Sudhakar Prasad (University of New Mexico), Xuan Luo (University of New Mexico)
Keywords: Imaging
Abstract:
A finite support of the object brightness distribution causes band-limited optical image data to contain information about that distribution at spatial frequencies above the optical band edge. Such information can be extracted from the image data, thus enabling optical superresolution (OSR). OSR attained in this way, as we shall show, is largely independent of the degree of detector undersampling, or digital sub-resolution. We shall present a unified Fisher-information-theoretic analysis of the processes of support-induced digital and optical superresolution of a sequence of sub-pixel-shifted low-resolution images in one (1D) and two (2D) spatial dimensions. We first express the object spatial spectrum as a linear superposition over its samples via a sampling type of expansion. This expansion captures quantitatively the extent of the coupling of information pertaining to superresolving (SR) frequency samples into the measurement band and thus into the image data. We then use the concept of Fisher information to determine the minimum error bounds on an unbiased estimation of these SR frequency samples, or equivalently the maximum possible fidelity of recovery of such information. The Fourier-domain sampling function is the usual sinc function in 1D, a direct product of two sinc functions for a 2D rectangular support, and closely related to the Fourier-Bessel function for a 2D circular support. We analyze these three cases in great detail with the help of FI and associated Cramer-Rao bounds under a variety of noise and other operating conditions. In this work we also verify a number of conclusions reached by previous researchers, including the extreme difficulty of obtaining even a modest frequency extrapolation with the help of support alone and a spatially-varying resolution improvement for bright sources.
This work attempts to clarify a number of foundational issues about digital and optical superresolution from an estimation-theoretic viewpoint, particularly the essential role of prior knowledge and the enormous difficulty in achieving any significant bandwidth extension in an unbiased reconstruction. We shall also present our preliminary results on a general computational approach for addressing arbitrarily complicated support geometries in our FI-based analysis.
Date of Conference: September 1-4. 2009
Track: Imaging