CONE-BEAM TOMOGRAPHY WITH DISCRETE-DATA SETS

Sufficientcy conditions for cone-beam data are well known for the case of continuous data collection along a cone-vertex curve with continuo us detectors. These continuous conditions are inadequate for real-worl d data where discrete vertex geometries and discrete detector arrays a re used. In this paper we present a theoretical formulation of cone-be am tomography with arbitrary discrete arrays of detectors and vertices . The theory models the imaging system as a linear continuous-to-discr ete mapping and represents the continuous object exactly as a Fourier series. The reconstruction problem is posed as the estimation of some subset of the Fourier coefficients. The main goal of the theory is to determine which Fourier coefficients can be reliably determined from t he data delivered by a specific discrete design. A Fourier component w ill be well determined by the data if it satisfies two conditions: it makes a strong contribution to the data, and this contribution is rela tively independent of the contribution of other Fourier components. To make these considerations precise, we introduce a concept called the cross-talk matrix. A diagonal element of this matrix measures the stre ngth of a Fourier component in the data, while an off-diagonal element quantifies the dependence or aliasing of two different components. On e reasonable approach to system design is to attempt to make the diago nal elements of this matrix large and the off-diagonal elements small for some set of Fourier components. If this goal can be achieved, simp le linear reconstruction algorithms are available for estimating the F ourier coefficients. To illustrate the usefulness of this approach, nu merical results on the cross-talk matrix are presented for different d iscrete geometries derived from a continuous helical vertex orbit, and simulated images reconstructed with two linear algorithms are present ed.