A new formulation of an iterative restoration algorithm is presented in this dissertation for the solution of noisy, non-linear restoration problems. The constraints and distortions are assumed to be known and are modeled as general (not necessarily linear) projections. The constraints are divided into two fundamentally different types, system constraints and signal constraints, of which only the latter are susceptible to noise. Imposing the system constraints and imposing the signal constraints are performed as two separate steps. Comparison of the signals produced by each step enables one to determine whether or not the generated sequence of signals is converging. Without the separation of steps certain types of errors would be undetectable. The problem caused by additive noise is overcome by modifying the replacement step. This simple change improves the quality of the reconstructed or restored signal and the convergence properties of the algorithm. The replacement step is changed from the direct substitution of given signal values to the operation of projecting onto a set surrounding these given values. Projection onto a surrounding hypersphere, called partial replacement, is the simplest case. Projection onto a surrounding hyperellipsoid, called windowed replacement, is a more complicated non-linear case that can take into account specific properties of the noise to more effectively restore the original signal. Both of these cases are developed and examined.