S in R³, H³(K^r) = H³(K) + rΦ²_K + r²Φ¹_K + r³Φ⁰_K

Φ¹_K = tot. mean curvature

Φ⁰_K = tot. Gaussian curvature Theorem (Weyl): If K ⊆ R^d is a domain with smooth boundary M,

then r 7→ vol^d(K^r) is a degree d polynomial on [0, R] for some R > 0.

Φ²_K = area(S)

Federer’s local tube formula

x

K^r K

p_K(x) M(K)

y

Definition: The medial axis of K ⊆ R^d is M(K) := {x ∈ R^d; # proj_K(x) > 1}

proj_K(x) = arg min_p∈K kx − pk

Federer’s local tube formula

K^r K

Definition: The medial axis of K ⊆ R^d is M(K) := {x ∈ R^d; # proj_K(x) > 1}

Definition: reach(K) ≥ R iff K^R does not intersect M(K).

proj_K(x) = arg min_p∈K kx − pk

Federer’s local tube formula

K^r K

Definition: The medial axis of K ⊆ R^d is M(K) := {x ∈ R^d; # proj_K(x) > 1}

Definition: reach(K) ≥ R iff K^R does not intersect M(K).

(i) Motzkin’s theorem: reach(K) = +∞ iff K is convex ;

proj_K(x) = arg min_p∈K kx − pk

Federer’s local tube formula

K^r K

Definition: The medial axis of K ⊆ R^d is M(K) := {x ∈ R^d; # proj_K(x) > 1}

Definition: reach(K) ≥ R iff K^R does not intersect M(K).

(i) Motzkin’s theorem: reach(K) = +∞ iff K is convex ;

(ii) If M is smooth, min. curvature radius of M ≥ reach(M) > 0.

proj_K(x) = arg min_p∈K kx − pk

Federer’s local tube formula

K^r K

B p⁻¹_K (B)

Definition: The medial axis of K ⊆ R^d is

Projection function p_K : R^d \ M(K) → K . M(K) := {x ∈ R^d; # proj_K(x) > 1}

Definition: reach(K) ≥ R iff K^R does not intersect M(K).

proj_K(x) = arg min_p∈K kx − pk

Federer’s local tube formula

Definition: The medial axis of K ⊆ R^d is

Projection function p_K : R^d \ M(K) → K . M(K) := {x ∈ R^d; # proj_K(x) > 1}

Definition: reach(K) ≥ R iff K^R does not intersect M(K).

Federer’s tube formula: Suppose R := reach(K) > 0. For all subset B of K, the map

K^r

K B

p⁻¹_K (B)

K^r0

r 7→ H^d(K^r ∩ p⁻¹_K (B)) is a polynomial of degree d on [0, reach(K)].

proj_K(x) = arg min_p∈K kx − pk

Boundary measures

E := K^r B K

p⁻¹_K (B)

µ_K,E(B) := H^d(E ∩ p⁻¹_K (B)) Definition: The boundary measure of K wrt a domain E is defined for B ⊆ K by

Boundary measures

E := K^r B K

p⁻¹_K (B)

Definition: The boundary measure of K wrt a domain E is defined for B ⊆ K by

µ_K,E := p_K# H^d E

Boundary measures

E := K^r B K

p⁻¹_K (B)

Definition: The boundary measure of K wrt a domain E is defined for B ⊆ K by

Federer’s tube formula: if reach(K) > R, ∃ signed meas. (Φ_i(K))_0≤i≤d st

∀r ∈ [0, R], µ_K,K^r = Pd

i=0 Φ^d−i_K rⁱ

µ_K,E := p_K# H^d E

Boundary measures

E := K^r B K

p⁻¹_K (B)

Definition: The boundary measure of K wrt a domain E is defined for B ⊆ K by

K

p_K(x) x

E P

p_P (x) Example: x

Federer’s tube formula: if reach(K) > R, ∃ signed meas. (Φ_i(K))_0≤i≤d st

∀r ∈ [0, R], µ_K,K^r = Pd

i=0 Φ^d−i_K rⁱ

µ_K,E := p_K# H^d E

Boundary measures

E := K^r B K

p⁻¹_K (B)

