Conductivity Imaging from Minimal Current Density Data Alexandru - PowerPoint PPT Presentation

Conductivity Imaging from Minimal Current Density Data Alexandru Tamasan University of Central Florida Joint work with: A. Nachman, A. Timonov, Z. Nashed, A. Moradifam, J. Veras

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 1/15 Motivation: Current density impedance imaging Goal: Determine the conductivity of human tissue by combining • electrical (voltage/current) measurements on the boundary (EIT) • magnitude of one current density field inside (CDI) Current Density Imaging ( Scott& Joy ’91) Very low frequency/ direct current ⇒ stationary Maxwell Current Density Field J := ∇ × H (two rotations of the object) MR measurements ⇒ Magnetic field H produced by the applied current can be identified from the total field produced by the coils+fixed magnet

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 2/15 1-Laplacian in the conformal metric g ij = | J | 2 / ( n − 1) δ ij σ − ≤ σ ( x ) ≤ σ + = isotropic conductivity of a body • Ohm’s Law: J = − σ ∇ u ⇒ σ = | J | / |∇ u | . • Conservation of charge (absence of sources/sinks inside): ∇ · J = 0 . 1-Laplacian (Seo et al., ’02): ∇ · ( | J | |∇ u |∇ u ) = 0 . Level sets of smooth, regular solutions are minimal surfaces in the metric | J | 2 / ( n − 1) δ ij g = � � .

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 3/15 Admissible Data: ( f, a ) ∈ H 1 / 2 ( ∂ Ω) × L 2 (Ω) ∃ σ ( x ) with 0 < c − ≤ σ ( x ) ≤ σ + , such that, if u σ is weak solution of ∇ · σ ∇ u σ = 0 , u σ | ∂ Ω = f, then a = | σ ∇ u σ | . σ = generating conductivity for the pair ( f, a ) , u = corresponding potential .

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 4/15 Sternberg-Ziemer example (for Dirichlet data) Sternberg& Ziemer � � 1 ∇ · |∇ u ( x ) |∇ u ( x ) = 0 , x ∈ D ≡ unit disk, u ( x ) = ( x 1 ) 2 − ( x 2 ) 2 , x ∈ ∂D. has a one parameter family of viscosity solutions u λ , λ ∈ ( − 1 , 1) , with u λ ≡ λ in inscribed rectangles. Remark : u λ s are NOT voltage potentials of some σ ∈ L ∞ + (Ω) : 1 ≡ | J | � = σ |∇ u λ | ≡ 0 .

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 5/15 Admissibility and the minimum weighted gradient problem • If ( f, a ) is admissible, say generated by some conductivity σ 0 then the corresponding voltage potential �� � a |∇ u | dx : u ∈ H 1 (Ω) , u | ∂ Ω = f u 0 ∈ argmin . Ω • If u 0 ∈ argmin �� Ω a |∇ u | dx : u ∈ H 1 (Ω) , u | ∂ Ω = f � and | J | / |∇ u 0 | ∈ L ∞ + (Ω) , then ( f, a ) is admissible. Notes : • Formally (not smooth) the Euler-Lagrange for � Ω a |∇ u | dx is the 1-Laplacian. • In the example before only u 0 (for λ = 0 ) is a minimizer of � Ω |∇ u ( x ) | dx .

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 6/15 Unique determination Theorem (Nachman-T-Timonov ’09, Moradifam-Nachman-T’ 11) ( f, | J | ) ∈ C 1 ,α ( ∂ Ω) × C α (Ω) = admissible pair, | J | > 0 a.e. in Ω . min � Ω | J ||∇ u | dx Then � � � u ∈ W 1 , 1 (Ω) over C (Ω) , |∇ u | > 0 a.e., u | ∂ Ω = f has a unique solution, say u 0 ; σ = | J | / |∇ u 0 | is the unique conductivity generating ( f, | J | ) . Note (joint with A. Moradifam and A. Nachman): Uniqueness carries to over { u ∈ BV (Ω) , u | ∂ Ω = f } Implies stability in the minimization problem!

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 7/15 Equipotential surfaces are (globally) area minimizing Theorem (Nachman-T-Timonov ’11) Let Ω ⊂ R n , n ≥ 2 , be Lipschitz domain, σ ∈ C 1 ,δ (Ω) , and f ∈ C 2 ,δ ( ∂ Ω) . Let | J | = σ |∇ u σ | , where u σ solves ∇ · σ ∇ u σ = 0 with u | ∂ Ω = f . Assume | J | > 0 in Ω . Then, for a.e. λ ∈ R and any v ∈ C 2 (Ω) with v | ∂ Ω = f and |∇ v | > 0 , � � | J ( x ) | dS x ≤ | J ( x ) | dS x ; u − 1 ( λ ) ∩ Ω v − 1 ( λ ) ∩ Ω dS = induced Euclidean surface measure. Note: the integrals also represent the surface area induced from the metric g = | J | ( n − 1) / 2 δ ij .

