The use of a Riesz fractional differential-based approach for texture enhancement in image processing

Abstract: Texture enhancement is an important component of image processing, with extensive application in science and engineering. The quality of medical images, quantified using the texture of the images, plays a significant role in the routine diagnosis performed by medical practitioners. Previously, image texture enhancement was performed using classical integral order differential mask operators. Recently, first order fractional differential operators were implemented to enhance images. Experiments conclude that the use of the fractional differential not only maintains the low frequency contour features in the smooth areas of the image, but also nonlinearly enhances edges and textures corresponding to high-frequency image components. However, whilst these methods perform well in particular cases, they are not routinely useful across all applications. To this end, we applied the second order Riesz fractional differential operator to improve upon existing approaches of texture enhancement. Compared with the classical integral order differential mask operators and other fractional differential operators, our new algorithms provide higher signal to noise values, which leads to superior image quality.

A comparison of finite difference and finite volume methods for solving the space fractional advection-dispersion equation with variable coefficients

Transport processes within heterogeneous media may exhibit non-classical diffusion or dispersion; that is, not adequately described by the classical theory of Brownian motion and Fick's law. We consider a space fractional advection-dispersion equation based on a fractional Fick's law. The equation involves the Riemann-Liouville fractional derivative which arises from assuming that particles may make large jumps. Finite difference methods for solving this equation have been proposed by Meerschaert and Tadjeran. In the variable coefficient case, the product rule is first applied, and then the Riemann-Liouville fractional derivatives are discretised using standard and shifted Grunwald formulas, depending on the fractional order. In this work, we consider a finite volume method that deals directly with the equation in conservative form. Fractionally-shifted Grunwald formulas are used to discretise the fractional derivatives at control volume faces. We compare the two methods for several case studies from the literature, highlighting the convenience of the finite volume approach.

Spectral approximations to the fractional integral and derivative

In this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods.

Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative

Rayleigh–Stokes problems have in recent years received much attention due to their importance in physics. In this article, we focus on the variable-order Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative. Implicit and explicit numerical methods are developed to solve the problem. The convergence, stability of the numerical methods and solvability of the implicit numerical method are discussed via Fourier analysis. Moreover, a numerical example is given and the results support the effectiveness of the theoretical analysis.

The analytical solution and numerical solution of the fractional diffusion-wave equation with damping

Fractional partial differential equations have been applied to many problems in physics, finance, and engineering. Numerical methods and error estimates of these equations are currently a very active area of research. In this paper we consider a fractional diffusionwave equation with damping. We derive the analytical solution for the equation using the method of separation of variables. An implicit difference approximation is constructed. Stability and convergence are proved by the energy method. Finally, two numerical examples are presented to show the effectiveness of this approximation.

A computationally effective alternating direction method for the space and time fractional Bloch–Torrey equation in 3-D

The space and time fractional Bloch–Torrey equation (ST-FBTE) has been used to study anomalous diffusion in the human brain. Numerical methods for solving ST-FBTE in three-dimensions are computationally demanding. In this paper, we propose a computationally effective fractional alternating direction method (FADM) to overcome this problem. We consider ST-FBTE on a finite domain where the time and space derivatives are replaced by the Caputo–Djrbashian and the sequential Riesz fractional derivatives, respectively. The stability and convergence properties of the FADM are discussed. Finally, some numerical results for ST-FBTE are given to confirm our theoretical findings.

Enduring powers of attorney : promoting attorneys' accountability as substitute decision makers

Aim The misuse and abuse of Enduring Powers of Attorney (EPAs) by attorneys, particularly in relation to financial decision-making, is a growing concern. This paper explores the opportunities to enhance accountability of attorneys at the time of the execution of the document in Queensland. Method A four stage multi-method design comprised a critical reference group; semi-structured interviews with 32 principals or potential principals, attorneys and witnesses; two focus groups with service providers and a state wide survey of 76 principals, attorneys and witnesses. Results Across all methods and user groups, understanding the role and obligations of the attorney in an EPA was consistently identified as problematic. Conclusions Promoting accountability and understanding can be addressed by greater attention to the role of the attorney in the forms/ guidelines and in the structure and witnessing of the forms, increased direction about record keeping and access to appropriate advice and support.

