Surface reconstruction via mean curvature flow Emre Baspinar - - PowerPoint PPT Presentation

Surface reconstruction via mean curvature flow

Emre Baspinar

supervised by prof. dr. Giovanna Citti

Department of Mathematics, University of Bologna

December 6, 2015

Physical motivation

Outline

Part I Preliminaries

Sub-Riemannian geometry Vanishing viscosity

Literature on uniqueness Uniqueness in sub-Riemannian mean curvature flow Part II Bence-Merriman-Osher algorithm Citti-Sarti diffusion driven motion A new diffusion driven motion

in Euclidean setting in the sub-Riemannian setting

Summary and future work

PART I Uniqueness in the sub-Riemannian mean curvature flow

Preliminaries: SE(2) sub-Riemannian geometry

Elements: (x, y, θ) ∈ SE(2) Horizontal plane: span{X1 = cos(θ)∂x + sin(θ)∂y, X2 = ∂θ} For u : SE(2) → R

horizontal gradient: ∇hu = (X1u, X2u) horizontal divergence: divh ν = X1ν1 + X2ν2 horizontal unit normal: νh =

∇hu |∇hu| = (X1u, X2u)

√

(X1u)2+(X2u)2

horizontal Laplacian: ∆hu = X 2

1 u + X 2 2 u

horizontal mean curvature: Kh = divh(νh) = divh

• ∇hu

|∇hu|

Preliminaries: SE(2) sub-Riemannian geometry

X3 = − sin(θ)∂x + cos(θ)∂y Elements: (x, y, θ) ∈ SE(2) Horizontal plane: span{X1 = cos(θ)∂x + sin(θ)∂y, X2 = ∂θ} For u : SE(2) → R

full gradient: ∇u = (X1u, X2u, X3u) full divergence: div ν = X1ν1 + X2ν2+X3ν3 full unit normal: ν =

∇u |∇u| = (X1u, X2u, X3u)

√

(X1u)2+(X2u)2+(X3u)2

full Laplacian: ∆u = X 2

1 u + X 2 2 u+X 2 3 u

full mean curvature: K = div(ν) = div

• ∇u

|∇u|

• Degenerate!

Preliminaries: SE(2) sub-Riemannian geometry

X3 = − sin(θ)∂x + cos(θ)∂y Elements: (x, y, θ) ∈ SE(2) Horizontal plane: span{X1 = cos(θ)∂x + sin(θ)∂y, X2 = ∂θ} For u : SE(2) → R

full gradient: ∇u = (X1u, X2u, X3u) full divergence: div ν = X1ν1 + X2ν2+X3ν3 full unit normal: ν =

∇u |∇u| = (X1u, X2u, X3u)

√

(X1u)2+(X2u)2+(X3u)2

full Laplacian: ∆u = X 2

1 u + X 2 2 u+X 2 3 u

full mean curvature: K = div(ν) = div

• ∇u

|∇u|

• Degenerate!

Non-commutative Lie algebra: [X1, X2] = −X3 = sin(θ)∂x − cos(θ)∂y Challenging but satisfies H¨

• rmander condition!

Preliminaries: Sub-Riemannian mean curvature flow

     ut =

2

• i,j=1
• δij − XiuXju

|∇hu|2

• Xiju

in SE(2) × (0, ∞)

u = u0

• n SE(2) × {0}

Characteristic points: |∇hu| =

• (X1u)2 + (X2u)2 = 0

Global description BUT... Not defined when ∇hu = 0! Requires regularization

Preliminaries: Vanishing viscosity

Regularized equation      uǫ

t = 2

• i,j=1
• δij −

XiuǫXjuǫ ǫ2+|∇huǫ|2

• Xijuǫ

uǫ(., 0) = u0(.) No characteristic points! Degenerate equation      ut =

2

• i,j=1
• δij − XiuXju

|∇hu|2

• Xiju

u(., 0) = u0(.) Not defined when ∇hu = 0!

