Efficient Solution Methods for Inverse Problems with Application to Tomography Radon Transform and Friends
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1 Efficient Solution Methods for Inverse Problems with Application to Tomography Radon Transform and Friends Alfred K. Louis Institut für Angewandte Mathematik Universität des Saarlandes Saarbrücken and Novosibirsk NSU, October, 2011 Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 1 / 37
2 Content 1 Radon Transform and Basic Properties 2 Ray Transforms 3 Inversion Formulas for Radon Transform 4 Resolution, Stability 5 Algorithms Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 2 / 37
3 Content Radon Transform and Basic Properties 1 Radon Transform and Basic Properties 2 Ray Transforms 3 Inversion Formulas for Radon Transform 4 Resolution, Stability 5 Algorithms Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 3 / 37
4 Radon Transform and Basic Properties X-Ray Tomography Assumptions: X-Rays travel on straight lines Attenuation proportional to path length Attenuation proportional to number of photons Radon Transform I = I t f g L = ln(i 0 /I L ) = L f dt Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 4 / 37
5 Radon Transform and Basic Properties X-Ray Tomography Assumptions: X-Rays travel on straight lines Attenuation proportional to path length Attenuation proportional to number of photons Radon Transform I = I t f g L = ln(i 0 /I L ) = L f dt Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 4 / 37
6 Radon Transform and Basic Properties Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 5 / 37
7 Radon Transform and Basic Properties Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 6 / 37
8 Radon Transform Radon Transform and Basic Properties Rf (θ, s) = θ S 1 and s R. Transform for fixed direction = f (x)δ(s x θ)dx R 2 f (sθ + tθ )dt R R θ f (s) = Rf (θ, s) Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 7 / 37
9 Radon Transform and Basic Properties Radon Transform in Hihger Dimensions Rf (θ, s) = f (x)δ(s x θ)dx R N Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 8 / 37
10 Radon Transform and Basic Properties Properties of the Radon Transform Assume all integrals exist Rf (θ, s)ψ(s)ds = R f (x)ψ(x θ)dx R N Proof: Rf (θ, s)ψ(s)ds = f (x)δ(s x θ)dxψ(s)ds R R R N = f (x) ψ(s)δ(s x θ)dsdx R N R Example: Put ψ(s) = (2π) 1/2 exp( ısσ) Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 9 / 37
11 Application Radon Transform and Basic Properties Put ψ(s) = s m, then ( Rf (θ, s)s m ds = f (x) x θ ) m dx R R N = p m (θ) where p m is a polynomial of degree m in θ. See Consistency Conditions. Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 10 / 37
12 Radon Transform and Basic Properties Projection Theorem Fourier Transform ˆf (ξ) = (2π) N/2 f (x) exp( ıx ξ)dx R N Projection Theorem Rf (θ, σ) = (2π) (N 1)/2ˆf (σθ) Basis for inversion formulas, continuity results etc. Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 11 / 37
13 Exercise Radon Transform and Basic Properties Compute the 2D Radon transform of Example 1: Example 2: f (x) = f (x) = { 1 : x 1 0 : otherwise { 1 : x 1and x : otherwise Example 3: { 1 : ( x 1 f (x) = a ) 2 + ( x 2 b ) : otherwise Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 12 / 37
14 Radon Transform and Basic Properties Invariance Properties Translation: Let T1 a f (x) = f (x a). Then RT a 1 = T a θ 2 R with T2 t g(θ, s) = g(θ, s t). Rotation: Let T 1 Uf (x) = f (Ux) where U is a unitary matrix. Then RT U 1 = T U 2 R where T2 U g(θ, s) = g(uθ, s). Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 13 / 37
15 Radon Transform and Basic Properties Invariance Properties, Cont d Dilation: r > 0 Rf (rθ, rs) = r 1 Rf (θ, s) General Transform: Let T1 A f (x) = f (Ax) where A is a regular matrix. Then RT1 A = T 2 A R where ( A T2 A T g(θ, s) = det θ A A T θ g A T θ, s ) A T. θ Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 14 / 37
