Backward Diffusion Research Articles

Lead, zinc and copper fine powder with controlled size and shape

This study describes the preparation of lead, zinc and copper powders by hydrometallurgy from secondary resources. Chloride, sulphate and acetate salts of zinc, copper and lead were prepared. The powders were prepared by reducing the ionic species of these metals by hydrazine hydrate or ascorbic acid. The effect of addition of some water soluble polar organic solvents to the aqueous salt solutions on the morphology and particle size of the prepared powder was studied. Findings were explained on the basis of the transition state theory and according to the Hughes and Ingold’s rule. Aqueous solutions alone produce metal powder having different size and irregular shape. The presence of polar organic solvents with high molecular weight and polarity produce powders having controlled size and regular morphology. The reason was because solvent polarity enhances the rate of red-ox reactions between metal ions and the reducing agent. The mean particle size of the powder was 60 um with zinc, 80 um with copper, and 90 um with lead. The extent of productivity was ≥98%. Results highlighted that the chemical reduction of the ionic species took place in a sequence steps. The first is a diffusion of the reactants across a boundary layer established at the polar site of the organic solvent molecules. The next step is the direct contact of the reactants. The third step involved reduction to yield powder. The last is the backward diffusion of the powder outside the boundary layer. Results showed that addition of water-miscible solvents having high dielectric constant increased the polarity of the medium. This energizes and enhances the one or more t step of the model to be more rapid to yield particles with small size and symmetrical shape.

Locating the source of spreading in temporal networks

Abstract The topological structure of many real networks changes with time. Thus, locating the sources of a temporal network is a creative and challenging problem, as the enormous size of many real networks makes it unfeasible to observe the state of all nodes. In this paper, we propose an algorithm to solve this problem, named the backward temporal diffusion process. The proposed algorithm calculates the shortest temporal distance to locate the transmission source. We assume that the spreading process can be modeled as a simple diffusion process and by consensus dynamics. To improve the location accuracy, we also adopt four strategies to select which nodes should be observed by ranking their importance in the temporal network. Our paper proposes a highly accurate method for locating the source in temporal networks and is, to the best of our knowledge, a frontier work in this field. Moreover, our framework has important significance for controlling the transmission of diseases or rumors and formulating immediate immunization strategies.

An innovative image encryption scheme based on chaotic map and Vigenère scheme

Confidentiality of images during data communication is a prime concern of many researchers, as a result, several mechanisms have been proposed for image encryption. An encryption for image is said to be effective if it has large key space, aperiodic nature and sensitive to the initial conditions. In addition, it is also desired that the mechanism must have a good combination of computational speed, security and time complexity. These features can be induced in the encryption schemes by incorporation of chaos during encryption/decryption process. In this paper, a novel image encryption scheme based on chaotic maps and Vigenere Scheme is proposed. This scheme has one round consisting of two steps: diffusion and confusion. The former step involves three stages: forward diffusion, matching process using Vigenere scheme and backward diffusion. In later part, position permutation using chaotic map is used to swap pixel positions. The proposed as well as other mechanisms in literature are implemented in Matlab-2015 and their efficacy are compared using several performance metrics. The simulation results show that our scheme is better in terms of time complexity while keeping Peak Signal to Noise Ratio (PSNR) values same and having almost ideal entropy, Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI).

Inverse problem for nonlinear backward space-fractional diffusion equation

Abstract In this paper, a backward diffusion problem for a space-fractional diffusion equation (SFDE) with nonlinear source in a strip is investigated. This problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a Riesz–Feller derivative of order γ ∈ ( 0 , 2 ] {\gamma\in(0,2]} . We show that such a problem is severely ill-posed and further propose a new modified regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under a priori bound assumptions for the exact solution. Our method improves some results of a previous paper, including the earlier paper [28] and some other papers. A general case of nonlinear terms for this problem is also considered.

Automatic Object-Based Hyperspectral Image Classification Using Complex Diffusions and a New Distance Metric

This paper proposes an approach for the development of automatic object-based techniques used for hyperspectral image classification. The proposed approach employs an adaptive smoothing step that utilizes an extension of partial differential equations (PDEs) from real domain (RPDE) to complex domain (CPDE). This idea results in generalized PDEs that simultaneously have the properties of both forward and backward diffusions. The genetic algorithm and an innovative fitness function are applied for adaptively tuning the CPDE parameters. The smoothed data are then fed into a Pixon-based object extraction process, which is itself an adaptive process. We also propose a novel distance metric for the Pixon creation step in order to facilitate the use of textural information which exists in the data. The spectral features of the extracted objects are used for a support vector machine classifier.

