De l’inf´erence g´eom´etrique au transport optimal num´erique

De l’inf´erence g´eom´etrique

au transport optimal num´erique

Quentin M´ erigot

Laboratoire Jean Kuntzmann, CNRS / Universit´e de Grenoble

Soutenance d’habilitation

Overview

1. Geometric inference from noisy data 2. Computational optimal transport

3. Far-Field Reflector Problem

4. Discretization of Functionals Involving the Monge-Amp` ere Operator.

1. Geometric inference from noisy data

PhD: Geometric Inference from Point Clouds

d_K = distance function to K p_K = projection on K

K ⊆ R^d

PhD: Geometric Inference from Point Clouds

d_K = distance function to K p_K = projection on K

K ⊆ R^d

K^R = d⁻¹_K ([0, R])

PhD: Geometric Inference from Point Clouds

µ_K,R = p_K# H^d

K^r d_K = distance function to K

p_K = projection on K K ⊆ R^d

K^R = d⁻¹_K ([0, R])

Boundary measure:

PhD: Geometric Inference from Point Clouds

I Estimation of Federer’s curvature measures from Hausdorff approximation.

[Chazal, Cohen-Steiner, M. 2008]

µ_K,R = p_K# H^d

K^r d_K = distance function to K

p_K = projection on K K ⊆ R^d

K^R = d⁻¹_K ([0, R])

Boundary measure:

P =

PhD: Geometric Inference from Point Clouds

I Estimation of Federer’s curvature measures from Hausdorff approximation.

[Chazal, Cohen-Steiner, M. 2008]

µ_K,R = p_K# H^d

K^r d_K = distance function to K

p_K = projection on K K ⊆ R^d

K^R = d⁻¹_K ([0, R])

Boundary measure:

P =

PhD: Geometric Inference from Point Clouds

I Anisotropic version: Voronoi covariance measures

I Estimation of Federer’s curvature measures from Hausdorff approximation.

[Chazal, Cohen-Steiner, M. 2008]

[M., Ovsjanikov, Guibas 2009]

I Handling outliers −→ distance to a probability measure [Chazal, Cohen-Steiner, M. 2010]

µ_K,R = p_K# H^d

K^r d_K = distance function to K

p_K = projection on K K ⊆ R^d

K^R = d⁻¹_K ([0, R])

Boundary measure:

P =

Simplifying the Distance to a Measure

d²_P,k(x) = min_{pi6=...6=pk∈P k}¹ Pk

i=1 kx − p_ik² k-Distance: Let P ⊆ R^d, k ∈ {1, . . . , |P|}

Simplifying the Distance to a Measure

d²_P,k(x) = min_{pi6=...6=pk∈P k}¹ Pk

i=1 kx − p_ik²

d²_P,k(x) = min_b∈Bary^k

P kx − bk² + w_b k-Distance: Let P ⊆ R^d, k ∈ {1, . . . , |P|}

Simplifying the Distance to a Measure

d²_P,k(x) = min_{pi6=...6=pk∈P k}¹ Pk

i=1 kx − p_ik²

d²_P,k(x) = min_b∈Bary^k

P kx − bk² + w_b

−→ 1-semiconcave k-Distance: Let P ⊆ R^d, k ∈ {1, . . . , |P|}

−→ sublevel sets = ∪ balls

−→ exponential # of barycenters relation to weighted Voronoi diagrams

Simplifying the Distance to a Measure

d²_P,k(x) = min_{pi6=...6=pk∈P k}¹ Pk

i=1 kx − p_ik²

d²_P,k(x) = min_b∈Bary^k

P kx − bk² + w_b k-Distance: Let P ⊆ R^d, k ∈ {1, . . . , |P|}

Simplification problem: ∃ ”small” B ⊆ Bary^k_P s.t. k d_P,k −d_Bk_∞ is ”small” ? d²_B(x) = min_b∈B kx − bk² + w_b

Simplifying the Distance to a Measure

d²_P,k(x) = min_{pi6=...6=pk∈P k}¹ Pk

i=1 kx − p_ik²

d²_P,k(x) = min_b∈Bary^k

P kx − bk² + w_b k-Distance: Let P ⊆ R^d, k ∈ {1, . . . , |P|}

Simplification problem: ∃ ”small” B ⊆ Bary^k_P s.t. k d_P,k −d_Bk_∞ is ”small” ? d²_B(x) = min_b∈B kx − bk² + w_b

∀x ∈ K, ∀r ≤ diam(K), µ_K(B(x, r)) ≥ α_Kr<sup>`</sup> Ex: Area measure on a compact surface → ` = 2.

K := spt(µ ) x

B(x, r)

Assume µ_K is supported on K and ∃α_K > 0 such that I Positive result:

[Guibas, M., Morozov, 2011]

Simplifying the Distance to a Measure

d²_P,k(x) = min_{pi6=...6=pk∈P k}¹ Pk

i=1 kx − p_ik²

d²_P,k(x) = min_b∈Bary^k

P kx − bk² + w_b k-Distance: Let P ⊆ R^d, k ∈ {1, . . . , |P|}

Simplification problem: ∃ ”small” B ⊆ Bary^k_P s.t. k d_P,k −d_Bk_∞ is ”small” ? d²_B(x) = min_b∈B kx − bk² + w_b

∀x ∈ K, ∀r ≤ diam(K), µ_K(B(x, r)) ≥ α_Kr^` Assume µ_K is supported on K and ∃α_K > 0 such that I Positive result:

K µ_K

P −→ if W₂(µ_P , µ_K) is small, then one can construct a good approximation of d_P,k involving only |P| barycenters.

