Hat Function Research Articles

On the void explanation of the Cold Spot

Abstract The integrated Sachs–Wolfe (ISW) contribution induced on the cosmic microwave background by the presence of a supervoid as the one detected by Szapudi et al. (2015) is reviewed in this letter in order to check whether it could explain the Cold Spot (CS) anomaly. Two different models, previously used for the same purpose, are considered to describe the matter density profile of the void: a top hat function and a compensated profile produced by a Gaussian potential. The analysis shows that, even enabling ellipticity changes or different values for the dark-energy equation of state parameter ω, the ISW contribution due to the presence of the void does not reproduce the properties of the CS.

Generalization of FEM Using Node-Based Shape Functions

In standard FEM, the stiffness of an element is exclusively influenced by nodes associated with the element via its element-based shape functions. In this paper, the authors present a method that can be viewed as a generalization of FEM for which the influence of a node is not limited by a hat function around the node. Shape functions over an element can be interpolated over a predefined set of nodes around the element. These node-based shape functions employ Kriging Interpolations commonly found in geostatistical technique. In this study, a set of influencing nodes are covered by surrounding layers of elements defined as its domain of influence (DOI). Thus, the element stiffness is influenced by not only the element nodes, but also satellite nodes outside the element. In a special case with zero satellite nodes, the method is specialized to the conventional FEM. This method is referred to as Node-Based Kriging FEM or K-FEM. The K-FEM has been tested on 2D elastostatic, Reissner-Mindlin’s plate and shell problems. In all cases, exceptionally accurate displacement and stress fields can be achieved with relatively coarse meshes. In addition, the same set of Kringing shape functions can be used to interpolate the mesh geometry. This property is very useful for representing the curved geometry of shells. The distinctive advantage of the K-FEM is its inheritance of the computational procedure of FEM. Any existing FE code can be easily extended to K-FEM; thus, it has a higher chance to be accepted in practice.

Fourth-root summation of contrast over area: No end in sight when spatially inhomogeneous sensitivity is compensated by a witch's hat.

Measurements of area summation for luminance-modulated stimuli are typically confounded by variations in sensitivity across the retina. Recently we conducted a detailed analysis of sensitivity across the visual field (Baldwin, Meese, & Baker, 2012) and found it to be well described by a bilinear "witch's hat" function: Sensitivity declines rapidly over the first eight cycles or so, but more gently thereafter. Here we multiplied luminance-modulated stimuli (4 cycles/degree gratings and "Swiss cheeses") by the inverse of the witch's hat function to compensate for the inhomogeneity. This revealed summation functions that were straight lines (on double log axes) with a slope of -1/4 extending to ≥33 cycles, demonstrating fourth-root summation of contrast over a wider area than has previously been reported for the central retina. Fourth-root summation is typically attributed to probability summation, but recent studies have rejected that interpretation in favor of a noisy energy model that performs local square-law transduction of the signal, adds noise at each location of the target, and then sums over signal area. Modeling shows our results to be consistent with a wide field application of such a contrast integrator. We reject a probability summation model, a quadratic model, and a matched template model of our results under the assumptions of signal detection theory. We also reject the high threshold theory of contrast detection under the assumption of probability summation over area.

Two-dimensional Amari neural field model with periodic microstructure: Rotationally symmetric bump solutions

We investigate existence and stability of rotationally symmetric bump solutions to a homogenized two-dimensional Amari neural field model with periodic micro-variations built in the connectivity strength and by approximating the firing rate function with unit step function. The effect of these variations is parameterized by means of one single parameter, called the degree of heterogeneity. The bumps solutions are assumed to be independent of the micro-variable. We develop a framework for study existence of bumps as a function of the degree of heterogeneity as well as a stability method for the bumps. The former problem is based on the pinning function technique while the latter one uses spectral theory for Hilbert–Schmidt integral operators. We demonstrate numerically these procedures for the case when the connectivity kernel is modeled by means of a Mexican hat function. In this case the generic picture consists of one narrow and one broad bump. The radius of the narrow bumps increases with the heterogeneity. For the broad bumps the radius increases for small and moderate values of the activation threshold while it decreases for large values of this threshold. The stability analysis reveals that the narrow bumps remain unstable while the broad bumps are destabilized when the degree of heterogeneity exceeds a certain critical value.