Literature on uniqueness

Euclidean, Evans-Spruck and Chen-Giga-Goto Euclidean, Deckelnick Heisenberg group, existence of graph, Capogna-Citti Heisenberg group, axisymmetricity, Ferrari-Liu-Manfredi Problematic with characteristic points! What about general setting?

Uniqueness in vanishing viscosity sense

1. sup

ξ∈SE(2),0≤t≤T

• (uǫ1 − uǫ2)(ξ, t)
• attainable?
• 2. Argue by contradiction:

For all M ≥ 0, there exist ǫ1(M) and ǫ2(M) s.t. sup

ξ∈SE(2),0≤t≤T

• (uǫ1 − uǫ2)(ξ, t)
• ≥ M(ǫ1 − ǫ2)α,

employing ω(ξ, η, t) = uǫ1(ξ, t) − uǫ2(η, t) − φ(ξ, η, t), with penalization φ(ξ, η, t) = µ γ (ǫ1 − ǫ2)1− γ

2 |ξ − η|γ

0 + M

2T (ǫ1 − ǫ2)αt.

Remarks on ω and φ

Contradictory hypothesis sup

ξ∈SE(2),0≤t≤T

• (uǫ1 − uǫ2)(ξ, t)
• ≥ M(ǫ1 − ǫ2)α

Test function and penalization ω(ξ, η, t) = uǫ1(ξ, t) − uǫ2(η, t) − φ(ξ, η, t) φ(ξ, η, t) = µ

γ (ǫ1 − ǫ2)1− γ

2 |ξ − η|γ

0 + M 2T (ǫ1 − ǫ2)αt

1 Parameters doubled: Derivatives of |ξ − η|γ 2 Penalization with large γ: |ξ − η|0 → 0 3 Attainability of sup ω: |ξ| → ∞ or |η| → ∞ 4 Opposite derivatives: Dξφ = −Dηφ 5 Estimates on uǫ1 and uǫ2 derivatives at (ˆ

ξ, ˆ η, ˆ t) where sup ω = ω(ˆ ξ, ˆ η, ˆ t)

Conclusions from uniqueness

sup

ξ∈SE(2),0≤t≤T

• (uǫ1 − uǫ2)(ξ, t)
• ≤ M(ǫ1 − ǫ2)α

1

sup

ξ∈SE(2),0≤t≤T

• (uǫ1 − uǫ2)(ξ, t)
• ≤ M(ǫ1 − ǫ2)α

= sup

ξ∈SE(2),0≤t≤T

• (uǫ1 − u)(ξ, t)
• ≤ M(ǫ1)α

as ǫ2 → 0 = ⇒ uǫ1 → u as ǫ1 → 0

2 Not dependent on u0 3 Dependence only on Γ0 = {ξ ∈ SE(2) | u0(ξ) = 0}

PART II A new diffusion driven motion

Bence-Merriman-Osher algorithm

• ut − ∆u = 0

in Rn × (0, ∞) u = χC0 in C0 × {t = 0}

Citti-Sarti diffusion driven motion

A new Euclidean diffusion driven motion

x =

• x1, . . . , xn−1, xn(x1, . . . , xn−1)
• = x0 + tvν ∈ Rn

New surface definition ∂Ct ≡ {x ∈ Rn | ∇u(x, t), r = 0} Gradient along unit normal r = ν = ⇒ v ≈ K as t → 0 Gradient along fixed direction r v ≈ r, en ν, enν, rK as t → 0

A new sub-Riemannian diffusion driven motion

ξ =

• x, y, θ(x, y)
• = ξ0 + tvνh ∈ SE(2),

X2 = ∂θ New surface definition ∂Ct ≡ {x ∈ SE(2)

• ∇hu(x, t), r = 0}

Gradient along unit normal r = νh = ⇒ v ≈ Kh as t → 0 Gradient along fixed direction r v ≈ r, X2 ν, X2ν, rKh as t → 0

Summary and future work

Summary and future work

Main findings Uniqueness of vanishing viscosity solutions A new diffusion driven motion Future work Implementation of sub-Riemannian mean curvature flow Extension to other Lie groups

Thank you!