Anisotropic Diffusion Equation Research Articles

A random walk on small spheres method for solving transient anisotropic diffusion problems

Abstract A meshless stochastic algorithm for solving anisotropic transient diffusion problems based on an extension of the classical Random Walk on Spheres method is developed. Direct generalization of the Random Walk on Spheres method to anisotropic diffusion equations is not possible, therefore, we have derived approximations of the probability densities for the first passage time and the exit point on a small sphere. The method can be conveniently applied to solve diffusion problems with spatially varying diffusion coefficients and is simply implemented for complicated three-dimensional domains. Particle tracking algorithm is highly efficient for calculation of fluxes to boundaries. We present some simulation results in the case of cathodoluminescence and electron beam induced current in the vicinity of a dislocation in a semiconductor material.

A global sparse gradient based coupled system for image denoising

We consider a coupled system for image denoising, which consists of an anisotropic diffusion equation and a global sparse gradient model (GSG). The global sparse gradient model is used to estimate the edge map from the noisy image accurately and robustly. The anisotropic diffusion process is guided by the edge map. Furthermore, we prove the existence and uniqueness of the solutions of the coupled system with Dirichlet initial–boundary value problem. Finally, comparing with the edge indicators based on local methods, experimental results demonstrate that our proposed method has the state-of-the-art performance among the PDE-based anisotropic diffusion approaches both in objective measurement and visual evaluation.

Robust and effective automatic parameter choice for medical image filtering.

ABSTRACTThe analysis of medical image data currently requires the interpretation of a trained and experienced user. The technological advances in imaging machinery and the understanding of disease onset, as well as medical planning, all favour the need for ever more automatic and robust methods for evaluating the health state of a subject. Here, we concentrate on methods for processing medical image data, as currently provided by existing imaging technologies, in particular the effectiveness of automatic image filtering in order to remove noise and improve the sharpness of distinct objects. The filtering approach is based on a partial differential equation, namely the Perona–Malik anisotropic diffusion equation. The approach adopted for terminating the iterative filtering procedure is based on image quality descriptors. In specific, we observe the rate of change of these to infer the transient effects of the filtering process. The entire pipeline is demonstrated to work effectively on different sets of medical image data, including MRI, CTA and CT, both in individual 2-D images in a stack, as well as treating the complete 3D volumetric dataset.

Reflection-Waveform Inversion Regularized with Structure-Oriented Smoothing Shaping

Restricted by the limited length of the receiver arrays, estimating the seismic properties of deep targets requires inverting reflection data. Reflection-waveform inversion (RWI) utilizes reflection data to recover the background velocity. RWI splits the Earth model into a long-wavelength background velocity and short-wavelength reflectors. Because of the limited apertures of reflection data illuminating targets, and the trade-off between the depth of reflectors and the average velocity above the reflectors, RWI is inherently ill-posed and contains a large null space. One manifestation of the ill-posedness is the existence of characteristic artificial blob-shaped or column-like anomalies in the velocity models estimated with RWI. That erroneous background velocity, which still fits the travel time of reflection arrivals in the recorded data, can lead to inaccurate velocity models and distorted migration images in the subsequent processes. In order to mitigate those artifacts and improve the conditioning of RWI, we incorporate a piece of prior information into RWI, i.e. slow property variation along geological structures. This prior information is expressed mathematically as structure-oriented smoothing. We achieved this smoothing by solving an anisotropic diffusion equation with a finite-difference method. Thus, we formulated a method for RWI regularized with structure-oriented smoothing shaping by alternating the following three steps: building temporary reflectors, updating background velocity, and solving the diffusion equation. We successfully tested this scheme on the Marmousi model and a modified overthrust model. The results show that the regularized RWI yields a stable and accurate background velocity, and it is more effective than conventional Tikhonov regularization approaches. The estimated background model leads to a high-quality final velocity model when used as an initial model in conventional full-waveform inversion.

A new tailored finite point method for strongly anisotropic diffusion equation on misaligned grids

This paper presents a new tailored finite point method (TFPM) for the strongly anisotropic diffusion equation on Cartesian coordinates. The novelty is that the scheme is constructed by using the interface conditions for each cell. Several numerical experiments are presented to show the performance of this new scheme. Numerically, the method can not only achieve good accuracy, but also sharply capture internal layers.

Noise-Driven Anisotropic Diffusion Filtering for X-Ray Low Contrast Imaging

We describe theoretically a possible application of advanced noise-filtering technique to X-ray imaging of low contrast objects, using the example of X-ray studies of laser fusion capsules. Based on the concept of the nonlinear anisotropic diffusion equation, we develop a computer code which uses noise-driven parameterization to optimize the image denoising process. The computation results demonstrate an effective noise filtering while preserving the edge pattern of noisy images obtained with the original contrast as low as ~1%.

