Saturation Rates for Filtered Back Projection

Transcription

1 Saturation Rates for Filtered Back Projection Matthias Beckmann, Armin Iske Fachbereich Mathematik, Universität Hamburg 3rd IM-Workshop on Applied Approximation, Signals and Images Bernried, 21st February 2018

2 Outline 1 Filtered Back Projection Basic Reconstruction Problem Approximate Reconstruction Formula 2 Analysis of the Reconstruction Error in the L 2 -Norm Error Estimate Convergence 3 L 2 -Error Analysis for C k -Window Functions Numerical Observations Error Estimate for C k -Windows 4 Asymptotic L 2 -Error Analysis Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 2 / 23

3 Basic Reconstruction Problem Problem formulation: Let Ω R 2 be bounded. Reconstruct a bivariate function f f x, y) with support suppf ) Ω from given Radon data {Rf t, θ) t R, θ [0, π)}, where the Radon transform Rf of f L 1 R 2 ) is defined as Rf t, θ) = f x, y) dx dy for t, θ) R [0, π). {x cosθ)+y sinθ)=t} Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 3 / 23

4 Basic Reconstruction Problem Problem formulation: Let Ω R 2 be bounded. Reconstruct a bivariate function f f x, y) with support suppf ) Ω from given Radon data {Rf t, θ) t R, θ [0, π)}, where the Radon transform Rf of f L 1 R 2 ) is defined as Rf t, θ) = f x, y) dx dy for t, θ) R [0, π). {x cosθ)+y sinθ)=t} y n θ = sinθ), cosθ) ) nθ = cosθ), sinθ) ) xl = t cosθ),tsinθ) ) lt,θ x Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 3 / 23

5 Basic Reconstruction Problem Problem formulation: Let Ω R 2 be bounded. Reconstruct a bivariate function f f x, y) with support suppf ) Ω from given Radon data {Rf t, θ) t R, θ [0, π)}, where the Radon transform Rf of f L 1 R 2 ) is defined as Rf t, θ) = f x, y) dx dy for t, θ) R [0, π). {x cosθ)+y sinθ)=t} Analytical solution: The inversion of R involves the back projection Bh of h L 1 R [0, π)), Bhx, y) = 1 π π 0 hx cosθ) + y sinθ), θ) dθ for x, y) R 2, and is given, for f L 1 R 2 ) CR 2 ), by the filtered back projection formula f x, y) = 1 2 B F 1 [ S FRf )S, θ)] ) x, y) x, y) R 2. Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 3 / 23

6 t The Shepp-Logan Phantom and its Radon Transform The Shepp-Logan phantom f θ The Radon transform Rf Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 4 / 23

7 The Back Projection of the Shepp-Logan Phantom Original Shepp Logan Phantom Unfiltered backprojection Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 5 / 23

8 The Filtered Back Projection Formula OBS! The filtered back projection formula f x, y) = 1 2 B F 1 [ S FRf )S, θ)] ) x, y) is ugly and unstable. Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 6 / 23

9 The Filtered Back Projection Formula OBS! The filtered back projection formula f x, y) = 1 2 B F 1 [ S FRf )S, θ)] ) x, y) is ugly and unstable. Stabilization: Replace S by a low-pass filter A L S) = S W S /L) with finite bandwidth L > 0 and even window W of compact suppw ) [ 1, 1]. Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 6 / 23

10 Approximate Reconstruction Stabilization: Replace the factor S by a low-pass filter A L : R R, A L S) = S W S /L) with finite bandwidth L > 0 and an even window function W : R R with compact support suppw ) [ 1, 1]. Approximate reconstruction formula: We can express the resulting approximate FBP reconstruction f L as f L = 1 2 B F 1 A L Rf ) = f K L, where we rely for f L 1 R 2 ) and g L 1 R [0, π)) on the standard relation Bg f = Bg Rf ) and define the convolution kernel K L : R 2 R as K L x, y) = 1 2 B F 1 A L ) x, y) for x, y) R 2. Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 7 / 23