Wavelet analysis of spring climate characteristics in arid aeolian area of agro-pastoral ecotone in China

The unique regional climate characteristics are among the main reasons for the frequent wind-sand activity in arid and cold areas in the agro-pastoral ecotone in Inner Mongolia, China. This paper focuses on the time series of temperature and precipitation in spring when sandstorms often occur in the area. Based on meteorological data for a 46-year period from 1959 to 2004, multi-scale variations and abrupt changes in temperature and precipitation were analyzed with the Mexican hat function (MHF) wavelet method, showing the multi-scale variation characteristics of temperature and precipitation, as well as the periods and change points at different time scales. The relationship between temperature and precipitation was obtained using the wavelet analysis method. Obvious staggered features of the variations of spring temperature and precipitation were observed in this agro-pastoral ecotone. The strongest oscillation periods of spring temperature variations were 1 and 22 years, while for precipitation, the strongest oscillation periods of variations were 2, 8, and 22 years. In addition, lower spring temperature corresponded to lower precipitation, whereas higher temperature yielded higher precipitation rate.

Extraction of time and frequency features from grip force rates during dexterous manipulation.

The time course of grip force from object contact to onset of manipulation has been extensively studied to gain insight into the underlying control mechanisms. Of particular interest to the motor neuroscience and clinical communities is the phenomenon of bell-shaped grip force rate (GFR) that has been interpreted as indicative of feedforward force control. However, this feature has not been assessed quantitatively. Furthermore, the time course of grip force may contain additional features that could provide insight into sensorimotor control processes. In this study, we addressed these questions by validating and applying two computational approaches to extract features from GFR in humans: 1) fitting a Gaussian function to GFR and quantifying the goodness of the fit [root-mean-square error, (RMSE)]; and 2) continuous wavelet transform (CWT), where we assessed the correlation of the GFR signal with a Mexican Hat function. Experiment 1 consisted of a classic pseudorandomized presentation of object mass (light or heavy), where grip forces developed to lift a mass heavier than expected are known to exhibit corrective responses. For Experiment 2, we applied our two techniques to analyze grip force exerted for manipulating an inverted T-shaped object whose center of mass was changed across blocks of consecutive trials. For both experiments, subjects were asked to grasp the object at either predetermined or self-selected grasp locations ("constrained" and "unconstrained" task, respectively). Experiment 1 successfully validated the use of RMSE and CWT as they correctly distinguished trials with versus without force corrective responses. RMSE and CWT also revealed that grip force is characterized by more feedback-driven corrections when grasping at self-selected contact points. Future work will examine the application of our analytical approaches to a broader range of tasks, e.g., assessment of recovery of sensorimotor function following clinical intervention, interlimb differences in force control, and force coordination in human-machine interactions.

Nonlinear Langevin Equations for Wandering Patterns in Stochastic Neural Fields

We analyze the effects of additive, spatially extended noise on spatiotemporal patterns in continuum neural fields. Our main focus is how fluctuations impact patterns when they are weakly coupled to an external stimulus or another equivalent pattern. Showing the generality of our approach, we study both propagating fronts and stationary bumps. Using a separation of time scales, we represent the effects of noise in terms of a phase-shift of a pattern from its uniformly translating position at long time scales, and fluctuations in the pattern profile around its instantaneous position at short time scales. In the case of a stimulus-locked front, we show that the phase-shift satisfies a nonlinear Langevin equation (SDE) whose deterministic part has a unique stable fixed point. Using a linear-noise approximation, we thus establish that wandering of the front about the stimulus-locked state is given by an Ornstein--Uhlenbeck (OU) process. Analogous results hold for the relative phase-shift between a pair of mutually coupled fronts, provided that the coupling is excitatory. On the other hand, if the mutual coupling is given by a Mexican hat function (difference of exponentials), the linear-noise approximation breaks down due to the coexistence of stable and unstable phase-locked states in the deterministic limit. Similarly, the stochastic motion of mutually coupled bumps can be described by a system of nonlinearly coupled SDEs, which can be linearized to yield a multivariate OU process. As in the case of fronts, large deviations can cause bumps to temporarily decouple, leading to a phase-slip in the bump positions.

Dynamics of Moment Neuronal Networks with Intra- and Inter-Interactions

A framework of moment neuronal networks with intra- and inter-interactions is presented. It is to show how the spontaneous activity is propagated across the homogeneous and heterogeneous network. The input-output firing relationship and the stability are first explored for a homogeneous network. For heterogeneous network without the constraint of the correlation coefficients between neurons, a more sophisticated dynamics is then explored. With random interactions, the network gets easily synchronized. However, desynchronization is produced by a lateral interaction such as Mexico hat function. It is the external intralayer input unit that offers a more sophisticated and unexpected dynamics over the predecessors. Hence, the work further opens up the possibility of carrying out a stochastic computation in neuronal networks.