Bayesian approach to inverse problems for functions with a variable-index Besov prior

The Bayesian approach has been adopted to solve inverse problems that reconstruct a function from noisy observations. Prior measures play a key role in the Bayesian method. Hence, many probability measures have been proposed, among which total variation (TV) is a well-known prior measure that can preserve sharp edges. However, it has two drawbacks, the staircasing effect and a lack of the discretization-invariant property. The variable-index TV prior has been proposed and analyzed in the area of image analysis for the former, and the Besov prior has been employed recently for the latter. To overcome both issues together, in this paper, we present a variable-index Besov prior measure, which is a non-Gaussian measure. Some useful properties of this new prior measure have been proven for functions defined on a torus. We have also generalized Bayesian inverse theory in infinite dimensions for our new setting. Finally, this theory has been applied to integer- and fractional-order backward diffusion problems. To the best of our knowledge, this is the first time that the Bayesian approach has been used for the fractional-order backward diffusion problem, which provides an opportunity to quantify its uncertainties.

Dynamic Simulation of Backward Diffusion Based on Random Walk Theory

Results of diffusion study in silicon showed that diffusion of the selfinterstitial and vacancy could be backward diffusion and their diffusivity could be negative [1]. The backward diffusion process and negative diffusivity is contrary to the fundamental laws of diffusion such as the law of Fick law, namely the diffusive flux of backward diffusion goes from regions of low concentration to regions of high concentration. The backward diffusion process have been explained [2]. In this paper, the backward diffusion process is simulated. Results is corresponding to theory and show that when thermal velocity of the low concentration area is greater than thermal velocity of the high concentration area, the backward diffusion can be occurred.

Current-Controlled Spin Precession of Quasistationary Electrons in a Cubic Spin-Orbit Field.

Space- and time-resolved measurements of spin drift and diffusion are performed on a GaAs-hosted two-dimensional electron gas. For spins where forward drift is compensated by backward diffusion, we find a precession frequency in the absence of an external magnetic field. The frequency depends linearly on the drift velocity and is explained by the cubic Dresselhaus spin-orbit interaction, for which drift leads to a spin precession angle twice that of spins that diffuse the same distance.

Edge and contrast preserving in total variation image denoising

Total variation (TV) regularization can very well remove noise and simultaneously preserve the sharp edges. But it has the drawback of the contrast loss in the restoration. In this paper, we first theoretically analyze the loss of contrast in the original TV regularization model, and then propose a forward-backward diffusion model in the framework of total variation, which can effectively preserve the edges and contrast in TV image denoising. A backward diffusion term based on a nonconvex and monotony decrease potential function is introduced in the TV energy, resulting in a forward-backward diffusion. In order to finely control the strength of the forward and backward diffusion, and separately design the efficient algorithm to numerically implement the forward and backward diffusion, we propose a two-step splitting method to iteratively solve the proposed model. We adopt the efficient projection algorithm in the dual framework to solve the forward diffusion in the first step, and then use the simple finite differences scheme to solve the backward diffusion to compensate the loss of contrast occurred in the previous step. At last, we test the models on both synthetic and real images. Compared with the classical TV, forward and backward diffusion (FBD), two-step methods (TSM), and TV-FF models, our model has the better performance in terms of peak signal-to-noise ratio (PSNR) and mean structural similarity (MSSIM) indexes.

On a Riesz–Feller space fractional backward diffusion problem with a nonlinear source

In this paper, a backward diffusion problem for a space-fractional diffusion equation with a nonlinear source in a strip is investigated. This problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a Riesz–Feller derivative of order α∈(0,2]. A nonlinear problem is severely ill-posed, therefore we propose two new modified regularization solutions to solve it. We further show that the approximated problems are well-posed and their solutions converge if the original problem has a classical solution. In addition, the convergence estimates are presented under a priori bounded assumption of the exact solution. For estimating the error of the proposed method, a numerical example has been implemented.

Volume enhancement with externally controlled anisotropic diffusion

This paper proposes a method to enhance volumetric data using anisotropic diffusion controlled by another voxel array representing the same object with different physical quantities. The main application of this approach is to enhance volumetric functional data (obtained e.g. with PET or SPECT) based on anatomic (e.g. CT or MRI) information. Enhancement includes noise removal, sharpening and resolution upsampling. As different modalities measure different physical quantities that may or may not be correlated, enhancement must be carefully designed not to introduce spurious features that are present only in one modality. Forward diffusion working with non-negative diffusivity guarantees this kind of causality but also limits the potential of enhancement. To allow the preservation or even the increase of the dynamic range, diffusion should also go backwards. Therefore, we propose a forward---backward diffusion scheme for the enhancement where stability and the avoidance of spurious features are provided by the automatic determination of parameters controlling the diffusion process.

