Projection based image restoration, super-resolution and error correction codes

Abstract

Super-resolution is the ability of a restoration algorithm to restore meaningful spatial frequency content beyond the diffraction limit of the imaging system. The Gerchberg-Papoulis (GP) algorithm is one of the most celebrated algorithms for super-resolution. The GP algorithm is conceptually simple and demonstrates the importance of using a priori information in the formation of the object estimate. In the first part of this dissertation the continuous GP algorithm is discussed in detail and shown to be a projection on convex sets algorithm. The discrete GP algorithm is shown to converge in the exactly-, over- and under-determined cases. A direct formula for the computation of the estimate at the kth iteration and at convergence is given. This analysis of the discrete GP algorithm sets the stage to connect super-resolution to error-correction codes. Reed-Solomon codes are used for error-correction in magnetic recording devices, compact disk players and by NASA for space communications. Reed-Solomon codes have a very simple description when analyzed with the Fourier transform. This signal processing approach to error-correction codes allows the error-correction problem to be compared with the super-resolution problem. The GP algorithm for super-resolution is shown to be equivalent to the correction of errors with a Reed-Solomon code over an erasure channel. The Restoration from Magnitude (RFM) problem seeks to recover a signal from the magnitude of the spectrum. This problem has applications to imaging through a turbulent atmosphere. The turbulent atmosphere causes localized changes in the index of refraction and introduces different phase delays in the data collected. Synthetic aperture radar (SAR) and hyperspectral imaging systems are capable of simultaneously recording multiple images of different polarizations or wavelengths. Each of these images will experience the same turbulent atmosphere and have a common phase distortion. A projection based restoration algorithm for the simultaneous restoration of pairs of images experiencing a common phase distortion is presented.