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Foot-mounted positioning devices are becoming more and more popular in the different application field. For example, inertial sensors are now embedded in safety shoes to monitor security. They allow positioning with zero velocity update to bound the error growth of foot-mounted inertial sensors. High positioning accuracy depends on robust zero velocity detector (ZVD). Existing Artificial Intelligent (AI)-based methods classify the pedestrian dynamics to adjust ZVD at the cost of high computation costs and error propagation from miss-classification. We propose a machine learning model to detect zero velocity moments without any pre-classification step, named Uniform AI Model for All pedestrian Motions (UMAM). Performance is evaluated by benchmarking on two new subjects of opposite gender and different size, not included in the training data set, over complex indoor/outdoor paths of 2 km for subject 1 and 2.1 km for subject 2. We obtain an average 2D loop closure error of less than 0.37%.

Wall shear stress (WSS), the spatial gradient of flow velocity at luminal surface, has been employed for aneurysmal hemodynamic analysis, but it is sensitive to surface irregularities and noise. We devised a volumetric approach to evaluate discriminant power of intra-dome flow velocity distribution and modal analysis in rupture status determination compared with previously described WSS analysis.Catheter three-dimensional rotational angiographic datasets matched for volume were segmented in 20 sidewall aneurysms (10 ruptured), computational fluid dynamics simulations were performed, and velocity distributions were extracted from mesh-independent isometric sampling followed by moment analysis (mean, variance, skewness, and kurtosis). Univariate and multivariate analysis was used to evaluate discriminant performance of velocity moments. Sensitivity of velocity moments and WSS was evaluated to bleb presence and surface irregularity using digital bleb removal and surface noise addition.Velocity moments of ruptured aneurysms showed higher skewness (2.45 ± 0.57 vs. 1.36 ± 0.82, P = 0.003) and kurtosis (11.83 ± 4.77 vs. 6.05 ± 4.65, P = 0.01) with lower mean (0.019 ± 0.01 vs. 0.038 ± 0.02, P = 0.03) compared with unruptured lesions; in multivariate modeling, skewness alone emerged as best predictor (area under the curve = 0.88). Bleb removal increased low WSS by 548%, and surface noise decreased it by 85.8% while having a smaller (7%) effect on velocity skewness and kurtosis.High aneurysm dome flow velocity skewness and kurtosis suggest an exponential distribution in ruptured lesions, with high peaks at low velocities, consistent with areas of slow flow. In contrast to WSS-based techniques, this approach is robust against surface variations, with promising improved rupture status discriminant performance that requires further validation in expanded future studies.

The current study performed the laboratory flume to examine the velocity distribution and the velocity moments of turbulent flow over a sand-gravel mixture bed. The 3D instantaneous velocity data of water is collected by using acoustic doppler velocimeter (ADV) at the test section which will provide an important data related to the flow turbulence. The parameters of turbulence measured that the lower value of longitudinal velocity (velocity along flow direction) is observed close to the bed surface and increases with the flow depth. The lower magnitude of von Karman constant is achived than the universal value. The damping of Reynolds shear stress observed close to the boundary wall indicating lower exchange of flow energy towards the boundary and vice versa. The sand-gravel bed influenced the turbulence intensities with higher magnitude in the streamwise than those in vertical direction. The velocity moments which shows the structure function of fluctuating components of velocity, has been analyzed. The present study also analysed the flow anisotropy, and the flatness factor in the flow.

I present a flexible solution for the axisymmetric Jeans equations of stellar hydrodynamics under the assumption of an anisotropic (three-integral) velocity ellipsoid aligned with the spherical polar coordinate system. I describe and test a robust and efficient algorithm for its numerical computation. I outline the evaluation of the intrinsic velocity moments and the projection of all first and second velocity moments, including both the line-of-sight velocities and the proper motions. This spherically-aligned Jeans Anisotropic Modelling (JAM_sph) method can describe in detail the photometry and kinematics of real galaxies. It allows for a spatially-varying anisotropy, or stellar mass-to-light ratios gradients, as well as for the inclusion of general dark matter distributions and supermassive black holes. The JAM_sph method complements my previously derived cylindrically-aligned JAM_cyl and spherical Jeans solutions, which I also summarize in this paper. Comparisons between results obtained with either JAM_sph or JAM_cyl can be used to asses the robustness of inferred dynamical quantities. As an illustration, I modelled the Atlas3D sample of 260 early-type galaxies with high-quality integral-field spectroscopy, using both methods. I found that they provide statistically indistinguishable total-density logarithmic slopes. This may explain the previously-reported success of the JAM method in recovering density profiles of real or simulated galaxies. A reference software implementation of JAM_sph is included in the publicly-available JAM software package., Comment: 19 pages, 12 figures, LaTeX. Accepted for publication in MNRAS. The method described in this paper is included in the JAM software package at https://purl.org/cappellari/software