”Witnessed k-distance”

[Guibas, M., Morozov, 2011]

Simplifying the Distance to a Measure

d²_P,k(x) = min_{pi6=...6=pk∈P k}¹ Pk

i=1 kx − p_ik²

d²_P,k(x) = min_b∈Bary^k

P kx − bk² + w_b k-Distance: Let P ⊆ R^d, k ∈ {1, . . . , |P|}

Simplification problem: ∃ ”small” B ⊆ Bary^k_P s.t. k d_P,k −d_Bk_∞ is ”small” ? d²_B(x) = min_b∈B kx − bk² + w_b

∀x ∈ K, ∀r ≤ diam(K), µ_K(B(x, r)) ≥ α_Kr^` Assume µ_K is supported on K and ∃α_K > 0 such that I Positive result:

µ_K

P −→ if W₂(µ_P , µ_K) is small, then one can construct a good approximation of d_P,k involving only |P| barycenters.

”Witnessed k-distance”

−→ Application to reconstruction and topological data analysis.

[Guibas, M., Morozov, 2011]

Simplifying the Distance to a Measure

d²_P,k(x) = min_{pi6=...6=pk∈P k}¹ Pk

i=1 kx − p_ik²

d²_P,k(x) = min_b∈Bary^k

P kx − bk² + w_b k-Distance: Let P ⊆ R^d, k ∈ {1, . . . , |P|}

Simplification problem: ∃ ”small” B ⊆ Bary^k_P s.t. k d_P,k −d_Bk_∞ is ”small” ? d²_B(x) = min_b∈B kx − bk² + w_b

∀x ∈ K, ∀r ≤ diam(K), µ_K(B(x, r)) ≥ α_Kr^` Assume µ_K is supported on K and ∃α_K > 0 such that I Positive result:

I Negative result: exponential lower bound when P is drawn randomly from S^d−1. K

µ_K

P −→ if W₂(µ_P , µ_K) is small, then one can construct a good approximation of d_P,k involving only |P| barycenters.

”Witnessed k-distance”

−→ Application to reconstruction and topological data analysis.

[Guibas, M., Morozov, 2011]

6

Sharp Feature Estimation

Input: 50k point set P

I Application of the witnessed k-distance to curvature estimation and edge recovery

6

Sharp Feature Estimation

Input: 50k point set P Output: estimated edges and directions I Application of the witnessed k-distance to curvature estimation and edge recovery

[Cuel, Lachaud, M., Thibert 2014]

6

Sharp Feature Estimation

Input: 50k point set P Output: estimated edges and directions I Application of the witnessed k-distance to curvature estimation and edge recovery

[Cuel, Lachaud, M., Thibert 2014]

From curvature estimation to the inverse problem: Monge-Amp`ere equations.

2. Computational optimal transport

Computational optimal transport

Hungarian algorithm linear programming

Discrete source and target

Bertsekas’ auction algorithm

α_i β_j

Computational optimal transport

Hungarian algorithm linear programming

Discrete source and target

Bertsekas’ auction algorithm

α_i β_j

Source and target with density:

Benamou-Brenier ’00 Loeper-Rapetti ’05

Benamou-Froese-Oberman ’12

Computational optimal transport

Hungarian algorithm linear programming

Discrete source and target

Bertsekas’ auction algorithm

α_i β_j

Source with density, discrete target:

Kitagawa ’12

Source and target with density:

Benamou-Brenier ’00 Loeper-Rapetti ’05

Benamou-Froese-Oberman ’12

Theoretical use: Alexandrov, Pogorelov, etc.

Aurenhammer, Hoffmann, Aronov ’98 Cullen ’89

Computational optimal transport

Hungarian algorithm linear programming

Discrete source and target

Bertsekas’ auction algorithm

α_i β_j

Source with density, discrete target:

Kitagawa ’12

Source and target with density:

Benamou-Brenier ’00 Loeper-Rapetti ’05

Benamou-Froese-Oberman ’12

Flexibility for the cost function but computationally expensive

Theoretical use: Alexandrov, Pogorelov, etc.

Aurenhammer, Hoffmann, Aronov ’98 Cullen ’89

Computational optimal transport

Hungarian algorithm linear programming

Discrete source and target

Bertsekas’ auction algorithm

α_i β_j

Source with density, discrete target:

Kitagawa ’12

Source and target with density:

Benamou-Brenier ’00 Loeper-Rapetti ’05

Benamou-Froese-Oberman ’12

Theoretical use: Alexandrov, Pogorelov, etc.

