Geometric Aspects Of Tomographic Signal Processing
Tomographic signal processing encompasses a wide variety of applications in such areas as medical imaging, geophysical and oceanographic signal processing, machine vision, and astrophysics. All such applications, however, share a common underlying structure. An object is to be reconstructed from its noisy, and possibly sparse!, line-integral projections. Mathematically, the problem is that of inversion of the Radon-transform of a function from noisy data. If one assumes that the object in question is binary, i.e. has constant intensity, then the line-integral projection data take on a very rich geometxic structure. Specifically, the projection data can be interpreted as chord-length measurements across the interior of the object. In this case, the problem can be regarded as one of reconstructing the boundary of a binary object given noisy, and possibly sparse, chord-length measurements across the object. In this paper we describe those geometric constraints that arise from the above interpretation. These constraints include: 1. Constraints arising from prior assumptions on the shape of the object such as convexity, smoothness, etc; 2. Constraints on the number of projections needed so that a unique reconstruction is obtained. These constraints may differ according to the type of projection being applied (fan-beam, or parallel-beam), and are critical in identifying redundancy in the data and to the design of efficient processing algorithms; 3. A geometric interpretation of the Radon-Transform Constraints. We will show that the incorporation of these constraints is critical to the reconstruction when the data is sparse and/or the signal-to-noise ration is low. Furthermore, we demonstrate that the inclusion of these constraints in the reconstruction algorithms will aid the reconstruction process even if the data is not sparse or excessively noisy. lThis work was supported by the National Science Foundation under Grant 9015281-MIP, the Office of Naval Research under Grant N00014-91-J-1004, and the US Army Research Office under Contract DAAL0386-K-0171.