Numerical Methods for geodesic X-ray transforms and applications to open theoretical questions

Transcription

1 Numerical Methods for geodesic X-ray transforms and applications to open theoretical questions François Monard Department of Mathematics, University of Washington. Nov. 13, 2014 UW Numerical Analysis Research Club 1 / 36

2 Outline 1 The 2D Radon transform 2 Geodesic X-ray transforms 3 Reconstruction formulas 4 Implementation 5 Application to the study of the impact of conjugate points

3 The 2D Radon transform Radon transform - definition First considered and inverted by Johann Radon in 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. 2 / 36

4 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 is given by log I in = a(l) dl. I out a 0: attenuation coefficient. L 3 / 36

5 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 4 / 36

6 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. 5 / 36

7 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. 5 / 36

8 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. 5 / 36

9 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. 6 / 36

10 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. 6 / 36

11 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. 6 / 36

12 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. 6 / 36

13 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. 6 / 36

14 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. 6 / 36

15 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. 7 / 36

16 Outline 1 The 2D Radon transform 2 Geodesic X-ray transforms 3 Reconstruction formulas 4 Implementation 5 Application to the study of the impact of conjugate points

17 Geodesic X-ray transforms Geodesic X-Ray transforms in two dimensions (M, g) non-trapping Riemannian manifold with boundary with unit tangent bundle SM and influx boundary + SM = {(x, v) SM : x M, g(v, ν x ) > 0}, ν x : unit inner normal Geodesic flow: φ t(x, v) = (γ x,v (t), γ x,v (t)), (x, v) SM, t [τ(x, v), τ(x, v)]. Geodesic X-ray transform of a function f : If (x, v) = τ(x,v) 0 f (γ x,v (t)) dt, (x, v) + SM. Restrict to unit-speed geodesics by homogeneity arguments. 8 / 36

18 Geodesic X-ray transforms Applications 1 Transmission tomography in media with variable index of refraction. 2 Geophysical imaging. 3 Linearized Calderón s problem (Λ γ? γ). 4 Linearized (conformal) boundary rigidity ({d g (x, x ), (x, x ) ( M) 2 }? g). 5 Higher-dimensional tensors: Ultrasound Doppler tomography, linearized boundary rigidity, elastography in slightly anisotropic media [Sharafutdinov 94]. 9 / 36

19 Geodesic X-ray transforms GXRT (versus Radon) In general non-euclidean geometry Scanning geometries: GXRT can only be parameterized in fan-beam coordinates (from the boundary). 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 / 36

20 Geodesic X-ray transforms GXRT (versus Radon) In general non-euclidean geometry Scanning geometries: GXRT can only be parameterized in fan-beam coordinates (from the boundary). 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 / 36

21 Geodesic X-ray transforms GXRT (versus Radon) In general non-euclidean geometry Scanning geometries: GXRT can only be parameterized in fan-beam coordinates (from the boundary). 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 / 36

22 Geodesic X-ray transforms GXRT (versus Radon) In general non-euclidean geometry Scanning geometries: GXRT can only be parameterized in fan-beam coordinates (from the boundary). 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 / 36

23 Geodesic X-ray transforms GXRT (versus Radon) In general non-euclidean geometry Scanning geometries: GXRT can only be parameterized in fan-beam coordinates (from the boundary). 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 / 36

24 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 point Non-convex No conj. points Conj. points Simple = almost as nice as Euclidean. 11 / 36

25 Geodesic X-ray transforms Conjugate points Conjugate points do not violate the uniqueness theorem for ODEs: In SM M S 1 Projected onto M 12 / 36

26 Geodesic X-ray transforms Literature 1/3 - simple case Classically: Radial metrics: [Herglotz 1905], [Wiechert-Zoeppritz 1907] Symmetric spaces: [Radon 1917] (Euclidean), [Funk 1916] (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. 13 / 36

27 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 13] 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] general caustics. 14 / 36

28 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, P-U reconstructions [M. 14] 15 / 36

29 Outline 1 The 2D Radon transform 2 Geodesic X-ray transforms 3 Reconstruction formulas 4 Implementation 5 Application to the study of the impact of conjugate points

