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Direct Computation of Inverse Problems for Biomedical Imaging Isaac Harari∗ Tel Aviv University July 2011 ∗ w/Uri Albocher, TAU; Assad A. Oberai & Yixiao Zhang, RPI; Paul E. Barbone & Carlos E. Rivas, BU Motivation Breast cancer study ’identifies tumourcausing enzyme’ — BBC News (2009) Breast cancers are surrounded by stiffer, more fibrous tissue. These properties have helped doctors to detect breast cancers. Variations in mechanical properties can signify disease. Nondestructive/noninvasive material characterization. 1 Inverse Problems Typical solution procedure: iterative constrained optimization Example: elastic modulus reconstruction Additional Info Math Model Computer Model Measured u m(x) Yes u(x) u(x)=u m (x)? µ(x) Done! No Modulus µ(x) Update µ(x) Robust: accommodates partial data, noisy data. . . 2 Applications with Full-field Data Shear-flexural test [Avril et al., 2008] 3 Aquifer management Given measurements of hydraulic head H, determine transmissivity T (∝ hyd. conductivity = permeability), where 100˚ 99˚ 98˚ GILLESPIE TRAVIS BLANCO EXPLANATION Ground-water-model area Barton Springs LAKE TRAVIS Active model area Inactive model area HAYS Boundary of recharge zone (outcrop) (modified from Puente, 1978) Boundary separating flow units (Maclay and Land, 1988, fig. 22, table 4) 560 Generalized direction of ground-water flow 30˚ 6800 66 40 6 0 62 COMAL 0 60 Well in which water-level measurement made KENDALL Spring BANDERA REAL San Marcos Springs CALDWELL 660 640 620 EDWARDS CANYON LAKE Hueco Springs it low BEXAR ter un Comal Springs nf s Ea 0 680 MEDINA LAKE 76 S∂tH = ∇ · (T ∇H) + R 0 KERR Water-level contour—Shows altitude at which water level would have stood in tightly cased wells open to Edwards aquifer. Dashed where inferred. Interval 20 feet. Datum is NGVD 29 (Roberto Esquilin, Edwards Aquifer Authority, written commun., 2004) 58 700 Here S is storativity R is a source. GUADALUPE 820 0 0 78 0 76 860 840 860 840 0 72 0 San Pedro Springs it 0 w un al flo entr th-c Sou it 88 San Antonio Springs it un flow tral -cen 760 GA 900 74 72 h Nort A IPP KN P Las Moras Springs 0 80 78 820 800 0 780 760 UVALDE 700 KINNEY GONZALES 680 w un ern flo 740 -south ern West Leona Springs MEDINA 29˚ MAVERICK Approximate area of Upper Cretaceous volcanic activity WILSON [Lindgren et al., 2004] ATASCOSA 0 ZAVALA 10 20 30 40 MILES FRIO 4 Biomechanical imaging Ultrasound provides movie of interior of deforming medium.1 Tissue mimicking phantom Bmode ultrasound 1 T. Hall, Ultrasound Lab, U. Wisc. Axial strain 5 Displ. & strain 6 Model selection Soft tissue, starting point for inverse problem: simplified isotropic, inhomogeneous, linear viscoelasticity. Compressible or incompressible? Easier to solve for compressible case. Reconstructions based on compressible models become unstable in