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 { X 1 = cos( θ ) ∂ x + sin( θ ) ∂ y , X 2 = ∂ θ } For u : SE(2) → R horizontal gradient: ∇ h u = ( X 1 u , X 2 u ) horizontal divergence: div h ν = X 1 ν 1 + X 2 ν 2 ( X 1 u , X 2 u ) ∇ h u √ horizontal unit normal: ν h = |∇ h u | = ( X 1 u ) 2 +( X 2 u ) 2 horizontal Laplacian: ∆ h u = X 2 1 u + X 2 2 u � � ∇ h u horizontal mean curvature: K h = div h ( ν h ) = div h |∇ h u |

Preliminaries: SE(2) sub-Riemannian geometry X 3 = − sin( θ ) ∂ x + cos( θ ) ∂ y Elements: ( x , y , θ ) ∈ SE(2) Horizontal plane: span { X 1 = cos( θ ) ∂ x + sin( θ ) ∂ y , X 2 = ∂ θ } For u : SE(2) → R full gradient: ∇ u = ( X 1 u , X 2 u , X 3 u ) full divergence: div ν = X 1 ν 1 + X 2 ν 2 + X 3 ν 3 ( X 1 u , X 2 u , X 3 u ) ∇ u √ full unit normal: ν = |∇ u | = ( X 1 u ) 2 +( X 2 u ) 2 +( X 3 u ) 2 full Laplacian: ∆ u = X 2 1 u + X 2 2 u + X 2 3 u � � ∇ u full mean curvature: K = div( ν ) = div |∇ u | Degenerate!

Preliminaries: SE(2) sub-Riemannian geometry X 3 = − sin( θ ) ∂ x + cos( θ ) ∂ y Elements: ( x , y , θ ) ∈ SE(2) Horizontal plane: span { X 1 = cos( θ ) ∂ x + sin( θ ) ∂ y , X 2 = ∂ θ } For u : SE(2) → R full gradient: ∇ u = ( X 1 u , X 2 u , X 3 u ) full divergence: div ν = X 1 ν 1 + X 2 ν 2 + X 3 ν 3 ( X 1 u , X 2 u , X 3 u ) ∇ u √ full unit normal: ν = |∇ u | = ( X 1 u ) 2 +( X 2 u ) 2 +( X 3 u ) 2 full Laplacian: ∆ u = X 2 1 u + X 2 2 u + X 2 3 u � � ∇ u full mean curvature: K = div( ν ) = div |∇ u | Degenerate! Non-commutative Lie algebra: [ X 1 , X 2 ] = − X 3 = sin( θ ) ∂ x − cos( θ ) ∂ y Challenging but satisfies H¨ ormander condition!

Preliminaries: Sub-Riemannian mean curvature flow  2 � � δ ij − X i uX j u u t = �  X ij u in SE(2) × (0 , ∞ )  |∇ h u | 2 i , j =1  u = u 0 on SE(2) × { 0 }  ( X 1 u ) 2 + ( X 2 u ) 2 = 0 � Characteristic points: |∇ h u | = Global description BUT... Not defined when ∇ h u = 0! Requires regularization

Preliminaries: Vanishing viscosity Regularized equation Degenerate equation   2 2 � � � � X i u ǫ X j u ǫ δ ij − X i uX j u u ǫ X ij u ǫ t = � δ ij − �   u t = X ij u   ǫ 2 + |∇ h u ǫ | 2 |∇ h u | 2 i , j =1 i , j =1 u ǫ ( ., 0) = u 0 ( . )   u ( ., 0) = u 0 ( . )   No characteristic points! Not defined when ∇ h u = 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 � � � ( u ǫ 1 − u ǫ 2 )( ξ, t ) 1. sup attainable? � � � ξ ∈ SE(2) , 0 ≤ t ≤ T 2. Argue by contradiction: For all M ≥ 0 , there exist ǫ 1 ( M ) and ǫ 2 ( M ) s.t. � � � ( u ǫ 1 − u ǫ 2 )( ξ, t ) � ≥ M ( ǫ 1 − ǫ 2 ) α , sup � � ξ ∈ SE(2) , 0 ≤ t ≤ T employing ω ( ξ, η, t ) = u ǫ 1 ( ξ, t ) − u ǫ 2 ( η, t ) − φ ( ξ, η, t ) , with penalization φ ( ξ, η, t ) = µ 0 + M γ ( ǫ 1 − ǫ 2 ) 1 − γ 2 | ξ − η | γ 2 T ( ǫ 1 − ǫ 2 ) α t .