30 Reconstruction formulas A transport problem on the unit sphere bundle WLOG, parameterize (M, g) in isothermal coordinates where M R 2 and g = e 2λ(x,y) Id, and the unit circle bundle SM := {(x, v) TM : g(v, v) = 1}. is parameterized by (x, y, θ) M S 1 (i.e. speed vector has the form v = e λ(x,y) ˆθ). Geodesics: integral curves of the vector field X = e λ (cos θ x + sin θ y + ( sin θ x λ + cos θ y λ) θ ). Then If can be seen as the ingoing trace of the solution to the transport equation Xu(x, θ) = f (x) (SM), u SM = / 36

31 Reconstruction formulas The Pestov-Uhlmann reconstruction formulas Fourier analysis w.r.t. θ: u(x, θ) =! u k (x)e ikθ, u L 2 (SM). k Z In this decomposition, define the fiberwise Hilbert transform H by ihu(x, θ) = u k (x)e ikθ u k (x)e ikθ. k>0 k<0 H and X satisfy the commutator [Pestov-Uhlmann 05] [H, X ]u = X u 0 + (X u) 0, u 0 := 1 2π 2π Fredholm equation: [Pestov-Uhlmann 04, Theorem 5.4] 0 u(x, θ) dθ. (Id + W 2 )f = (X w (f ) ψ ) 0, w (f ) := H(If ), where we have defined the linear operator Wf := (X u f ) / 36

32 Reconstruction formulas Results and consequences Next: when the metric is simple, it is proved in [Pestov-Uhlmann 04, Prop. 5.1] that the operator W is smoothing, hence compact from L 2 (M) to itself. Consequences: The formula stably reconstructs the singularities of f. Invertibility holds modulo the finite-dimensional space ker(id + W 2 ) of smooth ghosts. If the metric has constant curvature, then W 0, so the formula is exact. Improvement: [Krishnan 10] establishes W L 2 L 2 C κ : if κ small enough, the Neumann series below is justified: f = ( W 2 ) k (X w (f ) ψ ) 0, w (f ) := H(If ). k=0 18 / 36

33 Outline 1 The 2D Radon transform 2 Geodesic X-ray transforms 3 Reconstruction formulas 4 Implementation 5 Application to the study of the impact of conjugate points

34 Implementation Setting and parameterization of the data space Work in isothermal coordinates. Degrees of freedom: the isotropic metric, g(x, y) = e 2λ(x,y) Id. the boundary (domain is star-shaped w.r.t. (0, 0) by construction), defined by a function r(β). Fan-beam coordinates: The data space is an equispaced discretized version of (β, α) S 1 [ π 2, π 2 ]. β: initial boundary point r(β) ( cos β sin β), α: angle between initial speed and inner normal ν(β). Solving geodesics: discretize using you favorite method ẋ = e λ cos θ, ẏ = e λ sin θ, θ = e λ [ˆθ, λ]. Having (λ, x λ, y λ) as function handles helps accuracy (no need for interpolation of a metric on an underlying cartesian grid). 19 / 36

35 Implementation Examples of phantoms, domains and metrics Examples of phantoms and domains: Constant curvature metrics (left: positive; right: negative). 20 / 36

36 Implementation Forward map and Hilbert transform f (x, y) is either defined on an underlying cartesian grid (necessary for iterative reconstruction) or by function handles. Computation of forward data: Riemann sums, where each access to f is either by interpolation or handle call. Hilbert transform: FFT in α. β-slice by β-slice. Example of forward data and Hilbert transforms. Metric: constant positive curvature with R = / 36

37 Implementation Approximate inversion Fredholm equation: f + Kf = (X w (f ) ψ ) 0, where K compact, w (f ) = 1 2 H(If ) and Simplification: X := e λ (ˆθ (ˆθ λ) θ ), w ψ (x, v) := w(φ τ(x, v) (x, v)), (x, v) SM. ) (X w (f ) ψ ) 0 = (e e 2λ(x) 2π λ ( sin θ ) (f ) S 1 cos θ w ψ (x, θ) dθ. RHS can be implemented much faster than LHS. Can be vectorized in (x, y). Riemann sum in θ. by FD on the cartesian grid. Bottleneck: O(N 3 ) calls to the basepoint function. 22 / 36

38 Implementation One-shot inversion - constant positive curvature g(x, y) = 4R 4 (x 2 +y 2 +R 2 ) 2. R = 1.2 (left) and R = 2 (right). 23 / 36

