Reconstructing thin shapes by a level set technique presented by: - - PowerPoint PPT Presentation

UNIVERSITY OF MANCHESTER

School of Mathematics

Reconstructing thin shapes by a level set technique presented by: Oliver Dorn joint with:

• D. Alvarez, N. Irishina, M. Moscoso

Universidad Carlos III de Madrid

Minisymposium on ’New Developments in Geometric Inverse Problems (1)’ Conference AIP 2009, Vienna, July 20-24, 2009.

OUTLINE New application for level sets: crack detection Classical shape evolution Representing thin shapes by two level set functions Deforming thin shapes with level sets The numerical algorithm Numerical experiments Summary and future work Joint work with Diego ´ Alvarez and Miguel Moscoso, UC3M.

I

CLASSICAL SHAPE EVOLUTION

plane z=0 x y z shape S F(S) F(S) shape S F(S) level set function f=f(S)

II

CLASSICAL SHAPE EVOLUTION

plane z=0 x y z shape S δS S+δS + level set function f(S+δS) = f(S)+ δ f (F(S))

III

CLASSICAL SHAPE EVOLUTION

source Ω domain receiver receiver F(S) F(S)=F(u(S),z(S)) shape S S +δ

IV

CLASSICAL SHAPE EVOLUTION Level set approach: In order to evolve the level set function f such that the zero level set follows the flow F , we need to numerically solve the following Hamilton-Jacobi type equation (Osher and Sethian, 1988): ∂f ∂t + F |∇f| = 0.

V

CRACK-TYPE STRUCTURES

Real

5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50

VI

PROBLEMS OF CLASSICAL LEVEL SET TECHNIQUES Classical level set techniques describe ’volumetric’ shapes Cracks are ’thin’ shapes, ideally without volume Cracks are not ’closed’ Cracks might be ’broken’, consisting of several pieces ’Crack evolution’: Propagating cracks vs. changing topology

• f cracks

Idea: use two level set functions for describing cracks (thin shapes)!

VII

First level set function ϕ

Ω Γ ϕ

ϕ

n(x)

VIII

Second level set function ψ

• Ω

ψ Γϕ B S

IX

THE PHYSICAL PROBLEM The electric potential uj satisfies ∇ · b(x)∇uj = 0 in Ω, (1) and the boundary condition uj = γj

• n

∂Ω. (2) The corresponding physical measurements are Aj(b) = gj = ∂uj ∂n (dl) (3) taken at positions dl ∈ ∂Ω for l = 1, . . . , l. Residual operator Rj(b) = Aj(b) − ˜ gj. (4)

X

CLASSICAL GRADIENT DIRECTION Least squares cost functional: Jj(b) = 1 2Rj(b)2

Z

(5) Classical gradient direction gradbJj(x) = R′

j(b)∗Rj(b)

(6) Adjoint formulation: gradbJj(x) = ∇uj · ∇zj, (7) Adjoint equation: uj solves (1)-(2) and zj solves the following adjoint equation ∇ · b∇zj = 0 in Ω, (8) bzj = Rj(b)

• n ∂Ω.

(9)

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REPRESENTING THIN SHAPES (CRACKS)

thin region

1

Γ2 be b i be R

2 \ D

R

2 \ D

ε n ε exterior D exterior D Γ

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REPRESENTING THIN SHAPES (CRACKS) First level set function ϕ(x, y): Ω1 = {(x, y) ∈ I R2 : ϕ(x, y) ≤ 0}, (10) Γ1 = {(x, y) ∈ I R2 : ϕ(x, y) = 0}. (11) Γ1 is the zero level set. The outward normal n to Γ1 is n(x) = ∇ϕ(x) |∇ϕ(x)|

• n

Γ1. (12) Thin region D of width > 0 : D = Ω1 ∩ {y ∈ I R2 : there exist x ∈ Γ1 such that y = x − αn(x), for some 0 ≤ α ≤ }. (13)

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REPRESENTING THIN SHAPES (CRACKS)

y

1

Γ

2

D S S ψ>0 ψ<0 ψ>0 ψ<0 ψ>0 ϕ>0 ϕ<0 ϕ=0 B B x Γ

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REPRESENTING THIN SHAPES (CRACKS) Second level set function ψ(x) defines pieces of the crack: B = {(x, y) ∈ I R2 : ψ(x, y) < 0 }. (14) The thin shapes S are given as S = (D ∩ B) ∩ Ω. (15) The conductivity distribution in Ω is given as b(x) =

          

bi for x ∈ S[ϕ, ψ], be for x ∈ Ω\S[ϕ, ψ]. (16)

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SHAPE EVOLUTION (CRACKS)

D’ e

b = be

b = bi

Γ1’ Γ 2’ Γ1 Γ 2 x1 = x−εn(x) (x) ζ (x) ζ x ε D

b = b

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Propagating the first level set function

S+ζ S+ζ S S

+ − − +

b −>b b −>b

e i e i

δϕJj ≈

• S+ (bi − be)gradbJj(x)ζ(x)n(x)ds(x) −

(17)

• S− (bi − be)gradbJj(x)ζ(x)n(x)ds(x) .

