Tomographic Terahertz Imaging Using Sequential Subspace Optimization - PowerPoint PPT Presentation

Tomographic Terahertz Imaging Using Sequential Subspace Optimization Thomas Schuster, Anne Wald Department of Mathematics Saarland University Saarbr¨ ucken Workshop on New Trends in Parameter Identification for Mathematical Models Rio de Janeiro, 31 October 2017 T. Schuster, A. Wald (UdS) THz tomography using SESOP Rio de Janeiro 2017 1 / 23

Overview Introduction 1 Sequential subspace optimization 2 Notation and basic definitions Motivation - SESOP for linear problems SESOP and RESESOP for nonlinear problems Convergence and regularization Terahertz tomography 3 Physical background The forward operator The inverse problem of 2D THz tomography Numerical results 4 Conclusion and outlook 5 T. Schuster, A. Wald (UdS) THz tomography using SESOP Rio de Janeiro 2017 2 / 23

Introduction Introduction Sequential subspace optimization (SESOP) Iterative reconstruction technique for inverse problems here: SESOP for nonlinear inverse problems in Hilbert spaces Extension (acceleration) of Landweber’s method Regularization technique (RESESOP) reduction of the number of iterations Terahertz tomography novel imaging technique in nondestructive testing for plastics and ceramics approach via scattering problems: nonlinear inverse problem solution of the inverse problem of THz tomography with an adapted RESESOP method T. Schuster, A. Wald (UdS) THz tomography using SESOP Rio de Janeiro 2017 3 / 23

Introduction Introduction Sequential subspace optimization (SESOP) Iterative reconstruction technique for inverse problems here: SESOP for nonlinear inverse problems in Hilbert spaces Extension (acceleration) of Landweber’s method Regularization technique (RESESOP) reduction of the number of iterations Terahertz tomography novel imaging technique in nondestructive testing for plastics and ceramics approach via scattering problems: nonlinear inverse problem solution of the inverse problem of THz tomography with an adapted RESESOP method T. Schuster, A. Wald (UdS) THz tomography using SESOP Rio de Janeiro 2017 3 / 23

Sequential subspace optimization Notation and basic definitions Notation and definitions Let X , Y be Hilbert spaces and M F ( x )= y := { x ∈ X : F ( x ) = y } the solution set for a nonlinear operator F : D ( F ) ⊆ X → Y . We assume to have noisy data y δ with noise level δ ≥ � y − y δ � . Hyperplanes, halfspaces and stripes Let u ∈ X \ { 0 } and α, ξ ∈ R , ξ ≥ 0. We define the (affine) hyperplane H ( u , α ) := { x ∈ X : � u , x � = α } , the halfspace H ≤ ( u , α ) := { x ∈ X : � u , x � ≤ α } and the stripe H ( u , α, ξ ) := { x ∈ X : |� u , x � − α | ≤ ξ } . T. Schuster, A. Wald (UdS) THz tomography using SESOP Rio de Janeiro 2017 4 / 23

Sequential subspace optimization Notation and basic definitions Notation and definitions Let X , Y be Hilbert spaces and M F ( x )= y := { x ∈ X : F ( x ) = y } the solution set for a nonlinear operator F : D ( F ) ⊆ X → Y . We assume to have noisy data y δ with noise level δ ≥ � y − y δ � . Hyperplanes, halfspaces and stripes Let u ∈ X \ { 0 } and α, ξ ∈ R , ξ ≥ 0. We define the (affine) hyperplane H ( u , α ) := { x ∈ X : � u , x � = α } , the halfspace H ≤ ( u , α ) := { x ∈ X : � u , x � ≤ α } and the stripe H ( u , α, ξ ) := { x ∈ X : |� u , x � − α | ≤ ξ } . T. Schuster, A. Wald (UdS) THz tomography using SESOP Rio de Janeiro 2017 4 / 23

Sequential subspace optimization Motivation - SESOP for linear problems SESOP iteration for linear problems Ax = y in Hilbert spaces � t n , i A ∗ w n , i , x n +1 := x n − i ∈ I n I n a finite index set, w n , i ∈ Y for all i ∈ I n , and the parameters t n = ( t n , i ) i ∈ I n minimize 2 � � h n ( t ) := 1 � � � � t i A ∗ w n , i � � � � x n − + t i w n , i , y . � � 2 � � i ∈ I n i ∈ I n � � Lemma [Sch¨ opfer, S., Louis (2008)] The minimization of h n ( t ) is equivalent to computing the metric projection � x n +1 = P H n ( x n ) , H n := H n , i , i ∈ I n onto the intersection of hyperplanes A ∗ w n , i , x H n , i := � x ∈ X : � � = � � � w n , i , y ⊇ M Ax = y . T. Schuster, A. Wald (UdS) THz tomography using SESOP Rio de Janeiro 2017 5 / 23

Sequential subspace optimization Motivation - SESOP for linear problems SESOP iteration for linear problems Ax = y in Hilbert spaces � t n , i A ∗ w n , i , x n +1 := x n − i ∈ I n I n a finite index set, w n , i ∈ Y for all i ∈ I n , and the parameters t n = ( t n , i ) i ∈ I n minimize 2 � � h n ( t ) := 1 � � � � t i A ∗ w n , i � � � � x n − + t i w n , i , y . � � 2 � � i ∈ I n i ∈ I n � � Lemma [Sch¨ opfer, S., Louis (2008)] The minimization of h n ( t ) is equivalent to computing the metric projection � x n +1 = P H n ( x n ) , H n := H n , i , i ∈ I n onto the intersection of hyperplanes A ∗ w n , i , x H n , i := � x ∈ X : � � = � � � w n , i , y ⊇ M Ax = y . RESESOP for linear operators and noisy data Project sequentially onto intersections of stripes H ( u , α, ξ ), where ξ = ξ ( δ ) , M Ax = y ⊆ H ( u , α, ξ ) . T. Schuster, A. Wald (UdS) THz tomography using SESOP Rio de Janeiro 2017 5 / 23

