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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.


Related in: MedlinePlus

Nonlinear parabolic blurring of sharp MRI brain image g(x, y), by using it as initial values in Eq. (11) with two different sets of values for the constants a,b.
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f3-jres.118.010: Nonlinear parabolic blurring of sharp MRI brain image g(x, y), by using it as initial values in Eq. (11) with two different sets of values for the constants a,b.

Mentions: In Fig. 3, the original sharp MRI brain image (A) is blurred to form image (B), by applying the finite difference scheme in Eq. (18) to the parabolic equation Eq. (11), with coefficients a=b = 0. A different blurred image is then obtained, image (C), by repeating this process with coefficients a = 1.25, b = 0.6. Images (B) and (C) appear very similar in quality, and, from Table 1, both these images have almost the same values for ‖f‖1, ‖f‖2 and ‖∇f‖1. In particular, ‖∇f‖1 has been reduced by almost a factor of two from its original value in image (A), reflecting substantial blurring. The PSNR value in image (C) is noticeably smaller than in image (B), indicating greater degradation in image (C). However, since the PSNR metric requires knowledge of the original sharp image, in practice, such increased degradation in image (C) would not be known to a user. In fact, both images (B) and (C) appear to have been blurred, more or less equally, by convolution with a type of Gaussian-like point spread function.


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)

Nonlinear parabolic blurring of sharp MRI brain image g(x, y), by using it as initial values in Eq. (11) with two different sets of values for the constants a,b.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f3-jres.118.010: Nonlinear parabolic blurring of sharp MRI brain image g(x, y), by using it as initial values in Eq. (11) with two different sets of values for the constants a,b.
Mentions: In Fig. 3, the original sharp MRI brain image (A) is blurred to form image (B), by applying the finite difference scheme in Eq. (18) to the parabolic equation Eq. (11), with coefficients a=b = 0. A different blurred image is then obtained, image (C), by repeating this process with coefficients a = 1.25, b = 0.6. Images (B) and (C) appear very similar in quality, and, from Table 1, both these images have almost the same values for ‖f‖1, ‖f‖2 and ‖∇f‖1. In particular, ‖∇f‖1 has been reduced by almost a factor of two from its original value in image (A), reflecting substantial blurring. The PSNR value in image (C) is noticeably smaller than in image (B), indicating greater degradation in image (C). However, since the PSNR metric requires knowledge of the original sharp image, in practice, such increased degradation in image (C) would not be known to a user. In fact, both images (B) and (C) appear to have been blurred, more or less equally, by convolution with a type of Gaussian-like point spread function.

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.


Related in: MedlinePlus