Definition: The boundary measure of K wrt a domain E is defined for B ⊆ K by

Question: What is the dependence of µ_K,E on K ? For what distance ? K

p_K(x) x

E P

p_P (x) Example: x

Federer’s tube formula: if reach(K) > R, ∃ signed meas. (Φ_i(K))_0≤i≤d st

∀r ∈ [0, R], µ_K,K^r = Pd

i=0 Φ^d−i_K rⁱ

µ_K,E := p_K# H^d E

Boundary measures

E := K^r B K

p⁻¹_K (B)

Definition: The boundary measure of K wrt a domain E is defined for B ⊆ K by

Question: What is the dependence of µ_K,E on K ? For what distance ? K

p_K(x) x

E Example:

P

Federer’s tube formula: if reach(K) > R, ∃ signed meas. (Φ_i(K))_0≤i≤d st

∀r ∈ [0, R], µ_K,K^r = Pd

i=0 Φ^d−i_K rⁱ

µ_K,E := p_K# H^d E

Bounded-Lipschitz distance

BL₁ := {χ : R^d → R; 1-Lipschitz, kχk_∞ ≤ 1}

Bounded-Lipschitz distance

BL₁ := {χ : R^d → R; 1-Lipschitz, kχk_∞ ≤ 1}

d_bL(µ, ν) := sup_χ∈BL

1 | R

χ d µ − R

χ d ν|

Bounded-Lipschitz distance: For µ, ν = measures with finite mass,

Bounded-Lipschitz distance

BL₁ := {χ : R^d → R; 1-Lipschitz, kχk_∞ ≤ 1}

d_bL(µ, ν) := sup_χ∈BL

1 | R

χ d µ − R

χ d ν|

Bounded-Lipschitz distance: For µ, ν = measures with finite mass,

W₁(µ, ν)/r ≤ d_bL(µ, ν) ≤ W₁(µ, ν) where W₁ is the Wasserstein distance.

I If X ⊆ B(0, r) with r ≥ 1, and µ, ν are probability measures on X,

Bounded-Lipschitz distance

BL₁ := {χ : R^d → R; 1-Lipschitz, kχk_∞ ≤ 1}

d_bL(µ, ν) := sup_χ∈BL

1 | R

χ d µ − R

χ d ν|

Bounded-Lipschitz distance: For µ, ν = measures with finite mass,

W₁(µ, ν)/r ≤ d_bL(µ, ν) ≤ W₁(µ, ν) where W₁ is the Wasserstein distance.

I If X ⊆ B(0, r) with r ≥ 1, and µ, ν are probability measures on X,

I Lemma: d_bL(µ_K,E, µ_L,E) ≤ kp_K − p_Lk_L¹_(E).

Bounded-Lipschitz distance

sup_χ∈BL

1 | R

K χ(p) d µ_K,E(p) − R

K χ(p) d µ_L,E(p)|

BL₁ := {χ : R^d → R; 1-Lipschitz, kχk_∞ ≤ 1}

d_bL(µ, ν) := sup_χ∈BL

1 | R

χ d µ − R

χ d ν|

Bounded-Lipschitz distance: For µ, ν = measures with finite mass,

W₁(µ, ν)/r ≤ d_bL(µ, ν) ≤ W₁(µ, ν) where W₁ is the Wasserstein distance.

I If X ⊆ B(0, r) with r ≥ 1, and µ, ν are probability measures on X,

I Lemma: d_bL(µ_K,E, µ_L,E) ≤ kp_K − p_Lk_L¹_(E).

Bounded-Lipschitz distance

sup_χ∈BL

1 | R

K χ(p) d µ_K,E(p) − R

K χ(p) d µ_L,E(p)|

= sup_χ∈BL

1 | R

E χ(p_K(x)) − χ(p_L(x)) d H^d(x)|

BL₁ := {χ : R^d → R; 1-Lipschitz, kχk_∞ ≤ 1}

d_bL(µ, ν) := sup_χ∈BL

1 | R

χ d µ − R

χ d ν|

Bounded-Lipschitz distance: For µ, ν = measures with finite mass,

W₁(µ, ν)/r ≤ d_bL(µ, ν) ≤ W₁(µ, ν) where W₁ is the Wasserstein distance.

