Iterative analytic extension in tomographic imaging

Gengsheng L Zeng

doi:10.1186/s42492-021-00099-5

Abstract

If a spatial-domain function has a finite support, its Fourier transform is an entire function. The Taylor series expansion of an entire function converges at every finite point in the complex plane. The analytic continuation theory suggests that a finite-sized object can be uniquely determined by its frequency components in a very small neighborhood. Trying to obtain such an exact Taylor expansion is difficult. This paper proposes an iterative algorithm to extend the measured frequency components to unmeasured regions. Computer simulations show that the proposed algorithm converges very slowly, indicating that the problem is too ill-posed to be practically solvable using available methods.

Highlights

In medical or industrial imaging, a stable image reconstruction depends on sufficient data acquisition
We propose an iterative algorithm trying to extend the measured frequency components to unmeasured frequency components
The algorithm is in the projections onto convex sets” (POCS) form, alternating between the spatial and frequency domains

Summary

Introduction

In medical or industrial imaging, a stable image reconstruction depends on sufficient data acquisition. The data acquisition geometry required for a stable reconstruction is different from that demanded theoretically. As pointed out in Naterer’s book [1], a stable reconstruction in parallel-beam imaging requires the angular detector coverage of 180°. It is theoretically possible to perform limited-angle tomography, for example, with a smaller angular coverage of just 10°. [1], it is an extremely ill-posed problem to reconstruct an image from a very small angular coverage. It is practically impossible to stably reconstruct an image with a very small angular coverage with noisy measurements. The ill-condition characters were mathematically established by studying the spectrum of the singular values of the limited data tomography [1]

Methods

Results

Discussion

Conclusion