Sampling the 2-D Radon transform

Abstract

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 2M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /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.