Bochner subordination, logarithmic diffusion equations, and blind deconvolution of hubble space telescope imagery and other scientific data

Abstract

Abstract. Generalized Linnik processes and associated logarithmic diffusion equations can be constructed by appropriate Bochner randomization of the time variable in Brownian motion and the related heat conduction equation. Remarkably, over a large but finite frequency range, generalized Linnik characteristic functions can exhibit almost Gaussian behavior near the origin, while behaving like low exponent isotropic Lévy stable laws away from the origin. Such behavior matches Fourier domain behavior in a large class of real blurred images of considerable scientific interest, including Hubble space telescope imagery and scanning electron micrographs. This paper develops a powerful blind deconvolution procedure based on postulating system optical transfer functions (otf) in the form of generalized Linnik characteristic functions. The system otf and 'true' sharp image are then reconstructed by solving a related logarithmic diffusion equation backwards in time, using the blurred image as data at time t = 1. The present methodology significantly improves upon previous work based on system otfs in the form of Lévy stable characteristic functions. Such improvement is validated by the substantially smaller image Lipschitz exponents that ensue, confirming increased fine structure recovery. These results resolve the unexplained appearance of exceptionally low Lévy stable exponents in previous work on the same class of images. The paper is illustrated with striking enhancements of gray scale and colored images.

Reconstructing the past from imprecise knowledge of the present :

“…Indeed, successful recoveries of contaminant plumes in hydrology, as well as striking enhancement of Hubble telescope galaxy images, have been documented [5], [6], [14]. Nevertheless, the behavior of the Hölder exponents in (13) reflects a basic underlying truth.…”

“…Using f (x) in (14), the Van Cittert procedure was applied for 1000 iterations and resulted in h 1000 (x) = w green 0 (x), shown as the green trace in Figure 2 (bottom), with a final L ∞ residual of 1.4E-3, and an L 2 relative error at t = 1, f − S[h 1000 ] 2 / f 2 = 0.023%. Both of these values are noticeably smaller than was the case in Example 1.…”

“…With initial data w red 0 (x), the parabolic problem w t = Lw is first integrated up to time t = 1, to produce a presumed good approximation f (x) to the unknown true solution w red (x, 1). This calculated f (x), shown as a black curve in the figures, is then viewed as given data in an applied science or engineering context and is used in the Van Cittert procedure (14) to recover the unknown initial value w f (x, 0) that gave rise to f (x) at t = 1. However, it will turn out that f (x) is also a close approximation to an unsuspected additional solution w green (x, 1) corresponding to initial data w green 0 (x) that can be vastly different from w red 0 (x).…”

“…To implement the iterative process in (14), use is made of an efficient, highly accurate parabolic equation solver. This method of lines procedure is discussed in [8] and is implemented as subroutine D03PDF/D03PDA in the NAG Mathematical Software Library.…”

“…Such notions have also been found useful in image deblurring. In [14], and references therein, backward in time problems for fractional and/or logarithmic diffusion equations,…”