Remarks on ω and φ Contradictory hypothesis � � � ( u ǫ 1 − u ǫ 2 )( ξ, t ) � ≥ M ( ǫ 1 − ǫ 2 ) α sup � � ξ ∈ SE(2) , 0 ≤ t ≤ T Test function and penalization ω ( ξ, η, t ) = u ǫ 1 ( ξ, t ) − u ǫ 2 ( η, t ) − φ ( ξ, η, t ) φ ( ξ, η, t ) = µ γ ( ǫ 1 − ǫ 2 ) 1 − γ 2 | ξ − η | γ 0 + M 2 T ( ǫ 1 − ǫ 2 ) α t 1 Parameters doubled: Derivatives of | ξ − η | γ 0 2 Penalization with large γ : | ξ − η | 0 → 0 3 Attainability of sup ω : | ξ | �→ ∞ or | η | �→ ∞ 4 Opposite derivatives: D ξ φ = − D η φ 5 Estimates on u ǫ 1 and u ǫ 2 derivatives at (ˆ η, ˆ ξ, ˆ t ) where sup ω = ω (ˆ η, ˆ ξ, ˆ t )

Conclusions from uniqueness � � � ( u ǫ 1 − u ǫ 2 )( ξ, t ) � ≤ M ( ǫ 1 − ǫ 2 ) α sup � � ξ ∈ SE(2) , 0 ≤ t ≤ T 1 � � � ( u ǫ 1 − u ǫ 2 )( ξ, t ) � ≤ M ( ǫ 1 − ǫ 2 ) α sup � � ξ ∈ SE(2) , 0 ≤ t ≤ T = � � � ( u ǫ 1 − u )( ξ, t ) � ≤ M ( ǫ 1 ) α sup as ǫ 2 → 0 � � ξ ∈ SE(2) , 0 ≤ t ≤ T = ⇒ u ǫ 1 → u as ǫ 1 → 0 2 Not dependent on u 0 3 Dependence only on Γ 0 = { ξ ∈ SE(2) | u 0 ( ξ ) = 0 }

PART II A new diffusion driven motion

Bence-Merriman-Osher algorithm � in R n × (0 , ∞ ) u t − ∆ u = 0 u = χ C 0 in C 0 × { t = 0 }

Citti-Sarti diffusion driven motion

A new Euclidean diffusion driven motion = x 0 + tv ν ∈ R n � � x = x 1 , . . . , x n − 1 , x n ( x 1 , . . . , x n − 1 ) New surface definition ∂ C t ≡ { x ∈ R n | �∇ u ( x , t ) , r � = 0 } Gradient along unit normal r = ν = ⇒ v ≈ K as t → 0 Gradient along fixed direction r � r , e n � v ≈ � ν, e n �� ν, r � K as t → 0

A new sub-Riemannian diffusion driven motion � � ξ = x , y , θ ( x , y ) = ξ 0 + tv ν h ∈ SE(2) , X 2 = ∂ θ New surface definition � �∇ h u ( x , t ) , r � = 0 } � ∂ C t ≡ { x ∈ SE(2) Gradient along unit normal r = ν h = ⇒ v ≈ K h as t → 0 Gradient along fixed direction r � r , X 2 � v ≈ � ν, X 2 �� ν, r � K h 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!