39 Implementation One-shot inversion - constant positive curvature g(x, y) = 4R 4 (x 2 +y 2 +R 2 ) 2. R = 1.2 (left) and R = 2 (right). 24 / 36

40 Implementation One-shot inversion - constant negative curvature g(x, y) = 4R 4 (x 2 +y 2 R 2 ) 2. R = 2 (left) and R = 1.2 (right). 25 / 36

41 Implementation One-shot inversion - constant negative curvature g(x, y) = 4R 4 (x 2 +y 2 R 2 ) 2. R = 2 (left) and R = 1.2 (right). 26 / 36

42 Implementation Constant curvature. Non-simple case (a) Phantom f and domain (b) Sample geodesics for g R,+ with R = 1 (c) Forward data (d) Pointwise error f f rc Figure: Non-simple domain with constant positive curvature. 27 / 36

43 Implementation General (non-constant curvature) case Goal: from the equation f + Kf = AIf, compute a finite number of terms of the Neumann series f = ( K) p AIf = (Id AI ) p AIf. p=0 Experiments: Metric: focusing lens with parameter 0 k < e. ( g(x, y) = exp k exp ( r 2 )) 2σ 2, σ = 0.25, r 2 = (x 0.2) 2 + y 2. p=0 k = 0.3 k = 0.6 k = / 36

44 Implementation Reconstruction of a function (t. to b.: k = 0.3, 0.6, 1.2) 29 / 36

45 Implementation Other types of integrands [Pestov-Uhlmann 05] also provide reconstruction of integrands of the type (solenoidal vector fields) X f, f = f (x). [M. 14] provides reconstruction formulas for integrands of the type (symmetric differentials) f (x)e ikθ, X [f (x)e ikθ ], k Z. All these formulas were implemented with the present code. 30 / 36

46 Outline 1 The 2D Radon transform 2 Geodesic X-ray transforms 3 Reconstruction formulas 4 Implementation 5 Application to the study of the impact of conjugate points

47 Application to the study of the impact of conjugate points Location of the artifacts in the non-simple case Heuristics: artifacts appear at the conjugate locus of the initial singularities. Joint work with Plamen Stefanov and Gunther Uhlmann. 31 / 36

48 Application to the study of the impact of conjugate points Loss of stability in the presence of conjugate points Theorem (M., Stefanov, Uhlmann 14) Suppose p 1, p 2 are conjugate along γ 0 and let f 1 supported near p 1 such that (p 1, ξ 1 ) WF (f 1 ). Then there exists f 2 supported near p 2 with (p 2, ξ 2 ) WF (f 2 ) such that I (f 1 f 2 ) C. GXRT as an FIO: both singularities (p 1, ξ 1 ) and (p 2, ξ 2 ) are mapped to the same singularity in data space. In other words, the data I (f 1 f 2 ) does not allow us to resolve the singularities at ξ 1 or ξ 2, and the inversion problem becomes severely ill-posed. 32 / 36

49 Application to the study of the impact of conjugate points Illustration of cancellation of singularities in the presence of caustics 1/2. (a) f 1 (b) If 1 (on +SM 1) (c) If 1 remapped to +SM 2 33 / 36

50 Application to the study of the impact of conjugate points Illustration of cancellation of singularities in the presence of caustics 2/2. (d) f 1 f 2 (right: with geodesics) (e) If 1 (f) I (f 1 f 2) 34 / 36

51 Conclusion Open questions/future projects Question: How to sample + SM appropriately? Question: How to find reconstruction algorithms including a regularization parameter? I.e. find a formula reflecting W b x f = R t (w b s Rf ) in the Euclidean case [Natterer 01]. Understand better the settings when the Neumann series does and does not converges (when is Id + W 2 injective?). Parallelize! Write a package. Issues when considering non-linear problems (e.g. boundary rigidity). Go to 3D. Start over. Thermostat/magnetic flows, attenuated case. Start over. 35 / 36

52 Conclusion Thank you References available at F.M., Numerical implementation of two-dimensional geodesic X-ray transforms and their inversion, SIIMS 7(2): (2014). F.M., On reconstruction formulas for the ray transform acting on symmetric differentials on surfaces, Inverse Problems 30: (2014). F.M., P. Stefanov and G.Uhlmann, The geodesic X-ray transform on Riemannian surfaces with conjugate points, Communications in Mathematical Physics, to appear (2014). arxiv: / 36