On the reconstruction of convex bodies from random normal measurements

(1)

On the reconstruction of convex bodies from random normal measurements

Quentin M´ erigot

LJK / CNRS / Universit´e de Grenoble

Joint work with Hiba Abdallah, Universit´e de Grenoble

(2)

Motivation

I CEA-Leti has designed small sensors that can measure their orientation accelerometer + magnetometer =⇒ oriented normal

(3)

Motivation

I CEA-Leti has designed small sensors that can measure their orientation

I Applications in monitoring the deformations of a known surface S

accelerometer + magnetometer =⇒ oriented normal

already a non-convex inverse problem

Biard, Lacolle, Szafran (LJK) Huard, Sprynski (CEA)

(4)

Motivation

I What happens if we throw sensors on S and measure normals only ?

input data = unorganized normals without positional information.

(5)

Motivation

I What happens if we throw sensors on S and measure normals only ?

input data = unorganized normals without positional information.

(6)

1. Minkowski problem

(7)

Surface area measure

n_K : ∂K → S^d−1 K

n_K(x) x

K = convex body in R^d

(8)

Surface area measure

n_K : ∂K → S^d−1 K

n_K(x) x

K = convex body in R^d Definition: The surface area measure µ_K is a measure on S^d−1 defined by:

µ_K(B) = area({x ∈ ∂K; n_K(x) ∈ B})

(9)

Surface area measure

n_K : ∂K → S^d−1 K

n_K(x) x

In particular, µ_K(S^d−1) = area(∂K).

µ_K(B) = area({x ∈ ∂K; n_K(x) ∈ B})

= area(n⁻¹_K (B))

(10)

Surface area measure

n_K : ∂K → S^d−1 K

n_K(x) x

µ_K(B) = area({x ∈ ∂K; n_K(x) ∈ B})

F_i P

Example: P ⊆ R^d is a polyhedron

F_i = ith face

n_i = unit normal

(11)

Surface area measure

n_K : ∂K → S^d−1 K

n_K(x) x

µ_K(B) = area({x ∈ ∂K; n_K(x) ∈ B})

F_i µ_P = PN

i=1 area(F_i)δ_n_i P

Example: P ⊆ R^d is a polyhedron

F_i = ith face

n_i = unit normal

(12)

Minkowski-Alexandrov theorem

Minkowski problem: Given a measure µ on S^d−1, find K with µ_K = µ.

(13)

Minkowski-Alexandrov theorem

Properties of µ_K: K = bounded convex body with non-empty interior Minkowski problem: Given a measure µ on S^d−1, find K with µ_K = µ.

(14)

Minkowski-Alexandrov theorem

Properties of µ_K: K = bounded convex body with non-empty interior (non-degeneracy) for all hyperplane H, µ_K(S^d−1 \ H) 6= 0

(15)

Minkowski-Alexandrov theorem

Avoids this situation:

If L ⊆ R^d−1, and K = L × R, then n_K(∂K) ⊆ S^d−2 × {0}

L

K

(16)

Minkowski-Alexandrov theorem

(zero-mean) mean(µ_K) := R

S^d−1 v d µ_K(v) = 0

(17)

Minkowski-Alexandrov theorem

S^d−1 v d µ_K(v) = 0

u

Indeed, hmean(µ_K))|ui = R

S^d−1hx|ui d µ_K(v)

(18)

Minkowski-Alexandrov theorem

S^d−1 v d µ_K(v) = 0

u

= R

∂Khn_K(x)|ui d x

(19)

Minkowski-Alexandrov theorem

S^d−1 v d µ_K(v) = 0

u

H = {u}^⊥

= area(π_H(∂K)) − area(π_H(∂K))

= R

∂Khn_K(x)|ui d x

(20)

Minkowski-Alexandrov theorem

S^d−1 v d µ_K(v) = 0

u

H = {u}^⊥

= area(π_H(∂K)) − area(π_H(∂K))

= 0

= R

∂Khn_K(x)|ui d x

(21)

Minkowski-Alexandrov theorem

S^d−1 v d µ_K(v) = 0

Theorem (Minkowski-Alexandrov): Given any measure µ on S^d−1, which satisfies (non-degeneracy) and (zero-mean), there exists

a convex body K with µ = µ_K. This body is unique up to translation.