Aurenhammer, Hoffmann, Aronov ’98 Cullen ’89

Optimal transport: Monge problem

ρ ∈ P^ac(X) X = d-manifold ν = P

y∈Y ν_yδ_y ∈ P(Y ) Y finite

y

prob. measure with density

X

Y

prob. measure

Optimal transport: Monge problem

ρ ∈ P^ac(X) X = d-manifold ν = P

y∈Y ν_yδ_y ∈ P(Y ) Y finite

T⁻¹(y) y

Transport map: T_#ρ = ν iff

∀y ∈ Y, ρ(T⁻¹({y})) = ν_y

T : X → Y

X

Y

Optimal transport: Monge problem

ρ ∈ P^ac(X) X = d-manifold ν = P

y∈Y ν_yδ_y ∈ P(Y ) Y finite

T⁻¹(y) y

Transport map: T_#ρ = ν iff

∀y ∈ Y, ρ(T⁻¹({y})) = ν_y

Cost function: c : X × Y → R T : X → Y

X

Y

Optimal transport: Monge problem

ρ ∈ P^ac(X) X = d-manifold ν = P

y∈Y ν_yδ_y ∈ P(Y ) Y finite

Monge problem: T_c(ρ, ν) := min{R

c(x, T(x)) d ρ(x); T_#ρ = ν}

T⁻¹(y) y

Transport map: T_#ρ = ν iff

∀y ∈ Y, ρ(T⁻¹({y})) = ν_y

Cost function: c : X × Y → R T : X → Y

X

Y

Optimal transport: Monge problem

ρ ∈ P^ac(X) X = d-manifold ν = P

y∈Y ν_yδ_y ∈ P(Y ) Y finite

Monge problem: T_c(ρ, ν) := min{R

X c(x, T(x)) d ρ(x); T_#ρ = ν}

T⁻¹(y) y

Transport map: T_#ρ = ν iff

∀y ∈ Y, ρ(T⁻¹({y})) = ν_y

Cost function: c : X × Y → R T : X → Y

X

Y

ex: X, Y ⊆ R^d, c = ¹₂k.k² Wasserstein distance W₂ = T_c^1/2

Weighted Voronoi and Optimal Transport

We assume (Twist): ∀x ∈ X, the map y ∈ Y 7→ ∇_xc(x, y) is injective.

Weighted Voronoi and Optimal Transport

Generalized Voronoi diagram: Given ψ : Y → R,

We assume (Twist): ∀x ∈ X, the map y ∈ Y 7→ ∇_xc(x, y) is injective.

Vor^ψ_c (y) = {x ∈ X; ∀z ∈ Y, c(x, y) + ψ(y) ≤ c(x, z) + ψ(z)}

y z

Weighted Voronoi and Optimal Transport

Generalized Voronoi diagram: Given ψ : Y → R,

We assume (Twist): ∀x ∈ X, the map y ∈ Y 7→ ∇_xc(x, y) is injective.

Vor^ψ_c (y) = {x ∈ X; ∀z ∈ Y, c(x, y) + ψ(y) ≤ c(x, z) + ψ(z)}

y

z I the Voronoi cells partition X up to a negligible set

Weighted Voronoi and Optimal Transport

Generalized Voronoi diagram: Given ψ : Y → R,

T_c^ψ(x) = arg min_y∈Y c(x, y) + ψ(y)

We assume (Twist): ∀x ∈ X, the map y ∈ Y 7→ ∇_xc(x, y) is injective.

Vor^ψ_c (y) = {x ∈ X; ∀z ∈ Y, c(x, y) + ψ(y) ≤ c(x, z) + ψ(z)}

y z

is uniquely defined H^d-almost everywhere

I the Voronoi cells partition X up to a negligible set I the ”generalized nearest neighbor” map

Weighted Voronoi and Optimal Transport

Generalized Voronoi diagram: Given ψ : Y → R,

T_c^ψ(x) = arg min_y∈Y c(x, y) + ψ(y)

We assume (Twist): ∀x ∈ X, the map y ∈ Y 7→ ∇_xc(x, y) is injective.

Vor^ψ_c (y) = {x ∈ X; ∀z ∈ Y, c(x, y) + ψ(y) ≤ c(x, z) + ψ(z)}

y z

is uniquely defined H^d-almost everywhere

Lemma: Given ψ : Y → the map T^ψ is a c-optimal transport map between I the Voronoi cells partition X up to a negligible set

I the ”generalized nearest neighbor” map

Optimal transport as concave maximization

Theorem: Finding a c-O.T between ρ with density and ν = P

Y ν_yδ_y

⇐⇒ finding ψ : Y → R such that T_c#^ψ ρ = ν

⇐⇒ maximizing the concave function Φ

Aurenhammer, Hoffman, Aronov ’98

Φ(ψ) := P

y

R

Vor^ψ_c (y)[c(x, y) + ψ(y)] d ρ(x) − P

y ψ(y)ν_y

Optimal transport as concave maximization

Theorem: Finding a c-O.T between ρ with density and ν = P

Y ν_yδ_y

⇐⇒ finding ψ : Y → R such that T_c#^ψ ρ = ν

⇐⇒ maximizing the concave function Φ

Aurenhammer, Hoffman, Aronov ’98

Φ(ψ) := P

y

R

Vor^ψ_c (y)[c(x, y) + ψ(y)] d ρ(x) − P

y ψ(y)ν_y

I Byproduct of Kantorovich duality.

Optimal transport as concave maximization

Theorem: Finding a c-O.T between ρ with density and ν = P

Y ν_yδ_y

⇐⇒ finding ψ : Y → R such that T_c#^ψ ρ = ν

⇐⇒ maximizing the concave function Φ

Aurenhammer, Hoffman, Aronov ’98

Φ(ψ) := P

y

R

Vor^ψ_c (y)[c(x, y) + ψ(y)] d ρ(x) − P

y ψ(y)ν_y

I Byproduct of Kantorovich duality.