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Propagating the first level set function Descent direction for first level set function: Fϕ(x) = −(bi−be)

• gradbJj(x) − gradbJj(x − n(x))
• χB(x)

with x = (x, y). Hamilton-Jacobi type formulation of shape evolution: ∂ϕ ∂t + Fϕ|∇ϕ| = 0 (18)

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Propagating the second level set function (example)

y

1

Γ

2

D S S ψ>0 ψ<0 ψ>0 ψ<0 ψ>0 ϕ>0 ϕ<0 ϕ=0 B B x Γ

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Propagating the second level set function (example) Descent direction for second level set function: Fψ(x) = −(bi − be)

gradbJj(x)δΓB(x)χD(x) dy

with x = (x, y). Hamilton-Jacobi type formulation of shape evolution: ∂ψ ∂t + Fψ|∇ψ| = 0 (19)

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SHAPE EVOLUTION (CRACKS) Iteration rule for ϕ(n) (moving the cracks): ϕ(n+1) = ϕ(n) + τϕF (n)

ϕ

|∇ϕ(n)| (20) ϕ(0) = ϕ0. (21) Iteration rule for ψ(n) (breaking and merging cracks, changing lengths): ψ(n+1) = ψ(n) + τψF (n)

ψ

|∇ψ(n)|, (22) ψ(0) = ψ0 . (23)

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Step n of the numerical algorithm

• 1. For each source γj, we calculate the residuals

ζj = g(n)

j

− gj for the electrical conductivity b(n)(x).

• 2. We solve the forward and adjoint problem and calculate

R′

j(b)∗Rj(b) = ∇uj · ∇zj .

• 3. We calculate Fϕ(x) and Fψ(x) as described before.
• 4. We choose appropriate extension velocities and apply some

additional regularization.

• 5. We correct the level set functions ϕ(n) and ψ(n) according

to the above derived iteration rules. The step-sizes are chosen empirically prior to starting the algorithm.

• 6. We determine the new parameter function b(n+1)(x).

XXII

Numerical examples

20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 50 100 150 200 50 100 150 200 5 10 15 0.02 0.04 0.06

Figura 1: First numerical experiment: reconstruction of a single crack. Top row (from left to right): Initial profile, profile after 10 source activations, and profile after 20 source activations. Center row (from left to right): profiles after 144, 252 and 324 source activations. Bottom row (from left to right): Reconstructed profile (after 540 source activations), true profile, and evolution of the cumulative cost Jloop versus number of loops.

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Numerical examples

0.5 1 0.5 1 −1 −0.5 0.5 0.5 1 0.5 1 −1 −0.5 0.5 0.5 1 0.5 1 −10 −5 5 0.5 1 0.5 1 −0.5 0.5 1

Figura 2: Initial and final level set function for the reconstruction of a single crack. Initial on top row: ϕ(0)(x) and ψ(0)(x). Final on bottom row: ϕ(f)(x) and ψ(f)(x)

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Numerical examples

20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 50 100 150 200 50 100 150 200 10 20 30 0.05 0.1 0.15 0.2

Figura 3: Second numerical experiment: reconstruction of three cracks. On the two first rows from left to right: initial guess, and recons- truction after 10,40,252,360,504 iterations. On the third row from left to right: final reconstruction (900 iteration), real crack and evolution

• f Jloop versus the number of loops.

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Numerical examples

0.5 1 0.5 1 −2 2 .5 1 1 .5 .5 1 1 .5 0.5 1 0.5 1 −10 −5 5

Figura 4: Final level set function, in the case of reconstructing three cracks. Left column: surface and contour plot of sign of ϕ(x). Right column: surface and contour plot of sign of ψ(x)

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Numerical examples

20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 50 100 150 200 50 100 150 200 5 10 15 0.05 0.1 0.15 0.2

Figura 5: Thrid numerical experiment: Reconstructing a closed curve. On the two first rows from left to right: initial guess, reconstruction after 10,30,40,80,160 iterations. On the third row from left to right: final reconstruction (360 iteration), real crack and evolution of Jloop versus the number of loops.

XXVII

Numerical examples

0.5 1 0.5 1 −1 1 0.5 1 1 0.5 0.5 1 1 0.5 0.5 1 0.5 1 −10 −5 5

Figura 6: Final level set function, in the case of reconstructing three cracks. Left column: surface and contour plot of sign of ϕ(x). Right column: surface and contour plot of sign of ψ(x)

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Numerical examples

20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 20 40 60 80 100 50 100 150 200 50 100 150 200 20 40 0.02 0.04 0.06 0.08 0.1

Figura 7: Forth numerical experiment: a pitchfork shape. On the two first rows from left to right: initial guess, reconstruction after 72,144,360,684,864 iterations. On the third row from left to right: final reconstruction ( 1552 iteration), real crack and evolution of Jloop versus the number of loops.

XXIX

Numerical examples Novel level set technique for finding and characterizing thin shapes (cracks) from electrical boundary data. Two level set functions employed, one being responsible for the location and form of the thin shape, the other one for the length and connectivity Adjoint formulation for gradient calculation Applicable to the reconstruction of penetrable cracks Future extensions to the simultaneous recovery of interior parameter values and thickness of cracks seem possible. Thanks!

XXX