nearly incompressible range [Barbone-Oberai, 2007]. Compressible model can lead to wrong solution of inverse problem. =⇒ Incompressible 7 Outline Formulation Quasi-static Elasticity • Model problem: plane stress – – – – Strong form, compatibility. Proposed approach: AWE Well-posedness & convergence. Numerical examples. • Plane strain and three dimensions Time-harmonic Viscoelasticity • Complex adjoint weighted equations. • Regularization. • Computation. 8 Formulation Eqn’s of motion, freq. ω ∇ · σ + ρω 2u = 0 u(x) displ. σ(x) Cauchy stress ρ dens. (const.) Isotropic media (in mixed form, p = −λǫii) σij = −pδij + 2µǫij p(x) press. µ(x) shear modulus ǫ(x) inf. strain Forward prob., µ specified + bc’s, solve for u & p. Inverse prob., u specified + ?, solve for µ (& p). 9 Quasi-static Elasticity All quantities ∈ R Incompressible plane stress σij = 2µ (δij ǫkk + ǫij ) {z } | Eij Domain Ω ⊂ R2, w/boundary Γ. Given measurement(s) of ǫij (x), x ∈ Ω: Find unknown shear modulus µ(x) such that ∇ · (µE) = 0 10 Our approach: Direct (non-iterative) solution of (2) advection-type equations. Data st det(E) 6= 0 in Ω, avoid uniaxial stress. Two equations for single unknown. Overdetermined unless strains satisfy compatibility conditions. Calibration condition on µ (shear modulus) for unique solution: • Specified point value • Specified mean value Modulus dist. known in calibration region Ωcal ⊂ Ω. 11 Specified Point Problem Given E(x) non-singular everywhere in Ω ∇ · (Eµ) = 0 in Ω µ(x0) = µ0 x0 ∈ Ωcal, µ0 > 0 prescribed. Re-write system as ∇µ + E−1(∇ · E)µ = 0 Assume: • E is differentiable (displ’s are twice differentiable). −1 • Compatibility: ∇ × E ∇·E =0 Solution [Barbone-Oberai, 2007]. 12 Specified Mean Problem Given E(x) non-singular everywhere in Ω 1 meas(Ωcal) ∇ · (Eµ) = 0 in Ω Z µ dΩ = µ̄ Ωcal µ̄ > 0 prescribed. The cont. ft’ns µpt & µmean are scalar multiples, e.g. µpt(x) = µ0 mean µ (x) mean µ (x0) 13 Zero Mean Problem Substitute µ ← µ − µ̄ Given E(x) non-singular everywhere in Ω ∇ · (Eµ) = f in Ω Z µ dΩ = 0 Ωcal Here f = − (∇ · E) µ̄ 14 Standard Variational Formulations Function space Z 1 V = v ∈ H (Ω) Ωcal v dΩ = 0 Weak form (weighted residual): Stability Small variations in data can lead to large variations in solution. Two-equation system • Scalar test ft’n can’t enforce two eqn’s independently. • Vector test ft’n leads to overconstrained, rectangular matrix. 15 Least-squares Variational Formulation Standard formulation ∇ · (E w), ∇ · (E µ) = ∇ · (E w), f , ∀w ∈ V Robust and stable [Bochev-Choi, 2001]. Stability: depends on the data. holds for sufficiently small ∇ · E. Well-posed problem w/relatively weak assumptions. Galerkin discretization: overly dissipative, negative norms [Bochev, 1999]. 