11 Three Popular Examples for Low-Pass Filters The Ram-Lak filter. { S iff S L; A L S) = S L S) = 0 iff S > L. The Shepp-Logan filter. A L S) = S ) sinπs/2l)) L S) = πs/2l) { 2L π sinπs/2l)) iff S L; 0 iff S > L. The low-pass cosine filter. { S cosπs/2l)) iff S L; A L S) = S cosπs/2l)) L S) = 0 iff S > L. Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 8 / 23

12 Three Popular Examples for Low-Pass Filters 10 Ram Lak filter 7 Shepp Logan filter 4 low pass cosine filter Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 9 / 23

13 Analysis of the Reconstruction Error Aim Analyse the FBP reconstruction error e L = f f L depending on the window function W and the bandwidth L > 0. Previous results: Pointwise and L -error estimates by [Munshi et al., 1991, Munshi, 1992] L p -error estimates in terms of L p -moduli of continuity by [Madych, 1990] Our approach: L 2 -error estimates for target functions f from Sobolev spaces of fractional order, i.e., f H α R 2 ) = { g S R 2 ) g α < } for α > 0, where g 2 α = x 2 + y 2) α Fgx, y) 2 dx dy 2π R R L 2 -convergence rates L ) in terms of bandwidth L and smoothness α Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 10 / 23

14 A first L 2 Error Estimate Theorem A first L 2 -error estimate; Beckmann & I., 2015) Let f L 1 R 2 ) H α R 2 ), for some α > 0, W L R) and K L L 1 R 2 ). Then, the L 2 -norm of the FBP reconstruction error e L = f f L is bounded above by e L L2 R 2 ) 1 W,[ 1,1] f L2 R 2 ) + L α f α. Proof: For f L 1 R 2 ) L 2 R 2 ) and by the Rayleigh Plancherel theorem, we get e L 2 L 2 R 2 ) = f f K L 2 L 2 R 2 ) = 1 2π Ff Ff FK L 2 L 2 R 2 ;C) = 1 2π Ff W L Ff 2 L 2 R 2 ;C) = I 1 + I 2, where we used W L x, y) 2 ) = FK L x, y) for K L L 1 R 2 ), and where we let I 1 := 1 Ff W L Ff )x, y) 2 dx, y), 2π x,y) 2 L I 2 := 1 Ff x, y) 2 dx, y). 2π x,y) 2>L Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 11 / 23

15 A first L 2 Error Estimate Proof continued): For W L R), the first integral I 1 = 1 Ff W L Ff )x, y) 2 dx, y), 2π can be bounded above by x,y) 2 L I 1 1 2π 1 W L 2,[ L,L] Ff 2 L 2 R 2 ;C) = 1 W 2,[ 1,1] f 2 L 2 R 2 ). For f H α R 2 ), with α > 0, the second integral I 2 = 1 Ff x, y) 2 dx, y) 2π x,y) 2>L can be bounded above by I x 2 + y 2) α L 2α Ff x, y) 2 dx, y) L 2α f 2 2π α. x,y) 2>L Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 12 / 23

16 Refined L 2 -Error Analysis Theorem Refined L 2 -error estimate; Beckmann & I., 2016) Let f L 1 R 2 ) H α R 2 ) for some α > 0, let W L R) and K L L 1 R 2 ). Then, the L 2 -norm of the FBP reconstruction error e L = f f L is bounded above by e L L2 R 2 ) Φ 1 /2 α,w L) + L α) f α, where Φ α,w L) = 1 W S)) 2 sup S [ 1,1] 1 + L 2 S 2 ) α for L > 0. Proof: For f H α R 2 ), with α > 0, we bound integral I 2 as before and I 1 by I 1 = 1 2π x,y) 2 L sup S [ L,L] 1 W L x, y) 2 ) x 2 + y 2 ) α 1 + x 2 + y 2) α Ff x, y) 2 dx, y) ) x 2 + y 2) α Ff x, y) 2 dx dy. 2π 1 W L S)) S 2 ) α R R Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 13 / 23

