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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.


Ill behavior in non self adjoint problem. Accurate data f (x) (black curve), approximates two distinct solutions wred (x, t), (wgreen x, t) at time t = 1, with an L2 relative error of 0.023 %, and a pointwise accuracy of 1.4×10−3. These solutions originate from the vastly different initial values , and .
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f2-jres.118.010: Ill behavior in non self adjoint problem. Accurate data f (x) (black curve), approximates two distinct solutions wred (x, t), (wgreen x, t) at time t = 1, with an L2 relative error of 0.023 %, and a pointwise accuracy of 1.4×10−3. These solutions originate from the vastly different initial values , and .

Mentions: Let , shown as the red trace in Fig. 2, denote the initial data in Eq. (9), and let wred (x, t) be the corresponding solution. An accurate approximation to wred (x, 1) can be obtained numerically by integrating up to time t = 1. That approximation, denoted by f (x) is shown as the black trace in Fig. 2. The green trace in Fig. 2, , represents entirely different initial values in Eq. (9). However, the corresponding solution at t = 1, wgreen (x, 1), can also be well-approximated by the black trace f (x). Indeed, wgreen (x, 1) agrees with f (x) to within 1.4×10−3 pointwise, with an L2 relative error of 0.023% Also, , while . Therefore, both solutions wred (x, t) and wgreen (x, t) satisfy(10)‖w(.,1)−f‖2≤δ≤0.00023‖f‖2,‖w(.,3)‖2≤M=3.0.


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)

Ill behavior in non self adjoint problem. Accurate data f (x) (black curve), approximates two distinct solutions wred (x, t), (wgreen x, t) at time t = 1, with an L2 relative error of 0.023 %, and a pointwise accuracy of 1.4×10−3. These solutions originate from the vastly different initial values , and .
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f2-jres.118.010: Ill behavior in non self adjoint problem. Accurate data f (x) (black curve), approximates two distinct solutions wred (x, t), (wgreen x, t) at time t = 1, with an L2 relative error of 0.023 %, and a pointwise accuracy of 1.4×10−3. These solutions originate from the vastly different initial values , and .
Mentions: Let , shown as the red trace in Fig. 2, denote the initial data in Eq. (9), and let wred (x, t) be the corresponding solution. An accurate approximation to wred (x, 1) can be obtained numerically by integrating up to time t = 1. That approximation, denoted by f (x) is shown as the black trace in Fig. 2. The green trace in Fig. 2, , represents entirely different initial values in Eq. (9). However, the corresponding solution at t = 1, wgreen (x, 1), can also be well-approximated by the black trace f (x). Indeed, wgreen (x, 1) agrees with f (x) to within 1.4×10−3 pointwise, with an L2 relative error of 0.023% Also, , while . Therefore, both solutions wred (x, t) and wgreen (x, t) satisfy(10)‖w(.,1)−f‖2≤δ≤0.00023‖f‖2,‖w(.,3)‖2≤M=3.0.

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.