I If X ⊆ B(0, r) with r ≥ 1, and µ, ν are probability measures on X,

I Lemma: d_bL(µ_K,E, µ_L,E) ≤ kp_K − p_Lk_L¹_(E). change of variable formula

Bounded-Lipschitz distance

sup_χ∈BL

1 | R

K χ(p) d µ_K,E(p) − R

K χ(p) d µ_L,E(p)|

= sup_χ∈BL

1 | R

E χ(p_K(x)) − χ(p_L(x)) d H^d(x)|

≤ kp_K − p_Lk_L¹_(E)

BL₁ := {χ : R^d → R; 1-Lipschitz, kχk_∞ ≤ 1}

d_bL(µ, ν) := sup_χ∈BL

1 | R

χ d µ − R

χ d ν|

Bounded-Lipschitz distance: For µ, ν = measures with finite mass,

W₁(µ, ν)/r ≤ d_bL(µ, ν) ≤ W₁(µ, ν) where W₁ is the Wasserstein distance.

I If X ⊆ B(0, r) with r ≥ 1, and µ, ν are probability measures on X,

I Lemma: d_bL(µ_K,E, µ_L,E) ≤ kp_K − p_Lk_L¹_(E).

χ is 1-Lipschitz

Nonquantitative stability of curvature measures

Proposition: Let K_n, K be compact subsets of R^d s.t. K_n −→ K and R := min(reach(K), reach(K_n)) > 0

d_H

Then, for any r < R, and E ⊆ K^r,

lim_n→∞ d_bL(µ_Kn,E, µ_K,E) = 0

[Federer 1959]

lim_n→∞ kp_K − p_Knk_L^∞_(E) = 0 in particular:

Nonquantitative stability of curvature measures

Proposition: Let K_n, K be compact subsets of R^d s.t. K_n −→ K and R := min(reach(K), reach(K_n)) > 0

d_H

Then, for any r < R, and E ⊆ K^r,

lim_n→∞ d_bL(µ_Kn,E, µ_K,E) = 0

I Does not apply to a sequence of finite sets K_n converging to K.

[Federer 1959]

lim_n→∞ kp_K − p_Knk_L^∞_(E) = 0 in particular:

Nonquantitative stability of curvature measures

Proposition: Let K_n, K be compact subsets of R^d s.t. K_n −→ K and R := min(reach(K), reach(K_n)) > 0

d_H

Then, for any r < R, and E ⊆ K^r,

lim_n→∞ d_bL(µ_Kn,E, µ_K,E) = 0

I Does not apply to a sequence of finite sets K_n converging to K.

[Federer 1959]

lim_n→∞ kp_K − p_Knk_L^∞_(E) = 0

I Controlling kp_K − p_Lk_L^∞_(E) requires a lower bound on the reach.

in particular:

Nonquantitative stability of curvature measures

Proposition: Let K_n, K be compact subsets of R^d s.t. K_n −→ K and R := min(reach(K), reach(K_n)) > 0

d_H

Then, for any r < R, and E ⊆ K^r,

lim_n→∞ d_bL(µ_Kn,E, µ_K,E) = 0

I Does not apply to a sequence of finite sets K_n converging to K.

I Based on Arzela-Ascoli’s theorem =⇒ not quantitative.

[Federer 1959]

lim_n→∞ kp_K − p_Knk_L^∞_(E) = 0

I Controlling kp_K − p_Lk_L^∞_(E) requires a lower bound on the reach.

in particular:

Nonquantitative stability of curvature measures

Proposition: Let K_n, K be compact subsets of R^d s.t. K_n −→ K and

Corollary: For all 1 ≤ i ≤ d, lim_n→∞ d_bL(Φⁱ_K

n, Φⁱ_K) = 0.

R := min(reach(K), reach(K_n)) > 0

d_H

Then, for any r < R, and E ⊆ K^r,

lim_n→∞ d_bL(µ_Kn,E, µ_K,E) = 0

I Does not apply to a sequence of finite sets K_n converging to K.

