HIGHER ACCURACY ORDER IN DIFFERENTIATION-BY-INTEGRATION

In this text explicit forms of several higher precision order kernel functions (to be used in the differentiation-by-integration procedure) are given for several derivative orders. Also, a system of linear equations is formulated which allows to construct kernels with an arbitrary precision for an arbitrary derivative order. A computer study is realized and it is shown that numerical differentiation based on higher precision order kernels performs much better (w.r.t. errors) than the same procedure based on the usual Legendre-polynomial kernels. Presented results may have implications for numerical implementations of the differentiation-by-integration method.

Relationship between structural and functional changes in glaucomatous eyes: A multifocal electroretinogram study

Background: The association between the structure of the macular region and its function as measured by multifocal electroretinography (mfERG) and the mean thresholds (MT) of the visual field (VF) is not well-understood. Methods: The macular retinal nerve fiber layer (mRNFL) and the ganglion cell and inner plexiform layer (GCIPL) in six regions were measured by optical coherence tomography (OCT). For functional assessment, MT and mfERG scans with parameters of the second-order kernel responses within the central 5°, nasal to temporal amplitudes ratio (N/T), and the multifocal photopic negative response to B-wave ratio (mfPhNR/B) were measured. Forty-one glaucoma patients underwent OCT, mfERG, and MT measurement and 55 healthy subjects underwent mfERG. Results: The mfPhNR/B was significantly smaller ( P < 0.01) and the N/T was significantly larger ( P < 0.01) in glaucoma patients than in normal subjects. In glaucoma patients, the N/T is significantly correlated with the thickness of inferior and inferotemporal GCIPL ( r = -0.317 and -0.360, respectively) and MT of corresponding VF areas ( r = -0.330 and -0.334, respectively) (all P values < 0.05). The mfPhNR/B was significantly correlated with the thickness of mRNFL in the central area ( r = 0.365, P = 0.02) and with the MT of all corresponding VF areas ( r rages between 0.330 and 0.460, all P values < 0.04), except for the inferotemporal area. However, correlation was not observed between the N/T and the mfPhNR/B in any location. Conclusions: Significant differences exist between glaucoma and healthy participants in the N/T and mfPhNR/B. Correlations were observed between two mfERG parameters and OCT parameters or MT in glaucoma patients. Further research should seek to demonstrate whether the N/T and the mfPhNR/B should be applied in a complementary fashion.

Hysteretic behavior of spatially coupled phase-oscillators

Motivated by phenomena related to biological systems such as the synchronously flashing swarms of fireflies, we investigate a network of phase oscillators evolving under the generalized Kuramoto model with inertia. A distance-dependent, spatial coupling between the oscillators is considered. Zeroth and first order kernel functions with finite kernel radii were chosen to investigate the effect of local interactions. The hysteretic dynamics of the synchronization depending on the coupling parameter was analyzed for different kernel radii. Numerical investigations demonstrate that (1) locally locked clusters develop for small coupling strength values, (2) the hysteretic behavior vanishes for small kernel radii, (3) the ratio of the kernel radius and the maximal distance between the oscillators characterizes the behavior of the network.

A comparative analysis of grey ranking approaches

Purpose Ranking and comparing grey numbers represent a very important decision-making procedure in any given grey environment. The purpose of this paper is to study the existing approaches of ordering interval grey numbers in the context of decision making by surveying existing definitions. Design/methodology/approach Different methods developed for comparing grey numbers are presented along with their disadvantages and advantages in terms of comparison outcomes. Practical examples are employed to show the importance and necessity of using appropriate methods to compare grey numbers. Findings Most the available methods are not suitable for pointing out which number is larger when the nuclei of the grey numbers of concern are the same. Also, these available methods are also considered in terms of partial order and total order. Kernel and degree of greyness of grey numbers method is more advantageous than other methods and almost eliminates the shortcomings of other methods. Originality/value Different methods for ranking grey numbers are presented where each of them has disadvantages and advantages. By using different methods, grey interval numbers are compared and the results show that some methods cannot make grey number comparisons in some situations. The authors intend to find a method that can compare grey numbers in any situation. The findings of this research can prevent errors that may occur based on inaccurate comparisons of grey numbers in decision making. There are various research studies on the comparison of grey numbers, but there is no research on the comparison of these methods and their disadvantages, advantages or their total or partial order.

On Hybridizations of Fourth Order Kernel of the Beta Polynomial Family.

The usual second order nonparametric kernel estimators are of wide uses in data analysis and visualization but constrained with slow convergence rate. Higher order kernels provide a faster convergence rates and are known to be bias reducing kernels. In this paper, we propose a hybrid of the fourth order kernel which is a merger of two successive fourth order kernels and the statistical properties of these hybrid kernels were study. The results of our simulation reveals that the proposed higher order hybrid kernels outperformed their corresponding parent’s kernel functions using the asymptotic mean integrated squared error.

3D MR Image Denoising using higher Order Kernel Regression

Noise removal from images is among the challenging processes for researches. Image denoising is a crucial step to improve 3D image conspicuity and to enhance the performance of all the processing needs of quantitative image analysis. Magnetic Resonance (MR) imaging has an increasing importance in the field of medical diagnosis. MR 3D image de-noising has two features (i) tri-dimensional structure of images and (ii) the nature of the noise, which are Rician & Gaussian. Kernel regression is one of 3D non-parametric noise level estimation technique which is effective than other denoising experimental filters. The proposed Fourth order Kernel Regression (FKR) algorithm builds an efficient and robust estimator and improves the accuracy of noise and it further improves the finer estimations of pixel value and its gradients. Experimental results demonstrate positively by achieving better performance, with respect to other de-noising filters.