16 Radon Transform and Basic Properties Radon Transform and Derivatives Proof: Use Projection Theorem RD α f (θ, s) = θ α α Rf (θ, s) s α R = 2 s 2 R R θ D α f (σ) = (2π) (N 1)/2 (D α f )(σθ) Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 15 / 37
17 Radon Transform and Basic Properties Radon Transform and Derivatives Proof: Use Projection Theorem RD α f (θ, s) = θ α α Rf (θ, s) s α R = 2 s 2 R R θ D α f (σ) = (2π) (N 1)/2 (D α f )(σθ) = (2π) (N 1)/2 ı α (σθ) αˆf (σθ) Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 15 / 37
18 Radon Transform and Basic Properties Radon Transform and Derivatives Proof: Use Projection Theorem RD α f (θ, s) = θ α α Rf (θ, s) s α R = 2 s 2 R R θ D α f (σ) = (2π) (N 1)/2 (D α f )(σθ) = (2π) (N 1)/2 ı α (σθ) αˆf (σθ) = ı α σ α θ α Rθ f (σ) Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 15 / 37
19 Radon Transform and Basic Properties Radon Transform and Derivatives Proof: Use Projection Theorem RD α f (θ, s) = θ α α Rf (θ, s) s α R = 2 s 2 R R θ D α f (σ) = (2π) (N 1)/2 (D α f )(σθ) = (2π) (N 1)/2 ı α (σθ) αˆf (σθ) = ı α σ α θ α Rθ f (σ) = θ α D α R θ f (σ) Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 15 / 37
20 Continuity Radon Transform and Basic Properties Let Ω R N be bounded, C N 1 = S N 1 R R : L 2 (Ω) L 2 (C N 1 ) compact R : L 2 (Ω) H (N 1)/2 (C N 1 ) bounded, Natterer, 1979 Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 16 / 37
21 Content Ray Transforms 1 Radon Transform and Basic Properties 2 Ray Transforms 3 Inversion Formulas for Radon Transform 4 Resolution, Stability 5 Algorithms Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 17 / 37
22 Ray Transforms Ray Transforms Transverse Ray Transform (P w)(ξ, s) = Longitudinal Ray Transform (Pw)(ξ, s) = 1 s 2 w(sξ + tξ ), ξ dt. 1 s 2 1 s 2 w(sξ + tξ ), ξ dt. 1 s 2 Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 18 / 37
23 Ray Transforms Ray Transforms Transverse Ray Transform (P w)(ξ, s) = Longitudinal Ray Transform (Pw)(ξ, s) = 1 s 2 w(sξ + tξ ), ξ dt. 1 s 2 1 s 2 w(sξ + tξ ), ξ dt. 1 s 2 For fixed direction ξ the transforms can be rewritten as (P w)(ξ, s) = R ξ (w ξ)(s), (Pw)(ξ, s) = R ξ (w ξ )(s). Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 18 / 37
24 Ray Transforms Ray Transforms Transverse Ray Transform (P w)(ξ, s) = Longitudinal Ray Transform (Pw)(ξ, s) = 1 s 2 w(sξ + tξ ), ξ dt. 1 s 2 1 s 2 w(sξ + tξ ), ξ dt. 1 s 2 For fixed direction ξ the transforms can be rewritten as (P w)(ξ, s) = R ξ (w ξ)(s), (Pw)(ξ, s) = R ξ (w ξ )(s). Questions: Ask Derevtsov, Sharafutdinov! Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 18 / 37
25 Decompositions Ray Transforms w can be decomposed uniquely in a sum of potential, solenoidal and harmonic parts (Helmholtz-Hodge decomposition), w = ϕ + w s + h, div w s = 0, ϕ B = 0, w s, ν B = 0, where ϕ is a potential vector field, ϕ H 1 0 (B), w s is a solenoidal vector field, h is a harmonic vector field, with h to be harmonic function, and ν is the vector of outer to B normal. Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 19 / 37
26 Null Spaces Ray Transforms Lemma For potentials ϕ H0 1 (B) the following relations hold P ( ϕ ) = P ( ϕ ) = s ( Rϕ ) and P ( ϕ ) = P ( ϕ ) = 0. Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 20 / 37
27 Null Spaces Ray Transforms Lemma For potentials ϕ H0 1 (B) the following relations hold P ( ϕ ) = P ( ϕ ) = s ( Rϕ ) and P ( ϕ ) = P ( ϕ ) = 0. P ( ϕ ) (ξ, s) = R ξ ( ϕ ξ )(s) = ξ ξ s ( Rξ ϕ ) (s) = 0 Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 20 / 37
28 Content Inversion Formulas for Radon Transform 1 Radon Transform and Basic Properties 2 Ray Transforms 3 Inversion Formulas for Radon Transform 4 Resolution, Stability 5 Algorithms Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 21 / 37
29 Inversion Formulas for Radon Transform Inversion Formula Let α < N. Then R 1 = 1 2 (2π)1 N I α R I α+1 N where R is the adjoint operator from L 2 to L 2 known as backprojection R g(x) = g(θ, x θ)dθ S N 1 and the Riesz potential I α is defined with the Fourier transform Î α g(θ, σ) = σ α ĝ(θ, σ) Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 22 / 37