On a nonlocal analog of the Kuramoto–Sivashinsky equation

We study a nonlocal equation, analogous to the Kuramoto–Sivashinsky equation, in which short waves are stabilized by a possibly fractional diffusion of order less than or equal to two, and long waves are destabilized by a backward fractional diffusion of lower order. We prove the global existence, uniqueness, and analyticity of solutions of the nonlocal equation and the existence of a compact attractor. Numerical results show that the equation has chaotic solutions whose spatial structure consists of interacting travelling waves resembling viscous shock profiles.

A tutorial on inverse problems for anomalous diffusion processes

Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately described by fractional differential equations, which involves fractional derivatives in time or/and space. The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level. The new physics can change dramatically the behavior of the forward problems. For example, the solution operator of the time fractional diffusion diffusion equation has only limited smoothing property, whereas the solution for the space fractional diffusion equation may contain weak singularity. Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest. In this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leffler function and singular value decomposition, to examine the degree of ill-posedness of several ‘classical’ inverse problems for fractional differential equations involving a Djrbashian–Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order differential equations. We discuss four inverse problems, i.e., backward fractional diffusion, sideways problem, inverse source problem and inverse potential problem for time fractional diffusion, and inverse Sturm–Liouville problem, Cauchy problem, backward fractional diffusion and sideways problem for space fractional diffusion. It is found that contrary to the wide belief, the influence of anomalous diffusion on the degree of ill-posedness is not definitive: it can either significantly improve or worsen the conditioning of related inverse problems, depending crucially on the specific type of given data and quantity of interest. Further, the study exhibits distinct new features of ‘fractional’ inverse problems, and a partial list of surprising observations is given below. (a) Classical backward diffusion is exponentially ill-posed, whereas time fractional backward diffusion is only mildly ill-posed in the sense of norms on the domain and range spaces. However, this does not imply that the latter always allows a more effective reconstruction. (b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but numerically can be nearly well-posed. (c) The classical Sturm–Liouville problem requires two pieces of spectral data to uniquely determine a general potential, but in the fractional case, one single Dirichlet spectrum may suffice. (d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart, depending on the location of the lateral Cauchy data. In many cases, the precise mechanism of these surprising observations is unclear, and awaits further analytical and numerical exploration, which requires new mathematical tools and ingenuities. Further, our findings indicate fractional diffusion inverse problems also provide an excellent case study in the differences between theoretical ill-conditioning involving domain and range norms and the numerical analysis of a finite-dimensional reconstruction procedure. Throughout we will also describe known analytical and numerical results in the literature.

Improved Diffusion Rate with Triple Well Potential for Stem Cell Image Segmentation

Segmentation of stem cells is a challenging process in image processing where level set method is used extensively. This level set criterion with re-initialisation process reduces its wider range of application. Thus re-initialisation is eradicated by a method called distance regularisation in which the signed distance property is maintained near the zero level set via forward and backward (FAB) diffusion effect. A potential function is habituated to sustain the desired shape of the level set function. The initial potential function used in distance regularisation is single-well potential technique that causes excess backward diffusion and oscillation in LSF, so it is succeeded by double-well potential. But this double well technique losses the capability to conserve the regularity at some points. This problem is eliminated by a new novel method called improved diffusion rate with triple-well potential. The proposed diffusion rate can maintain the signed distance property of the level set function (LSF) and overcome the side effect of double-well potential with reduced number of iterations.

Joint reconstruction of PET-MRI by exploiting structural similarity

Recent advances in technology have enabled the combination of positron emission tomography (PET) with magnetic resonance imaging (MRI). These PET-MRI scanners simultaneously acquire functional PET and anatomical or functional MRI data. As function and anatomy are not independent of one another the images to be reconstructed are likely to have shared structures. We aim to exploit this inherent structural similarity by reconstructing from both modalities in a joint reconstruction framework. The structural similarity between two modalities can be modelled in two different ways: edges are more likely to be at similar positions and/or to have similar orientations. We analyse the diffusion process generated by minimizing priors that encapsulate these different models. It turns out that the class of parallel level set priors always corresponds to anisotropic diffusion which is sometimes forward and sometimes backward diffusion. We perform numerical experiments where we jointly reconstruct from blurred Radon data with Poisson noise (PET) and under-sampled Fourier data with Gaussian noise (MRI). Our results show that both modalities benefit from each other in areas of shared edge information. The joint reconstructions have less artefacts and sharper edges compared to separate reconstructions and the ℓ2-error can be reduced in all of the considered cases of under-sampling.