17 Refined L 2 -Error Analysis Proof continued): Since 1 W L S)) 2 1 W sup S [ L,L] 1 + S 2 ) α = sup S /L)) 2 1 W S)) 2 S [ L,L] 1 + S 2 ) α = sup S [ 1,1] 1 + L 2 S 2 ) α, and with letting we find and so e L 2 L 2 R 2 ) I 1 Φ α,w L) = sup S [ 1,1] sup S [ 1,1] 1 W S)) 2 sup S [ 1,1] 1 + L 2 S 2 ) α for L > 0 ) 1 W S)) L 2 S 2 ) α f 2 α = Φ α,w L) f 2 α ) 1 W S)) L 2 S 2 ) α + L 2α f 2 α = Φ α,w L) + L 2α) f 2 α. Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 14 / 23

18 Refined L 2 -Error Analysis Theorem Convergence of Φ α,w ; Beckmann & I., 2016) Let W be continuous on [ 1, 1] and satisfy W 0) = 1. Then, for all α > 0, 1 W S)) 2 Φ α,w L) = max S [0,1] 1 + L 2 S 2 ) α 0 for L. Proof sketch): Let Sα,W,L [0, 1] be the smallest maximizer in [0, 1]of Φ α,w,l S) := 1 W S))2 1 + L 2 S 2 ) α for S [0, 1]. Case 1: S α,w,l uniformly bounded away from 0, S α,w,l c c α,w > 0. Then, 0 Φ α,w,l S α,w,l ) = 1 W S α,w,l ) ) L2 S α,w,l )2) α 1 W 2,[ 1,1] 1 + L 2 c 2 ) α L 0. Case 2: Sα,W,L 0 for L. Then, 0 Φ α,w,l S α,w,l ) = 1 W S α,w,l ) ) L2 S α,w,l )2) α 1 W S α,w,l) ) 2 L 0. Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 15 / 23

19 Convergence of the FBP Reconstruction Corollary L 2 convergence of FBP reconstruction; Beckmann & I., 16) Let f L 1 R 2 ) H α R 2 ) for some α > 0, K L L 1 R 2 ) and W C[ 1, 1]) with W 0) = 1. Then, the L 2 -norm of the FBP reconstruction error e L = f f L satisfies e L L2 R 2 ) = o1) for L. Basic Assumption Let S α,w,l be uniformly bounded away from 0, i.e., there is c α,w > 0 satisfying S α,w,l c α,w L > 0. OBS! Under the basic assumption we have Φ α,w L) c 2α α,w 1 W 2,[ 1,1] L 2α = OL 2α ) for L. Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 16 / 23

20 Rate of Convergence Theorem Rate of convergence; Beckmann & I., 2016) Let f L 1 R 2 ) H α R 2 ) for some α > 0, let K L L 1 R 2 ) and W C[ 1, 1]) with W 0) = 1. Further, let the basic assumption be satisfied for a constant c α,w. Then, the L 2 -norm of the FBP reconstruction error e L = f f L is bounded by ) e L L2 R 2 ) c α α,w 1 W,[ 1,1] + 1 L α f α. Therefore, e L L2 R 2 ) = OL α ) for L, i.e., the decay rate is determined by the smoothness α of the target function f. OBS! Let the window function W C[ 1, 1]), W χ [ 1,1], satisfy W S) = 1 S ε, ε) for some 0 < ε < 1. Then, the basic assumption is fulfilled with c α,w = ε. Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 17 / 23

21 Numerical Observations We investigated the behaviour of Sα,W,L numerically for standard low-pass filters: Shepp-Logan filter: W S) = sinc ) πs 2 χ[ 1,1] S), Cosine filter: W S) = cos ) πs 2 χ[ 1,1] S), Hamming filter for β [ 1 2, 1] ): W S) = β + 1 β) cosπs)) χ [ 1,1] S), Gaussian filter for β > 1): W S) = exp πs /β) 2) χ [ 1,1] S). For α < 2, we found that the basic assumption is fulfilled whereby c α,w > 0 L > 0 : S α,w,l c α,w Φ α,w L) = OL 2α ) for L. But for α 2, we observed Sα,W,L 0 for L. Moreover, in this case the convergence rate of Φ α,w stagnates at Φ α,w L) = OL 4 ) for L. Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 18 / 23