I ∇Φ(ψ) = (ρ(Vor^ψ_c (y)) − ν_y)_y∈Y . Hence,

∇Φ = 0 ⇐⇒ discrete Monge-Amp`ere equation:

∀y ∈ Y, ρ(Vor^ψ_c (y)) = ν_y.

Optimal transport as concave maximization

Theorem: Finding a c-O.T between ρ with density and ν = P

Y ν_yδ_y

⇐⇒ finding ψ : Y → R such that T_c#^ψ ρ = ν

⇐⇒ maximizing the concave function Φ

Aurenhammer, Hoffman, Aronov ’98

Φ(ψ) := P

y

R

Vor^ψ_c (y)[c(x, y) + ψ(y)] d ρ(x) − P

y ψ(y)ν_y

I Byproduct of Kantorovich duality.

I ∇Φ(ψ) = (ρ(Vor^ψ_c (y)) − ν_y)_y∈Y . Hence,

∇Φ = 0 ⇐⇒ discrete Monge-Amp`ere equation:

I Existing numerical methods: coordinate-wise increment with minimum step,

N³

∀y ∈ Y, ρ(Vor^ψ_c (y)) = ν_y.

Optimal transport as concave maximization

Theorem: Finding a c-O.T between ρ with density and ν = P

Y ν_yδ_y

⇐⇒ finding ψ : Y → R such that T_c#^ψ ρ = ν

⇐⇒ maximizing the concave function Φ

Aurenhammer, Hoffman, Aronov ’98

Φ(ψ) := P

y

R

Vor^ψ_c (y)[c(x, y) + ψ(y)] d ρ(x) − P

y ψ(y)ν_y

I Byproduct of Kantorovich duality.

I ∇Φ(ψ) = (ρ(Vor^ψ_c (y)) − ν_y)_y∈Y . Hence,

∇Φ = 0 ⇐⇒ discrete Monge-Amp`ere equation:

I Existing numerical methods: coordinate-wise increment with minimum step,

[Oliker–Prussner]

with complexity O(^N_ε³ log(N)), ε = precision.

Contribution: Efficient heuristic to optimize Φ −→ quasi-Newton and multiscale

∀y ∈ Y, ρ(Vor^ψ_c (y)) = ν_y.

Multiscale Approach to Semi-discrete OT, c = ¹ ₂ k.k ²

Φ_ν(ψ) := P

y

R

Vor^ψ_c (y)[c(x, y) + ψ(y)] d ρ(x) − P

y ψ(y)ν_y I Single scale

−→ Evaluation of Φ: Voronoi cells (CGAL) + Exact integration

−→ low-storage quasi-Newton method

[M., Comput. Graph. Forum / SGP 2011]

Multiscale Approach to Semi-discrete OT, c = ¹ ₂ k.k ²

I Multiscale. (A) Simplification of target measure using Lloyd’s algorithm

ν₀

Φ_ν(ψ) := P

y

R

Vor^ψ_c (y)[c(x, y) + ψ(y)] d ρ(x) − P

y ψ(y)ν_y I Single scale

−→ Evaluation of Φ: Voronoi cells (CGAL) + Exact integration

−→ low-storage quasi-Newton method

[M., Comput. Graph. Forum / SGP 2011]

Multiscale Approach to Semi-discrete OT, c = ¹ ₂ k.k ²

I Multiscale. (A) Simplification of target measure using Lloyd’s algorithm

ν₀ ν₁

π₀₁ Φ_ν(ψ) := P

y

R

Vor^ψ_c (y)[c(x, y) + ψ(y)] d ρ(x) − P

y ψ(y)ν_y I Single scale

−→ Evaluation of Φ: Voronoi cells (CGAL) + Exact integration

−→ low-storage quasi-Newton method

[M., Comput. Graph. Forum / SGP 2011]

Multiscale Approach to Semi-discrete OT, c = ¹ ₂ k.k ²

I Multiscale. (A) Simplification of target measure using Lloyd’s algorithm

ν₀ ν₁ ν₂

π₀₁ π₁₂

Φ_ν(ψ) := P

y

R

Vor^ψ_c (y)[c(x, y) + ψ(y)] d ρ(x) − P

y ψ(y)ν_y I Single scale

−→ Evaluation of Φ: Voronoi cells (CGAL) + Exact integration

−→ low-storage quasi-Newton method

[M., Comput. Graph. Forum / SGP 2011]

Multiscale Approach to Semi-discrete OT, c = ¹ ₂ k.k ²

I Multiscale. (A) Simplification of target measure using Lloyd’s algorithm

ν₀ ν₁ ν₂

π₀₁ π₁₂

ψ₂ = arg max Φ_ν2 starting from ψ₂⁰ = 0

(B) Coarse-to-fine

Φ_ν(ψ) := P

y

R

Vor^ψ_c (y)[c(x, y) + ψ(y)] d ρ(x) − P

y ψ(y)ν_y I Single scale

resolution:

−→ Evaluation of Φ: Voronoi cells (CGAL) + Exact integration

−→ low-storage quasi-Newton method

[M., Comput. Graph. Forum / SGP 2011]

