Trevor J. Hall

Image restoration and resolution enhancement

The ill-posed problem of restoring object information from finitely many measurements of its spectrum can be solved by using the best approximation in Hilbert spaces appropriately designed to include a priori information about object extent and shape and noise statistics. The procedures that are derived are noniterative, the linear ones extending the minimum-energy band-limited extrapolation methods (and thus related to Gerchberg–Papoulis iteration) and the nonlinear ones generalizing Burg’s maximum-entropy reconstruction of nonnegative objects.

Stable, noniterative object reconstruction from incomplete data using a priori knowledge

Object reconstruction in weighted Hilbert space is related to the Miller regularization theory [ MillerK., SIAM J. Math. Anal.1, 52 ( 1970)]. Experimental results illustrating the power of the method are presented.

Nonuniqueness of superresolution techniques applied to sampled data

It is demonstrated that, given a finite number of samples of a function, the constraints of square integrability and band limitation are not sufficient to determine that function uniquely. Hence existing algorithms for band-limited interpolation and extrapolation (e.g., for super resolution) require additional constraints such as those provided by an appropriate model.

Image compression and restoration incorporating prior knowledge

It is shown how to incorporate prior knowledge into the restoration of an image from a finite number of its expansion coefficients in some basis. The significant image enhancement obtained for simple prior knowledge suggests an application to image-transform coding. A computer simulation is presented that demonstrates that compression ratios greater than 13:1 are possible.

On coupled-wave theory of two-beam self-diffraction

Coupled-wave equations are derived, which describe, in the geometrical-optics limit, the interaction between two beams in a medium having intensity dependent permittivity. These equations are interpreted physically and a solution is presented for the special case of non-uniform plane waves. New results on energy exchange and beam profile distortion are obtained and are contrasted with known results for static grating.

Sampling and interpolation in two dimensions

The aim of this paper is to consider the interpretation of limited data sets to represent functions which are bandlimited. It is shown that limited data leads to a degree of non-uniqueness of interpolation and extrapolation that only the imposition of a priori constraints can reduce. An approach is described to enable a priori information to be incorporated in a Fourier reconstruction scheme.

Optical phase conjugation for the study of the analytic properties of optical fields

The importance of the analytic character of optical fields is discussed, in particular the encoding of information by complex zeros. The properties of the real part of the complex field are outlined and its usefulness in determining the zero distribution for a field from intensity data is presented. The real part of an optical field may be formed by optical phase conjugation.