Starting from a generalized population balance equation and the Boltzmann–Enskog collision model for hard spheres, a kinetic theory model for polydisperse gas–particle flows is presented. Here, polydispersity results from spherical particles with the same material density but different diameters. The particle size distribution (PSD) of the particles is allowed to evolve in space and time due to physical processes such as mixing. In order to treat a continuous PSD, the particle-phase model is formulated in terms of the moments of the PSD, and velocity moments conditioned on the particle size. Velocity moments up to second order are included, resulting in transport equations for the mass, momentum and granular temperature, all conditioned on the particle size. In the numerical implementation, the PSD is represented using quadrature-based moment methods (QBMM). With QBMM, a continuous PSD can be treated using a relatively small number of moments as compared to class or sectional methods. Here, a realizable numerical algorithm for solving the moment system in a finite-volume code is proposed, valid for dilute systems wherein frictional forces are negligible. The ability of the proposed model to describe polydisperse gas–particle systems is demonstrated using cluster-induced turbulence and riser flow.

In this paper, we prove a quantitative version of the semiclassical limit from the Hartree to the Vlasov equation with singular interaction, including the Coulomb potential. To reach this objective, we also prove the propagation of velocity moments and weighted Schatten norms which implies the boundedness of the space density of particles uniformly in the Planck constant., 42 pages, typos added, clarifications of hypotheses in theorems 3 and 4 and missing sup_{[0,T]} added

We consider weak solutions to the three dimensional Vlasov–Poisson system in the periodic case. We show that the velocity moments of order >3 can be propagated.

The pairwise velocity generating function $G$ has deep connection with both the pairwise velocity probability distribution function and modeling of redshift space distortion (RSD). Its implementation into RSD modeling is often faciliated by expansion into series of pairwise velocity moments $\langle v_{12}^n\rangle$. Motivated by the logrithmic transformation of the cosmic density field, we investigate an alternative expansion into series of pairwise velocity cumulants $\langle v_{12}^n\rangle_c$ . We numerically evaluate the convergence rate of the two expansions, with three $3072^3$ particle simulations of the CosmicGrowth N-body simulation series. (1) We find that the cumulant expansion performs significantly better, for all the halo samples and redshifts investigated. (2) For modeling RSD at $k_{\|}, 19 pages, 11 figures

In the literature different so-called $\kappa$-distribution functions are discussed to fit and model the velocity (or energy) distributions of solar wind species, pickup ions or magnetospheric particles. Here we introduce a generalized (isotropic) $\kappa$-distribution as a "cookbook", which admits as special cases, or "recipes", all the other known versions of $\kappa$-models. A detailed analysis of the generalized distribution function is performed, providing general analytical expressions for the velocity moments, Debye length, and entropy, and pointing out a series of general requirements that plasma distribution functions should satisfy. From a contrasting analysis of the recipes found in the literature, we show that all of them lead to almost the same macroscopic parameters with a small standard deviation between them. However, one of these recipes called the regularized $\kappa$-distribution provides a functional alternative for macroscopic parameterization without any constraint for the power-law exponent $\kappa$., Comment: 20 pager, 9 Figures

This experimental study investigated the higher-order velocity moments, turbulence time and length scales, and energy dissipation rates around an intermediately submerged boulder within a wake-interference flow regime in a rock-ramp fish passage. The results show a noticeable variation in the studied parameters in the wake of the boulder, as well as near the bed and boulder crest. The higher-order velocity moments show the presence of infrequent strong ejections downstream of the boulder, which may lead to higher sediment deposition and vertical mixing. The eddy length scales and the volumetric energy dissipation in this experimental model were discussed in relation to fish behavior for both the experimental model and a prototype. Relationships were proposed to roughly estimate integral length scales and energy dissipation rates around the boulder over the flow depth. The findings of this study may improve the design of rock-ramp fish passages considering the effects of turbulence on fish swimming performance and sediment transport.