Multiscale Approach to Semi-discrete OT, c = ¹ ₂ k.k ²

I Multiscale. (A) Simplification of target measure using Lloyd’s algorithm

ν₀ ν₁ ν₂

π₀₁ π₁₂

ψ₂ = arg max Φ_ν2 ψ₁ = arg min Φ_ν1

w. ψ₁⁰ = π₁₂⁻¹ ◦ ψ2 starting from ψ₂⁰ = 0

(B) Coarse-to-fine

Φ_ν(ψ) := P

y

R

Vor^ψ_c (y)[c(x, y) + ψ(y)] d ρ(x) − P

y ψ(y)ν_y I Single scale

resolution:

−→ Evaluation of Φ: Voronoi cells (CGAL) + Exact integration

−→ low-storage quasi-Newton method

[M., Comput. Graph. Forum / SGP 2011]

Multiscale Approach to Semi-discrete OT, c = ¹ ₂ k.k ²

I Multiscale. (A) Simplification of target measure using Lloyd’s algorithm

ν₀ ν₁ ν₂

π₀₁ π₁₂

ψ₂ = arg max Φ_ν2 ψ₁ = arg min Φ_ν1

w. ψ₁⁰ = π₁₂⁻¹ ◦ ψ2 starting from ψ₂⁰ = 0 ψ₀ = arg min Φ_ν0

w. ψ₀⁰ = π₀₁⁻¹ ◦ ψ1

(B) Coarse-to-fine

Φ_ν(ψ) := P

y

R

Vor^ψ_c (y)[c(x, y) + ψ(y)] d ρ(x) − P

y ψ(y)ν_y I Single scale

resolution:

−→ Evaluation of Φ: Voronoi cells (CGAL) + Exact integration

−→ low-storage quasi-Newton method

[M., Comput. Graph. Forum / SGP 2011]

Multiscale Approach to Semi-discrete OT, c = ¹ ₂ k.k ²

I Multiscale. (A) Simplification of target measure using Lloyd’s algorithm

ν₀ ν₁ ν₂

π₀₁ π₁₂

ψ₂ = arg max Φ_ν2 ψ₁ = arg min Φ_ν1

w. ψ₁⁰ = π₁₂⁻¹ ◦ ψ2 starting from ψ₂⁰ = 0 ψ₀ = arg min Φ_ν0

w. ψ₀⁰ = π₀₁⁻¹ ◦ ψ1

(B) Coarse-to-fine

Φ_ν(ψ) := P

y

R

Vor^ψ_c (y)[c(x, y) + ψ(y)] d ρ(x) − P

y ψ(y)ν_y I Single scale

resolution:

−→ Evaluation of Φ: Voronoi cells (CGAL) + Exact integration

First numerical method able to handle large semi-discrete OT problems

−→ low-storage quasi-Newton method

[M., Comput. Graph. Forum / SGP 2011]

Some Pictures of Optimal Transport Plans, c = ¹ ₂ k.k ²

Source: picture ”Cameraman”

Target: Lloyd sampling of picture ”Peppers” (k = 625)

McCann ’97 Displacement interpolation

[M., Comput. Graph. Forum / SGP 2011]

Some Pictures of Optimal Transport Plans, c = ¹ ₂ k.k ²

The mass of Dirac at p is spread onto Vor^ψ_c (p)

Source: picture ”Cameraman”

Target: Lloyd sampling of picture ”Peppers” (k = 625)

McCann ’97 Displacement interpolation

ψ = (1 − t) · ψ^sol + t · 0 t = ¹₄

[M., Comput. Graph. Forum / SGP 2011]

Some Pictures of Optimal Transport Plans, c = ¹ ₂ k.k ²

Source: picture ”Cameraman”

Target: Lloyd sampling of picture ”Peppers” (k = 625)

McCann ’97 Displacement interpolation

[M., Comput. Graph. Forum / SGP 2011]

Some Pictures of Optimal Transport Plans, c = ¹ ₂ k.k ²

The mass of Dirac at p is spread onto Vor^ψ_c (p)

Source: picture ”Cameraman”

Target: Lloyd sampling of picture ”Peppers” (k = 625)

McCann ’97 Displacement interpolation

ψ = (1 − t) · ψ^sol + t · 0 t = ³₄

[M., Comput. Graph. Forum / SGP 2011]

Some Pictures of Optimal Transport Plans, c = ¹ ₂ k.k ²

Source: picture ”Cameraman”

Target: Lloyd sampling of picture ”Peppers” (k = 625)

McCann ’97 Displacement interpolation

[M., Comput. Graph. Forum / SGP 2011]

Some Pictures of Optimal Transport Plans, c = ¹ ₂ k.k ²

k = 625

k = 15000

[M., Comput. Graph. Forum / SGP 2011]

Some Pictures of Optimal Transport Plans, c = ¹ ₂ k.k ²

k = 625

k = 15000

[M., Comput. Graph. Forum / SGP 2011]

Some Pictures of Optimal Transport Plans, c = ¹ ₂ k.k ²

k = 625

k = 15000

[M., Comput. Graph. Forum / SGP 2011]

3D version of the algorithm by B. L´evy (2014), with k up to 1M points.

Hessian of Kantorovich functional, c = ¹ ₂ k.k ²

∇Φ(ψ) = (ρ(Vor^ψ_c (y)) − ν_y)_y∈Y

Assume X ⊆ R^d is open, bounded and connected / ρ ∈ P^ac(X) ∩ C⁰(Ω), ρ ≥ ε.