Sequential subspace optimization Motivation - SESOP for linear problems SESOP iteration for linear problems Ax = y in Hilbert spaces � t n , i A ∗ w n , i , x n +1 := x n − i ∈ I n I n a finite index set, w n , i ∈ Y for all i ∈ I n , and the parameters t n = ( t n , i ) i ∈ I n minimize 2 � � h n ( t ) := 1 � � � � t i A ∗ w n , i � � � � x n − + t i w n , i , y . � � 2 � � i ∈ I n i ∈ I n � � Lemma [Sch¨ opfer, S., Louis (2008)] The minimization of h n ( t ) is equivalent to computing the metric projection � x n +1 = P H n ( x n ) , H n := H n , i , i ∈ I n onto the intersection of hyperplanes A ∗ w n , i , x H n , i := � x ∈ X : � � = � � � w n , i , y ⊇ M Ax = y . RESESOP for linear operators and noisy data Project sequentially onto intersections of stripes H ( u , α, ξ ), where ξ = ξ ( δ ) , M Ax = y ⊆ H ( u , α, ξ ) . T. Schuster, A. Wald (UdS) THz tomography using SESOP Rio de Janeiro 2017 5 / 23

Sequential subspace optimization SESOP and RESESOP for nonlinear problems SESOP for nonlinear inverse problems F ( x ) = y We consider nonlinear inverse problems F ( x ) = y , F : D ( F ) ⊆ X → Y . Assume that F satisfies the tangential cone condition � ≤ c tc � F ( x ) − F (˜ � x ) − F ′ ( x )( x − ˜ � � F ( x ) − F (˜ x ) x ) � for x , ˜ x ∈ B ρ ( x 0 ) and 0 < c tc < 1. Furthermore, � F ′ ( x ) � ≤ c F . sup x ∈ B ρ ( x 0 ) Use stripes also in F ( x ) case of exact data y . x n Take into account the local character as well as the nonlinearity of operators. H ( u n , α n , ξ n ) T. Schuster, A. Wald (UdS) THz tomography using SESOP Rio de Janeiro 2017 6 / 23

Sequential subspace optimization SESOP and RESESOP for nonlinear problems From SESOP for linear problems to SESOP for nonlinear problems Stripes in the linear case � n , i , y δ �� � � H δ A ∗ w δ w δ � ≤ δ � w δ � � � n , i := x ∈ X : n , i , x − n , i � � � � Stripes in the nonlinear case � i ) − y δ �� � H δ F ′ ( x δ i ) ∗ w δ n , i , x δ w δ n , i , F ( x δ � � � n , i := x ∈ X : i − x − � � � � � � � ≤ � w δ � R δ � � n , i � i � + δ + δ ⊇ M F ( x )= y c tc RESESOP algorithm n ) − y δ fulfills � R δ As long as the residual R δ n := F ( x δ n � > τδ , compute � � x δ n +1 = x δ t δ n , i F ′ ( x δ i ) ∗ w δ H ( u δ n , i , α δ n , i , ξ δ n − ∈ n , i ) n , i i ∈ I δ i ∈ I δ n n n +1 � 2 ≤ � z − x δ n � 2 − C and � z − x δ � � R δ � n � , δ, c tc , c F for z ∈ M F ( x )= y . T. Schuster, A. Wald (UdS) THz tomography using SESOP Rio de Janeiro 2017 7 / 23

Sequential subspace optimization Convergence and regularization Convergence of the SESOP algorithm Let n ∈ I n for each n ∈ N , I n ⊆ { n − N + 1 , ..., n } ∩ N , N ≥ 1 fixed w n , i := R i = F ( x i ) − y for each i ∈ I n , x + ∈ B ρ / 2 ( x 0 ) be the unique solution in B ρ ( x 0 ). Theorem [Wald, S. (2016)] The sequence of iterates { x n } n ∈ N , generated by the SESOP algorithm, converges strongly to x + , if the optimization parameters t n , i are bounded, | t n , i | ≤ t for some t > 0 for all i ∈ I n and n ∈ N . T. Schuster, A. Wald (UdS) THz tomography using SESOP Rio de Janeiro 2017 8 / 23

Sequential subspace optimization Convergence and regularization Convergence of the SESOP algorithm Let n ∈ I n for each n ∈ N , I n ⊆ { n − N + 1 , ..., n } ∩ N , N ≥ 1 fixed w n , i := R i = F ( x i ) − y for each i ∈ I n , x + ∈ B ρ / 2 ( x 0 ) be the unique solution in B ρ ( x 0 ). Theorem [Wald, S. (2016)] The sequence of iterates { x n } n ∈ N , generated by the SESOP algorithm, converges strongly to x + , if the optimization parameters t n , i are bounded, | t n , i | ≤ t for some t > 0 for all i ∈ I n and n ∈ N . T. Schuster, A. Wald (UdS) THz tomography using SESOP Rio de Janeiro 2017 8 / 23