A Direct D-bar Reconstruction Algorithm for Recovering a Complex - PowerPoint PPT Presentation

A Direct D-bar Reconstruction Algorithm for Recovering a Complex Conductivity in 2-D Jennifer Mueller Department of Mathematics and School of Biomedical Engineering Colorado State University Fort Collins, CO

Collaborators on this Project Raul Gonzalez-Lima, USP Marcelo Amato, USP Natalia Herrera, USP Sarah Hamilton, CSU Alan Von Hermann, CSU

The EIT Problem Current is applied on electrodes on the surface of the body and the resulting voltage is measured. Let γ = σ + i ωǫ . ∇ · γ ∇ u = 0 in Ω u = f on ∂ Ω Ω e l γ ∂ u Λ γ : f → ∂ν | ∂ Ω Mathematically, this is governed by the inverse admittivity problem: Can the admittivity γ be recovered from measurements of the Dirichlet-to-Neuman (DN) map Λ γ ?

Applications of EIT Medical Applications in 2-D: • Monitoring ventilation and perfusion in ARDS patients • Detection of pneumothorax • Diagnosis of pulmonary edema and pulmonary embolus

Clinical Applications What does the silent zone represent? • Pneumothorax? • Pulmonary edema? • Hyperinflation? • Atelectasis? 0Figure courtesy of M. Amato from Real-time detection of pneumothorax using electrical impedance tomography , Crit Care Med 2008 (Costa et al)

Global Uniqueness Result: Brown, Uhlmann A classic result showed that once differentiable conductivities are uniquely determined by knowledge of the DN map Λ σ : Theorem Let Ω ∈ R 2 be a bounded domain with Lipschitz boundary and σ be a measurable function bounded away from zero and infinity. If σ 1 and σ 2 are two conductivities with ∇ σ i in L p (Ω) , p > 2 , and Λ σ 1 = Λ σ 2 , then σ 1 = σ 2 . 0 Brown and Uhlmann, Comm PDEs, 1997

Global Uniqueness Result of Francini Assume there exist positive constants σ 0 and E such that σ > σ 0 in Ω , � σ � W 2 , ∞ (Ω) ≤ E , � ǫ � W 2 , ∞ (Ω) ≤ E Theorem Let Ω be an open bounded domain in R 2 with Lipschitz boundary. Let σ j and ǫ j satisfy the conditions above. Then there exists a constant ω 0 such that if γ j = σ j + i ωǫ j for j=1,2 and ω < ω 0 and if Λ γ 1 = Λ γ 2 , then γ 1 = γ 2 . 0 Francini, Inverse Problems, 2000

General Overview: D-bar Methods for EIT D-bar reconstruction methods capitalize on the direct relationship between the conductivity and CGO solutions to a PDE related to the inverse conductivity problem (possibly through a transformation). Λ γ − → Scattering transform − → CGO solutions − → γ They are • Mesh independent • Trivially parallelizable

General Overview: D-bar Methods for EIT The CGO solution depends on an auxilliary variable k ∈ C . Typically, a ¯ ∂ equation in z for the CGO solution leads to a direct formula for γ . The link between the DN map and the CGO solution is through a nonlinear Fourier transform known as the scattering transform. A ¯ ∂ equation in the auxilliary variable k for the CGO solution involves the scattering transform and completes the constructive steps.

General Overview: D-bar Methods for EIT The CGO solution depends on an auxilliary variable k ∈ C . Typically, a ¯ ∂ equation in z for the CGO solution leads to a direct formula for γ . The link between the DN map and the CGO solution is through a nonlinear Fourier transform known as the scattering transform. A ¯ ∂ equation in the auxilliary variable k for the CGO solution involves the scattering transform and completes the constructive steps.

