An optimal technique for constraint-based image restoration and reconstruction

A new technique for finding an optimal feasible solution to the general image reconstruction and restoration problem is described. This method allows the use of prior knowledge of the properties of both the solution and any noise present on the data. The problem is formulated as the optimization of a cost function over the intersection of a number of convex constraint sets; each set being defined as containing those solutions consistent with a particular constraint. A duality theorem is then applied to yield a dual problem in which the unknown image is replaced by a model defined in terms of a finite dimensional parameter vector and the kernels of the integral equations relating the data and solution. The dual problem may then be solved for the model parameters using a gradient descent algorithm. This method serves as an alternative to the primal constrained optimization and projection onto convex sets (POCS) algorithms. Problems in which this new approach is appropriate are discussed. An example is given for image reconstruction from noisy projection data; applying the dual method results in a fast nonlinear algorithm. Simulation results demonstrate the superiority of the optimal feasible solution over one obtained using a suboptimal approach.