16 Proposed approach Adjoint Weighted Equation for thermal conductivity reconstruction [Barbone-Oberai-H., 2007]. AWE for linear elasticity: plane, 3d, anisotropy. b(w, µ) = l(w), ∀w ∈ V Incompressible plane stress b(w, µ) = E∇w, ∇ · (E µ) l(w) = E∇w, f Weighted by “adjoint” operator (coincides w/LS for ∇ · E = 0). Motivated by VMS stabilization, USFEM, NOPG. ∼ FOSLL∗ [Manteuffel-McCormick, 2001]. 17 AWE Properties Euler-Lagrange equations: ∇ · (E × PDE′s) in Ω n · E × PDE′s on Γ 2nd-order elliptic equation for µ (assuming E2 > 0). By construction, if µ is a sol’n of the strong form, then it is a sol’n of AWE. 18 Assumptions • ∇ · E ∈ L2(Ω) • The E-norm kwkE = kE∇wk is a norm on V. kwkE ≤ C1 kwk1 • Generalized Poincaré inequality k(∇ · E)wk2 ≤ CPE kE∇wk2 For stability (holds for sufficiently small ∇ · E, dep. on data). CPE < 1 19 Well posedness Uniqueness, by coercivity. Equivalence between AWE and strong form: • Recall, if µ is a sol’n of the strong form, then it is a sol’n of AWE. • Reverse, by uniqueness (if strong form is well posed). Existence, by Lax-Milgram w/coercivity. Remark Sol’n exists for E’s that violate compatibility condition. =⇒ AWE less sensitive to noisy data than strong form. 20 Discretization Galerkin fe approx. µh ≈ µ cont. piecewise poly’s (complete to order p). Support def. by mesh partition of Ω, mesh size h. Set of ft’ns V h ⊂ V b(wh, µh) = l(wh), Convergence, error ∀wh ∈ V h e = µ − µh Consistency b(wh, e) = 0, ∀wh ∈ V h Error estimate kekE ≤ C hp 21 Numerical Results Convergence study x−y 2 + 0.2 E= symm. y−x − 0.7 2 x−y + 1.2 2 Note −0.3 0 −0.4 −0.5 −0.5 −0.6 εxx −0.2 −0.7 −1 −0.8 −0.9 −1.5 1 ∇ · E 6= 0 −1 1 Y axis Exponentially varying solution −1.1 0.5 0.5 0 0 X axis ǫ11 µ = exp(2(x + y) ) 22 Convergence study Meshes (bilin. quads), unif. refin.: 2 × 2 −→ 256 × 256 2 10 AWE L2 norm AWE H1 semi−norm LS L2 norm 1 10 LS H1 semi−norm 0 Error 10 1 −1 10 2 1 −2 10 2 −3 10 −3 10 −2 −1 10 10 0 10 h Similar studies on randomly distorted meshes (10%), nearly identical. 23 Circular Inclusions Smooth circular inclusion in homo. background, 5:1 (max.) contrast. “Measurement” (forward problem, tan. displ. bc’s): Data computed on fine mesh & downsampled (∇ · E 6= 0). 1.1 1.2 1 0.9 0.8 0.8 εxy 1 0.6 0.7 0.4 0.6 0.2 1 0.5 1 0.5 0.4 0.5 0.3 Y axis 0 0 X axis 24 Reconstruction (smooth) 50 × 50 unif. mesh 1 0.9 0.8 AWE LS Exact 5 4.5 4.5 4 4 0.7 3.5 3.5 0.6 µ 3 0.5 2.5 0.4 0.3 2 0.2 1.5 0.1 3 2.5 2 1.5 1 1 0 0 0.2 0.4 0.6 0.8 1 0 0.5 1 Distance along diagonal 1.5 25 Stepped profile Homo. circular inclusion in homo. background, 5:1 contrast. “Measurement” (forward problem, tan. displ. bc’s): Data computed on fine mesh & downsampled (∇ · E 6= 0). 1.3 1.2 1.5 1.1 1 1 εxy 0.9 0.8 0.5 0.7 0.6 0 1 0.5 1 0.5 Y axis 0.5 0 0 0.4 0.3 X axis 26 Reconstruction (stepped) 50 × 50 unif. mesh 5.5 1 4.5 0.9 0.8 4 0.7 AWE LS Exact 5 4.5 3.5 0.6 4 3.5 3 µ 0.5 2.5 0.4 0.3 2 0.2 1.5 0.1 3 2.5 2 1.5 1 1 0 0 0.2 0.4 0.6 0.8 1 0.5 0 0.5 1 Distance along diagonal 1.5 27 Estimate Stability Reconstruction more difficult w/rough data. Recall gen. Poincaré inequality k(∇ · E)wk2 ≤ CPE kE∇wk2 For stability, CPE 10 < 1. stepped smooth 9 8 Approx. E downsampled from 40×40. Coeff. matrix of inverse prob. > 0 to almost 4 (stepped) and 14 (smooth). 