22 Numerical Observations Φα,W L 2α Φα,W Φα,W Φα,W L a) α = L b) α = L c) α = 2 Φα,W L Φα,W Φα,W Φα,W L d) α = L e) α = L f) α = 4 Fig.: Decay rate of Φ α,w for the Shepp-Logan filter Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 19 / 23

23 L 2 -Error Analysis for C k -Windows Theorem Convergence rate of Φ α,w for W C k ; Beckmann & I., 16) For k 2, let W C k [ 1, 1] satisfy Then, we have where the constant W 0) = 1, W j) 0) = 0 1 j k 1 Φ α,w L) { c 2 α,k for α > k k!) W k) 2 2,[ 1,1] L 2k 1 k!) W k) 2 2,[ 1,1] L 2α for α k, c α,k = k α k ) k/2 α k ) α/2 α is strictly monotonically decreasing in α > k. In particular, we have Φ α,w L) = O L 2 min{k,α}) for L. Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 20 / 23

24 L 2 -Error Analysis for C k -Windows Corollary L 2 -error estimate for C k -windows; Beckmann & I., 2016) Let f L 1 R 2 ) H α R 2 ), for some α > 0, let K L L 1 R 2 ) and W C k [ 1, 1]), for k 2, with W 0) = 1, W j) 0) = 0 1 j k 1. Then, the L 2 -norm of the FBP reconstruction error e L = f f L is bounded by cα,k k! W k),[ 1,1] L k + L ) f α α for α > k e L L 2 R 2 ) 1 k! W k),[ 1,1] L α + L ) f α α for α k. In particular, we have e L L 2 R 2 ) c W k),[ 1,1] L min{k,α} + L α) f α = O L min{k,α}). In conclusion, the rate of convergence saturates at OL k ). Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 21 / 23

25 Asymptotic L 2 -Error Analysis Theorem Asymptotic L 2 -error estimate; Beckmann & I., 2016) Let f L 1 R 2 ) H α R 2 ), for some α > 0, K L L 1 R 2 ) and let W L R) be k-times differentiable at the origin, k 2, with W 0) = 1, W j) 0) = 0 1 j k 1. Then, the L 2 -norm of the FBP reconstruction error e L = f f L is bounded by 2 k! c α,k W k) 0) L k + L ) f α α + ol k ) for α > k e L L2 R 2 ) 2 k! W k) 0) L α + L ) f α α + ol α ) for α k with the strictly monotonically decreasing constant k ) k/2 α k ) α/2 c α,k = for α > k. α k α In particular, we have e L L2 R 2 ) c W k) 0) L min{k,α} + L α) f α + o L min{k,α}). Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 22 / 23

26 References M. Beckmann, A. Iske: Error estimates and convergence rates for filtered back projection, 33 pages, to appear in Mathematics of Computation. M. Beckmann, A. Iske: Sobolev error estimates for filtered back projection reconstructions. IEEE 2017 International Conference Sampling Theory and Applications SampTA2017), 2017, M. Beckmann, A. Iske: Approximation of bivariate functions from fractional Sobolev spaces by filtered back projection. Hamburger Beiträge zur Angewandten Mathematik , 2017, M. Beckmann, A. Iske: Analysis of the inherent reconstruction error in filtered back projection. Hamburger Beiträge zur Angewandten Mathematik , 2016, M. Beckmann, A. Iske: Error estimates for filtered back projection. IEEE 2015 International Conference Sampling Theory and Applications SampTA2015), 2015, W.R. Madych: Summability and approximate reconstruction from Radon transform data. Contemporary Mathematics, Volume 113, AMS, Providence, 1990, P. Munshi: Error analysis of tomographic filters. I: theory. NDT & E International 254/5), 1992, P. Munshi, R.K.S. Rathore, K.S. Ram, M.S. Kalra: Error estimates for tomographic inversion. Inverse Problems 73), 1991, F. Natterer: The Mathematics of Computerized Tomography. Classics in Applied Mathematics 32, SIAM, Philadelphia, Matthias Beckmann University of Hamburg) Saturation Rates for Filtered Back Projection 23 / 23