I Based on Arzela-Ascoli’s theorem =⇒ not quantitative.

[Federer 1959]

lim_n→∞ kp_K − p_Knk_L^∞_(E) = 0

I Controlling kp_K − p_Lk_L^∞_(E) requires a lower bound on the reach.

in particular:

Nonquantitative stability of curvature measures

Proposition: Let K_n, K be compact subsets of R^d s.t. K_n −→ K and

Corollary: For all 1 ≤ i ≤ d, lim_n→∞ d_bL(Φⁱ_K

n, Φⁱ_K) = 0.

Goal: Show kp_K − p_Lk_L¹_(E) = O(d_H(K, L)^α) for arbitrary compact sets R := min(reach(K), reach(K_n)) > 0

d_H

Then, for any r < R, and E ⊆ K^r,

lim_n→∞ d_bL(µ_Kn,E, µ_K,E) = 0

I Does not apply to a sequence of finite sets K_n converging to K.

I Based on Arzela-Ascoli’s theorem =⇒ not quantitative.

[Federer 1959]

lim_n→∞ kp_K − p_Knk_L^∞_(E) = 0

I Controlling kp_K − p_Lk_L^∞_(E) requires a lower bound on the reach.

in particular:

Optimal stability for boundary measures 1/3

d_bL(µ_K,E, µ_L,E) ≤ c_K,Ep

d_H(K, L) assuming that d_H(K, L) ≤ diam(K).

[Chazal, Cohen-Steiner, M. 2007]

Theorem: Let K, L ⊆ R^d be compact sets and E a bounded domain

Optimal stability for boundary measures 1/3

d_bL(µ_K,E, µ_L,E) ≤ c_K,Ep

d_H(K, L) assuming that d_H(K, L) ≤ diam(K).

[Chazal, Cohen-Steiner, M. 2007]

Theorem: Let K, L ⊆ R^d be compact sets and E a bounded domain

c_K,E = 4(H^d(E)H^d−1(∂E) diam(K))^1/2

Optimal stability for boundary measures 1/3

d_bL(µ_K,E, µ_L,E) ≤ c_K,Ep

d_H(K, L) assuming that d_H(K, L) ≤ diam(K).

[Chazal, Cohen-Steiner, M. 2007]

Theorem: Let K, L ⊆ R^d be compact sets and E a bounded domain

c_K,E = 4(H^d(E)H^d−1(∂E) diam(K))^1/2

Corollary: Given compact sets K and L in R^d, and R > 0, d_bL(µ_K,K^R, µ_L,L^R) ≤ c_K,Rp

d_H(K, L)

Optimal stability for boundary measures 1/3

d_bL(µ_K,E, µ_L,E) ≤ c_K,Ep

d_H(K, L) assuming that d_H(K, L) ≤ diam(K).

[Chazal, Cohen-Steiner, M. 2007]

Theorem: Let K, L ⊆ R^d be compact sets and E a bounded domain

c_K,E = 4(H^d(E)H^d−1(∂E) diam(K))^1/2

Corollary: Given compact sets K and L in R^d, and R > 0, d_bL(µ_K,K^R, µ_L,L^R) ≤ c_K,Rp

d_H(K, L)

Corollary: Assume reach(K) ≥ R, and L is any compact set. Then,

∀i ∈ {1, . . . , d}, d_bL( ˜Φⁱ_L, Φⁱ_K) ≤ c_K,Rp

d_H(K, L).

Optimal stability for boundary measures 1/3

d_bL(µ_K,E, µ_L,E) ≤ c_K,Ep

d_H(K, L) assuming that d_H(K, L) ≤ diam(K).

[Chazal, Cohen-Steiner, M. 2007]

Theorem: Let K, L ⊆ R^d be compact sets and E a bounded domain

c_K,E = 4(H^d(E)H^d−1(∂E) diam(K))^1/2

Corollary: Given compact sets K and L in R^d, and R > 0, d_bL(µ_K,K^R, µ_L,L^R) ≤ c_K,Rp

d_H(K, L)

Corollary: Assume reach(K) ≥ R, and L is any compact set. Then,

∀i ∈ {1, . . . , d}, d_bL( ˜Φⁱ_L, Φⁱ_K) ≤ c_K,Rp

d_H(K, L).

defined through polynomial fitting, i.e.

