A Level-Set Framework for Shape OptimisationInternational Research rainingT Group IGDK 1754 A Level-Set Framework for Shape Optimisation Daniel Kraft November 30th, 2015 Daniel KraftA

International Research Training Group IGDK 1754

A Level-Set Framework for Shape Optimisation

Daniel Kraft

November 30th, 2015

Daniel Kraft A Level-Set Framework for Shape Optimisation November 30th, 2015 1


Overview

1. The Level-Set MethodLevel Sets and the Speed MethodThe Hopf-Lax Formula

2. Gradient-Descent MethodsShape CalculusSteepest-Descent Directions

3. Numerical ResultsImage SegmentationPDE-Constrained Shape Optimisation



The Level-Set Method



The Level-Set Function

Geometries as level sets of φ : D ⊂ Rn → R:

Ω = Ω(φ) = x ∈ D | φ(x) < 0Γ = ∂Ω = x ∈ D | φ(x) = 0

Also irregular shapes are possible:





Ω = Ω(φ) = x ∈ D | φ(x) < 0Γ = ∂Ω = x ∈ D | φ(x) = 0


φ(x , y) =√x2 + y2 − 1 or φ(x , y) = x2 + y2 − 1

D Γ

Ω





Ω = Ω(φ) = x ∈ D | φ(x) < 0Γ = ∂Ω = x ∈ D | φ(x) = 0


φ(x , y) = max (|x | − 1, |y | − 1)

D Γ

Ω





Ω = Ω(φ) = x ∈ D | φ(x) < 0Γ = ∂Ω = x ∈ D | φ(x) = 0


φ(x , y) = min(√

(x − 2)2 + y2 − 1,√(x + 2)2 + y2 − 2

)− c

D D D

c = 0 c = 12 c = 1



The Speed Method in Normal Direction

Speed eld F : D → R in normal direction:

a

b

c

Level-Set Equation

φt(x , t) + F (x) |∇φ(x , t)| = 0, φ0(x , 0) = φ0(x)

It has a unique viscosity solution, see Crandall, Ishii, Lions [3] and Giga [5].Original work by Osher, Sethian [6].



The Speed Method in Normal Direction

Speed eld F : D → R in normal direction:

a

b

c

Level-Set Equation

φt(x , t) + F (x) |∇φ(x , t)| = 0, φ0(x , 0) = φ0(x)

It has a unique viscosity solution, see Crandall, Ishii, Lions [3] and Giga [5].Original work by Osher, Sethian [6].



Changes in Topology Are Possible

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Changes in Topology Are Possible

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The Sign of F

Important is the sign of F :

Theorem

Let F = F+ − F− be the decomposition in positive and negative parts, and

φ±t (x , t) + F±(x)∣∣∇φ±(x , t)∣∣ = 0, φ±(x , 0) = ±φ0(x).

Then:

φ(x , t) =

φ+(x , t) F (x) > 0φ0(x) F (x) = 0

−φ−(x , t) F (x) < 0

F (x) = 0 means φ(x , ·) is constant: → Geometric constraints!

One can reduce all considerations to the case F ≥ 0.



The Sign of F

Important is the sign of F :

Theorem

Let F = F+ − F− be the decomposition in positive and negative parts, and

φ±t (x , t) + F±(x)∣∣∇φ±(x , t)∣∣ = 0, φ±(x , 0) = ±φ0(x).

Then:

φ(x , t) =

φ+(x , t) F (x) > 0φ0(x) F (x) = 0

−φ−(x , t) F (x) < 0

F (x) = 0 means φ(x , ·) is constant: → Geometric constraints!

One can reduce all considerations to the case F ≥ 0.



Mayer's Problem

Paths suited to F ≥ 0:

St(x) =ξ ∈W 1,∞([0, t]) | ξ(0) = x , |ξ′(τ)| ≤ F (ξ(τ)) for all τ ∈ [0, t]

Reachable set:

Rt(x) = ξ(t) | ξ ∈ St(x)

Mayer's Problem

V (x , t) = infξ∈St (x)

φ0(ξ(t)) = infy∈Rt (x)

φ0(y)

Hamilton-Jacobi-Bellman: The level-set equation!