30 Proof Inversion Formulas for Radon Transform I α f (x) = (2π) N/2 R N e ıx ξ ξ αˆf (ξ)dξ Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 23 / 37
31 Proof Inversion Formulas for Radon Transform I α f (x) = (2π) R N/2 e ıx ξ ξ αˆf (ξ)dξ N = (2π) N/2 e ıσx θ σ N 1 αˆf (σθ)dσdθ S N 1 0 Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 23 / 37
32 Proof Inversion Formulas for Radon Transform I α f (x) = (2π) R N/2 e ıx ξ ξ αˆf (ξ)dξ N = (2π) N/2 e ıσx θ σ N 1 αˆf (σθ)dσdθ S N 1 0 = (2π) N+1/2 e ıσx θ σ N 1 α Rf (θ, σ)dσdθ S N 1 0 Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 23 / 37
33 Proof Inversion Formulas for Radon Transform I α f (x) = (2π) R N/2 e ıx ξ ξ αˆf (ξ)dξ N = (2π) N/2 e ıσx θ σ N 1 αˆf (σθ)dσdθ S N 1 0 = (2π) N+1/2 e ıσx θ σ N 1 α Rf (θ, σ)dσdθ S N 1 0 = 1 2 (2π) N+1/2 e ıσx θ S N 1 σ N 1 α Rf (θ, σ)dσdθ Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 23 / 37
34 Proof Inversion Formulas for Radon Transform I α f (x) = (2π) R N/2 e ıx ξ ξ αˆf (ξ)dξ N = (2π) N/2 e ıσx θ σ N 1 αˆf (σθ)dσdθ S N 1 0 = (2π) N+1/2 e ıσx θ σ N 1 α Rf (θ, σ)dσdθ S N 1 0 = 1 2 (2π) N+1/2 e ıσx θ S N 1 σ N 1 α Rf (θ, σ)dσdθ = 1 2 S (2π) N+1 I α+1 N Rf (θ, x θ)dθ N 1 Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 23 / 37
35 Inversion Formulas for Radon Transform Local vs. Nonlocal Formulas N even N odd where H Hilbert transform I 1 N g = ( 1) (N 2)/2 Hg N 1 I 1 N g = ( 1) (N 1)/2 g N 1 Hg(s) = 1 π g(t) s t dt Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 24 / 37
36 Content Resolution, Stability 1 Radon Transform and Basic Properties 2 Ray Transforms 3 Inversion Formulas for Radon Transform 4 Resolution, Stability 5 Algorithms Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 25 / 37
37 Resolution, Stability Singular Value Decomposition Consider the Radon transform for w(s) = (1 s 2 ) 1/2 R : L 2 (Ω) L 2 (C 1, w 1 ) Then π σ ml = 2, l = 0,, m, m 0 m + 1 Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 26 / 37
38 Resolution, Stability Singular Value Decomposition Consider the Radon transform for w(s) = (1 s 2 ) 1/2 Then Problem midly ill-posed R : L 2 (Ω) L 2 (C 1, w 1 ) π σ ml = 2, l = 0,, m, m 0 m + 1 Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 26 / 37
39 Resolution, Stability Singular Value Decomposition Consider the Radon transform for w(s) = (1 s 2 ) 1/2 Then R : L 2 (Ω) L 2 (C 1, w 1 ) π σ ml = 2, l = 0,, m, m 0 m + 1 Problem midly ill-posed Singular functions: Zernike and Chebychev polynomials and spherical harmonics Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 26 / 37
40 Idea of Derivation Resolution, Stability U(α) rotation, θ(ϕ) S 1, then U(α)θ(ϕ) = θ(ϕ α) Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 27 / 37
41 Idea of Derivation Resolution, Stability U(α) rotation, θ(ϕ) S 1, then U(α)θ(ϕ) = θ(ϕ α) Spherical harmonics Y l (θ(ϑ)) = e ılϑ, decompose f (x) = f (rθ(ϑ)) = f l (r)y l (θ(ϑ)) Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 27 / 37
42 Idea of Derivation Resolution, Stability U(α) rotation, θ(ϕ) S 1, then U(α)θ(ϕ) = θ(ϕ α) Spherical harmonics Y l (θ(ϑ)) = e ılϑ, decompose f (x) = f (rθ(ϑ)) = f l (r)y l (θ(ϑ)) R(r r f ) = (s s 1)Rf R(r f ) = (s2 r 2 s 2 2s s + 2)Rf Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 27 / 37
43 Resolution, Stability Idea of Derivation U(α) rotation, θ(ϕ) S 1, then U(α)θ(ϕ) = θ(ϕ α) Spherical harmonics Y l (θ(ϑ)) = e ılϑ, decompose f (x) = f (rθ(ϑ)) = f l (r)y l (θ(ϑ)) R(r r f ) = (s s 1)Rf R(r f ) = (s2 r 2 s 2 2s s + 2)Rf Spherical Harmonics are eigenfunctions of Laplace-Beltrami operator, then RD r (m) = D s (m)r Eigenfunctions of the differential operators give radial part. Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 27 / 37
44 Resolution, Stability Limited Angle Transform Data missing for ϕ Φ < π/2 π σ ml = 2 λ(m, l, Φ), l = 0,, m, m 0 m + 1 Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 28 / 37
45 Resolution, Stability Limited Angle Transform Data missing for ϕ Φ < π/2 π σ ml = 2 λ(m, l, Φ), l = 0,, m, m 0 m + 1 Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 28 / 37