Edge preservation image enlargement and enhancement method based on the adaptive Perona–Malik non‐linear diffusion model

In this study, the authors have proposed a new super resolution (SR) model based on the Perona–Malik regularisation scheme. The new model integrates into its regularisation component an adaptive exponential term which automatically adjusts itself depending on the local image features. This lends more sensitivity and adaptability to the proposed model, thereby making the reconstruction process much less punishing against semantically important features. Therefore, regularisation is stronger in homogeneous regions, and weaker in the neighbourhood of boundaries. The proposed method has a promising capability of supressing noise more effectively, while preserving important image features. The approach used differs significantly from the available methods, especially in the manner in which adaptability has been deployed. Noting that SR methods are less sensitive to the local image topography, a factor that causes the super-resolved images to be visually poor, the new method sensitively probes the local features of the image, and determines the necessary level of reconstruction and regularisation. Additionally, the formulation robustly introduces a backward diffusion, a phenomenon proved from literature to have a tendency of sharpening edges. The authors have included empirical reconstruction results to demonstrate that their model produces better images in comparison with other classical methods.

A new a posteriori parameter choice strategy for the convolution regularization of the space-fractional backward diffusion problem

In this paper, we consider a backward space-fractional diffusion problem. We propose an a posteriori parameter choice rule for the regularization method given in Zheng and Wei (2010), where the authors proposed a regularization method called convolution regularization method, and gave an a priori parameter choice strategy. In this paper, we study the same problem but give a new a posteriori parameter choice based on a modified version of the discrepancy principle, and obtain a log-type error estimate under an additional source condition. Numerical results show that our method is feasible.

The Equation of Backward Diffusion and Negative Diffusivity

Up to now, all the diffusion theories have shown that the direction of diffusion flux is from high concentration area to lower concentration area (forward diffusion) and diffusivity is positive. In some research on simultaneous diffusion of boron and point defects in Si, the results show that the diffusivity of interstitials can be negative and diffusion process can be backward diffusion. In backward diffusion process, the direction of diffusion flux is from low concentration area to higher concentration area. The phenomenon of backward diffusion and negative diffusivity should be explained. Based on thermodynamic theory, the equation of backward diffusion and negative diffusivity are presented and discussed.

Ultrasound image denoising using backward diffusion and framelet regularization

This paper introduces a novel variational method for ultrasound image denoising for speckle suppression and edge enhancement. This method is designed to utilize the favorable denoising properties of framelet regularization and edge enhancement of backward diffusion technique. The sparsity and multiresolution properties of the framelet is well suited for speckle noise reduction. The fidelity term of the method can be obtained by Maximum a Posteriori (MAP). The introduction of backward diffusion and framelet regularization makes it difficult to solve the variational energy function. To simplify minimization problem, the Split Bregman algorithm for the proposed model is proposed and then we use it for ultrasound image denoising. Experiment results validate the usefulness of the proposed method for ultrasound image denoising.

Enhanced distance regularization for re-initialization free level set evolution with application to image segmentation

In level set methods, the re-initialization remedy is widely applied to periodically replace the degraded level set function (LSF) with a signed distance function to maintain its regularity. Due to its various limitation, the energy functional regularization based methods using variational technique (e.g. distance, Gaussian, and reaction–diffusion based regularization methods) are recently introduced to replace this remedy. However, the relationship among them seems to be less investigated. In this paper, an enhanced distance regularized level set evolution (DRLSE-E) completely free of the re-initialization procedure is proposed based on analyzing these recent regularization models. DRLSE-E has an intrinsic capability of maintaining LSF׳s regularity, particularly the desirable signed distance property in a vicinity of the zero level set and the flat property out of this vicinity, which ensures accurate computation and stable level set evolution. Like other re-initialization free methods, DRLSE-E has simple and efficient numerical scheme in implementation, flexible initialization. Furthermore, DRLSE-E has the advantage of faster evolving speed and more numerical accuracy than distance regularized methods because of its forward and backward diffusion rate considering two competing components during the evolution. As an application example, DRLSE-E is used to typical edge-based and region-based active contour models for image segmentation and shows its competitiveness. Considering DRLSE-E is general, it can be easily incorporated into various existing level set models for image segmentation, filtering, and other tasks.