Fractional models for the migration of biological cells in complex spatial domains

Travelling wave phenomena are observed in many biological applications. Mathematical theory of standard reaction-diffusion problems shows that simple partial differential equations exhibit travelling wave solutions with constant wavespeed and such models are used to describe, for example, waves of chemical concentrations, electrical signals, cell migration, waves of epidemics and population dynamics. However, as in the study of cell motion in complex spatial geometries, experimental data are often not consistent with constant wavespeed. Non-local spatial models have successfully been used to model anomalous diffusion and spatial heterogeneity in different physical contexts. In this paper, we develop a fractional model based on the Fisher-Kolmogoroff equation and analyse it for its wavespeed properties, attempting to relate the numerical results obtained from our simulations to experimental data describing enteric neural crest-derived cells migrating along the intact gut of mouse embryos. The model proposed essentially combines fractional and standard diffusion in different regions of the spatial domain and qualitatively reproduces the behaviour of neural crest-derived cells observed in the caecum and the hindgut of mouse embryos during in vivo experiments.

Dreams and nightmares of alternative powers' futures

This chapter argues the importance of the role and nature of other powers to world order. The author suggests that, if the US are not prepared to take a lead in creating a rules-based legal order, they should and can do so – and it is in their interests to do so. America should be a natural leader in this process, taking part in a global dialogue just as they did in the transatlantic dialogue during the late eighteenth century.

Gender-based undervaluation and the equal remuneration powers of Fair Work Australia

We investigate gender-based wage undervaluation in light of FairWork Australia’s major recent decision for social and community service workers. Using regression methods, we demonstrate that wages for employees in female-dominated occupations are significantly lower than for comparable employees in male-dominated and integrated occupations. This undervaluation is present for both male and female employees, and persists after controlling for industry of employment. We then estimate the undervaluation within industry and juxtapose the results with evidence on the industry distribution of award reliance, a proxy for Fair Work Australia’s equal remuneration powers. There is not a strong relationship within industries between the extent of gender-based undervaluation and award reliance. This suggests that ‘equal remuneration for work of equal or comparable value’ is unlikely to be achieved universally by Fair Work Australia without substantial spillovers between awards and non-award agreements.

The wrong tool for the job? The predictive powers of the DBQ in a sample of Queensland motorists

Introduction This study reports on the application of the Manchester Driver Behaviour Questionnaire (DBQ) to examine the self-reported driving behaviours (e.g., speeding, errors & aggressive manoeuvres) and predict crash involvement among a sample of general Queensland motorists. Material and Methods Surveys were completed by 249 general motorists on-line or via a pen-and-paper format. Results A factor analysis revealed a three factor solution for the DBQ which was consistent with previous Australian-based research. It accounted for 40.5% of the total variance, although some cross-loadings were observed on nine of the twenty items. The internal reliability of the DBQ was satisfactory. However, multivariate analysis using the DBQ revealed little predictive ability of the tool to predict crash involvement or demerit point loss e.g. violation notices. Rather, exposure to the road was found to be predictive of crashes, although speeding did make a small contribution to those who recently received a violation notice. Conclusions Taken together, the findings contribute to a growing body of research that raises questions about the predictive ability of the most widely used driving assessment tool globally. Ongoing research (which also includes official crash and offence outcomes) is required to better understand the actual contribution that the DBQ can make to understanding and improving road safety. Future research should also aim to confirm whether this lack of predictive efficacy originates from broader issues inherent within self-report data (e.g., memory recall problems) or issues underpinning the conceptualisation of the scale.

