Reconstructing conductivities in three dimensions using a - PowerPoint PPT Presentation

Reconstructing conductivities in three dimensions using a non-physical scattering transform Kim Knudsen Department of Mathematics Technical University of Denmark Joint with Jutta Bikowski and Jennifer Mueller, Colorado State University AIP2009 Vienna July 24 Direct reconstruction in 3D 1/26

Outline 1. Introduction to the problem 2. Calderon’s approximate reconstruction algorithm 3. Exact high complex frequency reconstruction algorithm 4. Implementation details and numerical results Direct reconstruction in 3D 2/26

1. Introduction to the problem Direct reconstruction in 3D 3/26

The conductivity equation Smooth domain Ω ⊂ R 3 ; conductivity γ ∈ L ∞ (Ω) , C − 1 ≤ γ ≤ C , for C > 0 . A voltage potential u in Ω : ∂ Ω ν ∇ · γ ∇ u = 0 in Ω . Ω γ ( x ) Measurements: f , g Apply voltage potential: f = u | ∂ Ω Measure current flux : g = γ∂ ν u | ∂ Ω . Dirichlet to Neumann (Voltage to Current) map Λ γ : f �→ g , weakly defined by � � v | ∂ Ω = h ∈ H 1 / 2 ( ∂ Ω) . � Λ γ f , h � = (Λ γ f ) hd σ ( x ) = γ ∇ u · ∇ vdx , ∂ Ω Ω Direct reconstruction in 3D 4/26

Inverse problem Consider the (non-linear) mapping Λ: γ �→ Λ γ . This mapping encodes the direct problem. The Calder´ on problem (the inverse conductivity problem): • Uniqueness: Is Λ injective? • Reconstruction: How can γ be computed from Λ γ ? Direct reconstruction in 3D 5/26

Transformation to Schr¨ odinger equation Suppose u solves ∇ · γ ∇ u = 0 in Ω , u | ∂ Ω = f . Then v = γ − 1 / 2 u solves v | ∂ Ω = γ − 1 / 2 f , (∆ + q ) v = 0 in Ω with q = − ∆ γ 1 / 2 /γ 1 / 2 ⇔ (∆ + q ) γ 1 / 2 = 0 . Dirichlet to Neumann map Λ q f = ∂ ν v . If γ = 1 near ∂ Ω then Λ q = Λ γ . Equivalent inverse problem: • Does Λ q determine q ? • How can q be computed from Λ q ? Direct reconstruction in 3D 6/26

Short and incomplete history 1980 Calder´ on: Problem posed, uniqueness for linearized problem, and linear, approximate reconstruction algorithm 1987 Sylvester and Uhlmann: Uniqueness for smooth conductivities. Implicit reconstruction algorithm 1987-88 Novikov, Nachman-Sylvester-Uhlmann, Nachman: Uniqueness for conductivities with 2 derivatives and explicit high frequency reconstruction algorithm. Multidimensional D-bar equation. 2003 Brown-Torres, P¨ aiv¨ arinta-Panchenko-Uhlmann: Uniqueness for conductivities with 3 / 2 derivatives. 2006 Cornean-Knudsen-Siltanen: Low frequency reconstruction algorithm 2009 Bikowski-Knudsen-Mueller: Numerical implementation of reconstruction algorithms Direct reconstruction in 3D 7/26

2. Calderon’s approximate reconstruction algorithm Direct reconstruction in 3D 8/26

Calder´ on’s reconstruction method Integration by parts � � (Λ γ − Λ 1 ) f , h � = ( γ − 1 ) ∇ u · ∇ vdx , Ω where ∇ · γ ∇ u = 0 , u | ∂ Ω = f ∆ v = 0 v | ∂ Ω = h . Idea: take h , f as restrictions of harmonic functions f = e ix · ζ h = e − ix · ( ξ + ζ ) with ξ ∈ R 3 and ζ ∈ C 3 s.t. ( ξ + ζ ) 2 = ζ 2 = 0 . The near-field scattering transform � (Λ γ − Λ 1 ) e ix · ζ , e − ix · ( ξ + ζ ) � t exp ( ξ, ζ ) = � ( γ − 1 ) ∇ u exp ( x , ζ ) · ∇ e − ix · ( ξ + ζ ) dx , = Ω Direct reconstruction in 3D 9/26 with

