Positivity constraint implementations for multiframe blind deconvolution reconstructions

Abstract

Reconstruction of imagery degraded by atmospheric turbulence is inherently an ill-posed problem. Multiple solutions can be found which satisfy the measured data to the extent allowed by the noise statistics. The symmetric nature of a spatially invariant imaging system gives rise to an ambiguity between object and distortion. Furthermore, portions of the object spatial frequency information are often attenuated beyond usefulness. Research in this field is primarily concerned with resolving this ambiguity and making a quality estimate of the object in a timely fashion. Many of the more advanced image reconstruction algorithms are iterative algorithms, seeking to minimize error/cost or maximize likelihood/conditional probability. The estimates, of course, are random variables, and regularization and constraints are often employed to guide solutions or reduce effects of overfitting. While positivity is an often employed constraint, one can conceive of various ways to achieve this beyond the typical approach of squaring the search parameters. We evaluate several functional parameterizations as well as the use of an optimizer employing search boundaries. We also consider the order of application in conjunction with a smoothness constraint. Performance is quantified by metrics, including RMS error (spatial domain), RMS phase error curves (spatial frequency domain), and timing to characterize computational demands. We find the RMS phase error curves to be the most valuable as they assess reconstruction accuracy as a function of level of detail. Of the parameterizations evaluated, implementation of object positivity through squaring the parameters prior to application of a smoothness constraint achieved the best blend of accuracy and runtime execution speed. Development of the theoretical background in a common framework contributed to reuse of constituent quantities, both in the development of the likelihood and many gradient expressions as well as implementation of those expressions into computer code. Such reuse promotes accuracy as well as ease of new development.