Hazardous Continuation Backward in Time in Nonlinear Parabolic Equations, and an Experiment in Deblurring Nonlinearly Blurred Imagery.

Carasso AS - J Res Natl Inst Stand Technol (2013)

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f1-jres.118.010: Behavior of Hölder exponent µ(t) in inequality (4) reflects rate at which the forward evolution equation wt=Lw has forgotten the past, as t increases from t = 0 to t=T = 1. Deviations away from a linear, time-independent, self adjoint spatial differential operator L, can lead to exponential decay in µ(t), t ↓ 0 and affect backward reconstruction from t=T.
Mentions: In many engineering or applied science contexts, only educated guesses would generally be available to estimate δ and M, rather than exact values. Typically, the L2 relative error(5)‖w(.,T)−f‖2/‖w(.,T)‖2≤δ/{‖f‖2−δ}≈δ/‖f‖2,might be expected to be on the order of 1% or thereabouts. Since the given data f (x) may simultaneously approximate several distinct solutions wp(x, t) of Eq. (2) at time T, there are, in general, infinitely many possible solutions of Eqs. (2) and (3). If δ is small, it is generally assumed that any two such solutions would differ only slightly. The extent to which this expectation is justified depends on the decay behavior in the Hölder exponent µ(t) as illustrated in Fig. 1. In the best possible case, that of a linear self adjoint elliptic operator L with time-independent coefficients, we have µ(t) = t/T, so that µ(t) decays linearly to zero as continuation progresses from t=T to t = 0. At t=T/2, we have µ (T/2) = 1/2, and . This indicates a loss of acccuracy from O(δ) to , while still only half way to t = 0. More typically, µ(t) is sublinear in t, possibly with rapid exponential decay. This can lead to much more severe loss of accuracy as reconstruction progresses to t = 0. Such rapid decay of µ to zero can be brought about by various factors, including nonlinearity, non self adjointness, diffusion coefficients that grow rapidly with time, or adverse spectral properties in the elliptic operator L. In all cases, Eq. (4) does not guarantee any accuracy at t = 0, but only provides the redundant information ‖w1(.,0) − w2(.,0)‖2 ≤ 2M.

Bottom Line: Visually and statistically indistinguishable blurred images are presented, with vastly different deblurring results.These examples indicate that how an image is nonlinearly blurred is critical, in addition to the amount of blur.The equations studied represent nonlinear generalizations of Brownian motion, and the blurred images may be interpreted as visually expressing the results of novel stochastic processes.

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Affiliation: National Institute of Standards and Technology, Gaithersburg, MD 20899.

ABSTRACT
Identifying sources of ground water pollution, and deblurring nanoscale imagery as well as astronomical galaxy images, are two important applications involving numerical computation of parabolic equations backward in time. Surprisingly, very little is known about backward continuation in nonlinear parabolic equations. In this paper, an iterative procedure originating in spectroscopy in the 1930's, is adapted into a useful tool for solving a wide class of 2D nonlinear backward parabolic equations. In addition, previously unsuspected difficulties are uncovered that may preclude useful backward continuation in parabolic equations deviating too strongly from the linear, autonomous, self adjoint, canonical model. This paper explores backward continuation in selected 2D nonlinear equations, by creating fictitious blurred images obtained by using several sharp images as initial data in these equations, and capturing the corresponding solutions at some positive time T. Successful backward continuation from t=T to t = 0, would recover the original sharp image. Visual recognition provides meaningful evaluation of the degree of success or failure in the reconstructed solutions. Instructive examples are developed, illustrating the unexpected influence of certain types of nonlinearities. Visually and statistically indistinguishable blurred images are presented, with vastly different deblurring results. These examples indicate that how an image is nonlinearly blurred is critical, in addition to the amount of blur. The equations studied represent nonlinear generalizations of Brownian motion, and the blurred images may be interpreted as visually expressing the results of novel stochastic processes.

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