Hessian of Kantorovich functional, c = ¹ ₂ k.k ²

Proposition: Φ is twice differentiable almost everywhere and

∂²Φ

∂z∂y (ψ) = R

Vor^ψ_c (y,z)

ρ(x) dx ky−zk

∂²Φ

∂y² (ψ) = − P

z6=y

∂²Φ

∂y∂z (ψ)

∇Φ(ψ) = (ρ(Vor^ψ_c (y)) − ν_y)_y∈Y

Assume X ⊆ R^d is open, bounded and connected / ρ ∈ P^ac(X) ∩ C⁰(Ω), ρ ≥ ε.

Vor^ψ_c (y, z) := Vor^ψ_c (y) ∩ Vor^ψ_c (z)

Hessian of Kantorovich functional, c = ¹ ₂ k.k ²

Proposition: Φ is twice differentiable almost everywhere and

∂²Φ

∂z∂y (ψ) = R

Vor^ψ_c (y,z)

ρ(x) dx ky−zk

∂²Φ

∂y² (ψ) = − P

z6=y

∂²Φ

∂y∂z (ψ)

∇Φ(ψ) = (ρ(Vor^ψ_c (y)) − ν_y)_y∈Y

Assume X ⊆ R^d is open, bounded and connected / ρ ∈ P^ac(X) ∩ C⁰(Ω), ρ ≥ ε.

(y, z) ∈ G ⇐⇒ _∂z∂y^∂2Φ (ψ) > 0

Vor^ψ_c (y, z) := Vor^ψ_c (y) ∩ Vor^ψ_c (z)

I D²Φ(ψ) is the Laplacian of a weighted graph G

Hessian of Kantorovich functional, c = ¹ ₂ k.k ²

Proposition: Φ is twice differentiable almost everywhere and

∂²Φ

∂z∂y (ψ) = R

Vor^ψ_c (y,z)

ρ(x) dx ky−zk

∂²Φ

∂y² (ψ) = − P

z6=y

∂²Φ

∂y∂z (ψ)

∇Φ(ψ) = (ρ(Vor^ψ_c (y)) − ν_y)_y∈Y

Assume X ⊆ R^d is open, bounded and connected / ρ ∈ P^ac(X) ∩ C⁰(Ω), ρ ≥ ε.

(y, z) ∈ G ⇐⇒ _∂z∂y^∂2Φ (ψ) > 0 ⇐⇒ Vor^ψ_c (y, z) ∩ X 6= ∅.

Vor^ψ_c (y, z) := Vor^ψ_c (y) ∩ Vor^ψ_c (z)

i.e. G = 1-skeleton of restricted regular triangulation I D²Φ(ψ) is the Laplacian of a weighted graph G

Hessian of Kantorovich functional, c = ¹ ₂ k.k ²

Proposition: Φ is twice differentiable almost everywhere and

∂²Φ

∂z∂y (ψ) = R

Vor^ψ_c (y,z)

ρ(x) dx ky−zk

∂²Φ

∂y² (ψ) = − P

z6=y

∂²Φ

∂y∂z (ψ)

∇Φ(ψ) = (ρ(Vor^ψ_c (y)) − ν_y)_y∈Y

Assume X ⊆ R^d is open, bounded and connected / ρ ∈ P^ac(X) ∩ C⁰(Ω), ρ ≥ ε.

(y, z) ∈ G ⇐⇒ _∂z∂y^∂2Φ (ψ) > 0 ⇐⇒ Vor^ψ_c (y, z) ∩ X 6= ∅.

Vor^ψ_c (y, z) := Vor^ψ_c (y) ∩ Vor^ψ_c (z)

i.e. G = 1-skeleton of restricted regular triangulation I D²Φ(ψ) is the Laplacian of a weighted graph G

I min_y∈Y ρ(Vor^ψ_c (y) ∩ X) > 0 =⇒ G is connected

Hessian of Kantorovich functional, c = ¹ ₂ k.k ²

Proposition: Φ is twice differentiable almost everywhere and

∂²Φ

∂z∂y (ψ) = R

Vor^ψ_c (y,z)

ρ(x) dx ky−zk

∂²Φ

∂y² (ψ) = − P

z6=y

∂²Φ

∂y∂z (ψ)

∇Φ(ψ) = (ρ(Vor^ψ_c (y)) − ν_y)_y∈Y

Assume X ⊆ R^d is open, bounded and connected / ρ ∈ P^ac(X) ∩ C⁰(Ω), ρ ≥ ε.

(y, z) ∈ G ⇐⇒ _∂z∂y^∂2Φ (ψ) > 0 ⇐⇒ Vor^ψ_c (y, z) ∩ X 6= ∅.