General Overview: D-bar Methods for EIT The CGO solution depends on an auxilliary variable k ∈ C . Typically, a ¯ ∂ equation in z for the CGO solution leads to a direct formula for γ . The link between the DN map and the CGO solution is through a nonlinear Fourier transform known as the scattering transform. A ¯ ∂ equation in the auxilliary variable k for the CGO solution involves the scattering transform and completes the constructive steps.

General Overview: D-bar Methods for EIT The CGO solution depends on an auxilliary variable k ∈ C . Typically, a ¯ ∂ equation in z for the CGO solution leads to a direct formula for γ . The link between the DN map and the CGO solution is through a nonlinear Fourier transform known as the scattering transform. A ¯ ∂ equation in the auxilliary variable k for the CGO solution involves the scattering transform and completes the constructive steps.

To learn more, see our forthcoming book: Linear and Nonlinear Inverse Problems with Practical Applications, by JM and Samuli Siltanen In production, SIAM 2012

The Potential Matrix Define the matrix potential Q by   − ∂γ 1 / 2 − 1 0 � � 0 2 ∂ log γ γ 1 / 2 Q = = 2 ¯ ¯ − 1  ∂γ 1 / 2  ∂ log γ 0 − 0 γ 1 / 2 and matrices D and D k by � ¯ ¯ ¯ � � � ∂ 0 ∂ ∂ − ik D = D k = 0 ∂ + ik ∂ ∂ � � � � where ¯ ∂ z = 1 ∂ x + i ∂ ∂ ∂ z = 1 ∂ x − i ∂ ∂ and . 2 ∂ y 2 ∂ y

Exponentially Growing Solutions Given a solution u ∈ H 1 (Ω) of ∇ · ( γ ( z ) ∇ u ( z )) = 0, the vector � v � � ∂ u � = γ 1 / 2 ¯ w ∂ u solves � � v � � D − Q = 0 (1) w For k = k 1 + ik 2 ∈ C , seek solutions ψ of ( ?? ) of the form � e izk � 0 ψ ( z , k ) = M ( z , k ) e − i ¯ zk 0 where M converges to the identity matrix as | z | → ∞ .

Exponentially Growing Solutions The CGO solutions M ( z , k ) satisfy ( D k − Q ) M = 0 Or in integral form 1 + 1 � Q 12 ( ζ ) M 21 ( ζ, k ) M 11 ( z , k ) = d ζ z − ζ π Ω 1 � e − k ( z − ζ ) Q 21 ( ζ ) M 11 ( ζ, k ) M 21 ( z , k ) = d ζ z − ¯ π ¯ ζ Ω and similarly for M 12 and M 22 , which are coupled. z ¯ Here e k ( z ) = exp ( i ( zk + ¯ k )) .

Computation of CGO Solutions Applying FFT’s on a suitable grid of meshsize h to the integral form of the equations 1 + h 2 IFFT ( FFT ( 1 M 11 ( z , k ) = π z ) FFT ( Q 12 ( z ) M 21 ( z , k ))) h 2 IFFT ( FFT ( e − k ( z − ζ ) M 21 ( z , k ) = ) FFT ( Q 21 ( z ) M 11 ( z , k ))) π z results in a linear system that can be solved by, eg, GMRES.

Reconstruction of Q Knowledge of the full matrix M results in a direct reconstruction formula for Q and hence γ . Theorem For any ρ > 0 , � k 0 →∞ µ ( B ρ ( 0 )) − 1 Q ( z ) = lim D k M ( z , k ) d µ ( k ) . k : | k − k 0 | < r This large k limit presents a problem for practical computation. 0 Theorem 6.2 of Francini, 2000

Reconstruction of Q The following is a direct reconstruction formula for Q and hence γ involving a small k limit: Theorem Define M + ( z , k ) ≡ M 11 ( z , k ) + e − k ( z ) M 12 ( z , k ) M − ( z , k ) ≡ M 22 ( z , k ) + e k ( z ) M 21 ( z , k ) . Then z M + ( z , 0 ) Q 21 ( z ) = ∂ z M − ( z , 0 ) Q 12 ( z ) = ∂ ¯ M − ( z , 0 ) M + ( z , 0 ) 0 Hamilton, 2012

