Mathematical extrapolation of image spectrum for constraint-set design and set-theoretic superresolution

Abstract

Several powerful iterative algorithms are being developed for the restoration and superresolution of diffraction-limited imagery data by use of diverse mathematical techniques. Notwithstanding the mathematical sophistication of the approaches used in their development and the potential for resolution enhancement possible with their implementation, the spectrum extrapolation that is central to superresolution comes in these algorithms only as a by-product and needs to be checked only after the completion of the processing steps to ensure that an expansion of the image bandwidth has indeed occurred. To overcome this limitation, a new approach of mathematically extrapolating the image spectrum and employing it to design constraint sets for implementing set-theoretic estimation procedures is described. Performance evaluation of a specific projection-onto-convex-sets algorithm by using this approach for the restoration and superresolution of degraded images is outlined. The primary goal of the method presented is to expand the power spectrum of the input image beyond the range of the sensor that captured the image.

Estimating missing information by maximum likelihood deconvolution

“…This regain of information seems to be increasing with the number of iterations. However, it remains unclear, whether this energy regain is based on anything related to analytical continuation (Bhattacharjee and Sundareshan, 2003;Sementilli et al, 1993;Conchello, 1998) or rather on implicit assumptions in the algorithm about ''typical objects''. Implicit assumptions are that allowed objects are restricted to be positive everywhere.…”

Mathematical concepts of optical superresolution

“…Comparison of the GP algorithm with a direct (non-iterative) restoration scheme based on the singular value decomposition and the least-squares solution is discussed by Walsh and Nielsen-Delaney [221]. More recently, an iterative algorithm based on POCS approach that ensures a priori specified level of spectral extrapolation has been proposed [222]. Another development of the GP algorithm is the prior discrete Fourier transformation (PDFT) algorithm [4].…”

Resolution-enhanced imaging using interferenceless coded aperture correlation holography with sparse point response

“…In the NLR technique, the magnitudes, Î OBJ and Î PSH , are raised to the power of o and r, respectively, where Î OBJ and Î PSH are the Fourier transforms of I OBJ and I PSH given in Eqs. (7) and (10), respectively. In the NLR optimization procedure, only the spectral magnitudes of the object, and of the reconstructing function, are raised to the power of o and r, respectively, while the phase information remains intact.…”

“…Object information in this context means the spatial spectrum of the object's image. Spatial filtering 6 , digital post-processing 7 and engineering the point spread function 8 are a few examples of resolution enhancement with the same spatial bandwidth used by the original systems. In general, the methods based on the same spatial bandwidth have been found less effective in a practical noisy environment.…”