µ_L,Lr` = Pd

i=0 Φ˜^d−i_L r_`ⁱ for fixed 0 < r₀ < . . . < r_d < R

Optimal stability for boundary measures 1/3

d_bL(µ_K,E, µ_L,E) ≤ c_K,Ep

d_H(K, L) assuming that d_H(K, L) ≤ diam(K).

Optimality of exponent:

K = unit disk ⊆ R² E = B(0, 1 + R)

R

E K

[Chazal, Cohen-Steiner, M. 2007]

Theorem: Let K, L ⊆ R^d be compact sets and E a bounded domain

Optimal stability for boundary measures 1/3

d_bL(µ_K,E, µ_L,E) ≤ c_K,Ep

d_H(K, L) assuming that d_H(K, L) ≤ diam(K).

Optimality of exponent:

K = unit disk ⊆ R² E = B(0, 1 + R)

L_{`</sub> = reg. polygon in K d<sub>H</sub>(K, L<sub>`}) = O(`²) R

E K

p_L`

`

L_`

[Chazal, Cohen-Steiner, M. 2007]

Theorem: Let K, L ⊆ R^d be compact sets and E a bounded domain

Optimal stability for boundary measures 1/3

d_bL(µ_K,E, µ_L,E) ≤ c_K,Ep

d_H(K, L) assuming that d_H(K, L) ≤ diam(K).

Optimality of exponent:

K = unit disk ⊆ R² E = B(0, 1 + R)

L_{`</sub> = reg. polygon in K d<sub>H</sub>(K, L<sub>`}) = O(`²) R

E K

A constant fraction of E is projected to the vertices of P_`.

p_L`

`

L_`

[Chazal, Cohen-Steiner, M. 2007]

Theorem: Let K, L ⊆ R^d be compact sets and E a bounded domain

Optimal stability for boundary measures 1/3

d_bL(µ_K,E, µ_L,E) ≤ c_K,Ep

d_H(K, L) assuming that d_H(K, L) ≤ diam(K).

Optimality of exponent:

K = unit disk ⊆ R² E = B(0, 1 + R)

L_{`</sub> = reg. polygon in K d<sub>H</sub>(K, L<sub>`}) = O(`²) R

E K

A constant fraction of E is projected to the vertices of P_`.

d_bL(µ_K,E, µ<sub>L`,E</sub>) = Ω(`) p_L`

`

L_`

[Chazal, Cohen-Steiner, M. 2007]

Theorem: Let K, L ⊆ R^d be compact sets and E a bounded domain

Optimal stability for boundary measures 1/3

d_bL(µ_K,E, µ_L,E) ≤ c_K,Ep

d_H(K, L) assuming that d_H(K, L) ≤ diam(K).

Optimality of exponent:

K = unit disk ⊆ R² E = B(0, 1 + R)

L_{`</sub> = reg. polygon in K d<sub>H</sub>(K, L<sub>`}) = O(`²) R

E K

A constant fraction of E is projected to the vertices of P_`.

d_bL(µ_K,E, µ<sub>L`,E</sub>) = Ω(`)

= Ω(p

d_H(K, L_{`</sub>)) p<sub>L`}

`

L_`

[Chazal, Cohen-Steiner, M. 2007]

Theorem: Let K, L ⊆ R^d be compact sets and E a bounded domain

Optimal stability for boundary measures 2/3

Theorem: If d_H(K, L) ≤ diam(K), d_bL(µ_K,E, µ_L,E) ≤ c_E,K d^1/2_H (K, L)

Optimal stability for boundary measures 2/3

Theorem: If d_H(K, L) ≤ diam(K), d_bL(µ_K,E, µ_L,E) ≤ c_E,K d^1/2_H (K, L) Step 1: d_bL(µ_K,E, µ_L,E) ≤ kp_K − p_Lk_L¹_(E) ≤ c_K,Ekp_K − p_Lk_L²_(E)