46 Resolution, Stability Problem extremely ill-posed for f H 1/2 Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 29 / 37
47 Resolution, Stability Problem extremely ill-posed for f H 1/2 But some parts can be well recovered Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 29 / 37
48 Resolution, Stability Problem extremely ill-posed for f H 1/2 But some parts can be well recovered See SVD or Wavefront Sets ( Quinto ) Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 29 / 37
49 Resolution, Stability Problem extremely ill-posed for f H 1/2 But some parts can be well recovered See SVD or Wavefront Sets ( Quinto ) Limited Angle Transform R Φ : L 2 (Ω) L 2 (S Φ [ 1, 1]) where S Φ = {θ(ϕ) : ϕ Φ} is not rotaional invariant, hence reconstuction kernel ψ x (θ, s) even for mollifier E( x y ). Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 29 / 37
50 Resolution, Stability Limited Angle Reconstructionn 0 in missing range: Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 30 / 37
51 Resolution, Stability Limited Angle Reconstructionn 0 in missing range: Reconstruction with full and limited data from Syncrotron Measurements Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 30 / 37
52 Resolution, Stability Resolution Sampling theorems, Shannon, Petersen-Middleton Null Space for finitely many data Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 31 / 37
53 Resolution, Stability Resolution Sampling theorems, Shannon, Petersen-Middleton Null Space for finitely many data CT: data Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 31 / 37
54 Resolution, Stability Resolution Sampling theorems, Shannon, Petersen-Middleton Null Space for finitely many data CT: data Resolution: Object size / 1000 Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 31 / 37
55 Content Algorithms 1 Radon Transform and Basic Properties 2 Ray Transforms 3 Inversion Formulas for Radon Transform 4 Resolution, Stability 5 Algorithms Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 32 / 37
56 Filter Algorithms Replace σ by σ F γ (σ) Shepp-Logan: for σ 1: F(σ/γ) = sinc(σπ/2) Filter with ψ γ (s) = u(s) = γ 2π 3 u(γs) π/2 s sin s π 2 /4 s 2 Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 33 / 37
57 Filter Algorithms Replace σ by σ F γ (σ) Shepp-Logan: for σ 1: Filter with F(σ/γ) = sinc(σπ/2) ψ γ (s) = u(s) = γ 2π 3 u(γs) π/2 s sin s π 2 /4 s 2 For s = lh and γ = π/h, h detector spacing ψ γ (s l ) = γ2 π l 2 Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 33 / 37
58 Algorithm Algorithms Based on Approximate Inverse and Trapezoidal Rule Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 34 / 37
59 Algorithm Algorithms Based on Approximate Inverse and Trapezoidal Rule Filtered Backprojection Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 34 / 37
60 Algorithm Algorithms Based on Approximate Inverse and Trapezoidal Rule Filtered Backprojection Filtering the data q(ω, s) = Rf (ω, t)ψ γ (s t)dt = h k ψ γ (s s k )Rf (ω l, s k ) Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 34 / 37
61 Algorithm Continued Algorithms Backprojection f γ (x) = q(ω, x ω)dω S 1 = π L (1 η)q(ω l, s k ) + ηq(ω l, s k+1 ) L l=1 kh x ω l < (k + 1)h η = x ω l /h k Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 35 / 37
62 Algorithms Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 36 / 37
63 Iterative Algorithms Algorithms ART and variants Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 37 / 37
64 Iterative Algorithms Algorithms ART and variants Preferable for incomplete data Laminography etc Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 37 / 37
65 Iterative Algorithms Algorithms ART and variants Preferable for incomplete data Laminography etc EM algorithms for very noisy data ( Emission Tomography: SPECT ) Alfred K. Louis Universität des Saarlandes Solution Methods & Tomography 37 / 37
Feature Reconstruction in Tomography
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