A one-population Amari model with periodic microstructure

We review the derivation of the homogenized one-population Amari equation by means of the two-scale convergence technique of Nguetseng in the case of periodic microvariation in the connectivity function. A key point in this derivation is Visintin's theorem for two-scale convergence of convolution integrals. We construct single bump solutions of the resulting homogenized equation using a pinning function technique for the case where the solutions are independent of the local variable and the firing rate function is modelled as a unit step function. The parameter measuring the degree of heterogeneity plays the role of a control parameter. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. A framework for analysing the stability of these structures is formulated. This framework is based on spectral theory for Hilbert–Schmidt integral operators and it deforms to the standard Evans function approach for the translational invariant case in the limit of no heterogeneity. The upper and lower bounds of the growth/decay rates of the perturbations imposed on the bump states can be expressed in terms of the operator norm of the actual Hilbert–Schmidt operator. Intervals for which the pinning function is increasing correspond to unstable bumps, while complementary intervals where the pinning function decreases correspond to stable bumps, just as in the translational invariant case. Examples showing the properties of the bumps are discussed in detail when the connectivity kernels are given in terms of an exponential decaying function, a wizard hat function and a damped oscillating function.

Dimension walks and Schoenberg spectral measures

Schoenberg (1938) identified the class of positive definite radial (or isotropic) functions φ{symbol}: ℝd → ℝ, φ{symbol}(0) = 1, as having a representation φ{symbol}(x) = ∫ℝ+ Ωd(tu)Gd(du), t = ||x||, for some uniquely identified probability measure Gd on ℝ+ and Ωd(t) = E(eit(e1,η)), where η is a vector uniformly distributed on the unit spherical shell Sd-1 ⊂ ℝd and e1 is a fixed unit vector. Call such Gd a d-Schoenberg measure, and let Φd denote the class of all functions f: ℝ+ → ℝ for which such a d-dimensional radial function φ{symbol} exists with f(t) = φ{symbol}(x) for t = ||x||. Matheron (1965) introduced operators Ĩ and D, called Montee and Descente, that map suitable f ∈ Φd into Φd' for some different dimension d': Wendland described such mappings as dimension walks. This paper characterizes Matheron's operators in terms of Schoenberg measures and describes functions, even in the class Φ∞ of completely monotone functions, for which neither Ĩf nor Df is well defined. Because f ∈ Φd implies f ∈ Φd' for d' < d, any f ∈ Φd has a d' Schoenberg measure Gd' for 1 ≤ d' < d and finite d ≥ 2. This paper identifies Gd' in terms of Gd via another 'dimension walk' relating the Fourier transforms Ωd' and Ωd that reflect projections on ℝd' within ℝd. A study of the Euclid hat function shows the indecomposability of Ωd. © 2014 American Mathematical Society.

Two bump solutions of a homogenized Wilson–Cowan model with periodic microstructure

We study the existence and stability of 2-bump solutions of the one-population homogenized Wilson–Cowan model, where the heterogeneity is built in the connectivity functions by assuming periodic modulations in both the synaptic footprint and in the spatial scale. The existence analysis reveals that the generic picture consists of two bumps states for each admissible threshold value for the case when the solutions are independent of the local variable and the firing rate function is modeled as a Heaviside function. A framework for analyzing the stability of 2-bumps is formulated, based on spectral theory for Fredholm integral operators. The stability method deforms to the standard Evans function approach for the translationally invariant case in the limit of no heterogeneity, in a way analogous to the single bump case for the homogenized model. The numerical study of the stability problem reveals that both the broad and narrow bumps are unstable just as in the translationally invariant case when the connectivity function is modeled by means of a wizard hat function. For the damped oscillating connectivity kernel, we give a concrete example of a 2-bump solution which is stable for all admissible values of the heterogeneity parameter.