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 8/15 Insulating and perfectly conductive embeddings V = Insulating “ σ = 0 ”, U =perfectly conductive “ σ = ∞ ”. Let k → ∞ in the equation: ∇ · ( χ U ( k ˜ σ − σ ) + σ ) ∇ u = 0 in Ω , ∂ ν u | ∂V = 0 , u | ∂ Ω = f. Still get ∇ · σ ∇ u = 0 in Ω \ ( U ∪ V ) ∇ u = 0 in U, | J | � = 0! but Further complications: In 3D+ and σ rough ⇒ Non-unique continuation for solutions of elliptic

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 9/15 Admissibility in the presence of insulating/infinitely conductive embeddings Admissibility of the data ( a, f ) • On Ω \ ( U ∪ V ) same as before (with u σ a solution of the limiting equation) • On U : � � � ∂u σ �� inf a |∇ v | dx − σ U + vdx = 0 � ∂ν � v ∈ W 1 , 1 ( U ) U ∂U • { x : a ( x ) = 0 } = V ∪ Γ ∪ E , where – V = one insulating connected component – Γ -negligible – E = Exotic = conductive region where ∇ u = 0

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 10/15 Admisibility is physical for infinitly conductive inclusions U ⊂ Ω open, σ ∈ L ∞ Ω \ U and a ∈ L ∞ (Ω) . Assume there exists J ∈ Lip ( U ; R n ) with ∇ · J = 0 , in U, | J | ≤ a, in U, J | ∂U = σ ∂u σ � . � ∂ν � ∂U Then � � � ∂u σ �� inf a |∇ v | dx − σ U + vdx = 0 � ∂ν � v ∈ W 1 , 1 ( U ) U ∂U � σ ∂u σ ∂ν ds = 0 . ∂U

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 11/15 What can be determined via the minimization problem? Step1: From minimization determine u outside the zero set of a . Step 2: Regions where u ≡ const. ⇒ PERFECT CONDUCTORS. Step 3: Determine σ outside the zeros of a and perfect conductors Step 4: Identify maximal open connected components within zeros of a . If at the boundary of such a set • u varies ⇒ INSULATOR • u = const. ⇒ Fake perfectly conductive (EXOTIC =only happen in 3D when data is rough than Lipschitz).

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 12/15 The least weighted total variation problem Would like solve: � a |∇ u | dx : u ∈ H 1 (Ω) , u | ∂ Ω = f } min { Ω Difficulties: • minimizing sequence { u n } is not necessarily bounded in H 1 (but merely in W 1 , 1 ). • Although u n converges in L 1 loc (Ω) , the limit is only BV . � min { a | Du | : u ∈ BV (Ω) , u | ∂ Ω = f } Ω New problem: if a solution lies in BV \ W 1 , 1 cannot be automatically approximated (in BV -norm) by smooth maps (otherwise they would be in W 1 , 1 ).

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 13/15 A regularized well-posed problem for the admissible case Theorem (Nashed-T’11)Consider � � |∇ u | 2 dx, u n ∈ argmin u ∈ H 1 0 F ǫ n [ u : a n ] := a n |∇ h f + ∇ u | dx + ǫ n Ω Ω where a n → a in L 2 (Ω) , and � a n − a � = o ( ǫ n ) . Then � lim inf[ F ǫ n [ u n : a n ]] = min a | Dv | v ∈ BV (Ω) ,v | ∂ Ω = f Ω If, in addition 0 < inf( a ) ≤ a ≤ sup( a ) < ∞ , then on a subsequence u n → v ∗ in L 1 , and v ∗ ∈ BV (Ω) is a minimizer. Moreover, provided v ∗ ∈ W 1 , 1 (Ω) , a σ = |∇ ( v ∗ + h f ) | .

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 14/15 Mixed Boundary Value Problem Figure 1: ∇ · | J | |∇ u |∇ u = 0 , u | Γ= f , ∂ ν u | Γ ± = g

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 15/15 Interior Data | J | and computed equipotential lines Computed characteristics from complete data (Brain) 1 0.9 0.8 0.7 0.6 0.5 y 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Figure 2:

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 16/15 Original and reconstructed conductivities via the equipotential lines Reconstruction of the brain Original conductivity (Brain) 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 y y 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x x Figure 3:

A conference in Inverse Problems in the honor of Gunther Uhlmann, Irvine, CA, June 19–23, 2012 17/15 Original and reconstructed conductivities via the minimization approach Error: L infty = 0.10289 and L 1 = 0.00830 Reconstructed Conductivity Map, #Iterations = 64 Target Conductivity Map 1.8 1 1 1.8 0.1 0.9 0.9 1.7 0.09 1.7 0.8 0.8 0.08 1.6 1.6 0.7 0.7 0.07 1.5 1.5 0.6 0.6 0.06 1.4 0.5 1.4 0.5 0.05 y y 0.4 0.4 1.3 0.04 1.3 0.3 0.3 0.03 1.2 1.2 0.2 0.2 0.02 1.1 1.1 0.1 0.1 0.01 1 0 0 1 0.5 1 0 0.5 1 0 0.5 1 x x x Figure 4: