Inversions of ray transforms on simple surfaces

Inversions of ray transforms on simple surfaces François Monard Department of Mathematics, University of Washington. June 09, 2015 Institut Henri Poincaré - Program on Inverse problems 1 / 42

Outline 1 Introduction 2 The 2D Radon transform 3 Geodesic X-ray transforms 4 Some weighted transforms 5 The attenuated geodesic X-ray transform Explicit holomorphic integrating factors Inversion procedure for functions Numerical examples

Introduction Geodesic X-Ray transforms in two dimensions (M, M, g) non-trapping Riemannian surface with strictly convex boundary. SM: unit tangent bundle. ± SM = {(x, v) SM : x M, ±g(v, ν x ) > 0}, Geodesic flow: φ t(x, v) = (γ x,v (t), γ x,v (t)), (ν x : unit inner normal) (x, v) SM, t [τ(x, v), τ(x, v)]. Geodesic X-ray transform of a function F L 2 (SM): I [F ](x, v) = τ(x,v) 0 F (φ t (x, v)) dt, (x, v) + SM. Contains the tensor tomography problem and weighted transforms. Underdetermined problem: 3D (unknown) > 2D (data). 2 / 42

Introduction Formally determined problems New goal: reconstruct a 2D function f (x) from such transforms, assuming the rest is known. Examples: F (x, v) = f (x) (unattenuated transform) Transmission tomography F (x, v) = X f (= df ) (solenoidal vector field) Doppler tomography F (x, v) = f (x)dz k (v k ) (symmetric differential) F (x, v) = f (x)w(x, v), w known (weighted, attenuated) SPECT More applications: Linearized Calderón s problem, deformation boundary rigidity, elastography in slightly anisotropic media [Sharafutdinov 94], Calderón problem for Hodge Laplacian (F. Chung s talk). 3 / 42

The 2D Radon transform Radon transform - definition First considered and inverted by Johann Radon in 1917. Let Ω R 2, bounded, f with support in Ω. R f (x) Rf (s, θ) = f (s ˆθ + t ˆθ ) dt, R ˆθ := ( cos θ sin θ ), (s, θ) R S1. 4 / 42

The 2D Radon transform History: application to transmission tomography An incoming beam of photons with intensity I in subject to Beer s law di dt + ai = 0 along its straight path through a body is such that the outgoing intensity measured gives log I in = a(l) dl. I out a 0: attenuation coefficient. L 5 / 42

The 2D Radon transform FST, Inversion formula and ill-conditioning Fourier Slice Theorem (motivates a first inversion formula) F s ρ [Rf ](ρ, θ) = F x ξ [f ](ρˆθ), (ρ, θ) R S 1. Second inversion (R t : backprojection) : f = 1 4π Rt H s Rf, Hg(t) = 1 π p.v. g(t) R t s ds. H s R t 6 / 42

The 2D Radon transform Regularization theory - FBP algorithms This transform is ill-posed of order 1/2: f H s C Rf H s+1/2, s 0. Proper regularization theory to deal with noisy data (d: rel. cutoff bandwidth): x ( ) W d f = R t s w d Rf, W d := R t w b, w d = w d (s). w d, d = 1.0 F s ρ w d ram-lak recons. 7 / 42

The 2D Radon transform Regularization theory - FBP algorithms This transform is ill-posed of order 1/2: f H s C Rf H s+1/2, s 0. Proper regularization theory to deal with noisy data (d: rel. cutoff bandwidth): W d x f = R t ( w d s Rf ), W d := R t w b, w d = w d (s). w d, d = 0.5 F s ρ w d ram-lak recons. 7 / 42

The 2D Radon transform Regularization theory - FBP algorithms This transform is ill-posed of order 1/2: f H s C Rf H s+1/2, s 0. Proper regularization theory to deal with noisy data (d: rel. cutoff bandwidth): W d x f = R t ( w d s Rf ), W d := R t w b, w d = w d (s). w d, d = 0.2 F s ρ w d ram-lak recons. 7 / 42

The 2D Radon transform Radon transform as an elliptic FIO 1 Well-described 1 2 correspondence between singularities in (x, y) and singularities in (s, θ). 2 In problems with partial data, one understands exactly which singularities are lost. 3 Example: exterior problem. 8 / 42

The 2D Radon transform Control of discretization errors [Natterer 01] Various choices of scanning geometry: Parallel, Fan-beam, PET, Interlaced parallel. Optimality of geometries well-understood. Nyquist criteria, estimates of sampling errors. 9 / 42

Geodesic X-ray transforms GXRT (versus Radon) In general non-euclidean geometry: Scanning geometries: GXRT can only be parameterized on + SM (no global parameterization). Fourier Slice Theorem: probably not in general. Regularization theory, control of discretization error: open. But... a reconstruction formula! (in the simple case) Last but not least... 10 / 42

Geodesic X-ray transforms Simple versus non-simple Definition: (M, g) is simple iff (i) M is strictly convex ( g( t t, ν) > c 0 > 0) and (ii) M has no conjugate points. Convex Non-convex No conj. points Conj. points Simple = almost as nice as Euclidean. 11 / 42

Geodesic X-ray transforms Literature 1/3 - simple case Classically: Radial metrics: [Herglotz 1905], [Wiechert-Zoeppritz 1907] Symmetric spaces: [Radon 1917] (Euclidean), [Funk 1916] (2-sphere), [Helgason ]. Injectivity: [Mukhometov 75] via energy estimates. Stability: [Stefanov-Uhlmann 04]: I t I is a ΨDO or order -1. Reconstruction algorithms: [Pestov-Uhlmann 04]: Fredholm equations on simple surfaces (+ range characterization) [Krishnan 10]: κ small = equations are invertible. 12 / 42

Geodesic X-ray transforms Literature 2/3 - The non-simple case S-Injectivity : [Sharafudtinov 97] on non-trapping spherically symmetric layers (n 2). [Stefanov-Uhlmann 08] real-analytic metrics satisfying additional conditions (n 3). [Uhlmann-Vasy, Zhou 13] [Stefanov-Uhlmann-Vasy 14] local inj. on manifolds satisfying a certain foliation condition. (n 3). Stability : [Stefanov-Uhlmann 12] effect of fold caustics (n 2). [M.-Stefanov-Uhlmann, 14], [Holman-Uhlmann 15] general caustics. 13 / 42

Geodesic X-ray transforms Literature 3/3 - Numerical methods Numerical Methods: Radon transform: [Natterer 01] thorough study. Attenuated case [Natterer 01, Kazantsev-Bukhgeim 07],.... Tensor tomography via polynomial bases [Derevtsov 05] Vector field tomography in a refractive medium [Svetov et al. 13] General metrics, Pestov-Uhlmann reconstructions [M. 14] 14 / 42

Geodesic X-ray transforms The geometry of the unit sphere bundle 1/2 Unit circle bundle: SM := {(x, v) TM : g(v, v) = 1}. Moving frame on SM: X (x, v) = d dt t=0φ t (x, v) (geodesic vector field), V = θ (vertical derivative), X = [X, V ] (horizontal derivative), with structure equations [X, V ] = X and [X, X ] κv. Metric on SM:, SM such that (X, X, V ) is orthonormal. L 2 (SM, dσ 3 ) with dσ 3 volume form. 15 / 42

Geodesic X-ray transforms The geometry of the unit sphere bundle 2/2 Orthogonal decomposition of L 2 (SM, dσ 3 ): L 2 (SM) = k Z H k, H k := ker(v ikid), Ω k = H k C (SM). u(x, θ) = k Z π k u(x, θ) = k Z π k u(x)e ikθ. Splitting: X = η + + η, η ± : Ω k Ω k±1. Hilbert transform: H s.t. H Ωk = i sign(k)id Ωk with sign(0) = 0. Commutator formula: [Pestov-Uhlmann 05] [H, X ]u = X π 0 + π 0 X. Even/odd decomposition: u = u + + u with u ± := k ev/od u k. 16 / 42

Geodesic X-ray transforms Transport equations Transport equations on SM: For f L 2 (SM), denote by u f the solution of Xu = f (SM), u SM = 0, so that u f +SM = If. For w L 2 µ( + SM), define w ψ the unique solution u to Two operators of interest: Xu = 0 (SM), u +SM = w. I 0 f = If for f L 2 (M). I h = I (X h) for h H0 1 (M) (solenoidal potential). 17 / 42

Geodesic X-ray transforms Pestov-Uhlmann inversions Fredholm equations: [Pestov-Uhlmann 04] (Id + W 2 )f = I A +H A I 0f, (Id + (W ) 2 )h = I0 A +H +A I h, where Wf := π 0 X u f and W h := π 0 u X h are L 2 (M) adjoints. Idea of proof (left eqn): 1 derive that f = π 0 Xu f = π 0 X Hu, f 2 use [H, X ] to derive a transport problem for Hu. f Facts: (M, g) simple = W, W smoothing [Pestov-Uhlmann 04] κ = cst = W 0. κ small = W, W L 2 (M)-contractions. [Krishnan 10] 18 / 42

Some weighted transforms A generalization to k-differentials Problem: For k Z, reconstruct f C (M) from I k f = I (fe ikθ ) or I k, f = I (X (fe ikθ )). Note the transport equation Xu = f (x)e ikθ, u SM = 0. Applications: Tensor tomography for certain kinds of tensors, not necessarily solenoidal. Buiding-block nature of I restricted to Ω k : was used in [P-S-U 13] to give a range characterization of geodesic ray transforms over m-tensors. Equivalent problems: weighted transform with weight e ikθ, or ray transforms with a unitary connection ( [P-S-U 13]): v := u f e ikθ satisfies Xv + ik(x λ)v = f (x), v SM = 0, 19 / 42

Some weighted transforms Injectivity [M., 14]: I k and I k, are injective, as a consequence of [P-S-U 12]. Idea for inversion: Hilbert transform shifted in frequency: For u L 2 (SM), define H (k) u := e ikθ H(e ikθ u). Equivalently, H (k) u = l Z i sign(l k)u l. Generalization of commutators in [Pestov-Uhlmann 05] : [H (k), X ] = X π k + π k X and [H (k), X ] = X π k π k X. (Proof : direct calculation in the decomposition of L 2 (SM)) 20 / 42

Some weighted transforms Fredholm equations [M., IP 14] Fredholm equations: (generalization of k = 0) (Id + W 2 k )f = I k, A +H (k) A σk I k f, (Id + (W k ) 2 )h = I k A +H (k) A σk I k, h, where σ k = ( 1) k+1, W k f := π k X u feikθ and W k f := π ku X (fe ikθ). Further facts: (M, g) simple = W k, Wk smoothing, L2 (M) L 2 (M) compact. W k L 2 L 2 C κ + kc κ. Inversion via Neumann series in some cases. Open: Is Id + W 2 k invertible for any simple (M, g)? 21 / 42

Some weighted transforms Numerical examples: I m f, m = 4, 7, 10 f geodesics: 22 / 42

Some weighted transforms Example of single inversion from I 4 f A + H (4)A I k, 23 / 42

Some weighted transforms Inversions from I m f, m = 4, 10 Reconstruction/error at first iteration Error at 5 iterations m = 4 m = 10 24 / 42

Some weighted transforms Numerical examples: I m, f, m = 4, 7, 10 f geodesics: 25 / 42

Some weighted transforms Inversions from I m, f, m = 4, 10 Reconstruction/error at first iteration Error at 5 iterations m = 4 m = 10 26 / 42

Some weighted transforms Numerical examples (non-simple): I m f, m = 4, 7, 10 f geodesics: 27 / 42

Some weighted transforms Non-simple case: inversions from I m, f, m = 4, 10 Reconstruction/error at first iteration Error at 5 iterations m = 4 m = 10 28 / 42

The attenuated geodesic X-ray transform The problem Transform of interest: a : M R known smooth attenuation. τ(x,v) ( t ) I af (x, v) = f (φ t(x, v)) exp a(φ s(x, v)) ds dt, (x, v) +SM. 0 0 Two integrands of interest: functions on M vector fields (not necessary solenoidal) Inverse problem: Can we recover f from I a f? Reduction to a PDE: I a f = u +SM, where u solves Xu + au = f (SM), u SM = 0. 29 / 42

The attenuated geodesic X-ray transform Litterature Euclidean/constant curvature case A-analytic functions [Arbuzov-Bukhgeim-Kazantsev 98] Complexification method [Novikov 02], [Natterer 01], [Boman-Strömberg 04], [Bal 04] [Hoell-Bal 13] Review paper: [Finch 04] and references there fan-beam: [Kazantsev-Bukhgeim 07] Riemannian case Injectivity/inversion: [Salo-Uhlmann 11] Connections/Higgs field [Paternain-Salo-Uhlmann 12] Tensor tomography on simple surfaces [PSU 13] Doppler transform [Holman-Stefanov 10] Stability: [Stefanov-Uhlmann 04] (weights) [M.-Stefanov-Uhlmann 14] Inversion for functions and vector fields [M., preprint 15] Higher dimensions: [Uhlmann-Vasy, Zhou 13] 30 / 42

The attenuated geodesic X-ray transform Euclidean case, past ideas Unattenuated case: In complex form, θ u = f can be rewritten as e iθ z u + e iθ z u = f. Approach 1 [Bukhgeim et al. 98]: Decompose into Fourier series, get a tridiagonal system: z u k 1 + z u k+1 = f k, k Z. If f k = 0 for all k < 0, reason on the sequence (u 0, u 1, u 2,... ). Attenuated case: Turn θ u + au = f into θ (ue h ) = fe h, where (fe h ) k = 0 for k < 0! Answer: h = 0 a(x + tθ) dt 1 2 (Id ih)pa(θ, x θ ). Approach 2 [Novikov 02]: complexify e iθ = λ, extend the problem to λ C and solve a Riemann-Hilbert problem. 31 / 42

The attenuated geodesic X-ray transform Present case Some ingredients: (Note: parallel geometry no longer available) Since SM M S 1, Fourier analysis on the fibers: u(x, θ) = k= u k (x, θ), u k (x, θ) = ũ k (x)e ikθ, u L 2 (SM). Splitting [Guillemin-Kazhdan 80]: X = η + + η, where η ± : Ω k Ω ±1 are elliptic (, operators). Projecting Xu = f onto Ω k yields the tridiagonal system: η + u k 1 + η u k+1 = f k, k Z. Hilbert transform: H(f (x)e ikθ ) = isgn (k)f (x)e ikθ with sgn (0) = 0. u is holomorphic iff u k = 0 for all k < 0. 32 / 42

The attenuated geodesic X-ray transform Holomorphic integrating factors (HIF) as a crucial tool [Salo-Uhlmann, JDG 11] : Important fact: Let u, f smooth such that Xu = f, f holomorphic and u SM = 0. If I 0, I injective, then u is holomorphic and u 0 = 0. Yields injectivity. Surjectivity of I (I 0 was done in [Pestov-Uhlmann 04]). Existence of HIF based on using preimages by I0 and I. First reconstruction procedure for functions. Questions: Except for constant curvature cases, how to explicitely construct HIFs? The procedure required solving 3D transport equations. Can we simplify this? 33 / 42

The attenuated geodesic X-ray transform Explicit holomorphic integrating factors An explicit construction of holo int. factors Recall unattenuated inversion formulas [Pestov-Uhlmann 04]: (Id + W 2 )f = I (... )I 0f, (Id + (W ) 2 )h = I 0 (... )I h. Besides providing inversion formulas for I 0 and I via Neumann series, they provide ways to construct preimages of I 0 and I. Main assumption: Id + W 2 is injective: so far, only possible under κ small [Krishnan 10]. When this is satisfied, w = 2πi(Id + ih)n ψ with I n = a is holo, odd solution to Xw = a. w = 2πi(Id + ih)p ψ with I0 p = a is holo, even solution to Xw = X a. these constructions are continuous in appropriate spaces. Note: In numerics, store n or p instead of w! 34 / 42

The attenuated geodesic X-ray transform Inversion procedure for functions Once a holomorphic solution w of Xw = a is constructed, work on the PDE for v = ue w : Xv = fe w (SM), v SM = e w I a f. One approach (among others): Introduce H w u := e w H(e w u) and derive that [H w, X ] = e w (X π 0 + π 0 X )e w. (works because Xw Ω 0 ). Follow procedure presented earlier. This time, it only works if v can be first made holomorphic. This can be done using, again, certain holomorphic integrating factors and using the fact that v SM is known. Then one can obtain f + W 2 w f = (... )I a f, where W w f = π 0 e w X u fe w is L 2 (M) L 2 (M)-compact. 35 / 42

The attenuated geodesic X-ray transform Inversion procedure for functions Theorem (M. 15) Let h L (SM) such that Vh C 1 (SM). Then K h : L 2 (M) L 2 (M) given by K h f := (X u hf ) 0 is continuous and satisfies K h L 2 L 2 C( κ h + Vh C 1). If h 1, K h is compact [Pestov-Uhlmann 04] and satisfies the estimate above [Krishnan 10]. Theorem (M. 15) If a C 2 and κ are small enough, one can reconstruct f from I a f via an explicit Neumann series. Numerics: smoothness requirement on a can probably be relaxed. 36 / 42

The attenuated geodesic X-ray transform Inversion procedure for functions Second inversion approach: à la Kazantsev-Bukhgeim Theorem (M. 15) Let (M, g) a simple Riemannian surface with boundary and a C (M, R). Then a function f C (M, R) is determined by I af via the reconstruction formula f = 2iη ( I(e w h ψ) ) 1, h = 1 2 (D w ) +SM, where w is an odd holomorphic solution of Xw = a, D = (Id ih)(e w I af ), and w is a holo solution of Xw = I 1 0, (A D). For f 1, f 2 C (M, R), at every x M where a(x) 0, the vector field V (x, θ) = f 1(x) cos θ + f 2(x) sin θ can be reconstructed from I av via the formula (f 1(x) if 2(x))e iθ = 2iη ( 1 a η ( (I(e w h ψ)) 1 ) ), with h, w, D, w constructed similarly as above. 37 / 42

The attenuated geodesic X-ray transform Numerical examples Setup Metric and geodesics: 38 / 42

The attenuated geodesic X-ray transform Numerical examples Inversion (low attenuation) 39 / 42

The attenuated geodesic X-ray transform Numerical examples Inversion (high attenuation) The Neumann series may diverge if attenuation is too high. 40 / 42

Conclusion Conclusion: We have presented inversion algorithms applicable to a broad range of ray transforms weighted or with unitary connections, attenuated. Attenuated case: explicit holomorphic integrating factors, inversion for functions and vector fields. Numerics work in more cases than what is theoretically proved. Prospects: Regularization theory on Riemannian surfaces? Magnetic case? Illustrate the [M.-Stefanov-Uhlmann, 15] results on surfaces with pairs of conjugate points. Understand error terms W, W k, W w better in all simple cases. 41 / 42

Conclusion Thank you! Numerical implementation of two-dimensional geodesic X-ray transforms and their inversion, SIIMS, 7(2): 1335 1357. On reconstruction formulas for the ray transform acting on symmetric differentials on surfaces, Inverse Problems 30:065001 (2014) Inversion of the attenuated geodesic X-ray transform over functions and vector fields on simple surfaces, submitted. (Mar. 2015) arxiv:1503.07190 References and slides available at http://www.math.washington.edu/~fmonard/research.html 42 / 42