Constraint optimization algorithms for digital image reconstruction from projections

The reconstructing of an image from its projections is formulated and solved as a constraint optimization problem. A general cost criterion is optimized and the result is applied to various specific cost functions such as entropy; variance and local discontinuity producing several relationships (models) between the image and the Lagrange multipliers. The frequency domain counterparts of these models create new interpolation schemes of which a Eessel type is used extensively. Derivative dependent cost criteria lead to smoother reconstructions. Employing the block successive relaxation iterative method and exploiting regions of equal elements in the new matrices of the linear system for the variance model, a stable algorithm is given and its convergence ib proven. This algorithm is very fast without employing any approximation of the matrices involved. Using the back projection model a non recursive algorithm is presented based on the block diagonalization of the permuted block circulant matrices introduced here. This diagonalization is achieved by the block Fourier transform. The optimization procedure was extended for the reconstruction from fan beam projections. The fan beam counterpart of the above non recursive algorithm is presented. The Fourier transform of a fan beam projection gives a slice of a new transform, the angular projection transform. A convolution algorithm based on this projection slice theorem is also presented.