Writing u exp = e ix · ζ + δ u yields � ( γ − 1 ) ∇ u exp ( x , ζ ) · ∇ e − ix · ( ζ + ξ ) dx t exp ( ξ, ζ ) = Ω = −| ξ | 2 � ( γ − 1 ) e − ix · ξ dx + R ( ξ, ζ ) . 2 Ω Estimate: | R ( ξ, ζ ) | ≤ C � γ − 1 � 2 L ∞ (Ω) ( 1 + | ζ | ) 2 e 2 R | ζ | . Ω ⊂ B R Apply low frequency filter χ K and invert Fourier transform to get Calder´ on approximation formula � t exp ( ξ, ζ ) 1 e ix · ξ χ K ( ξ ) d ξ. γ app ( x ) = 1 − 2 ( 2 π ) n | ξ | 2 Direct reconstruction in 3D 10/26

Writing u exp = e ix · ζ + δ u yields � ( γ − 1 ) ∇ u exp ( x , ζ ) · ∇ e − ix · ( ζ + ξ ) dx t exp ( ξ, ζ ) = Ω = −| ξ | 2 � ( γ − 1 ) e − ix · ξ dx + R ( ξ, ζ ) . 2 Ω Estimate: | R ( ξ, ζ ) | ≤ C � γ − 1 � 2 L ∞ (Ω) ( 1 + | ζ | ) 2 e 2 R | ζ | . Ω ⊂ B R Apply low frequency filter χ K and invert Fourier transform to get Calder´ on approximation formula � t exp ( ξ, ζ ) 1 e ix · ξ χ K ( ξ ) d ξ. γ app ( x ) = 1 − 2 ( 2 π ) n | ξ | 2 Remark: | ζ |→ 0 t exp ( ξ, ζ ) = −| ξ | 2 � γ 1 / 2 − 1 ( ξ ) . lim Direct reconstruction in 3D 10/26

Calder´ on reconstruction of potential With ξ ∈ R 3 and ζ ∈ C 3 with ( ξ + ζ ) 2 = ζ 2 = 0 , compute from Λ q � (Λ q − Λ 0 ) e ix · ζ , e − ix · ( ζ + ξ ) � t exp ( ξ, ζ ) = � qv exp ( x , ζ ) e − ix · ( ζ + ξ ) dx = Ω = ˆ q ( ξ ) + R ( ξ, ζ ) , where (∆ + q ) v exp = 0 in Ω and v exp | ∂ Ω = e ix · ξ . Estimate of R as before... Inversion formula � R 3 t exp ( ξ, ζ ) e ix · ξ χ R ( ξ ) d ξ. q ≈ q app ( x ) = Good approximation when q is small. Direct reconstruction in 3D 11/26

3. Exact high complex frequency reconstruction algorithm Direct reconstruction in 3D 12/26

Complex geometrical optics Complex geometrical optics: Let ζ ∈ C 3 such that ζ · ζ = 0 . For sufficiently large ζ there is a unique solution to the problem (∆ + q ) ψ ( x , ζ ) = 0 in R 3 , ψ ( x , ζ ) ∼ e ix · ζ for large | x | or | ζ | . ψ can be found by solving the Lippmann-Schwinger-Faddeev (LSF) equation � ψ ( x , ζ ) = e ix · ζ + G ζ ∼ e ix · ζ . G ζ ( x − y ) q ( y ) ψ ( y , ζ ) dx , ∆ G ζ = δ, Ω Moreover, ψ | ∂ Ω satisfies the solvable Fredholm equation � G ζ ( x − y )(Λ q − Λ 0 ) ψ ( y , ζ ) d σ ( y ) = e ix · ζ , ψ ( x , ζ ) + x ∈ ∂ Ω . Ω Direct reconstruction in 3D 13/26

Exceptional points Exceptional points: ζ ∈ C 3 for which there is no unique complex geometrical optics. • In 2D: q = − ∆ γ 1 / 2 /γ 1 / 2 if and only if there are no exceptional points. In 3D? • Theorem (Cornean - K - Siltanen 2003): If q is small there are no exceptional points, and γ can be reconstructed by a low complex frequency method. Direct reconstruction in 3D 14/26