(22)

Stability in Minkowski theorem

Definition: d_bL(µ, ν) = max_f_∈BL₁ | R

S^d−1 f d µ − R

S^d−1 f d ν|, where BL₁ = 1-Lipschitz functions bounded by 1 on S^d−1

(23)

Stability in Minkowski theorem

d_bL(µ, ν) = Wass₁(µ, ν) when µ(S^d−1) = ν(S^d−1).

(24)

Stability in Minkowski theorem

Theorem (Diskant; Hug-Schneider): Given convex bodies K and L with r ≤ min(inrad(K), inrad(L))

the following inequality holds:

min_x∈_R^d d_H(K, L) ≤ const(r, R, d)d_bL(µ_K, µ_L)^1/d R ≥ max(circumrad(K), circumrad(L))

(25)

Stability in Minkowski theorem

I Drop the assumptions on inrad/circumrad L (for inference) Our goals:

(26)

Stability in Minkowski theorem

I Drop the assumptions on inrad/circumrad L (for inference) I Replace the bounded-Lipschitz distance with a weaker one.

Our goals:

(27)

Stability in Minkowski theorem

I Drop the assumptions on inrad/circumrad L (for inference) I Replace the bounded-Lipschitz distance with a weaker one.

Our goals:

(28)

2. Improved stability theorem

(29)

A new distance between surface area measures

0

K

u

Definition: Support function of K, h_K : S^d−1 → R

(30)

A new distance between surface area measures

0

K

u

h_K(u)

Definition: Support function of K, h_K : S^d−1 → R h_K(u) := max_p∈Khu|pi

6= radial parameterization of ∂K

(31)

A new distance between surface area measures

0

K

u

h_K(u)

Lemma: K ⊆ B(0, r) ⇐⇒ h_K is r-Lipschitz and | h_K | ≤ r.

(32)

A new distance between surface area measures

0

K

u

h_K(u)

Corollary: C₁ := {h_K; K ⊆ B(0, 1)} ⊆ BL₁ 6= radial parameterization of ∂K

(33)

A new distance between surface area measures

0

K

u

h_K(u)

Corollary: C₁ := {h_K; K ⊆ B(0, 1)} ⊆ BL₁

Definition: convex-dual distance between measure µ and ν on S^d−1: d_C(µ, ν) := max_f_∈C₁ | R

S^d−1 f d ν| 6= radial parameterization of ∂K

(34)

A new distance between surface area measures

0

K

u

h_K(u)

S^d−1 f d ν|

I Since C₁ ⊆ BL₁, d_C ≤ d_bL; a stability result for d_C is stronger.

(35)

A new distance between surface area measures

0

K

u

h_K(u)

S^d−1 f d ν|

I d_C satisfies the triangle inequality, but d_C(µ, ν) = 0 6⇒ µ = ν. 6= radial parameterization of ∂K

(36)

A new distance between surface area measures

0

K

u

h_K(u)

S^d−1 f d ν|

I d_C satisfies the triangle inequality, but d_C(µ, ν) = 0 6⇒ µ = ν. 6= radial parameterization of ∂K

(37)

Stability theorem for the convex-dual distance

Theorem (Abdallah-M. ’13): Given a convex body K and µ a measure

min_x∈_R^d d_H(K, L) ≤ const(K)d_C(µ_K, µ_L)^1/d

on S^d−1 such that d_C(µ_K, µ) ≤ ε₀(K), there exists a convex body L such that µ = µ_L and moreover

(38)

Stability theorem for the convex-dual distance

I d_bL → d_C: same proof techniques as Diskant and Hug-Schneider.

(39)

Stability theorem for the convex-dual distance

I exponent _d¹ is most likely non-optimal (best possible: _d−1¹ )

(40)

Stability theorem for the convex-dual distance

I to drop the requirement r ≤ inrad(µ_L), R ≥ circumrad(µ_L), we

introduce the weak rotundity and exploit a lemma of Cheng and Yau.

