Introduction Inverse Problem Numerical Examples Conclusion On the identification of piecewise constant coefficients in optical diffusion tomography by level set. Juan Pablo Agnelli Universidad Nacional de C´ ordoba and CONICET. New Trends in Parameter Identification for Mathematical Model, IMPA 2017
Introduction Inverse Problem Numerical Examples Conclusion Joint work with: Adriano De Cezaro : Univ. Fed. Rio Grande. Antonio Leit˜ ao : Univ. Fed. Santa Catarina. Maicon Marques Alves : Univ. Fed. Santa Catarina.
Introduction Inverse Problem Numerical Examples Conclusion Introduction 1 2 Inverse Problem Numerical Examples 3 Conclusion 4
Introduction Inverse Problem Diffuse Optical Tomography Numerical Examples Conclusion What is Diffuse Optical Tomography (DOT)? DOT is a non-invasive technique that utilize light in the near infrared spectral region to measure the optical properties of a physical body. The object under study has to be light-transmitting or translucent, so it works best on soft tissues such as breast and brain tissue. By monitoring variations in the light absorption and scattering of the tissue, spatial maps of properties such as total hemoglobin concentration, blood oxygen saturation and scattering can be obtained. DOT has been applied in breast cancer imaging, brain functional imaging, stroke detection, muscle functional studies, etc.
Introduction Inverse Problem Diffuse Optical Tomography Numerical Examples Conclusion What is Diffuse Optical Tomography (DOT)? DOT is a non-invasive technique that utilize light in the near infrared spectral region to measure the optical properties of a physical body. The object under study has to be light-transmitting or translucent, so it works best on soft tissues such as breast and brain tissue. By monitoring variations in the light absorption and scattering of the tissue, spatial maps of properties such as total hemoglobin concentration, blood oxygen saturation and scattering can be obtained. DOT has been applied in breast cancer imaging, brain functional imaging, stroke detection, muscle functional studies, etc.
Introduction Inverse Problem Diffuse Optical Tomography Numerical Examples Conclusion The mathematical model A simplified equation to model the light propagation is the following: � − ∇ · ( a ( x ) ∇ u )+ c ( x ) u = 0 in Ω , ( DP ) a ( x ) ∂ u ∂ν = g on Γ . u photon density. a ( x ) diffusion coefficient. c ( x ) absorption coefficient. g Neumann boundary data. Ω open, bounded and connected with Lipschitz boundary Γ .
Introduction Inverse Problem Diffuse Optical Tomography Numerical Examples Conclusion Forward map Parameter-to-measurement (forward) map H 1 / 2 ( Γ ) F : = F g : D ( F ) → ( a , c ) �→ h : = u | Γ , where u = u ( g ) is the unique solution of ( DP ) given the boundary data g and the pair ( a , c ) . D ( F ) is the set of piecewise constant functions ( a , c ) ∈ [ L 1 ( Ω )] 2 s.t. a ≤ a ( x ) ≤ a , c ≤ c ( x ) ≤ c a . e . in Ω , where a , a , c and c are known non negative real numbers.
Introduction Inverse Problem Diffuse Optical Tomography Numerical Examples Conclusion Continuity of the forward map Theorem For each g ∈ H − 1 / 2 ( Γ ) , the corresponding forward map F g : D ( F ) → H 1 / 2 ( Γ ) is continuous in the [ L 1 ( Ω )] 2 − topology. The proof is based on a generalization of Meyers’ Theorem which prove that the solution u of ( DP ) belongs to W 1 , p ( Ω ) for some p > 2 (therefore better than the standard regularity u ∈ H 1 ( Ω ) ).
