We provide conditions for exact reconstruction of a bandlimited function from irregular polar samples of its Radon transform. First, we prove that the Radon transform is a continuous L2-operator for certain classes of bandlimited signals. We then show that the Beurling-Malliavin condition for the radial sampling density ensures existence and uniqueness of a solution. Moreover, Jaffard's density condition is sufficient for stable reconstruction.
For efficient experimental design of CT scans, i.e. determination of a suitable sampling geometry or a posteriori choice of function spaces for reconstruction, it is essential to understand the discretization effects due to sampling. Furthermore, the emergence of CT acquisition procedures involving incomplete or irregular data calls for irregular sampling theory. Past research has focused on functions that are simultaneously essentially space- and band-limited, i.e., functions for which decay exponentially with the radius @math of the ball @math . For these functions, interleaved sampling geometries are more efficient than regular sampling geometries @cite_14 @cite_9 .
@cite_9 @cite_14
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The Radon transform of a bivariate function, which has application in tomographic imaging, has traditionally been viewed as a parametrized univariate function. In this paper, the Radon transform is instead viewed as a bivariate function and two-dimensional sampling theory is used to address sampling and information content issues. It is Shown that the band region of the Radon transform of a function with a finite space-bandwidth product is a "finite-length bowtie." Because of the special shape of this band region. "Nyquist sampling" of the Radon transform is on a hexagonal grid. This sampling grid requires approximately one-half as many samples as the rectangular grid obtained from the traditional viewpoint. It is also shown that for a nonbandlimited function of finite spatial support, the bandregion of the Radon transform is an "infinite-length bowtie." Consequently, it follows that approximately 2M2 π independent pieces of information about the function can be extracted from M "projections." These results and others follow very naturally from the two-dimensional viewpoint presented. The Mathematics of Computerized Tomography covers the relevant mathematical theory of the Radon transform and related transforms and also studies more practical questions such as stability, sampling, resolution, and accuracy. Quite a bit of attention is given to the derivation, analysis, and practical examination of reconstruction algorithm, for both standard problems and problems with incomplete data.