Limited angle tomography and ridge functions

Over the years a number of tomographic applications have emerged in which only limited ranges of projection views are available, such as breast imaging PET scanners based on a pair of parallel planar or curved detectors. This paper addresses the topic of ridge functions and their possible usefulness for tomographic reconstruction from projections distributed in a limited range of views. In this work we derive the basis for speeding up the ridge functions procedure of reconstruction front equally spaced directions which constitute a part of the whole range of views. We describe the possibilities of fast reconstruction in terms of analytical inversion of a certain matrix, for which inversion was known only for the case of a complete range of views. A new inversion formula reducing the dimensionality of the reconstruction problem is derived. Two problems of reconstruction from truncated fan beam projections within the limited angle of view are considered with relevance to PET breast scanning geometries. Algebraic techniques for their solution are proposed. Numerical results of test reconstructions are presented and possible extensions to the 3D case are discussed.