7 lower bound of CE P Lower bound on CPE from alg. e-val. prob. on 20 × 20 mesh. 6 5 4 3 2 1 0 2 4 6 contrast 8 10 28 Gaussian white noise 0% 3% 1 10% 1 1 5 3.5 4.5 3 0.8 0.8 0.8 4 3 3.5 2.5 0.6 0.6 A A’ 2.5 A 2 0.4 0.6 3 A A’ 0.4 2 0.4 0.2 1.5 0.2 A’ 2.5 2 1.5 0.2 1.5 1 1 0 0 0.2 0.4 0.6 0.8 1 0 1 0 3.5 0.2 0.4 0.6 0.8 0 1 0 4 Exact AWE 0.4 0.6 0.8 1 3.5 Exact AWE 3.5 3 0.2 Exact AWE 3 3 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 1 0.5 0 1 1 0.2 0.4 0.6 0.8 1 0.5 0 0.2 0.4 0.6 0.8 1 0.5 0 0.2 0.4 0.6 0.8 1 29 Incompressible Plane Strain Given measurement of ǫ(x), x ∈ Ω: Find unknown µ(x) (& p(x) ) such that −∇p + 2∇ · (µǫ) = 0 in Ω Single loading requires large set of calibration data [Barbone-Bamber, 2002]. Consider measurements from two loading conditions ǫ1 & ǫ2 (lin. indep.). 4 1st-ord. PDE’s for 3 unknowns: µ, p1, & p2. 4 calibration conditions for µ & 2 conditions for 2 p’s = 6 total [Barbone-Gokhale, 2004]. 30 Calibration conditions Illustration: two homogeneous deformations (ε1,2 = const.) 0 ε2 ε1 0 , ǫ2 = ǫ1 = ε2 0 0 −ε1 2 µ = c1 + c2x + c3y + c4 x + y Most gen. dist. 2 Recall, modulus dist. known in calibration region µ(x) = µ̄(x), x ∈ Ωcal Modes suggest “moments” Z Ωcal xµ dΩ = Z Ωcal Z xµ̄ dΩ, Z Ωcal Z µ dΩ = Ωcal yµ dΩ = Ωcal (x2 + y 2)µ dΩ = Z µ̄ dΩ Ωcal Z y µ̄ dΩ Ωcal Z (x2 + y 2)µ̄ dΩ Ωcal 31 AWE Formulation Given measurements ǫ1(x) & ǫ2(x), x ∈ Ω: Find µ(x), (p1, & p2 ) satisfying 4 2 P (ǫl∇w1, ∇ · (µǫl)) − 2 (ǫ1∇w1, ∇p1) − 2 (ǫ2∇w1, ∇p2) = 0 l=1 2 (∇w2, ∇ · (µǫ1)) 2 (∇w3, ∇ · (µǫ2)) − (∇w2, ∇p1) =0 − (∇w3, ∇p2) =0 Lagrange multipliers enforce calibration conditions. Alternative: SVD. 32 Circular Inclusions 5:1 (max.) contrast 5 4 3 2 1 1 1 0.5 0.5 0 0 Loading conditions 33 Measurements 52 × 52 Shear −0.5 1 ε xy 1.5 yy 0 ε Smooth Compression −1 0.5 −1.5 1 0 1 1 0.5 1 0.5 0.5 0 0 0 0 0 1.5 −0.5 xy −1 ε yy 1 ε Stepped 0.5 0.5 −1.5 −2 1 0 1 1 0.5 0.5 0 0 1 0.5 0.5 0 0 34 Reconstruction (smooth 26 × 26) Lagrange mult. 5.5 µ AWE µ Exact 5 4.5 4 µ 3.5 3 2.5 2 1.5 1 0.5 0 0.5 1 1.5 Distance along diagonal 5 µ AWE µ Exact 4.5 4 SVD 3.5 µ 3 2.5 2 1.5 1 0.5 Ωcal = Ω/12 0 0.5 1 1.5 Distance along diagonal 35 Reconstruction (stepped 26 × 26) µ AWE µ Exact 5 4.5 4 3.5 µ Lagrange mult. 5.5 3 2.5 2 1.5 1 0.5 0 0.5 1 1.5 Distance along diagonal 5.5 µ AWE µ Exact 5 4.5 3.5 µ SVD 4 3 2.5 2 1.5 1 0.5 0 0.5 1 1.5 Distance along diagonal 36 Time-harmonic Viscoelasticity Eqn’s of motion ∇ · (aµ) = f Plane configurations: def. a & f . E.g. anti-plane shear a = ∇u f = −ρω 2u Two measured fields ui, w/ai & fi, i = 1, 2. 