Optimal stability for boundary measures 2/3

Theorem: If d_H(K, L) ≤ diam(K), d_bL(µ_K,E, µ_L,E) ≤ c_E,K d^1/2_H (K, L)

Step 2: p_K = ∇v_K a.e. where v_K(x) = ¹₂(kxk² − d_K(x)²) is convex Step 1: d_bL(µ_K,E, µ_L,E) ≤ kp_K − p_Lk_L¹_(E) ≤ c_K,Ekp_K − p_Lk_L²_(E)

Optimal stability for boundary measures 2/3

Theorem: If d_H(K, L) ≤ diam(K), d_bL(µ_K,E, µ_L,E) ≤ c_E,K d^1/2_H (K, L)

Step 2: p_K = ∇v_K a.e. where v_K(x) = ¹₂(kxk² − d_K(x)²) is convex indeed, v_K(x) = ¹₂ kxk² − min_p∈K ¹₂kx − pk²

= max_p∈K ¹₂kxk² − ¹₂kx − pk²

= max_p∈Khx|pi − ¹₂ kpk²

Step 1: d_bL(µ_K,E, µ_L,E) ≤ kp_K − p_Lk_L¹_(E) ≤ c_K,Ekp_K − p_Lk_L²_(E)

moreover, v_K is convex

Optimal stability for boundary measures 2/3

Theorem: If d_H(K, L) ≤ diam(K), d_bL(µ_K,E, µ_L,E) ≤ c_E,K d^1/2_H (K, L)

Step 2: p_K = ∇v_K a.e. where v_K(x) = ¹₂(kxk² − d_K(x)²) is convex indeed, v_K(x) = ¹₂ kxk² − min_p∈K ¹₂kx − pk²

= max_p∈K ¹₂kxk² − ¹₂kx − pk²

= max_p∈Khx|pi − ¹₂ kpk²

Step 1: d_bL(µ_K,E, µ_L,E) ≤ kp_K − p_Lk_L¹_(E) ≤ c_K,Ekp_K − p_Lk_L²_(E)

moreover, v_K is convex

−→ 1-semiconcavity of the distance function to a compact set.

Optimal stability for boundary measures 2/3

Theorem: If d_H(K, L) ≤ diam(K), d_bL(µ_K,E, µ_L,E) ≤ c_E,K d^1/2_H (K, L)

Step 2: p_K = ∇v_K a.e. where v_K(x) = ¹₂(kxk² − d_K(x)²) is convex Step 1: d_bL(µ_K,E, µ_L,E) ≤ kp_K − p_Lk_L¹_(E) ≤ c_K,Ekp_K − p_Lk_L²_(E)

Proposition: If u, v ∈ C²(E) are convex, and ∂E smooth,

k∇u − ∇vk_L²_(E) ≤ 2ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E)

Optimal stability for boundary measures 2/3

Theorem: If d_H(K, L) ≤ diam(K), d_bL(µ_K,E, µ_L,E) ≤ c_E,K d^1/2_H (K, L)

Step 2: p_K = ∇v_K a.e. where v_K(x) = ¹₂(kxk² − d_K(x)²) is convex Step 1: d_bL(µ_K,E, µ_L,E) ≤ kp_K − p_Lk_L¹_(E) ≤ c_K,Ekp_K − p_Lk_L²_(E)

Proposition: If u, v ∈ C²(E) are convex, and ∂E smooth,

k∇u − ∇vk_L²_(E) ≤ 2ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E) Step 3: With u = v_K, v = v_L, k∇v_Kk_L^∞_(E) + k∇v_Kk_L^∞_(E) = c_K

Optimal stability for boundary measures 2/3

Theorem: If d_H(K, L) ≤ diam(K), d_bL(µ_K,E, µ_L,E) ≤ c_E,K d^1/2_H (K, L)

Step 2: p_K = ∇v_K a.e. where v_K(x) = ¹₂(kxk² − d_K(x)²) is convex Step 1: d_bL(µ_K,E, µ_L,E) ≤ kp_K − p_Lk_L¹_(E) ≤ c_K,Ekp_K − p_Lk_L²_(E)