I exponent _d¹ is most likely non-optimal (best possible: _d−1¹ )

(41)

Weak rotundity

Definition: The weak rotundity of a measure µ on S^d−1 is:

rotund(µ) := min_y∈S^d−1 R

S^d−1 max(hy|vi, 0) d µ(v)

(42)

Weak rotundity

S^d−1 max(hy|vi, 0) d µ(v) I Lemma: rotund(µ) > 0 ⇐⇒ µ has non-degenerate support.

(43)

Weak rotundity

R ε

circumrad(K) ' R inrad(K) ' ε

area(∂K) ' R^d−1

rotund(µ_K) ' R^d−1ε I Lemma: rotund(µ) > 0 ⇐⇒ µ has non-degenerate support.

K = B^d−1(0, R) × [0, ε]

(44)

Weak rotundity

circumrad(K) ≤ const(d)area(∂K) ^d−1^d rotund(µ_K)⁻¹ inrad(K) ≥ const(d)area(∂K)^−d rotund(µ_K)^d

R ε

I Cheng-Yau Lemma:

K = B^d−1(0, R) × [0, ε]

(45)

Weak rotundity

circumrad(K) ≤ const(d)area(∂K) ^d−1^d rotund(µ_K)⁻¹ inrad(K) ≥ const(d)area(∂K)^−d rotund(µ_K)^d

R ε

I Cheng-Yau Lemma:

K = B^d−1(0, R) × [0, ε]

(46)

Stability of weak rotundity

Lemma: | rotund(µ) − rotund(ν)| ≤ d_C(µ, ν).

(47)

Stability of weak rotundity

Proof: Let y ∈ S^d−1, and S_y = [0, y] ⊆ B(0, 1). Then, h_S_y ∈ C₁, and Lemma: | rotund(µ) − rotund(ν)| ≤ d_C(µ, ν).

S_y y

(48)

Stability of weak rotundity

S_y y

h_S_y (v) = max

x∈S_yhx|vi =

( hv|yi if hv|yi ≥ 0 0 if hv|yi ≤ 0

v

(49)

Stability of weak rotundity

S_y y

h_S_y (v) = max

x∈S_yhx|vi =

= max(hv|yi, 0) v

(50)

Stability of weak rotundity

S_y y

h_S_y (v) = max

x∈S_yhx|vi =

= max(hv|yi, 0)

Letting f_µ(y) = R

S^d−1 max(hv|yi, 0) d µ(v), this gives:

v

(51)

Stability of weak rotundity

S_y y

h_S_y (v) = max

x∈S_yhx|vi =

= max(hv|yi, 0)

|f_µ(y) − f_ν(y)| ≤ d_C(µ, ν) := max_h∈C₁ | R

S^d−1 h d µ − R

S^d−1 h d ν| Letting f_µ(y) = R

v

(52)

Stability of weak rotundity

S_y y

h_S_y (v) = max

x∈S_yhx|vi =

= max(hv|yi, 0)

|f_µ(y) − f_ν(y)| ≤ d_C(µ, ν) := max_h∈C₁ | R

S^d−1 h d µ − R

S^d−1 h d ν| Letting f_µ(y) = R

v

(53)

Stability of weak rotundity

Corollary: Given K and µ on S^d−1 satisfying (zero mean), s.t.

d_C(µ_K, µ) ≤ ¹₂ rotund(µ_K)

Then there exists a convex set L such that µ_L = µ.

(54)

Stability of weak rotundity

Proof: rotund(µ_L) ≥ rotund(µ_K) − d_C(µ_K, µ) ≥ ¹₂ rotund(µ_K) > 0 Definition: The weak rotundity of a measure µ on S^d−1 is:

(55)

Stability of weak rotundity

Proof: rotund(µ_L) ≥ rotund(µ_K) − d_C(µ_K, µ) ≥ ¹₂ rotund(µ_K) > 0 Therefore µ satisfies (non-degeneracy) and (zero-mean).

The result follows from Minkowski’s theorem.

(56)

3. Reconstruction under random sampling

(57)

Reconstruction theorem

Definition: Given a convex body K, a random normal measurement is obtained by picking a random point x on ∂K and measuring n_K(x).

(58)

Reconstruction theorem

Theorem (Abdallah-M. ’13): Let K be a convex body w. area(∂K) = 1.

