Error estimates for tomographic inversion

Munshi, P.; Rathore, Ram K.S.; Ram, K.S.; Kalra, M. S.

doi:10.1088/0266-5611/7/3/007

Cited by 52 publications

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References 3 publications

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“…We remark at this point that pointwise and L ∞ -error estimates on e L were proven by Munshi et al in [7,8], supported by numerical experiments in [9]. Error bounds on the L p -norm of e L , in terms of an L p -modulus of continuity of f , were proven by Madych in [6].…”

Section: Error Analysissupporting

confidence: 55%

On the error behaviour of the filtered back projection

Beckmann

Iske

2016

Proc Appl Math and Mech

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The filtered back projection (FBP) formula allows us to reconstruct bivariate functions from given Radon samples. However, the FBP formula is numerically unstable and low-pass filters with finite bandwidth and a compactly supported window function are employed to make the reconstruction by FBP less sensitive to noise. In this paper we analyse the inherent reconstruction error which is incurred by the application of a low-pass filter with finite bandwidth. We present L 2 -error estimates on Sobolev spaces of fractional order along with asymptotic convergence rates, where the filter's bandwidth goes to infinity. Filtered back projectionThe term filtered back projection (FBP) refers to a well-known and commonly used reconstruction technique in computerized tomography (CT). The classical CT reconstruction problem can be formulated as follows.Thus, the CT reconstruction problem seeks for the inversion of the Radon transform R. For a comprehensive mathematical treatment of R and its inversion, we refer to [5,11].The inversion of R is well understood and given by the classical filtered back projection formula ([4, Theorem 6.2.])which holds for f ∈ L 1 (R 2 ) ∩ C(R 2 ), and where the back projectionfor (x, y) ∈ R 2 . Note that B is the adjoint operator of R. We remark that the FBP formula (1) is highly sensitive with respect to noise and, consequently, numerically unstable. To stabilize it, we follow a standard approach and replace the factor |S| in (1) by a low-pass filter A L of the formwith finite bandwidth L > 0 and an even window function W ∈ L ∞ (R) with compact support supp(W ) ⊆ [−1, 1].By applying the low-pass filter A L (S), the reconstruction of f is no longer exact, but the resulting approximate FBP reconstruction f L can be rewritten as Error analysisIn the following, we analyse the intrinsic FBP reconstruction error e L = f − f L which is incurred by the application of the low-pass filter A L . We remark at this point that pointwise and L ∞ -error estimates on e L were proven by Munshi et al. in [7,8], supported by numerical experiments in [9]. Error bounds on the L p -norm of e L , in terms of an L p -modulus of continuity of f , were proven by Madych in [6]. Our goal is to proof L 2 -error estimates on e L for the relevant case of target functions f from Sobolev spaces of fractional order (cf. [10]), i.e., we assumewhereThe presented results are refinements and extensions of our results in [2,3]. More details and proofs, along with numerical simulations, can be found in [1].and W ∈ C([−1, 1]) with W (0) = 1. Then, the L 2 -norm of the FBP reconstruction error e L = f − f L is bounded above by (1 − W (S)) 2 (1 + L 2 S 2 ) α −→ 0 for L −→ ∞.In particular, e L L 2 (R 2 ) −→ 0 for L −→ ∞.

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Section: Error Analysissupporting

confidence: 55%

On the error behaviour of the filtered back projection

Beckmann

Iske

2016

Proc Appl Math and Mech

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“…The inner integral in equation (7) is a one-dimensional convolution and the outer integral, corresponding to the averaging operation (over θ), is termed a backprojection-hence the name convolution backprojection for this particular implementation algorithm. The CBP method is also known as the filtered backprojection algorithm because of the "filtering" of the Fourier transform of the projection datap by the window (or filter) W (R) in the initial stages of the formulation given by equation (5). The function q(s), known as the convolving function, is evaluated once and stored for repeated use for different views (or different angles θ).…”

Section: Tomographic Imagingmentioning

confidence: 99%

“…The error theory [5][6][7] for the CBP algorithm provides an interesting alternative to the fractal theory characterization of composite materials [4]. There are two sets of signatures in this approach.…”

Section: Hamming Signaturementioning

confidence: 99%

“…Tomographic inversion using equation (7) depends on W (R), and the direct error theorem states [5][6] that the pointwise inherent error inf , CT image, is proportional to W (0), the second derivative of W (R) at the Fourier space origin. For a given data-ray spacing, the error versus W (0) graph is plotted and the slope/intercept of this linear graph is the "signature" S1 of the specimen.…”

Section: Signature S1mentioning

confidence: 99%

“…Here the characterization scheme does not use the concept of fractal dimension (as was done earlier). The theory of tomographic filters [5][6][7] plays a key role in this study. CT scans have been carried out for four different data-ray spacings, and the convolution backprojection (CBP) method has been used to reconstruct cross sections with five different Hamming filters [7].…”

Section: Introductionmentioning

confidence: 99%

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