An investigation on FBP reconstruction for attenuated radon transform with partial data

Abstract

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Objectives: Filtered backprojection (FBP) inversion of the Radon transform for half-scan data has been established, and its extension to inversion of the exponential Radon transform was proved mathematically by Rullgård. With the success of FBP-type inversion of the attenuated Radon transform for full-scan data as reported by Novikov, the stability of inverting the attenuated Radon transform for half scans was recently explored by Rullgård while the inversion still remains as an open problem. This work aims to answer the open problem by both theoretical analysis and numerical simulation. Methods: In theoretical aspect, we assume the half-scan problem as an inversion of partial data by the Novikov’s formula of full scans, and address the partial data inversion by band-limited signal extrapolation strategy. We propose an extrapolative FBP-type algorithm, which compensates for the non-uniform attenuation via the Novikov’s inversion formula and degenerates to the well-known iterative algorithm of Gerchberg and Papoulis when the attenuation vanishes. If the C1 norm of the attenuation map is sufficiently small (while still remains valid in most clinical situations), the algorithm is proven to be exponentially convergent for half-scan data. Results: The presented extrapolative FBP-type algorithm was tested by a modified Shepp-Logan digital phantom with non-uniform attenuating media. In noise-free cases, the extrapolative algorithm of half scans produced almost identical results as both the Novikov’ formula of full scans and the OS-EM method of half scans did. With Poisson noise presence, both the extrapolative FBP-type algorithm of half scans and the Novikov’ inversion formula of full scans generated similar results while the OS-EM method of half scans produced obvious checkerboard noise patterns. Conclusions: The solution for inversion of the attenuated Radon transform from half-scan data exists and is mathematically stable. While an explicit inversion formula has not been derived yet, our extrapolative approach proved a way converging to the solution and demonstrated the same performance as the full-scan inversion.