Fractional-order difference curvature-driven fractional anisotropic diffusion equation for image super-resolution

Image super-resolution methods-based existing edge indicating operators — namely Gauss curvature, mean curvature and gradient — cannot effectively identify the edges, ramps and flat regions and suffer from the loss of fine textures. To address these issues, this paper presents a fractional anisotropic diffusion equation based on a new edge indicator, named fractional-order difference curvature, which can characterize the intensity variations in images. We introduce the frequency-domain definition for fractional-order derivative by the Fourier transform, which is easy to implement numerically. The new edge indicator is better than the existing edge indicating operators in distinguishing between ramps and edges and can better handle the fine textures. Comparative results for natural images validate that the proposed method can yield a visually pleasing result and better values of MSSIM and PSNR.

On Computing Gradient of Products in Discretized Spaces and Its Effects in PDE Image Processing

Many digital signal and image processing methods involve computing the gradient of products of functions. However, the product rule for derivatives in continuous spaces, $\partial (fg)=g\partial f + f \partial g$ , does not generally hold in discretized spaces. Hence, computing the gradient of products becomes ambiguous as the results depend on whether to treat the product $fg$ as a single function or treat $f$ and $g$ as two separate functions and use the product rule. The two alternatives lead to different results, particularly for iterative solutions of differential equations since these small differences compound. Although this ambiguity is well-known, thus far a resolution has not been proposed in the literature. In this letter, we propose a mathematically rigorous procedure that selects the better alternative in the sense of yielding approximations that are closer to the continuous space results. As an example, we discuss the Perona-Malik anisotropic diffusion equation used in image processing.

Closed-form solution of stress state and stability analysis of wellbore in anisotropic permeable rocks

This work aims to develop a closed-form solution of stress state around wellbore drilled in the anisotropic permeable rocks taking into account the transient effect of fluid flow. For this purpose, a simplified method was adopted to solve the anisotropic hydraulic diffusion equation. Knowing the explicit transient evolution of pore pressure, the hydraulic and hydro-mechanical potentials were proposed through which stress state around wellbore are determined thanks to the use of the well-known complex potential approach. The validity of this closed-form solution was verified by comparing with the result obtained from the numerical simulation. By combining the analytical solution of stresses on well wall with a chosen anisotropic failure criterion of rock mass, the stability analysis of wellbore was carried out. Through the parametric study, we elucidated the influence of different factors on the safe mud pressure window notably the anisotropic poro-elastic properties, the anisotropic strength of the rock masses as well as the anisotropic initial stress state and the inclined bedding plane of formation.

Image enhancement with PDEs and nonconservative advection flow fields.

We propose a new method for the progressive enhancement of images degraded by noise and blur using shock filters. The originality of our approach lies in the iterative exploitation of nonconservative edge-based force fields normally designed for segmentation to determine both locations and amplitudes of the sharpening action. The proposed nonconservative shock filter (NCSF) produces images with strong discontinuities that can be used as an additional pre-processing step to facilitate higher level tasks such as edge detection or segmentation. NCSF is especially useful when coupled with an anisotropic diffusion equation in noisy configurations. Using the proposed method, edges that are not well defined prior to filtering can be iteratively recovered. We show processing results on various types of images, focusing on medical transrectal ultrasound and brain PET imaging.

The stability of the solutions of an anisotropic diffusion equation

This paper investigates an anisotropic diffusion equation with degeneracy on the boundary. A new kind of weak solution is introduced, and the existence of the nonnegative solution is proved by the parabolically regularized method. Moreover, if the interaction between the convection and the diffusion is considered, the stability of the solutions can be proved without the boundary value condition.

Single image fog removal algorithm in spatial domain using fractional order anisotropic diffusion

This paper presents a novel image defogging algorithm using fractional-order anisotropic diffusion equation. The proposed algorithm uses the airlight map extracted from the foggy model as the initial image in the anisotropic diffusion process. The iterative diffusion process improves this airlight map. The anisotropic diffusion process is generalized to the order of any real number between [1, 2) using the Riemann-Liouville definition of the fractional order derivatives. The formulation of the iterative process is carried out in the spatial domain to have a simple and computationally efficient implementation. Simulation results validate that the proposed algorithm is outperforming over few of the existing algorithms. The comparison study is carried out using different metrics like contrast gain, colorfulness index, contrast-to-noise ratio and visible edges ratio.

Coherent noise reduction of phase images in digital holographic microscopy based on the adaptive anisotropic diffusion.

