Project B11 (finished)

Project B11 - Error Estimates and Convergence Rates for Filtered Back Projection Reconstructions

Principal Investigator: Armin Iske

Background and Motivation

The method of filtered back projection (FBP) is a commonly used reconstruction technique in computerized tomography, which allows us to recover an unknown bivariate function from the knowledge of its Radon data. The reconstruction is based on the classical FBP formula, which yields an analytical inversion of the Radon transform provided that the complete Radon data is available. The FBP formula, however, is highly sensitive with respect to noise and, hence, numerically unstable. To overcome this problem, suitable low-pass filters of finite bandwidth and with compactly supported window functions are employed. This reduces the noise sensitivity, but only leads to an inexact approximation of the target function.

Aims and Objectives

The main objective of this project is to analyse the inherent FBP reconstruction error which is incurred by the application of the low-pass filter. To this end, we will focus on error estimates in Sobolev spaces of fractional order and, moreover, we will work on quantitative criteria to a priori evaluate the performance of the utilized low-pass filter by means of its window function. The error bounds are analyzed with respect to the bandwidth of the low-pass filter, the flatness of the filter’s window function at the origin, on the smoothness of the target function, and on the order of the considered Sobolev norm in which the reconstruction error is measured. Further, we aim to prove convergence for the approximate FBP reconstruction in the treated Sobolev norms along with asymptotic convergence rates as the filter’s bandwidth goes to infinity, where we in particular investigate on the saturation at fractional order depending on smoothness properties of the filter’s window function.
Finally, convergence rates for noisy data as the noise level goes to zero will be developed, where we prove estimates for the data error and combine these with our results for the approximation error. To this end, the filter’s bandwidth is coupled with the noise level to achieve the convergence. The theoretical results will be supported by numerical experiments.

PhD Student: Matthias Beckmann

Matthias Beckmann successfully defended his PhD on 17 May 2018

Publications

Matthias Beckmann and Armin Iske: Error Estimates and Convergence Rates for Filtered Back Projection.
Mathematics of Computation 88, 2019, 801-835.

M. Beckmann and A. Iske: Sobolev error estimates for filtered back projection reconstructions.
IEEE International Conference Sampling Theory and Applications (SampTA2017), 2017, 251-255.
DOI: 10.1109/SAMPTA.2017.8024439.

M. Beckmann and A. Iske: On the error behaviour of the filtered back projection.
Proc. Appl. Math. Mech. (PAMM) 16(1), 2016, 833-834.

M. Beckmann and A. Iske: Error Estimates for Filtered Back Projection.
IEEE International Conference Sampling Theory and Applications (SampTA2015), 2015, 553-557.
DOI: 10.1109/SAMPTA.2015.7148952.