Convergence Rates for Hölder-Windows in Filtered Back Projection

Beckmann, Matthias; Iske, Armin

doi:10.1109/sampta45681.2019.9030855

Cited by 3 publications

(9 citation statements)

References 12 publications

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“…If, for > 0 , W satisfies W ∈ C k−1 ([− , ]) with W(0) = 1 and W (j) (0) = 0 for all 1 ≤ j ≤ k − 1 as well as W (k) ∈ L p ([− , ]) for 1 < p ≤ ∞ , the H -norm of the FBP reconstruction error satisfies and, thus, the decay rate saturates at fractional order L −(k− 1 ∕p) . This generalizes our results in [3], where we considered windows W with supp(W) ⊆ [−1, 1] and assumed regularity of W on [−1, 1] instead of [− , ] for > 0 . Hence, our results substantiate the conclusion that the flatness of W at zero determines the decay rate of the inherent FBP reconstruction error.…”

Section: Discussionsupporting

confidence: 74%

“…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%

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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: Discussionsupporting

confidence: 74%

Section: Introductionsupporting

confidence: 88%

Section: Analysis Of the Inherent Fbp Reconstruction Errorsupporting

confidence: 81%

Section: Introductionsupporting

confidence: 82%

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Saturation rates of filtered back projection approximations

Beckmann

Iske

2020

Calcolo

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“…and, thus, existing error estimates [2,1,3,5] for the FBP approximation error f −f Ω carry over to the US-FBP error f − f λ Ω . For illustration, we apply [2, Theorem 5.5] and [1, Theorem 3] to obtain error estimates in Sobolev spaces of fractional order α > 0, given by…”

Section: Towards High-dynamic-range Tomographymentioning

confidence: 99%

The Modulo Radon Transform: Theory, Algorithms and Applications

Beckmann¹,

Bhandari²,

Krahmer³

2021

Preprint

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Recently, experiments have been reported where researchers were able to perform high dynamic range (HDR) tomography in a heuristic fashion, by fusing multiple tomographic projections. This approach to HDR tomography has been inspired by HDR photography and inherits the same disadvantages. Taking a computational imaging approach to the HDR tomography problem, we here suggest a new model based on the Modulo Radon Transform (MRT), which we rigorously introduce and analyze. By harnessing a joint design between hardware and algorithms, we present a singleshot HDR tomography approach, which to our knowledge, is the only approach that is backed by mathematical guarantees. On the hardware front, instead of recording the Radon Transform projections that my potentially saturate, we propose to measure modulo values of the same. This ensures that the HDR measurements are folded into a lower dynamic range. On the algorithmic front, our recovery algorithms reconstruct the HDR images from folded measurements. Beyond mathematical aspects such as injectivity and inversion of the MRT for different scenarios including band-limited and approximately compactly supported images, we also provide a first proof-of-concept demonstration. To do so, we implement MRT by experimentally folding tomographic measurements available as an open source data set using our custom designed modulo hardware. Our reconstruction clearly shows the advantages of our approach for experimental data. In this way, our MRT based solution paves a path for HDR acquisition in a number of related imaging problems.

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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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Convergence Rates for Hölder-Windows in Filtered Back Projection

Cited by 3 publications

References 12 publications

Saturation rates of filtered back projection approximations

Saturation rates of filtered back projection approximations

The Modulo Radon Transform: Theory, Algorithms and Applications

Error analysis for filtered back projection reconstructions in Besov spaces

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