Proposition: If u, v ∈ C²(E) are convex, and ∂E smooth,

k∇u − ∇vk_L²_(E) ≤ 2ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E) Step 3: With u = v_K, v = v_L, k∇v_Kk_L^∞_(E) + k∇v_Kk_L^∞_(E) = c_K

kp_K − p_Lk²_L2(E) = k∇v_K − ∇v_Lk²_L2(E) ≤ c_E,Kkv_K − v_Lk_L^∞_(E)

Optimal stability for boundary measures 2/3

Theorem: If d_H(K, L) ≤ diam(K), d_bL(µ_K,E, µ_L,E) ≤ c_E,K d^1/2_H (K, L)

Step 2: p_K = ∇v_K a.e. where v_K(x) = ¹₂(kxk² − d_K(x)²) is convex Step 1: d_bL(µ_K,E, µ_L,E) ≤ kp_K − p_Lk_L¹_(E) ≤ c_K,Ekp_K − p_Lk_L²_(E)

Proposition: If u, v ∈ C²(E) are convex, and ∂E smooth,

k∇u − ∇vk_L²_(E) ≤ 2ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E) Step 3: With u = v_K, v = v_L, k∇v_Kk_L^∞_(E) + k∇v_Kk_L^∞_(E) = c_K

kp_K − p_Lk²_L2(E) = k∇v_K − ∇v_Lk²_L2(E) ≤ c_E,Kkv_K − v_Lk_L^∞_(E)

= ¹₂k d²_K − d²_L k_L^∞_(E) ≤ ¹₂k d_K − d_L k_L^∞_(E) · k d_K + d_L k_L^∞_(E)

Optimal stability for boundary measures 2/3

Theorem: If d_H(K, L) ≤ diam(K), d_bL(µ_K,E, µ_L,E) ≤ c_E,K d^1/2_H (K, L)

Step 2: p_K = ∇v_K a.e. where v_K(x) = ¹₂(kxk² − d_K(x)²) is convex Step 1: d_bL(µ_K,E, µ_L,E) ≤ kp_K − p_Lk_L¹_(E) ≤ c_K,Ekp_K − p_Lk_L²_(E)

Proposition: If u, v ∈ C²(E) are convex, and ∂E smooth,

k∇u − ∇vk_L²_(E) ≤ 2ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E) Step 3: With u = v_K, v = v_L, k∇v_Kk_L^∞_(E) + k∇v_Kk_L^∞_(E) = c_K

kp_K − p_Lk²_L2(E) = k∇v_K − ∇v_Lk²_L2(E) ≤ c_E,Kkv_K − v_Lk_L^∞_(E)

= ¹₂k d²_K − d²_L k_L^∞_(E) ≤ ¹₂k d_K − d_L k_L^∞_(E) · k d_K + d_L k_L^∞_(E)

≤ c_K d_H(K, L)

Optimal stability for boundary measures 3/3

Proposition: If u, v ∈ C²(E) are convex, and ∂E smooth,

k∇u − ∇vk_L²_(E) ≤ 2ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E)

Optimal stability for boundary measures 3/3

Proposition: If u, v ∈ C²(E) are convex, and ∂E smooth,

k∇u − ∇vk_L²_(E) ≤ 2ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E)

R

E k∇u − ∇vk² = R

∂E(u − v)h∇u − ∇v|n_Ei − R

E(u − v)∆(u − v) Stokes

Optimal stability for boundary measures 3/3

Proposition: If u, v ∈ C²(E) are convex, and ∂E smooth,

k∇u − ∇vk_L²_(E) ≤ 2ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E)

R

E k∇u − ∇vk² = R

∂E(u − v)h∇u − ∇v|n_Ei − R

E(u − v)∆(u − v) Stokes

≤ ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E)

Optimal stability for boundary measures 3/3

Proposition: If u, v ∈ C²(E) are convex, and ∂E smooth,

k∇u − ∇vk_L²_(E) ≤ 2ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E)

R

E k∇u − ∇vk² = R

∂E(u − v)h∇u − ∇v|n_Ei − R

E(u − v)∆(u − v)