The Tarim picrite-basalt-rhyolite suite, a Permian flood basalt from Northwest China with contrasting rhyolites produced by fractional crystallization and anatexis

We report major and trace element composition, Sr–Nd isotopic and seismological data for a picrite–basalt–rhyolite suite from the northern Tarim uplift (NTU), northwest China. The samples were recovered from 13 boreholes at depths between 5,166 and 6,333 m. The picritic samples have high MgO (14.5–16.8 wt%, volatiles included) enriched in incompatible element and have high 87Sr/86Sr and low 143Nd/144Nd isotopic ratios (εNd (t) = −5.3; Sri = 0.707), resembling the Karoo high-Ti picrites. All the basaltic samples are enriched in TiO2 (2.1–3.2 wt%, volatiles free), have high FeOt abundances (11.27–15.75 wt%, volatiles free), are enriched in incompatible elements and have high Sr and low Nd isotopic ratios (Sri = 0.7049–0.7065; εNd (t) = −4.1 to −0.4). High Nb/La ratios (0.91–1.34) of basalts attest that they are mantle-derived magma with negligible crustal contamination. The rhyolite samples can be subdivided into two coeval groups with overlapping U–Pb zircon ages between 291 ± 4 and 272 ± 2 Ma. Group 1 rhyolites are enriched in Nb and Ta, have similar Nb/La, Nb/U, and Sr–Nd isotopic compositions to the associated basalts, implying that they are formed by fractional crystallization of the basalts. Group 2 rhyolites are depleted in Nb and Ta, have low Nb/La ratios, and have very high Sr and low Nd isotopic ratios, implying that crustal materials have been extensively, if not exclusively, involved in their source. The picrite–basalt–rhyolite suite from the NTU, together with Permian volcanic rocks from elsewhere Tarim basin, constitute a Large Igneous Province (LIP) that is characterized by large areal extent, rapid eruption, OIB-type chemical composition, and eruption of high temperature picritic magma. The Early Permian magmatism, which covered an area >300,000 km2, is therefore named the Tarim Flood Basalt.

Shifting Powers : Protection of Refugees and Displaced Persons in the Asia Pacific Region

The United Nations High Commissioner for Refugees' (UNHCR) 2011 statistics on refugee populations residing by region are a stark reminder of the challenge facing states and civil society in the Asia Pacific. In 2011, Africa hosted 2,149,000 refugees; the Americas, Europe, and Middle East and North Africa hosted 513 ,500, 1,605,500 and 1,889,900 respectively, while the Asia Pacific hosted a staggering 3,793,900. The fact that 35 per cent of the world's refugees reside in the Asia Pacific, coupled with the fact that 84 per cent of refugees displaced in Asia remain in the region,raises the questions why so few countries in the region are signatories to the Convention relating to the Status of Refugees ('Refugee Convention') or cognate rights instruments and why no formally binding regional agreement exists for the equitable sharing of responsibilities for refugees...

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

We present a finite volume method to solve the time-space two-sided fractional advection-dispersion equation on a one-dimensional domain. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes. We demonstrate how the finite volume formulation provides a natural, convenient and accurate means of discretising this equation in conservative form, compared to using a conventional finite difference approach. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

A comparison of finite difference and finite volume methods for solving the space-fractional advection-dispersion equation with variable coefficients

Transport processes within heterogeneous media may exhibit non- classical diffusion or dispersion which is not adequately described by the classical theory of Brownian motion and Fick’s law. We consider a space-fractional advection-dispersion equation based on a fractional Fick’s law. Zhang et al. [Water Resources Research, 43(5)(2007)] considered such an equation with variable coefficients, which they dis- cretised using the finite difference method proposed by Meerschaert and Tadjeran [Journal of Computational and Applied Mathematics, 172(1):65-77 (2004)]. For this method the presence of variable coef- ficients necessitates applying the product rule before discretising the Riemann–Liouville fractional derivatives using standard and shifted Gru ̈nwald formulas, depending on the fractional order. As an alternative, we propose using a finite volume method that deals directly with the equation in conservative form. Fractionally-shifted Gru ̈nwald formulas are used to discretise the Riemann–Liouville fractional derivatives at control volume faces, eliminating the need for product rule expansions. We compare the two methods for several case studies, highlighting the convenience of the finite volume approach.