Vor^ψ_c (y, z) := Vor^ψ_c (y) ∩ Vor^ψ_c (z)

−→ local quadratic convergence of Newton’s method i.e. G = 1-skeleton of restricted regular triangulation I D²Φ(ψ) is the Laplacian of a weighted graph G

=⇒ KerD²Ψ(ψ) = {cst}

I min_y∈Y ρ(Vor^ψ_c (y) ∩ X) > 0 =⇒ G is connected

Future work: Newton’s Method and Other Costs

I Quantitative lower bound on λ₁(−D²Φ) using a discrete Cheeger inequality:

λ₁(X) ≥ ¹₄h(X)² h(X) := min_{A min(H}^H_d^d−1_(A),H^(∂A)_d(X\A)

where A ⊆ X and ∂A smooth

[ongoing work with Kitagawa and Thibert]

Future work: Newton’s Method and Other Costs

I Quantitative lower bound on λ₁(−D²Φ) using a discrete Cheeger inequality:

λ₁(X) ≥ ¹₄h(X)² h(X) := min_{A min(H}^H_d^d−1_(A),H^(∂A)_d(X\A)

where A ⊆ X and ∂A smooth

I Possible improvement if one is able to control max_y diam(Vor^ψ_c (y))

[ongoing work with Kitagawa and Thibert]

' regularity result for discrete solutions.

Future work: Newton’s Method and Other Costs

I Quantitative lower bound on λ₁(−D²Φ) using a discrete Cheeger inequality:

λ₁(X) ≥ ¹₄h(X)² h(X) := min_{A min(H}^H_d^d−1_(A),H^(∂A)_d(X\A)

where A ⊆ X and ∂A smooth

I Possible improvement if one is able to control max_y diam(Vor^ψ_c (y))

I Extension to Ma-Trudinger-Wang costs, using work from Kitagawa? [Kitagawa ’12]

[ongoing work with Kitagawa and Thibert]

' regularity result for discrete solutions.

Future work: Newton’s Method and Other Costs

Towards the global convergence of a damped Newton method for MTW costs ? I Quantitative lower bound on λ₁(−D²Φ) using a discrete Cheeger inequality:

λ₁(X) ≥ ¹₄h(X)² h(X) := min_{A min(H}^H_d^d−1_(A),H^(∂A)_d(X\A)

where A ⊆ X and ∂A smooth

I Possible improvement if one is able to control max_y diam(Vor^ψ_c (y))

I Extension to Ma-Trudinger-Wang costs, using work from Kitagawa? [Kitagawa ’12]

[ongoing work with Kitagawa and Thibert]

' regularity result for discrete solutions.

3. Far-Field Reflector Problem

c(x, y ) = − log(1 − hx|y i) on S

in collaboration with P. Machado and B. Thibert

Far-Field Reflector Problem

S_∞² Punctual light at origin o, density f on S_o² Prescribed far-field: density g on S_∞²

S_o² o

o f

g

Far-Field Reflector Problem

S_∞² Punctual light at origin o, density f on S_o² Prescribed far-field: density g on S_∞²

S_o²

Goal: Find a surface R which sends (S_o², f) to (S_∞, g) under reflection by Snell-Descartes law.

o o

R

f

g

Far-Field Reflector Problem

S_∞² Punctual light at origin o, density f on S_o² Prescribed far-field: density g on S_∞²

S_o²

Goal: Find a surface R which sends (S_o², f) to (S_∞, g) under reflection by Snell-Descartes law.

o o

R

Generalized Monge-Amp`ere equation: g(T_R(u)) det(DT_R(u)) = f(u) f

g

T_R maps the direction of an emmited ray in S_o² to the direction of the reflected ray in S_∞² .

Far-Field Reflector Problem

S_∞² Punctual light at origin o, density f on S_o² Prescribed far-field: density g on S_∞²

S_o²

Goal: Find a surface R which sends (S_o², f) to (S_∞, g) under reflection by Snell-Descartes law.

o o

R

Generalized Monge-Amp`ere equation: g(T_R(u)) det(DT_R(u)) = f(u) f

g

T_R maps the direction of an emmited ray in S_o²

I Fully non linear PDE over the sphere with BV2 boundary conditions

to the direction of the reflected ray in S_∞² .

Far-Field Reflector Problem: Semi-discrete case

S_∞² y₁

y₂

y₃

Punctual light at origin o, µ ∈ P^ac(S_o²) Prescribed far-field: ν = P

i ν_iδ_yi ∈ P(S_∞² )

S_o² o

Far-Field Reflector Problem: Semi-discrete case

S_∞² y₁

y₂

y₃

Punctual light at origin o, µ ∈ P^ac(S_o²) Prescribed far-field: ν = P

i ν_iδ_yi ∈ P(S_∞² )

S_o²

Goal: Find a surface R which sends (S_o², µ) to (S_∞, ν) under reflection by Snell’s law.

o

R

Far-Field Reflector Problem: Semi-discrete case

y₁

y₂

y₃

Punctual light at origin o, µ ∈ P^ac(S_o²) Prescribed far-field: ν = P

i ν_iδ_yi ∈ P(S_∞² )

−→ P_i(κ_i) = solid paraboloid of revolution with

focal o, direction y_i and focal distance κ_i Goal: Find a surface R which sends (S_o², µ) to

(S_∞, ν) under reflection by Snell’s law.

o P₃

P₂

µ

P₁

S_o²

Far-Field Reflector Problem: Semi-discrete case

Punctual light at origin o, µ ∈ P^ac(S_o²) Prescribed far-field: ν = P

i ν_iδ_yi ∈ P(S_∞² )

−→ P_i(κ_i) = solid paraboloid of revolution with

focal o, direction y_i and focal distance κ_i

−→ PI^κ_i = π_S²

o ∩^N_j=1P_j(κ_j)

∩ ∂P_i(κ_i)

Goal: Find a surface R which sends (S_o², µ) to (S_∞, ν) under reflection by Snell’s law.

