Error estimates and convergence rates for filtered back projection

Beckmann, Matthias; Iske, Armin

doi:10.1090/mcom/3343

Cited by 10 publications

(17 citation statements)

References 18 publications

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“…Most notably, our results allow us to predict saturation of the order of convergence at fractional rates depending on smoothness properties of the filter's window function, which can be easily evaluated. The results presented in this paper generalize our previous findings in [1][2][3] and confirm the key observation that the flatness of the filter's window function at the origin determines the convergence behaviour of the approximate FBP reconstruction.…”

Section: Introductionsupporting

confidence: 88%

“…This in particular gives L 2 -error estimates, when = 0 . Moreover, this generalizes our previous results in [1][2][3], where we more restrictively deal with band-limited low-pass filters, i.e., compactly supported window functions.…”

Section: Analysis Of the Inherent Fbp Reconstruction Errorsupporting

confidence: 81%

“…Assuming certain regularity of the filter's window, we prove asymptotic convergence rates as the bandwidth goes to infinity in Sect. 4, where the presented results generalize our previous work [1][2][3] to more general filters and show that the rate of convergence saturates at fractional order depending on smoothness properties of the window. We finally provide numerical experiments in Sect.…”

Section: Introductionsupporting

confidence: 82%

“…Error bounds on the L p -norm of e L , in terms of L p -moduli of continuity of the target function f, are proven by Madych in [8]. But our approach is essentially different from previous approaches, see also [2,Section 3].…”

Section: Analysis Of the Inherent Fbp Reconstruction Errormentioning

confidence: 91%

“…For an analysis of the data error in the L 2 -norm in the case of band-limited low-pass filters we refer to [2,Section 8]. In this paper, we focus on the approximation error, i.e., we analyse the inherent error of the FBP method which is incurred by the application of the low-pass filter A L .…”

Section: Analysis Of the Inherent Fbp Reconstruction Errormentioning

confidence: 99%

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Saturation rates of filtered back projection approximations

Beckmann

Iske

2020

Calcolo

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This paper concerns the approximation of bivariate functions by using the well-known filtered back projection (FBP) formula from computerized tomography. We prove error estimates and convergence rates for the FBP approximation of target functions from Sobolev spaces $$\mathrm H^\alpha ({\mathbb {R}}^2)$$Hα(R2) of fractional order $$\alpha >0$$α>0, where we bound the FBP approximation error, which is incurred by the application of a low-pass filter, with respect to the weaker norms of the rougher Sobolev spaces $$\mathrm H^\sigma ({\mathbb {R}}^2)$$Hσ(R2), for $${0 \le \sigma \le \alpha }$$0≤σ≤α. In particular, we generalize our previous results to non band-limited filter functions and show that the decay rate of the error saturates at fractional order depending on smoothness properties of the filter’s window function at the origin. The theoretical results are supported by numerical simulations.

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Section: Introductionsupporting

confidence: 88%

Section: Analysis Of the Inherent Fbp Reconstruction Errorsupporting

confidence: 81%

Section: Introductionsupporting

confidence: 82%

Section: Analysis Of the Inherent Fbp Reconstruction Errormentioning

confidence: 91%

Section: Analysis Of the Inherent Fbp Reconstruction Errormentioning

confidence: 99%

See 3 more Smart Citations

Saturation rates of filtered back projection approximations

Beckmann

Iske

2020

Calcolo

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Error analysis for filtered back projection reconstructions in Besov spaces

2020

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Filtered back projection (FBP) methods are the most widely used reconstruction algorithms in computerized tomography (CT). The ill-posedness of this inverse problem allows only an approximate reconstruction for given noisy data. Studying the resulting reconstruction error has been a most active field of research in the 1990s and has recently been revived in terms of optimal filter design and estimating the FBP approximation errors in general Sobolev spaces. However, the choice of Sobolev spaces is suboptimal for characterizing typical CT reconstructions. A widely used model are sums of characteristic functions, which are better modelled in terms of Besov spaces B q α , p ( R 2 ) . In particular B 1 α , 1 ( R 2 ) with α ≈ 1 is a preferred model in image analysis for describing natural images. In case of noisy Radon data the total FBP reconstruction error ‖ f − f L δ ‖ ⩽ ‖ f − f L ‖ + ‖ f L − f L δ ‖ splits into an approximation error and a data error, where L serves as regularization parameter. In this paper, we study the approximation error of FBP reconstructions for target functions f ∈ L 1 ( R 2 ) ∩ B q α , p ( R 2 ) with positive α ∉ N and 1 ⩽ p, q ⩽ ∞. We prove that the L p -norm of the inherent FBP approximation error f − f L can be bounded above by ‖ f − f L ‖ L p ( R 2 ) ⩽ c α , q , W L − α | f | B q α , p ( R 2 ) under suitable assumptions on the utilized low-pass filter’s window function W. This then extends by classical methods to estimates for the total reconstruction error.

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MR. TOMP : Inversion of the Modulo Radon Transform (MRT) via Orthogonal Matching Pursuit (OMP)

Beckmann

Bhandari

2022

2022 IEEE International Conference on Image Processing (ICIP)

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In the recent years, practitioners in the area of tomography have proposed high dynamic range (HDR) solutions that are inspired by the multi-exposure fusion strategy in computational photography. To this end, multiple Radon Transform projections are acquired at different exposures that are algorithmically fused to facilitate HDR reconstruction. A single-shot alternative to multi-exposure fusion approach has been proposed in our recent line of work which is based on the Modulo Radon Transform (MRT). In this case, Radon Transform projections are folded via modulo non-linearity. This folding allows HDR values to be mapped into the dynamic range of the sensor and, thus, avoids saturation or clipping. The folded measurements are then mapped back to their ambient range using algorithms. The main goal of this paper is to introduce a novel, Fourier domain recovery method, namely, the OMP-FBP method, which is based on the Orthogonal Matching Pursuit (OMP) algorithm and Filtered Back Projection (FBP) formula. The proposed OMP-FBP method offers several advantages; it is agnostic to the modulo threshold or the number of folds, can handle much lower sampling rates than previous approaches and is empirically stable to noise and outliers. Computer simulations as well as hardware experiments in the paper validate the effectivity of the OMP-FBP recovery method.

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Error estimates and convergence rates for filtered back projection

Cited by 10 publications

References 18 publications

Saturation rates of filtered back projection approximations

Saturation rates of filtered back projection approximations

Error analysis for filtered back projection reconstructions in Besov spaces

MR. TOMP : Inversion of the Modulo Radon Transform (MRT) via Orthogonal Matching Pursuit (OMP)

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