Mayer's Problem

Paths suited to F ≥ 0:

St(x) =ξ ∈W 1,∞([0, t]) | ξ(0) = x , |ξ′(τ)| ≤ F (ξ(τ)) for all τ ∈ [0, t]

Reachable set:

Rt(x) = ξ(t) | ξ ∈ St(x)

Mayer's Problem

V (x , t) = infξ∈St (x)

φ0(ξ(t)) = infy∈Rt (x)

φ0(y)

Hamilton-Jacobi-Bellman: The level-set equation!



The Hopf-Lax Formula

Optimal-control theory implies a Hopf-Lax Formula:

Let d solve the Eikonal equation for the speed F :

F (x) |∇d(x)| = 1

d is the F -induced distance.

Theorem (Hopf-Lax Formula)

Let F ≥ 0 be Lipschitz continuous and have compact support in D. For all x

with F (x) > 0 and φ0(x) > 0, the solution of the level-set equation is given as:

φ(x , t) = inf φ0(y) | d(x , y) ≤ t

See also Falcone, Giorgi, Loreti [4] and Capuzzo-Dolcetta [2].



The Hopf-Lax Formula

Optimal-control theory implies a Hopf-Lax Formula:

Let d solve the Eikonal equation for the speed F :

F (x) |∇d(x)| = 1

d is the F -induced distance.

Theorem (Hopf-Lax Formula)

Let F ≥ 0 be Lipschitz continuous and have compact support in D. For all x

with F (x) > 0 and φ0(x) > 0, the solution of the level-set equation is given as:

φ(x , t) = inf φ0(y) | d(x , y) ≤ t

See also Falcone, Giorgi, Loreti [4] and Capuzzo-Dolcetta [2].



Representation of the Level-Set Domain

Distance to Initial Geometry

d0(x) = infy∈Ω0

d(x , y)

→ When does the advancing front hit x?

Ecient computation possible with Fast Marching (Sethian [7]).

Theorem (Representation Formula)

Let F ≥ 0. The time evolution of Ω0 is given by:

Ωt = x ∈ D | d0(x) < tΓt = x ∈ D | d0(x) = t

Generalisation is possible for arbitrary signs of F :→ Composite Fast Marching



Representation of the Level-Set Domain

Distance to Initial Geometry

d0(x) = infy∈Ω0

d(x , y)

→ When does the advancing front hit x?

Ecient computation possible with Fast Marching (Sethian [7]).

Theorem (Representation Formula)

Let F ≥ 0. The time evolution of Ω0 is given by:

Ωt = x ∈ D | d0(x) < tΓt = x ∈ D | d0(x) = t

Generalisation is possible for arbitrary signs of F :→ Composite Fast Marching



Demonstration

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Non-Fattening

Classical result for Γt = ∂Ωt : Non-fattening in a topological sense, consideringthe space-time. (See Barles, Soner, Souganidis [1].)

Based on our formula, one can easily deduce:

Theorem (Topological Non-Fattening)

Let F ≥ 0 and Ω0 = Ω0 ∪ Γ0. Then Ωt = Ωt ∪ Γt for all t ≥ 0.If F ≤ 0 and (Ω0 ∪ Γ0) = Ω0, then (Ωt ∪ Γt) = Ωt for all t ≥ 0.

Theorem (Measure-Theoretic Non-Fattening)

Let |Γ0| = 0. Then |Γt | = 0 for all t ≥ 0.



Non-Fattening









Non-Fattening









Gradient-Descent Methods



Shape Calculus

Theorem

Let f ∈ L1loc(D) and assume F ≥ 0. Then:

J(t) =

∫Ωt

f dx =

∫Ω0

f dx +

∫ t

0

∫Γs

Ff dσ ds

This yields the shape derivative of J:

dJ (Ωt ;F ) = J ′(t) =

∫Γt

Ff dσ

(In a weak sense, since J is absolutely continuous.)



Shape Calculus

Theorem

Let f ∈ L1loc(D) and assume F ≥ 0. Then:

J(t) =

∫Ωt

f dx =

∫Ω0

f dx +

∫ t

0

∫Γs

Ff dσ ds

This yields the shape derivative of J:

dJ (Ωt ;F ) = J ′(t) =

∫Γt

Ff dσ

(In a weak sense, since J is absolutely continuous.)



Total Shape Dierential

For a shape-dependent integrand:

J(Ω) =

∫Ω

f (x , Ω) dx ,

where for all xed x ∈ D:

f (x , Ωt) = f (x , Ω0) +

∫ t

0

f ′(x , Ωs) ds.

