Least-squares reconstruction of objects with missing high-frequency components

A least-squares estimation procedure was recently proposed for the restoration of an object that has had some high-frequency components removed [ J. Opt. Soc. Am.71, 95 ( 1981)]. We provide further discussion of the use of least-squares techniques for this purpose. We use the singular-value decomposition (SVD) of an appropriate matrix to explore the relationships among bandwidth, measurement noise, a priori constraints on the object, and the quality of the restoration. We show how the effects of ill conditioning, which arise as the bandwidth of the observation is reduced, can be mitigated by using an appropriate regularization technique. Finally, we describe a conjugate gradient descent (CGD) algorithm that yields a reconstruction nearly identical with that obtained by using the regularized SVD algorithm. The CGD algorithm has been adapted to two-dimensional objects for which the computational complexity of the SVD algorithm is impracticably high.