Data Consistency for Linograms and Planograms

Linograms are a well-known sampling scheme for the 2-D Radon transform, with several attractive mathematical properties, and planograms are a 3-D version of them. This paper describes necessary and sufficient consistency conditions for a given function to be the linogram or planogram of some density function. In the linogram case, for both parallel and divergent-beam geometries, the consistency conditions require the nth-order moments of the projections to be equal to polynomials of degree n in the projection index variable. For planograms, similar consistency conditions apply, but are expressed in terms of homogeneous polynomials. In both 2-D and 3-D cases, the parallel and divergent consistency conditions are not obtained by a simple reparameterization of the set of lines in space. The divergent cases present different relationships between the lines than do the parallel cases, and allow finite collections of divergent projections to be treated. Straight-forward mathematical demonstrations of these results are given, based on existing theorems in the literature. The remarkable symmetry of the parallel and divergent-beam consistency conditions implies the existence of “equivalent” object-pairs for which parallel projections of one object are equal to the divergent projections of the other. The relationship between these equivalent object-pairs and a theorem on divergent projections by Edholm and Danielsson is discussed.