The suppression of coherent noise can produce higher-quality reconstructed images in digital holographic microscopy. A robust and effective phase coherent noise denoising algorithm is proposed in this paper that combines the anisotropic diffusion equation and the phase quality map. In order to accurately identify the noise and signal pixels, we introduce the phase quality map and edge detection to quantify the quality of the pixel information. In addition, a synthetic diffusion function is established to control the speed of the anisotropic diffusion process based on the quality coefficient. Several experiments have been carried out to validate the effectiveness of the proposed algorithm for coherent noise reduction. The results demonstrate that the proposed algorithm can reduce coherent noise and preserve edge details well.

Respectively scaled HSS iteration methods for solving discretized spatial fractional diffusion equations

SummaryFor the discrete linear systems resulted from the discretization of the one‐dimensional anisotropic spatial fractional diffusion equations of variable coefficients with the shifted finite‐difference formulas of the Grünwald–Letnikov type, we propose a class of respectively scaled Hermitian and skew‐Hermitian splitting iteration method and establish its asymptotic convergence theory. The corresponding induced matrix splitting preconditioner, through further replacements of the involved Toeplitz matrices with certain circulant matrices, leads to an economic variant that can be executed by fast Fourier transforms. Both theoretical analysis and numerical implementations show that this fast respectively scaled Hermitian and skew‐Hermitian splitting preconditioner can significantly improve the computational efficiency of the Krylov subspace iteration methods employed as effective linear solvers for the target discrete linear systems.

Image Denoising Using Generalized Anisotropic Diffusion

Motivated by some recent works in fractional-order anisotropic diffusions, we introduce a class of generalized anisotropic diffusion equations for image denoising. We first define a new type of derivative (called G-derivative), which contains the fractional derivative as a special case, using the Fourier transform, then the generalized anisotropic diffusion equations are Euler–Lagrange equations of a cost functional which is an increasing function of the absolute value of the G-derivative of the image intensity function. All the G-derivative operators constitute a ring, and the semigroup property of the G-derivative consists with the semigroup property of the fractional derivative, so the resulting generalized anisotropic diffusions can be seen as generalizations of the fractional-order anisotropic diffusions. We also discuss the generalized Sobolev space described by the G-derivative and some variants of generalized anisotropic diffusions. The discretization of the G-derivative is computed in the frequency domain, and the stability analysis of the difference scheme is given. We list some generalized anisotropic diffusions and apply them to image denoising. Numerical results show that new models have great potentials in image denoising.

A Carleman estimate of some anisotropic space-fractional diffusion equations

This paper is concerned with Carleman estimates for some anisotropic space-fractional diffusion equations, which are important tools for investigating the corresponding control and inverse problems. By employing a special weight function and the nonlocal vector calculus, we prove a Carleman estimate and apply it to build a stability result for a backward diffusion problem.

Space-Time FE-DG Discretization of the Anisotropic Diffusion Equation in Any Dimension: The Spectral Symbol

The multidimensional heat equation, along with its more general version known as the (linear) anisotropic diffusion equation, is discretized by a discontinuous Galerkin (DG) method in time and a fi...

The use of principal spatial tensor to predict anisotropic fiber orientation in concentrated fiber suspensions

The Phelps-Tucker anisotropic rotary diffusion (ARD) equation constitutes an important development in relation to concentrated fiber suspension rheology and is significant in its ability to predict transient, flow-induced fiber orientation, especially for long fiber composites in the industry. Within this model, a critical spatial tensor was assumed to be a polynomial tensor-valued function depending on both the orientation tensor and the rate-of-strain tensor. However, it can be difficult to derive a large number of fitting ARD parameters. For simplification, we newly defined the spatial tensor as coaxial with the orientation tensor's principal directions, while the spatial tensor's principal components are to control anisotropic changes in the orientation tensor. Such a principal spatial tensor used in the Phelps-Tucker ARD model is demonstrated in accurately predicting the shell-core structure of fiber orientation distributions for injection molding simulation of long fiber composites, supported with e...

Uniformly Convergent Scheme for Strongly Anisotropic Diffusion Equations with Closed Field Lines

In magnetized plasma, the magnetic field confines particles around field lines. The ratio between the intensity of the parallel and perpendicular viscosities or heat conductions may reach the order...

Energy stable numerical methods for porous media flow type problems

Many problems arising in the context of multiphase porous media flows that take the form of degenerate parabolic equations have a dissipative structure, so that the energy of an isolated system is decreasing along time. In this paper, we discuss two approaches to tune a rather large family of numerical method in order to ensure a control on the energy at the discrete level as well. The first methodology is based on upwinding of the mobilities and leads to schemes that are unconditionally positivity preserving but only first order accurate in space. We present a second methodology which is based on the construction of local positive dissipation tensors. This allows to recover a second order accuracy w.r.t. space, but the preservation of the positivity is conditioned to some additional assumption on the nonlinearities. Both methods are based on an underlying numerical method for a linear anisotropic diffusion equation. We do not suppose that this building block is monotone.