≤ ku − vk_L^∞_(E) R

E(|∆u| + |∆v|) Stokes

≤ ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E)

Optimal stability for boundary measures 3/3

Proposition: If u, v ∈ C²(E) are convex, and ∂E smooth,

k∇u − ∇vk_L²_(E) ≤ 2ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E)

R

E k∇u − ∇vk² = R

∂E(u − v)h∇u − ∇v|n_Ei − R

E(u − v)∆(u − v)

≤ ku − vk_L^∞_(E) R

E(|∆u| + |∆v|) R

E |∆u| = R

E ∆u Stokes

convexity finally:

≤ ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E)

Optimal stability for boundary measures 3/3

Proposition: If u, v ∈ C²(E) are convex, and ∂E smooth,

k∇u − ∇vk_L²_(E) ≤ 2ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E)

R

E k∇u − ∇vk² = R

∂E(u − v)h∇u − ∇v|n_Ei − R

E(u − v)∆(u − v)

≤ ku − vk_L^∞_(E) R

E(|∆u| + |∆v|) R

E |∆u| = R

E ∆u Stokes

convexity Stokes finally:

≤ ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E)

= R

∂Eh∇u|n_Ei

Optimal stability for boundary measures 3/3

Proposition: If u, v ∈ C²(E) are convex, and ∂E smooth,

k∇u − ∇vk_L²_(E) ≤ 2ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E)

R

E k∇u − ∇vk² = R

∂E(u − v)h∇u − ∇v|n_Ei − R

E(u − v)∆(u − v)

≤ ku − vk_L^∞_(E) R

E(|∆u| + |∆v|) R

E |∆u| = R

E ∆u Stokes

convexity Stokes finally:

≤ ku − vk_L^∞_(E)(k∇uk_L^∞_(E) + k∇vk_L^∞_(E))H^d−1(∂E)

= R

∂Eh∇u|n_Ei ≤ k∇uk^∞_L (E)H^d−1(∂E)

Example of boundary measures

[Chazal-Cohen-Steiner-M. ’07]

2. Voronoi covariance measure

Voronoi-based normal estimation

Voronoi cell:

Vor_P (p) = {x ∈ R^d; ∀q ∈ P, P = {p₁, . . . , p_N } ⊆ R^d

kx − pk ≤ kx − qk}

Voronoi-based normal estimation

Voronoi cell:

Vor_P (p) = {x ∈ R^d; ∀q ∈ P, P = {p₁, . . . , p_N } ⊆ R^d

pole_C(p) := farthest point to p in Vor_C(p)

p

pole_C(p)

kx − pk ≤ kx − qk}

Voronoi-based normal estimation

Voronoi cell:

Vor_P (p) = {x ∈ R^d; ∀q ∈ P,

[Amenta, Bern 1999]

P = {p₁, . . . , p_N } ⊆ R^d

pole_C(p) := farthest point to p in Vor_C(p)

If C is a noiseless ε-sampling of a surface S, the angle between pole_C(p) − p and the normal of S at p is O(ε).

p

pole_C(p)

kx − pk ≤ kx − qk}

Voronoi-based normal estimation

Voronoi cell:

Vor_P (p) = {x ∈ R^d; ∀q ∈ P,

[Amenta, Bern 1999]

P = {p₁, . . . , p_N } ⊆ R^d

pole_C(p) := farthest point to p in Vor_C(p)

If C is a noiseless ε-sampling of a surface S, the angle between pole_C(p) − p and the normal of S at p is O(ε).

kx − pk ≤ kx − qk}

Voronoi-based normal estimation

Voronoi cell:

Vor_P (p) = {x ∈ R^d; ∀q ∈ P,

[Amenta, Bern 1999]

P = {p₁, . . . , p_N } ⊆ R^d

pole_C(p) := farthest point to p in Vor_C(p)

If C is a noiseless ε-sampling of a surface S, the angle between pole_C(p) − p and the normal of S at p is O(ε).

Need of an integral quantity to get stability under Hausdorff noise.

[Dey, Sun 2005]

kx − pk ≤ kx − qk}