37 Complex adjoint weighted equations Recall AWE b(w, µ) = l(w), ∀w ∈ V Complex-conjugated AWE (CAWE) b(w, µ) = (∇w, A∇µ + aµ) l(w) = (∇w, f ) Here A = X a∗i ⊗ ai i a = X a∗i ∇ · ai i f = X a∗i fi i 38 Poincaré ineq. =⇒ b(w, µ) bounded, coercive bilinear form & l(w) bounded linear form Lax-Milgram =⇒ CAWE solution exists and is unique. 39 Regularization Recall, quasi-statics, sharp variations in modulus dist. =⇒ large strain grad. which violate stability condition. In time-harmonic case violate stability even for smooth modulus when kL > 1 (k 2 = ρω 2/µref ). Total variation regularization [Rudin-Osher-Fatemi, 1992] ∇µ b(w, µ) + α Re ∇w, p |∇µ|2 + β 2 ! = l(w) Non-linear problem. 40 Computation Anti-plane shear. Non-dimensionalize: scale ui w/uref µ w/µref x w/L. Given ui, find µ ∇ · (µ∇ui) + k 2L2ui = 0, i = 1, 2 Compute w/bilin. quad’s. 41 Synthetic data Homo. rect. inclusion (2.5+0.35i) in homo. background (1+0.1i), kL = 30. “Measurement” (forward problem, 2 pt. sources, PML [Turkel-Yefet, 1998]): Data computed on 100 × 100 mesh & downsampled Reconstruction in reduced domain to exclude sources. 42 Imag. Real Reconstruction (40 × 40) 43 Imag. Real Reconstruction w/TV (α = 100) 44 Imag. Real Reconstruction w/noise (3%, α = 1000) 45 Gelatin phantom, MR 2 cylindrical inclusions in homogeneous background. MR measured displacement data, 300 Hz excitation2 Quadratic least-squares filter. 3 R.L. Ehman, Radiology, Mayo Clinic 46 Reconstruction (200 × 160) α = 6 × 1010 7 × 1010 8 × 1010 47 Summary Inverse problems with interior data: • Broad applications. • Well behaved. • Require care. AWE formulation: • Alternative variational framework, weighted by “adjoint” operator. • Analysis =⇒ AWE shares stability of LS. • Good numerical performance (discontinuities?). Current work: more gen. config’s (plane σ, noise, weights; 3d, etc.). 48 Background: Stabilized Methods Abstract Dirichlet problem, L is a 2nd-ord. diff. op. Lu = f in Ω u = 0 on Γ Variational formulation: u, v ∈ V, a(v, u) = (v, f ) a(v, u) = (v, Lu) = (L∗v, u) C 0(Ω) Galerkin method: uh, v h ∈ V h ⊂ V, a(v h, uh) = (v h, f ) Goal: high coarse-mesh accuracy for any L. 49 Best Approximation Optimality in “energy” norm, ∀U h ∈ V h h h a (e, e) ≤ a U − u, U − u e = uh − u. Laplacian L = −∆ energy norm a (v, v) ≡ (∇v, ∇v) H 1 semi-norm h h h h a v , e = 0 ⇐⇒ ∇v , ∇u = ∇v , ∇u Consistency H 1 proj. Good performance at any mesh refinement. Goal: best approximation in H 1 semi-norm for any L. 50 Petrov-Galerkin Formulation Optimal, but global [Givoli, 1988]: construct v̄ h ∈ V̄ h ⊂ V h a v̄ , u h h = v̄ , f L∗v̄ h = −∆v h h h v̄,n = v,n e = ∪Ωe in Ω on ∪ Γe is H 1 optimal, but finding v̄ h is very hard. NOPG [Barbone-H., 2001] Nearly opt. and local, change bc’s: v̄ h = v h on Γe v̄ h − v h are bubbles, but not residual free. Advection-diffusion Lu = −∇ · (κ∇u) + ∇ · (a u) [Nesliturk-H., 2003] 51 Variational