Tomographic Algorithms for General Line Integrals

This paper extends the constrained optimization image reconstruction techniques to curved-ray projections for a number of scanning systems and for 2-D and 3-D reconstructions. Teh treatment presented is general in that it includes the previous results on parallel-ray and divergent-ray geometries as special casss. Employing the projection data as constraints, the reconstruction problem is expressed as a generalized constrained optimization problem whose solution leads to a relationship (model) between the reconstruction and the associate Lagrange multiplier functions, thus introducting the models for a number of prescribed cost criteria and scanning systems involing curved-ray projections. Using these new models a category of subspaces of a Hirbert space, the elements of which are the functions (objects) for reconstruction, is presented. The projections and the minimum energy reconstruction are shown to be projections of the object into these new subspaces. The projection slice theorem is extended by introducing a new transform, a slice of which is equal to the Fourier series of the corresponding projection, irrespective of the projection geometry. The ART methods are extended to curved-ray scanning systems. Also a convergent iterative algorithms which evaluates the Lagrange multipliers from general line integrals and then calculates the reconstruction, is introduced. The iterations of this algorithm are identical to those of the additive ART, thus resulting in the same solution.