Then for η > 0, p ∈ (0, 1), and N random normal measurements with N ≥ const(K, d)η⁻¹² ^(3d+d²⁾ log(1/p),

one can construct a convex body L_N with P(d_H(L_N , K) ≥ η) ≤ 1 − p.

(59)

Reconstruction theorem

I For d = 2, N = Ω(η⁻⁵ log(1/p)), for d = 3, N = Ω(η⁻⁹ log(1/p)).

(60)

Reconstruction theorem

I For d = 2, N = Ω(η⁻⁵ log(1/p)), for d = 3, N = Ω(η⁻⁹ log(1/p)).

I Analysis using the bound involving d_bL would give η⁻⁶ and η⁻¹². Definition: Given a convex body K, a random normal measurement is obtained by picking a random point x on ∂K and measuring n_K(x).

(61)

Reconstruction theorem: sketch of proof

I µ_K,N = empirical measure constructed from the probability measure µ_K.

(62)

Reconstruction theorem: sketch of proof

Proposition: P(d_C(µ_K,N , µ_K) ≥ ε) ≤ 1 − 2 exp(const(d)ε^1−d² − N ε²)

(63)

Reconstruction theorem: sketch of proof

Proposition: P(d_C(µ_K,N , µ_K) ≥ ε) ≤ 1 − 2 exp(const(d)ε^1−d² − N ε²) Proof: Using Chernoff’s bound and the union bound, one has

P(d_C(µ_K,N , µ_K) ≥ ε) ≤ 1 − 2N (C₁, ε) exp(−2N ε²) where N (C₁, ε) = covering number of C₁ for k.k_∞. Moreover,

(64)

Reconstruction theorem: sketch of proof

Bronshtein’s theorem imply N (C₁, ε) ≤ exp(const(d)ε^1−d² ).

(NB: N (BL₁, ε) = Ω(exp(ε^1−d))

(65)

Reconstruction theorem: sketch of proof

I Conclusion uses the d_C-stability of the weak rotundity and mean, and the stability theorem using the convex-dual distance.

Bronshtein’s theorem imply N (C₁, ε) ≤ exp(const(d)ε^1−d² ).

(NB: N (BL₁, ε) = Ω(exp(ε^1−d))

(66)

Reconstruction theorem: an example

with a uniform noise of 0.05 on the normals.

I Reconstruction of a unit cube from N = 300 random normal measurements,

(67)

Reconstruction theorem: an example

with a uniform noise of 0.05 on the normals.

I Reconstruction of a unit cube from N = 300 random normal measurements,

(68)

Conclusion

(69)

Conclusion

Main result: In order to reconstruct a convex body K with Hausdorff error η

N ≥ const(K, d)η⁻ ¹² ^(3d+d²⁾.

and with probability 99%, one needs N random normal measurements with

(70)

Conclusion

N ≥ const(K, d)η⁻ ¹² ^(3d+d²⁾.

Open question: how to improve the exponent of η ?

(71)

Conclusion

N ≥ const(K, d)η⁻ ¹² ^(3d+d²⁾.

I Improve the exponent in the stability result. Seems difficult.

(72)

Conclusion

N ≥ const(K, d)η⁻ ¹² ^(3d+d²⁾.

I Improve the metric used between measures, e.g.

d_F(µ, ν) := max_f_∈F | R

S^d−1 f d(µ − ν)|

(73)

Conclusion

N ≥ const(K, d)η⁻ ¹² ^(3d+d²⁾.

S^d−1 f d(µ − ν)|

We replaced BL₁ with C₁, but an even smaller space would work:

F_K,L = {h_K, h_L} ∪ {h_S_y ; y ∈ S^d−1} ∪ {h_{y_}; y ∈ S^d−1}.

However, the space F_K,L depends on L. How to use this remark ?

(74)

Conclusion

N ≥ const(K, d)η⁻ ¹² ^(3d+d²⁾.

S^d−1 f d(µ − ν)|

We replaced BL₁ with C₁, but an even smaller space would work:

F_K,L = {h_K, h_L} ∪ {h_S_y ; y ∈ S^d−1} ∪ {h_{y_}; y ∈ S^d−1}.

However, the space F_K,L depends on L. How to use this remark ?