The scattering transform The key intermediate object, the non-physical scattering transform, � e − ix · ( ξ + ζ ) q ( x ) ψ ( x , ζ ) dx t ( ξ, ζ ) = Ω � ( ξ + ζ ) 2 = 0 . e − ix · ( ξ + ζ ) (Λ q − Λ 0 ) ψ ( x , ζ ) | ∂ Ω d σ ( x ) , = ∂ Ω Facts 1. ψ | ∂ Ω can be computed from boundary measurements by solving ψ + S ζ (Λ q − Λ 0 ) ψ = e ix · ζ , x ∈ ∂ Ω 2. t can be computed from boundary data 3. q can be computed by using the estimate | ˆ q ( ξ ) − t ( ξ, ζ ) | = O ( 1 / | ζ | ) Nonlinear, direct reconstruction algorithm: Λ q → t ( ξ, ζ ) → q ( x )( → γ ( x )) Direct reconstruction in 3D 15/26

Connection to Calder´ on reconstruction Near-field scattering transform: � (Λ q − Λ 0 ) e ix · ζ , e − ix · ( ζ + ξ ) � t exp ( ξ, ζ ) = � e − ix · ( ξ + ζ ) q ( x ) v exp ( x , ζ ) dx , = Ω with (∆ + q ) v exp = 0 in Ω and v exp | ∂ Ω = e ix · ζ . Scattering transform: � (Λ q − Λ 0 ) ψ, e − ix · ( ζ + ξ ) � t ( ξ, ζ ) = � e − ix · ( ξ + ζ ) q ( x ) ψ ( x , ζ ) dx , = Ω where (∆ + q ) ψ = 0 in R n and ψ ∼ e ix · ζ near ∞ . In 2D (Siltanen-Isaacson-Mueller, 2001) t was replaced by t exp . Direct reconstruction in 3D 16/26

4. Implementation details and numerical results Direct reconstruction in 3D 17/26

Implementation details t exp 1. Solve numerically (∆ + q ) v exp = 0 with v exp | ∂ Ω = e ix · ζ 2. Integrate numerically � qv exp e ix · ( ξ + ζ ) dx . Ω Spherically symmetric q ( x ) = q ( | x | ) in Ω = B ( 0 , 1 ) : e ix · ζ = e − ix · ( ξ + ζ ) = � � a l , k Y k b l , k Y k l , l k , l k , l Implies seperation of variables � a l , k R l ( r ) Y k v exp ( x ) = l ( x / r ) , r = | x | . k , l Finally � 1 � R l q ( r ) r 2 dr . t exp ( ξ, ζ ) = a l , k b l , k 0 k , l Direct reconstruction in 3D 18/26

Implementation details t Computation of Green’s function G ζ ( x ) = e ix · ζ g ζ ( x ) : � 1 g e 1 + ie 2 ( x ) = e − r + x 2 − ix 1 e − ru + x 2 − ix 1 − 1 � 1 − u 2 ) du , √ 1 − u 2 J 1 ( r | x | < 2 R 4 π r 4 π s from [Newton, 1989] + symmetry. Direct reconstruction in 3D 19/26

Implementation details t Computation of Green’s function G ζ ( x ) = e ix · ζ g ζ ( x ) : � 1 g e 1 + ie 2 ( x ) = e − r + x 2 − ix 1 e − ru + x 2 − ix 1 − 1 � 1 − u 2 ) du , √ 1 − u 2 J 1 ( r | x | < 2 R 4 π r 4 π s from [Newton, 1989] + symmetry. Computation of ψ : technique of Vainikko for solving Lippman-Schwinger eq. µ ( x , ζ ) = ψ ( x , ζ ) e − ix · ζ g ζ ( x ) = G ζ ( x ) e − ix · ζ . Then � µ ( x , ζ ) = 1 + g ζ ( x − y ) q ( y ) µ ( y , ζ ) dy . Ω Direct reconstruction in 3D 19/26

Implementation details t � µ ( x , ζ ) − g ζ ( x − y ) q ( y ) µ ( y , ζ ) dy = 1 . Ω Note • RHS is periodic • Integral is on bounded domain (compact support of q ) Periodic equation for µ p : � µ p ( x , ζ ) − ζ ( x − y ) q p ( y ) µ p ( y , ζ ) dy = 1 . p R 3 g • Periodic equation is uniquely solvable and on Ω µ p ( x , ζ ) = µ ( x , ζ ) • Solved using FFT and an GMRES Direct reconstruction in 3D 20/26