RESUMEN
This paper constructs an unconditionally stable explicit finite difference scheme, marching backward in time, that can solve an interesting but limited class of ill-posed, time-reversed, 2D incompressible Navier-Stokes initial value problems. Stability is achieved by applying a compensating smoothing operator at each time step to quench the instability. This leads to a distortion away from the true solution. However, in many interesting cases, the cumulative error is sufficiently small to allow for useful results. Effective smoothing operators based on (-∆) p , with real p > 2, can be efficiently synthesized using FFT algorithms. Similar stabilizing techniques were successfully applied in other ill-posed evolution equations. The analysis of numerical stabilty is restricted to a related linear problem. However, extensive numerical experiments indicate that such linear stability results remain valid when the explicit scheme is applied to a significant class of time-reversed nonlinear 2D Navier-Stokes initial value problems. Several reconstruction examples are included, based on the stream function-vorticity formulation, and focusing on 256 × 256 pixel images of recognizable objects. Such images, associated with non-smooth underlying intensity data, are used to create severely distorted data at time T > 0. Successful backward recovery is shown to be possible at parameter values exceeding expectations.
RESUMEN
This paper constructs an unconditionally stable explicit difference scheme, marching backward in time, that can solve a limited, but important class of time-reversed 2D Burgers' initial value problems. Stability is achieved by applying a compensating smoothing operator at each time step to quench the instability. This leads to a distortion away from the true solution. However, in many interesting cases, the cumulative error is sufficiently small to allow for useful results. Effective smoothing operators based on (-Δ) p , with real p > 2, can be efficiently synthesized using FFT algorithms, and this may be feasible even in non-rectangular regions. Similar stabilizing techniques were successfully applied in other ill-posed evolution equations. The analysis of numerical stabilty is restricted to a related linear problem. However, extensive numerical experiments indicate that such linear stability results remain valid when the explicit scheme is applied to a significant class of time-reversed nonlinear 2D Burgers' initial value problems. As illustrative examples, the paper uses fictitiously blurred 256 × 256 pixel images, obtained by using sharp images as initial values in well-posed, forward 2D Burgers' equations. Such images are associated with highly irregular underlying intensity data that can seriously challenge ill-posed reconstruction procedures. The stabilized explicit scheme, applied to the time-reversed 2D Burgers' equation, is then used to deblur these images. Examples involving simpler data are also studied. Successful recovery from severely distorted data is shown to be possible, even at high Reynolds numbers.
RESUMEN
This paper develops stabilized explicit marching difference schemes that can successfully solve a significant but limited class of multidimensional, ill-posed, backward in time problems for coupled hyperbolic/parabolic systems associated with vibrating thermoelastic plates and coupled sound and heat flow. Stabilization is achieved by applying compensating smoothing operators at each time step, to quench the instability. Analysis of convergence is restricted to the transparent case of linear, autonomous, selfadjoint spatial differential operators, and almost best-possible error bounds are obtained for backward in time reconstruction in that class of problems. However, the actual computational schemes can be applied to more general problems, including examples with variable time dependent coefficients, as well as nonlinearities. The stabilized explicit schemes are unconditionally stable, marching forward or backward in time, but the smoothing operation at each step leads to a distortion away from the true solution. This is the stabilization penalty. It is shown that in many problems of interest, that distortion is small enough to allow for useful results. Backward in time continuation is illustrated using 512×512 pixel images. Such images are associated with highly irregular non smooth intensity data that severely challenge ill-posed reconstruction procedures. Several computational experiments show that efficient FFT-synthesized smoothing operators, based on (-∆) p with real p > 2, can be successfully applied in a broad range of problems.
RESUMEN
Photoshop processing of latent fingerprints is the preferred methodology among law enforcement forensic experts, but that appproach is not fully reproducible and may lead to questionable enhancements. Alternative, independent, fully reproducible enhancements, using IDL Histogram Equalization and IDL Adaptive Histogram Equalization, can produce better-defined ridge structures, along with considerable background information. Applying a systematic slow motion smoothing procedure to such IDL enhancements, based on the rapid FFT solution of a Lévy stable fractional diffusion equation, can attenuate background detail while preserving ridge information. The resulting smoothed latent print enhancements are comparable to, but distinct from, forensic Photoshop images suitable for input into automated fingerprint identification systems, (AFIS). In addition, this progressive smoothing procedure can be reexamined by displaying the suite of progressively smoother IDL images. That suite can be stored, providing an audit trail that allows monitoring for possible loss of useful information, in transit to the user-selected optimal image. Such independent and fully reproducible enhancements provide a valuable frame of reference that may be helpful in informing, complementing, and possibly validating the forensic Photoshop methodology.
RESUMEN
This paper discusses a two step enhancement technique applicable to noisy Helium Ion Microscope images in which background structures are not easily discernible due to a weak signal. The method is based on a preliminary adaptive histogram equalization, followed by 'slow motion' low-exponent Lévy fractional diffusion smoothing. This combined approach is unexpectedly effective, resulting in a companion enhanced image in which background structures are rendered much more visible, and noise is significantly reduced, all with minimal loss of image sharpness. The method also provides useful enhancements of scanning charged-particle microscopy images obtained by composing multiple drift-corrected 'fast scan' frames. The paper includes software routines, written in Interactive Data Language (IDL),(1) that can perform the above image processing tasks.
RESUMEN
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.
RESUMEN
Helium ion microscopes (HIM) are capable of acquiring images with better than 1 nm resolution, and HIM images are particularly rich in morphological surface details. However, such images are generally quite noisy. A major challenge is to denoise these images while preserving delicate surface information. This paper presents a powerful slow motion denoising technique, based on solving linear fractional diffusion equations forward in time. The method is easily implemented computationally, using fast Fourier transform (FFT) algorithms. When applied to actual HIM images, the method is found to reproduce the essential surface morphology of the sample with high fidelity. In contrast, such highly sophisticated methodologies as Curvelet Transform denoising, and Total Variation denoising using split Bregman iterations, are found to eliminate vital fine scale information, along with the noise. Image Lipschitz exponents are a useful image metrology tool for quantifying the fine structure content in an image. In this paper, this tool is applied to rank order the above three distinct denoising approaches, in terms of their texture preserving properties. In several denoising experiments on actual HIM images, it was found that fractional diffusion smoothing performed noticeably better than split Bregman TV, which in turn, performed slightly better than Curvelet denoising.