The Gerchberg and POCS approaches in spectrum extrapolation of band-pass tomographic images

The aim of this study is to theoretically justify the use of a super-resolution algorithm, similar to the well-known Gerchberg-Papoulis (GP) method, for the enhancement of images acquired by means of a narrow-band electromagnetic tomographic system studied and experimented at IEI-CNR in recent years. Some experimental results are also reported in order to demonstrate the practical effectiveness of the algorithm. Reality and positivity constraints are included in the iterative enhancement scheme. Positivity is a nonlinear constraint and, although nonlinear GP-like techniques have been successfully applied in several fields, a theoretical problem arises in that the convergence of the method cannot be demonstrated following the Gerchberg's guide-lines, which are based on the linearity of the constraints. Following [17] and [19], an iterative Projection Onto Convex Sets (POCS) super-resolution algorithm suitable for band-pass images will be derived. This algorithm can be seen as a generalization of GP. The examples taken from practical applications presented here show that the proposed algorithm is fast and powerful, especially in reducing the ambiguity artifacts inherent in the particular type of images treated.