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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 (H) was successfully deblurred after 100 iterations. Visually indistinguishable image (I), blurred with stronger nonlinearities, could not be usefully deblurred.
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f8-jres.118.010: Nonlinearly blurred image (H) was successfully deblurred after 100 iterations. Visually indistinguishable image (I), blurred with stronger nonlinearities, could not be usefully deblurred.

Mentions: Figure 8 displays the results of deblurring these two images. Despite the strongly nonlinear blurring in image (H), through the function s(w) and the inclusion of both the and wwy terms in Eq. (13), reasonably good results are obtained after 100 Van Cittert iterations. The carrier’s command ‘island’ has been recovered, along with the two rows of planes on deck. As shown in Table 6, the values of ‖f‖1 and ‖f‖2 in the deblurred image (H), are very close to their original values, while ‖∇f‖1 has recovered 78 % of its value in image (G). Also, deblurring in image (H) has increased the PSNR from 20 to 23. It is noteworthy that the term in Eq. (13) did not preclude useful reconstruction in image (H). Surprisingly, deblurring in image (I) was not successful. There is no high frequency noise amplification in the deblurred image (I), even after 100 iterations, and the values of ‖f‖1 and ‖f‖2 are only about 3 % higher than their true values in image (G), as shown in Table 6. As was the case in the deblurred Marilyn Monroe image (F), substantial sharpening has occurred in the deblurred image (I), but the sharpened image is seriously marred by artifacts. Because of the moderating effect of the factor cos2w, it was not anticipated that the term 1.5w(cos2w)wy in Eq. (13) might be detrimental in image (I), since the term


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 (H) was successfully deblurred after 100 iterations. Visually indistinguishable image (I), blurred with stronger nonlinearities, could not be usefully deblurred.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f8-jres.118.010: Nonlinearly blurred image (H) was successfully deblurred after 100 iterations. Visually indistinguishable image (I), blurred with stronger nonlinearities, could not be usefully deblurred.
Mentions: Figure 8 displays the results of deblurring these two images. Despite the strongly nonlinear blurring in image (H), through the function s(w) and the inclusion of both the and wwy terms in Eq. (13), reasonably good results are obtained after 100 Van Cittert iterations. The carrier’s command ‘island’ has been recovered, along with the two rows of planes on deck. As shown in Table 6, the values of ‖f‖1 and ‖f‖2 in the deblurred image (H), are very close to their original values, while ‖∇f‖1 has recovered 78 % of its value in image (G). Also, deblurring in image (H) has increased the PSNR from 20 to 23. It is noteworthy that the term in Eq. (13) did not preclude useful reconstruction in image (H). Surprisingly, deblurring in image (I) was not successful. There is no high frequency noise amplification in the deblurred image (I), even after 100 iterations, and the values of ‖f‖1 and ‖f‖2 are only about 3 % higher than their true values in image (G), as shown in Table 6. As was the case in the deblurred Marilyn Monroe image (F), substantial sharpening has occurred in the deblurred image (I), but the sharpened image is seriously marred by artifacts. Because of the moderating effect of the factor cos2w, it was not anticipated that the term 1.5w(cos2w)wy in Eq. (13) might be detrimental in image (I), since the term

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.