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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)

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.

View Article: PubMed Central - PubMed

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.

No MeSH data available.


Nonlinearly blurred image (B) was successfully deblurred after 100 iterations. Visually similar image (C), blurred with additional nonlinearities, could not be deblurred.
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f4-jres.118.010: Nonlinearly blurred image (B) was successfully deblurred after 100 iterations. Visually similar image (C), blurred with additional nonlinearities, could not be deblurred.

Mentions: Figure 4 displays the results of backward in time continuation in Eq. (11), using the Van Cittert iteration in Eq. (17), with as in Eq. (19), and λ = 0.5. Remarkably, despite the strongly nonlinear blurring in image (B) through the function r(w) in Eq. (12), useful deblurring of that image is obtained after 100 iterations. From Table 2, we see that the values of‖f‖1 and ‖f‖2 in the deblurred image (B), are very close to their original values in image (A), while ‖∇f‖1 has recovered almost 90 % of its original value. Also, deblurring in image (B) has increased the PSNR from 25 to 34.


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)

Nonlinearly blurred image (B) was successfully deblurred after 100 iterations. Visually similar image (C), blurred with additional nonlinearities, could not be deblurred.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4487310&req=5

f4-jres.118.010: Nonlinearly blurred image (B) was successfully deblurred after 100 iterations. Visually similar image (C), blurred with additional nonlinearities, could not be deblurred.
Mentions: Figure 4 displays the results of backward in time continuation in Eq. (11), using the Van Cittert iteration in Eq. (17), with as in Eq. (19), and λ = 0.5. Remarkably, despite the strongly nonlinear blurring in image (B) through the function r(w) in Eq. (12), useful deblurring of that image is obtained after 100 iterations. From Table 2, we see that the values of‖f‖1 and ‖f‖2 in the deblurred image (B), are very close to their original values in image (A), while ‖∇f‖1 has recovered almost 90 % of its original value. Also, deblurring in image (B) has increased the PSNR from 25 to 34.

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.

View Article: PubMed Central - PubMed

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.

No MeSH data available.