The peculiar velocities of biased tracers of the cosmic density field contain important information about the growth of large scale structure and generate anisotropy in the observed clustering of galaxies. Using N-body data, we show that velocity expansions for halo redshift-space power spectra are converged at the percent-level at perturbative scales for most line-of-sight angles $\mu$ when the first three pairwise velocity moments are included, and that the third moment is well-approximated by a counterterm-like contribution. We compute these pairwise-velocity statistics in Fourier space using both Eulerian and Lagrangian one-loop perturbation theory using a cubic bias scheme and a complete set of counterterms and stochastic contributions. We compare the models and show that our models fit both real-space velocity statistics and redshift-space power spectra for both halos and a mock sample of galaxies at sub-percent level on perturbative scales using consistent sets of parameters, making them appealing choices for the upcoming era of spectroscopic, peculiar-velocity and kSZ surveys., Comment: 63 pages, 11 figures, updated to match version accepted by JCAP

We investigate the stellar and dynamical mass profiles in the centres of 25 brightest cluster galaxies (BCGs) at redshifts of 0.05 $\leq z \leq$ 0.30. Our spectroscopy enables us to robustly measure the Gauss-Hermite higher order velocity moments $h_{3}$ and $h_{4}$, which we compare to measurements for massive early-type galaxies, and central group galaxies. We measure positive central values for $h_{4}$ for all the BCGs. We derive the stellar mass-to-light ratio ($\Upsilon_{\star \rm DYN}$), and velocity anisotropy ($\beta$) based on a Multi-Gaussian Expansion (MGE) and axisymmetric Jeans Anisotropic Methods (JAM, cylindrically- and spherically-aligned). We explicitly include a dark matter halo mass component, which is constrained by weak gravitational lensing measurements for these clusters. We find a strong correlation between anisotropy and velocity dispersion profile slope, with rising velocity dispersion profiles corresponding to tangential anisotropy and decreasing velocity dispersion profiles corresponding to radial anisotropy. The rising velocity dispersion profiles can also indicate a significant contribution from the intracluster light (ICL) to the total light (in projection) in the centre of the galaxy. For a small number of BCGs with rising velocity dispersion profiles, a variable stellar mass-to-light ratio can also account for the profile shape, instead of tangential anisotropy or a significant ICL contribution. We note that, for some BCGs, a variable $\beta_{z}(r)$ (from radial to tangential anisotropy) can improve the model fit to the observed kinematic profiles. The observed diversity in these properties illustrates that BCGs are not the homogeneous class of objects they are often assumed to be., Comment: 25 pages, accepted for publication in MNRAS

The four-fifths law for third-order longitudinal moments is examined, using direct numerical simulation (DNS) data on three-dimensional (3D) forced incompressible magnetohydrodynamic (MHD) turbulence without a uniformly imposed magnetic field in a periodic box. The magnetic Prandtl number is set to one, and the number of grid points is ${512}^{3}$. A generalized K\'arm\'an-Howarth-Kolmogorov equation for second-order velocity moments in isotropic MHD turbulence is extended to anisotropic MHD turbulence by means of a spherical average over the direction of $\mathbit{r}$. Here, $\mathbit{r}$ is a separation vector. The viscous, forcing, anisotropic and nonstationary terms in the generalized equation are quantified. It is found that the influence of the anisotropic terms on the four-fifths law is negligible at small scales, compared to that of the viscous term. However, the influence of the directional anisotropy, which is measured by the departure of the third-order moments in a particular direction of $\mathbit{r}$ from the spherically averaged ones, on the four-fifths law is suggested to be substantial, at least in the case studied here.

Bank erosion in a sinuous alluvial channel is a continuous phenomenon resulting in bank instability and migration of sediment. In this study, flume experiments were conducted in a sinuous channel to investigate its morphological changes and hydrodynamics. High-order velocity fluctuation moments are analyzed at outer and inner banks to explain the morphological variation in a sinuous river channel. The variance of streamwise velocity fluctuations on both banks of the sinuous channel follows a logarithmic law from a particular depth. In the outer bend region, the magnitude of velocity fluctuation moment is significant, indicating erosion. The trend of velocity fluctuation at higher even-order moments is similar to the variance of streamwise velocity fluctuations where the outer bend magnitude is greater than the inner bend. The premultiplied probability density functions (PDFs) and the flatness factor show greater magnitude in the outer bend of the channel as compared to the inner bend.

The velocities of space plasma particles, often follow kappa distribution functions. The kappa index, which labels and governs these distributions, is an important parameter in understanding the plasma dynamics. Space science missions often carry plasma instruments on board which observe the plasma particles and construct their velocity distribution functions. A proper analysis of the velocity distribution functions derives the plasma bulk parameters, such as the plasma density, speed, temperature, and kappa index. Commonly, the plasma bulk density, velocity, and temperature are determined from the velocity moments of the observed distribution function. Interestingly, recent studies demonstrated the calculation of the kappa index from the speed (kinetic energy) moments of the distribution function. Such a novel calculation could be very useful in future analyses and applications. This study examines the accuracy of the specific method using synthetic plasma proton observations by a typical electrostatic analyzer. We analyze the modeled observations in order to derive the plasma bulk parameters, which we compare with the parameters we used to model the observations in the first place. Through this comparison, we quantify the systematic and statistical errors in the derived moments, and we discuss their possible sources.