Theorem (Total Shape Dierential)

Denote J(t) = J(Ωt). Then J is absolutely continuous and

dJ (Ωt ;F ) = J ′(t) =

∫Γt

Ff dσ +

∫Ωt

f ′ dx .



Total Shape Dierential

For a shape-dependent integrand:

J(Ω) =

∫Ω

f (x , Ω) dx ,

where for all xed x ∈ D:

f (x , Ωt) = f (x , Ω0) +

∫ t

0

f ′(x , Ωs) ds.

Theorem (Total Shape Dierential)

Denote J(t) = J(Ωt). Then J is absolutely continuous and

dJ (Ωt ;F ) = J ′(t) =

∫Γt

Ff dσ +

∫Ωt

f ′ dx .



A Chain Rule

Scalar, shape-dependent quantity:

J(Ω) =

∫Ω

f (x ,G (Ω)) dx

(Multiple G 's are also possible.)

Theorem (Chain Rule)

Let f ∈ C 1, then t 7→ J(t) = J(Ωt) is absolutely continuous and

dJ (Ωt ;F ) = J ′(t) =

∫Γt

Ff dσ +

∫Ωt

∂f

∂GG ′ dx .



A Chain Rule

Scalar, shape-dependent quantity:

J(Ω) =

∫Ω

f (x ,G (Ω)) dx

(Multiple G 's are also possible.)

Theorem (Chain Rule)

Let f ∈ C 1, then t 7→ J(t) = J(Ωt) is absolutely continuous and

dJ (Ωt ;F ) = J ′(t) =

∫Γt

Ff dσ +

∫Ωt

∂f

∂GG ′ dx .



Gradients in H1

Generic form of the shape derivative:

dJ (Ω;F ) =

∫Γ

f (. . .)F dσ

→ linear functional in H1(D), operating on F

dJ (Ω; ·) lives on the boundary Γ (Hadamard-Zolésio structure theorem)

H1 Shape Gradient

Riesz representative F ∈ H1(D) of dJ (Ω; ·) as gradient:

∀G ∈ H1(D) : 〈F ,G 〉

β

=

∫D

(FG +

β

〈∇F ,∇G 〉) dx = dJ (Ω;G )

→ the solution denes a speed eld on the domain D



Gradients in H1


dJ (Ω;F ) =

∫Γ

f (. . .)F dσ



H1 Shape Gradient


∀G ∈ H1(D) : 〈F ,G 〉

β

=

∫D

(FG +

β

〈∇F ,∇G 〉) dx = dJ (Ω;G )




Gradients in H1


dJ (Ω;F ) =

∫Γ

f (. . .)F dσ



H1 Shape Gradient


∀G ∈ H1(D) : 〈F ,G 〉β =

∫D

(FG + β 〈∇F ,∇G 〉) dx = dJ (Ω;G )




Gradient-Descent Method

One step in the gradient descent:

1. evaluate shape-dependent quantities for Ω

2. calculate functional dJ (Ω; ·)3. nd Riesz representative: gradient F

4. evolve Ω with a line search in direction −F

Repeat until no more changes are made.



Numerical Results



Image Segmentation

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Image Segmentation

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Cost and Shape Derivative

Consider D ⊂ Rn, e. g., D = [0, 1]2.Let u : D → R be a grey-scale image.

Denition (Cost Function)

J(Ω) =

∫Ω

(u − u)2 dx − 2γ · σ · |Ω| for ∅ 6= Ω ⊂ D

Shape Derivative

dJ (Ω;F ) =

∫Γ

((u − u)2

(1−

γ

σ

)− γσ

)F dσ






J(Ω) =

∫Ω

(u − u)2 dx − 2γ · σ · |Ω| for ∅ 6= Ω ⊂ D

Shape Derivative

dJ (Ω;F ) =

∫Γ

((u − u)2

(1−

γ

σ

)− γσ

)F dσ






J(Ω) =

∫Ω

(u − u)2 dx − 2γ · σ · |Ω| for ∅ 6= Ω ⊂ D

Shape Derivative

dJ (Ω;F ) =

∫Γ

((u − u)2

(1−

γ

σ

)− γσ

)F dσ



Eect of β

0 0.2 0.4 0.6 0.8 1

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Eect of β

β = 1

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Eect of β

β = 10−2

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-1

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Descent Run

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Descent Run

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Descent Run

0 20 40 60 80 1001e-7

1e-6

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

1e+1

Steps

Cost

Gradient Norm

Step Length



PDE-Constrained Shape Optimisation

Let D ⊂ R2 be compact, B ⊂ D, f ∈ L2(D) and ud ∈ L2(B).