Introduction Inverse Problem Diffuse Optical Tomography Numerical Examples Conclusion Inverse problem Since the optical properties within tissue are determined by the values of the diffusion and absorption coefficients, the problem of interest in DOT is the simultaneous identification of both coefficients from measurements of near-infrared diffusive light along the tissue boundary. Given a finite number of measurements h m , corresponding to inputs g m = ∂ u m ∂ν . Find ( a , c ) ∈ D ( F ) such that F m ( a , c ) = h m , for m = 1 ,..., M . (1)
Introduction Inverse Problem Diffuse Optical Tomography Numerical Examples Conclusion Inverse problem Given the nature of the measurements, we can not expect that exact data h m are available. Instead, one disposes only an approximate measured data h δ m satisfying � � � h m − h δ L 2 ( Γ ) ≤ δ , for m = 1 ,..., M � � m � where δ > 0 is the noise level. Find ( a , c ) ∈ D ( F ) such that F m ( a , c ) = h δ m , for m = 1 ,..., M . (2)
Introduction Level set approach Inverse Problem Level set approach: convergence analysis Numerical Examples Level set approach: numerical realization Conclusion Level set approach Level set functions φ a , φ c ∈ H 1 ( Ω ) are chosen in such a way that discontinuities of the coefficients ( a , c ) are located “along” its zero level sets Γ φ i : = { x ∈ Ω | φ i ( x ) = 0 } . The diffusion and absorption coefficients can be written as a 2 +( a 1 − a 2 ) H ( φ a ) , c 2 +( c 1 − c 2 ) H ( φ c ) = : P ( φ a , φ c ) � � ( a , c ) = Inverse problem: Find ( φ a , φ c ) ∈ [ H 1 ( Ω )] 2 such that F m ( P ( φ a , φ c )) = h δ m , m = 1 ,..., M . for
Introduction Level set approach Inverse Problem Level set approach: convergence analysis Numerical Examples Level set approach: numerical realization Conclusion Level set approach Level set functions φ a , φ c ∈ H 1 ( Ω ) are chosen in such a way that discontinuities of the coefficients ( a , c ) are located “along” its zero level sets Γ φ i : = { x ∈ Ω | φ i ( x ) = 0 } . The diffusion and absorption coefficients can be written as a 2 +( a 1 − a 2 ) H ( φ a ) , c 2 +( c 1 − c 2 ) H ( φ c ) = : P ( φ a , φ c ) � � ( a , c ) = Inverse problem: Find ( φ a , φ c ) ∈ [ H 1 ( Ω )] 2 such that F m ( P ( φ a , φ c )) = h δ m , m = 1 ,..., M . for
Introduction Level set approach Inverse Problem Level set approach: convergence analysis Numerical Examples Level set approach: numerical realization Conclusion Level set regularization A natural alternative to obtain stable solutions is to use a least-square approach combined with a Tikhonov-type regularization M � F m ( P ( φ a , φ c )) − h δ F α ( φ a , φ c ) : = ∑ m � 2 L 2 ( Γ ) + α R ( φ a , φ c ) (3) m = 1 where R ( φ a , φ c ) = � φ a − φ a 0 � 2 H 1 ( Ω ) + � φ c − φ c 0 � 2 H 1 ( Ω ) + β a | H ( φ a ) | BV ( Ω ) + β c | H ( φ c ) | BV ( Ω ) α > 0 plays the role of a regularization parameter and β j are scaling factors. The H 1 ( Ω ) -terms act as a control on the size of the norm of the level set function (key role to prove existence of minimizers). The BV ( Ω ) -seminorm terms penalize the length of the Hausdorff measure of the boundary of the sets Γ φ i 0 .
Introduction Level set approach Inverse Problem Level set approach: convergence analysis Numerical Examples Level set approach: numerical realization Conclusion Level set regularization A natural alternative to obtain stable solutions is to use a least-square approach combined with a Tikhonov-type regularization M � F m ( P ( φ a , φ c )) − h δ F α ( φ a , φ c ) : = ∑ m � 2 L 2 ( Γ ) + α R ( φ a , φ c ) (3) m = 1 where R ( φ a , φ c ) = � φ a − φ a 0 � 2 H 1 ( Ω ) + � φ c − φ c 0 � 2 H 1 ( Ω ) + β a | H ( φ a ) | BV ( Ω ) + β c | H ( φ c ) | BV ( Ω ) α > 0 plays the role of a regularization parameter and β j are scaling factors. The H 1 ( Ω ) -terms act as a control on the size of the norm of the level set function (key role to prove existence of minimizers). The BV ( Ω ) -seminorm terms penalize the length of the Hausdorff measure of the boundary of the sets Γ φ i 0 .
Introduction Level set approach Inverse Problem Level set approach: convergence analysis Numerical Examples Level set approach: numerical realization Conclusion Level set regularization A natural alternative to obtain stable solutions is to use a least-square approach combined with a Tikhonov-type regularization M � F m ( P ( φ a , φ c )) − h δ F α ( φ a , φ c ) : = ∑ m � 2 L 2 ( Γ ) + α R ( φ a , φ c ) (3) m = 1 where R ( φ a , φ c ) = � φ a − φ a 0 � 2 H 1 ( Ω ) + � φ c − φ c 0 � 2 H 1 ( Ω ) + β a | H ( φ a ) | BV ( Ω ) + β c | H ( φ c ) | BV ( Ω ) α > 0 plays the role of a regularization parameter and β j are scaling factors. The H 1 ( Ω ) -terms act as a control on the size of the norm of the level set function (key role to prove existence of minimizers). The BV ( Ω ) -seminorm terms penalize the length of the Hausdorff measure of the boundary of the sets Γ φ i 0 .
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