Multiscale Framework VMS [Hughes, 1995]: Direct sum Vh = VP ⊕ VE uP ∈ V P, “coarse scales” (resolved) = std. fe ft’ns. uE ∈ V E, “fine scales” (unresolved) = ???. [Franca-Farhat, 1995], [Cipolla, 1999]: linears ⊕ bubbles. a(v P, uP) + (L∗v P, uE) = (v P, f ) a(v E, uP) + a(v E, uE) = (v E, f ) =⇒ uE = M E(LuP − f ) Eliminate uE a(v P, uP) + (L∗v P, M ELuP) = (v P, f ) + (L∗v P, M Ef ) 52 Adjoint stabilized methods Local approx., τ ≈ M E, USFEM [Franca-Frey-Hughes, 1992] P ∗ P a(v P, uP) + (L∗v P, τ LuP)Ω e = (v , f ) + (L v , τ f )Ω e Residual-based method [Oberai-Pinsky, 2000]. Advection-diffusion Lu = −∇ · (κ∇u) + ∇ · (a u) Performs well in computation, recent review [Franca-Hauke-Masud, 2006]. τ =?, e.g., [Catabriga-Coutinho-Tezduyar, 2006]. NOPG =⇒ discard Galerkin in advective limit (L∗v, Lu) = (L∗v, f ) Novel variational formulation for pure advection [Oberai-Barbone-H., 2007]. 53 Sufficient conditions • Denote eigenvalues of E2(x) as γ1(x) ≤ γ2(x) 0 < γ0 ≤ γ1(x) ≤ γ2(x) ≤ γ∞ i.e. E2 > 0. =⇒ E-norm is equivalent to H 1 semi-norm | · |1 γ0|w|1 ≤ kwkE ≤ γ∞|w|1 • |∇ · E(x)| ≤ q0 =⇒ bound by std. Poincaré const. CPE ≤ CP q02/γ0 54 AWE stability Coercivity of b(·, ·) b(w, w) = = ≥ ≥ = >0 E∇w, ∇ · (E w) 2 kE∇wk + E∇w, (∇ · E) w 1 ǫ kwk2E − k(∇ · E)wk2, ∀ǫ > 0 1− 2 2ǫ 1 ǫ 1 − − CPE kwk2E 2 2ǫ q q 1 − CPE kwk2E select ǫ = CPE for CPE < 1. Depends on the data (holds for sufficiently small ∇ · E). 55 Convergence Error e = µ − µh Consistency b(wh, e) = 0, Interpolation error ei = µ − µi i i h e=µ − µ + µ − µ | {z } | {z } ei From consistency ∀wh ∈ V h eh 0 = b(wh, ei + eh) = b(wh, ei) + b(wh, eh) 56 Thus Select wh = eh b(wh, eh) = b(wh, ei) , h i b(e , e ) = b(e , e ) h b(·, ·) is continuous Coercivity ∀wh ∈ V h h h i b(e , e ) ≤ C2 kehkE keikE C3 kehk2E ≤ b(eh, eh) Combining C3 kehkE ≤ C2 keikE 57 Error estimate kekE h i = e +e E h i ≤ e E + e E (tri. ineq.) i ≤ (1 + C2/C3) e E i ≤ (1 + C2/C3) C1 e 1 ≤ C hp (interp. est.) Recall, h is mesh size p is order of (complete) poly. 58 Implementation: Specified mean Proper variational settings in terms of ft’ns w/specified (zero) mean µ̄. Specified mean ≡ global constraint: computationally cumbersome impairs efficiency Specified pt. is simple, but may degrade performance. (Similar to pressure Poisson eq’n for incomp. flow [Bochev-Lehoucq, 2005].) Continuous problems are equivalent cal meas(Ω ) µ̄ pt mean R µ (x) = µ (x) pt dΩ µ Ωcal Discrete eqn’s generally are not. 59 In AWE optimal performance is retained: For any w̃h, construct wh(x) = w̃h(x) − 1 meas(Ωcal) Z w̃h dΩ ∈ Vh Ωcal Recall AWE formulation b(wh, uh) = l(wh) Equation expressed in terms of ∇wh b(w̃h, uh) = b(wh, uh) l(w̃h) = l(wh) 60 Discrete AWE eqn’s are formally equivalent. Simple and “correct” procedure: Compute AWE sol’n for specified pt. problem and rescale. Does not hold for LS (req. more elaborate procedure). Accuracy? 61