Find Ω with B ⊂ Ω ⊂ D that minimises

J(Ω) =1

2‖u − ud‖2L2(B) + α |Γ | .

State Equation

u ∈ H1(Ω) solves the state equation:−∆u + u = f in Ω

∂u∂ν = 0 on Γ



PDE-Constrained Shape Optimisation

Let D ⊂ R2 be compact, B ⊂ D, f ∈ L2(D) and ud ∈ L2(B).


Find Ω with B ⊂ Ω ⊂ D that minimises

J(Ω) =1

2‖u − ud‖2L2(B) + α |Γ | .

State Equation

u ∈ H1(Ω) solves the state equation:−∆u + u = f in Ω

∂u∂ν = 0 on Γ



The Shape Derivative

Equations in Weak Form

Find u and p such that (for each v ∈ H1(D)):∫Ω

(〈∇u,∇v〉+ uv) dx =

∫Ω

fv dx∫Ω

(〈∇p,∇v〉+ pv) dx =

∫B

(u − ud)v dx

Shape Derivative

dJ (Ω;F ) =

∫Γ

(fp − 〈∇u,∇p〉 − up + ακ)F dσ

We require: F = 0 on B



The Shape Derivative

Equations in Weak Form

Find u and p such that (for each v ∈ H1(D)):∫Ω

(〈∇u,∇v〉+ uv) dx =

∫Ω

fv dx∫Ω

(〈∇p,∇v〉+ pv) dx =

∫B

(u − ud)v dx

Shape Derivative

dJ (Ω;F ) =

∫Γ

(fp − 〈∇u,∇p〉 − up + ακ)F dσ

We require: F = 0 on B



Descent Run

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Descent Run

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-1.5

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1.5



Descent Run

0 100 200 300 400 500 600 7001e-5

1e-4

1e-3

1e-2

1e-1

1e+0

1e+1

1e+2

Steps

Cost

Gradient Norm

Step Length



Conclusion

I Level sets allow a exible description of shapes.

I A Hopf-Lax formula can be employed for the time evolution.

I This yields a special shape calculus.

I One can formulate gradient methods for shape optimisation.

Thanks for your attention!



Conclusion

I Level sets allow a exible description of shapes.

I A Hopf-Lax formula can be employed for the time evolution.

I This yields a special shape calculus.

I One can formulate gradient methods for shape optimisation.

Thanks for your attention!



References I

I G. Barles, H. M. Soner, and P. E. Souganidis.Front Propagation and Phase Field Theory.SIAM Journal on Control and Optimization, 31(2):439469, March 1993.

I Italo Capuzzo-Dolcetta.A Generalized Hopf-Lax Formula: Analytical and Approximations Aspects.In Fabio Ancona, editor, Geometric Control and Nonsmooth Analysis,volume 76 of Series on Advances in Mathematics for Applied Sciences, pages136150. World Scientic, 2008.

I Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions.User's Guide to Viscosity Solutions of Second Order Partial DierentialEquations.Bulletin of the American Mathematical Society, 27(1):167, July 1992.



References II

I M. Falcone, T. Giorgi, and P. Loreti.Level Sets of Viscosity Solutions: Some Applications to Fronts andRendez-Vous Problems.SIAM Journal on Applied Mathematics, 54(5):13351354, 1994.

I Yoshikazu Giga.Surface Evolution Equations: A Level Set Approach, volume 99 ofMonographs in Mathematics.Birkhäuser, 2006.

I Stanley J. Osher and James A. Sethian.Fronts Propagating with Curvature-Dependent Speed: Algorithms Based onHamilton-Jacobi Formulations.Journal of Computational Physics, 79:1249, 1988.



References III

I James A. Sethian.A Fast Marching Level Set Method for Monotonically Advancing Fronts.Proceedings of the National Academy of Sciences, 93(4):15911595, 1996.


A Level-Set Framework for Shape OptimisationInternational Research rainingT Group IGDK 1754 A Level-Set Framework for Shape Optimisation Daniel Kraft November 30th, 2015 Daniel KraftA

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