The Scattering Transform The scattering transform matrix is defined by � e − i ¯ � kz Q 12 ( z ) ψ 22 ( z , k ) S ( k ) = i � 0 dz . − e i ¯ k ¯ z Q 21 ( z ) ψ 11 ( z , k ) π 0 R 2 The matrix M ( z , k ) satisfies the D-bar equation wrt k : � e ¯ � k ( z ) 0 ∂ k M ( z , k ) = M ( z , ¯ ¯ k ) S ( k ) , 0 e − k ( z ) 0 Francini, Inverse Problems, 2000

The Scattering Transform The scattering transform matrix is defined by � e − i ¯ � kz Q 12 ( z ) ψ 22 ( z , k ) S ( k ) = i � 0 dz . − e i ¯ k ¯ z Q 21 ( z ) ψ 11 ( z , k ) π 0 R 2 The matrix M ( z , k ) satisfies the D-bar equation wrt k : � e ¯ � k ( z ) 0 ∂ k M ( z , k ) = M ( z , ¯ ¯ k ) S ( k ) , 0 e − k ( z ) 0 Francini, Inverse Problems, 2000

Computation This results in two coupled systems. The first is ¯ M 12 ( z , ¯ ∂ k M 11 ( z , k ) = k ) e − k ( z ) S 21 ( k ) M 11 ( z , ¯ ¯ ∂ k M 12 ( z , k ) = k ) e ¯ k ( z ) S 12 ( k ) or in integral form M 11 ( z , k ) − 1 π k ∗ ( M 12 ( z , ¯ 1 = k ) e − k ( z ) S 21 ( k )) M 12 ( z , k ) − 1 π k ∗ ( M 11 ( z , ¯ 0 = k ) e ¯ k ( z ) S 12 ( k )) This can be discretized and a linear system results. Note that care must be taken with the conjugate with respect to k .

The Scattering Transform Denote the unit outer normal to ∂ Ω by ν = ν 1 + i ν 2 and its conjugate by ¯ ν = ν 1 − i ν 2 . Then i � e − i ¯ kz ψ 12 ( z , k ) ν ( z ) ds ( z ) S 12 ( k ) = 2 π ∂ Ω S 21 ( k ) = − i � e i ¯ z ψ 21 ( z , k ) ν ( z ) ds ( z ) . k ¯ 2 π ∂ Ω 0 A. Von Hermann, PhD thesis, Colorado State University, 2010

There exist CGO solutions u 1 and u 2 to the admittivity equation with asymptotic behavior u 2 ∼ e − ik ¯ u 1 ∼ e ikz z and as | z | , | k | → ∞ . ik − ik and the following connection to the DN map: e ikz � u 1 ( z , k ) = − G k ( z − ζ ) (Λ γ − Λ 1 ) u 1 ( ζ, k ) ds ( ζ ) ik ∂ Ω e − ik ¯ z � G k ( z − ζ ) (Λ γ − Λ 1 ) u 2 (¯ u 2 ( z , k ) = − ik − ζ, k ) ds ( ζ ) ∂ Ω 0 A. Von Hermann, PhD thesis, Colorado State University, 2010

where G k ( z ) is the Faddeev Green’s function G k ( z ) = e ikz e iz · ξ � ξ + 2 k ) d ξ k ∈ C \ 0 . ξ (¯ ( 2 π ) 2 R 2 These CGO solutions satisfy � Ψ 11 � ∂ z u 1 � Ψ 12 � ∂ z u 2 � � � � = γ 1 / 2 = γ 1 / 2 and , ¯ ¯ Ψ 21 ∂ z u 1 Ψ 22 ∂ z u 2 which leads to BIE’s for Ψ 12 and Ψ 21 ...