PI^κ₃

o P₃

P₂

µ

= partition of S_o² S_o²

Far-Field Reflector Problem: Semi-discrete case

Punctual light at origin o, µ ∈ P^ac(S_o²) Prescribed far-field: ν = P

i ν_iδ_yi ∈ P(S_∞² )

−→ P_i(κ_i) = solid paraboloid of revolution with

focal o, direction y_i and focal distance κ_i

−→ PI^κ_i = π_S²

o ∩^N_j=1P_j(κ_j)

∩ ∂P_i(κ_i)

Discrete Monge-Amp`ere equation: Find ~κ such that ∀i, µ(PI^κ_i ) = ν_i.

Goal: Find a surface R which sends (S_o², µ) to (S_∞, ν) under reflection by Snell’s law.

PI^κ₃

amount of light reflected in direction yi.

o P₃

P₂

µ

= partition of S_o² S_o²

Caffarelli-Kochengin-Oliker ’99

(FF)

Semi-Discrete Far-Field Reflector Problem as OT

Lemma: With c(x, y) = − log(1 − hx|yi), and ψ_i := log(κ_i), PI^κ_i = Vor^ψ_c (y_i).

PI^κ₃

o

P3(κ3)

Wang ’04

Glimm-Oliker ’03 [Machado, M., Thibert, Symp. Comp. Geom 2014]

Semi-Discrete Far-Field Reflector Problem as OT

Lemma: With c(x, y) = − log(1 − hx|yi), and ψ_i := log(κ_i), PI^κ_i = Vor^ψ_c (y_i).

PI^κ₃

o

P3(κ3)

Wang ’04

Glimm-Oliker ’03 [Machado, M., Thibert, Symp. Comp. Geom 2014]

(FF) Find κ such that ∀i, µ(PI^κ_i ) = ν_i. Corollary:

Semi-Discrete Far-Field Reflector Problem as OT

Lemma: With c(x, y) = − log(1 − hx|yi), and ψ_i := log(κ_i), PI^κ_i = Vor^ψ_c (y_i).

PI^κ₃

o

P3(κ3)

Wang ’04

Glimm-Oliker ’03 [Machado, M., Thibert, Symp. Comp. Geom 2014]

(FF) Find κ such that ∀i, µ(PI^κ_i ) = ν_i.

⇐⇒

(OT) Find ψ such that µ(Vor^ψ_c (y_i)) = ν_i Corollary:

Semi-Discrete Far-Field Reflector Problem as OT

Lemma: With c(x, y) = − log(1 − hx|yi), and ψ_i := log(κ_i), PI^κ_i = Vor^ψ_c (y_i).

PI^κ₃

o

P3(κ3)

Wang ’04

Glimm-Oliker ’03 [Machado, M., Thibert, Symp. Comp. Geom 2014]

I (FF) is a semi-discrete OT problem with c(x, y) = − log(1 − hx|yi)

(FF) Find κ such that ∀i, µ(PI^κ_i ) = ν_i.

⇐⇒

(OT) Find ψ such that µ(Vor^ψ_c (y_i)) = ν_i Corollary:

Semi-Discrete Far-Field Reflector Problem as OT

Lemma: With c(x, y) = − log(1 − hx|yi), and ψ_i := log(κ_i), PI^κ_i = Vor^ψ_c (y_i).

PI^κ₃

o

P3(κ3)

Wang ’04

Glimm-Oliker ’03 [Machado, M., Thibert, Symp. Comp. Geom 2014]

I (FF) is a semi-discrete OT problem with c(x, y) = − log(1 − hx|yi)

(FF) Find κ such that ∀i, µ(PI^κ_i ) = ν_i.

⇐⇒

(OT) Find ψ such that µ(Vor^ψ_c (y_i)) = ν_i Corollary:

Computation of the generalized Voronoi cells

Lemma: With ψ_i = log(κ_i), p_i := −_2κ^yi

i and ω_i := −k_2κ^yi

i k² − _κ¹

i , PI^κ_i = Vor^ψ_c (y_i) = Vor^ω_P,k.k²(p_i) ∩ S²

[Machado, M., Thibert, Symp. Comp. Geom 2014]

Computation of the generalized Voronoi cells

Lemma: With ψ_i = log(κ_i), p_i := −_2κ^yi

i and ω_i := −k_2κ^yi

i k² − _κ¹

i , PI^κ_i = Vor^ψ_c (y_i) = Vor^ω_P,k.k²(p_i) ∩ S²

I Computation combines Regular triangulation 3 from

[Machado, M., Thibert, Symp. Comp. Geom 2014]

CGAL with custom filtered predicates and data structures.

Computation of the generalized Voronoi cells

Lemma: With ψ_i = log(κ_i), p_i := −_2κ^yi

i and ω_i := −k_2κ^yi

i k² − _κ¹

i , PI^κ_i = Vor^ψ_c (y_i) = Vor^ω_P,k.k²(p_i) ∩ S²

I Computation combines Regular triangulation 3 from

[Machado, M., Thibert, Symp. Comp. Geom 2014]

I Applies to other geometric constructions involving confocal CGAL with custom filtered predicates and data structures.

quadrics of revolution appearing in geometric optics.