Error analysis of tomographic filters II: results

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

doi:10.1016/0963-8695(93)90353-v

Cited by 31 publications

(17 citation statements)

References 2 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: 54%

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: 54%

On the error behaviour of the filtered back projection

Beckmann

Iske

2016

Proc Appl Math and Mech

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“…Munshi et al [5] followed this work and carried out a precise quantification of these errors resulting in an approximate error formula for the CBP algorithm [6]. This formula was verified numerically [7] as well as experimentally on three different CT scanners located in Germany, Australia and England [8][9][10]. We thus now have precise prediction of error for the CBP algorithm.…”

Section: Convolution Back Projectionmentioning

confidence: 93%

Tomography

Munshi

2007

Reson

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Tomography is traditionally associated with medical 'CT' scanners that have been in use for over 30 years now. This mathematical concept is now applied widely by scientists and engineers in a variety of measurements involving solids, fluid, gases and plasmas. Instrumentation based on X-rays, gammarays, lasers and acoustics have been used successfully in these applications. Inherent error in such measurements can be also be estimated thus making tomographic imaging a very powerful non-invasive technique for field measurements.The concept of tomography involves (a) 'seeing' inside optically opaque solid objects without cutting them, and (b) making measurements inside a non-solid system in a non-invasive manner. The advent of computers resulted in renaming of this concept as 'computerized tomography' and is abbreviated as CT. The most popular application of CT has been in the field of medical diagnostics where it was known as X-ray 'CAT Scan' during 1970s. The inventor of this imaging machine, G.N. Hounsfield, was awarded the Nobel Prize in 1979 along with A. Cormack who demonstrated that the mathematics behind tomography is indeed applicable for real objects. The social contribution to X-ray CT scanners is immense as medical experts can image organs at a much higher spatial resolution compared to conventional X-ray scanners and to even diagnose cancer at an early stage.Hounsfield [1] used the techniques of linear algebra to obtain CT images from tomographic measurements, known as 'projections'. Fourier based methods of image reconstruction were not used due to difficulties in the numerical implementation of the algorithm. Around the same time, Ramachandran and Lakshminarayanan [2] published an alternative method for producing CT images from projection data (Figures 1 and 2). This landmark paper revolutionized the medical imaging field and all major manufacturers Prabhat Munshi is a Professor of MechanicalEngineering at IIT Kanpur. His major areas of his interest are computerized tomography and nuclear reactor safety analysis.

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“…The algorithm works with the one dimensional projection data of each plane of the three-dimensional field. The use of CBP in the present work has been preferred because of its significant advantages such as non-iterative character, availability of analytical results on convergence and established error estimates [18,19]. Here, the reconstructed function f(r,f), is evaluated by the integration formula [20,21]:…”

Section: Convolution Back Projectionmentioning

confidence: 99%