Discrimination of Visual Evoked Potentials Using Image Processing of Their Time-scale Representations

In this article, the continuous wavelet transform, based on the Mexican Hat function, is used to achieve an automatic processing and classification of visual evoked potentials. Indeed, the representations of the modulus of the wavelet coefficients in the time-scale plan, allowed us to define qualitative criterion for the discrimination of normal and pathological cases. In order to enhance the classification rate, we developed a new method that considers the visual evoked potential's scalogram as an image which is segmented, according to the maximal level of energy density, into many regions of significant interest. The analysis of the resulting segmented image permits to extract a vector of most significant features that will be used to classify normal and pathological signals through SVM classifier. The obtained results show that this technique not only, allows a more reliable distinction between normal and pathological cases, with a very high classification rate (93%) in comparison with the one of the conventional latency measurement (67%), but also, suggests that indications on the progression of the pathology can be provided.

An Image Mosaic Method Based on SIFT Feature Matching

This paper concerns the problem of image mosaic. An image matching method based on SIFT features and an image blending method of improved Hat function are proposed in the paper. SIFT feature is local feature and keeps invariant to scale zoom, rotation and illumination. It is also insensitive to noise, view point changing and so on. Because of this our method is insensitive to orientation, scale and illumination of input images, so it’s possible to accomplish image mosaic between arbitrary matching images and the Hat function blending algorithm with global intensity revise makes the mosaic image accepted by human eyes.

Traveling pulses in a stochastic neural field model of direction selectivity

We analyze the effects of extrinsic noise on traveling pulses in a neural field model of direction selectivity. The model consists of a one-dimensional scalar neural field with an asymmetric weight distribution consisting of an offset Mexican hat function. We first show how, in the absence of any noise, the system supports spontaneously propagating traveling pulses that can lock to externally moving stimuli. Using a separation of time-scales and perturbation methods previously developed for stochastic reaction-diffusion equations, we then show how extrinsic noise in the activity variables leads to a diffusive-like displacement (wandering) of the wave from its uniformly translating position at long time-scales, and fluctuations in the wave profile around its instantaneous position at short time-scales. In the case of freely propagating pulses, the wandering is characterized by pure Brownian motion, whereas in the case of stimulus-locked pulses, it is given by an Ornstein–Uhlenbeck process. This establishes that stimulus-locked pulses are more robust to noise.

A Petrov–Galerkin method for solving the generalized regularized long wave (GRLW) equation

The generalized regularized long wave (GRLW) equation is solved numerically by the Petrov–Galerkin method which uses a linear hat function as the trial function and a quintic B-spline function as the test function. Product approximation has been used in this method. A linear stability analysis of the scheme shows it to be conditionally stable. Test problems including the single soliton and the interaction of solitons are used to validate the suggested method, which is found to be accurate and efficient. Finally, the Maxwellian initial condition pulse is studied.

Transformed density rejection with inflection points

The acceptance-rejection algorithm is often used to sample from non-standard distributions. For this algorithm to be efficient, however, the user has to create a hat function that majorizes and closely matches the density of the distribution to be sampled from. There are many methods for automatically creating such hat functions, but these methods require that the user transforms the density so that she knows the exact location of the transformed density’s inflection points. In this paper, we propose an acceptance-rejection algorithm which obviates this need and can thus be used to sample from a larger class of distributions.

Numerical solutions of the modified Burgers’ equation by Petrov–Galerkin method

The modified Burgers’ equation (MBE) is solved numerically by the Petrov–Galerkin method using a linear hat function as the trial function and a cubic B-spline function as the test function. Product approximation has been used in this method. A linear stability analysis of the scheme shows it to be unconditionally stable. The accuracy of the presented method is demonstrated by two test problems. The numerical results are found in good agreement with the exact solutions.

A Petrov–Galerkin method for equal width equation

The equal width equation is solved numerically by Petrov–Galerkin method using linear hat function and quadratic B-spline function as trial and test functions respectively. Product approximation has been used in this method. A linear stability analysis of the scheme is shown to be conditionally stable. Test problems including the single soliton and the interaction of solitons are used to validate the suggested method, which is found to be accurate and efficient. The Maxwellian initial condition pulse and the development of an undular bore are also studied.

On the phi and psi transforms of Frazier and Jawerth

AbstractIn this article we study the phi and psi transforms of Frazier and Jawerth when either phi or psi is not band‐limited. Our main result concerns the invertibility of these transforms and is used to show that certain affine system generated by the Mexican hat function is complete in Lebesgue and Hardy spaces. © 2011 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim

Highly efficient rejection method for generating Nakagami-m sequences

A simple and highly efficient rejection method is presented for generating Nakagami- m sequences with arbitrary average power and fading factor. With a novel exponential decay hat function, the proposed method's rejection efficiency is up to 80% and the difference between theoretical and simulated distribution curves are neglectable.