We analyze the effects of the passing energetic particles on the resistive ballooning modes (RBM) and the energetic particle driven modes in JT-60SA plasma, which leads to the prediction of the stability in N-NBI heated plasma. The analysis is performed using the code FAR3d that solves the reduced MHD equations describing the linear evolution of the poloidal flux and the toroidal component of the vorticity in a full 3D system, coupled with equations of density and parallel velocity moments for the energetic particle (EP) species assuming an averaged Maxwellian EP distribution fitted to the slowing down distribution, including the effect of the acoustic modes. The simulations show the possible destabilization of a 3 / 2 − 4 / 2 TAE with a frequency (f) of 115 kHz, a 6 / 4 − 7 / 4 TAE with f = 98 kHz and a 6/4 or 7/4 BAE with f = 57 kHz in the ITER-like inductive scenario. If the energetic particle β increases, beta induced Alfven Eigenmodes (BAE), toroidal AEs (TAE) and elliptical AEs (EAE) are destabilized between the inner-middle plasma region, leading to the overlapping of AE of different toroidal families. If these instabilities coexist in the non-linear saturation phase the EP transport could be enhanced leading to a lower heating efficiency. For a hypothetical configuration based on the ITER-like inductive scenario but an center peaked EP profile, the EP β threshold increases and several BAEs are destabilized in the inner plasma region, indicating an improved AE stability with respect to the off-axis peaked EP profile. In addition, the analysis of a hypothetical JT-60SA scenario with a resonant q = 1 in the inner plasma region shows the destabilization of fishbones-like instabilities by the off-axis peaked EP profile. Also, the EPs have a stabilizing effect on the RBM, stronger as the population of EP with low energies (below 250 keV) increases at the plasma pedestal.

Cheniers are important for stabilising mud-dominated coastlines. A chenier is a body of wave-reworked, coarse-grained sediment consisting of sand and shells overlying a muddy substrate. In this paper we present and analyse a week of field observations of the dynamics of a single chenier along the coast of Demak, Indonesia. Despite relatively calm hydrodynamics during the one-week observational period, the chenier migrated surprisingly fast in the landward direction. The role of the tide and waves on the cross-shore chenier dynamics is explored using velocity moments as a proxy for the sediment transport. This approach shows that both tide and waves are capable of transporting the sediment of the chenier system. During calm conditions (representative for the south-east monsoon season), the tides generate a landward-directed sediment transport when the chenier crest is high relative to mean sea level. Waves only generate substantial sediment transport (direct, via skewness, and indirect, via stirring) when the chenier is submerged during periods with higher waves. The cross-shore chenier dynamics are very sensitive to the timing of tide and waves: most transport takes place when high water levels coincide with (relatively) high waves.

Problem of determination of isomer shift, all components of the electric field gradient and hyperfine magnetic field in case of mixed hyperfine interactions is presented. Orientation of hyperfine fields in the absorber Cartesian frame can be determined by few measurements with use of unpolarized radiation under different directions of wave vector with respect to the absorber. The method can be applied for absorbers with well separated absorption lines in their spectra. Explicit formulas for tensor components of hyperfine interactions derived from velocity moments formalism are presented.

We investigate the dynamics of the Vlasov-Poisson system in the presence of radiation damping. A propagation result for velocity moments of order $k>3$ is established in (Kunze and Rendall in Ann. Henri Poincare 2:857–886, 2001). In this paper, we prove existence of global solutions propagating velocity and velocity-spatial moments of order $k>2$ and establish an explicit polynomially growing in time bound on the moments.

Moment-based Angular Radial Transform, Legendre moment invariants and Zernike moments are a family of orthogonal functions which allow the generation of non-redundant descriptors by the projection of an image onto an orthogonal basis. These descriptors can be used for classification, such as in face recognition. Zernike moments and Legendre moments have already been used for this purpose.This paper proposes to use moment-based Angular Radial Transform for extracting the face characteristics that feed a Support Vector Machine or a Nearest Neighbor Classifier for face recognition. Facial images from the ORL database, Essex Faces94 database, Essex Faces96 database, and Yale database were used for testing the proposed approach. The experimental results obtained show that the proposed method is more efficient, in terms of recognition rate, than the methods based on Zernike and Legendre moments. It is also found that its performance is comparable to that of the best state-of-the-arts methods. Keywords: Angular Radial Transform (ART), Legendre moment invariants (LMI), Euclidean distance (ED), Pseudo-Zernike moments (PZM), Nearest Neighbor Classifier (NNC), Support Vector Machines (SVM)

Krawtchouk moments are discrete orthogonal moments which are effective for image representation. The translation and scale invariants of Krawtchouk moments are achieved either by normalizing the image or by using a combination of the corresponding invariants of geometric moments. However, the derivation of these functions is not based on Krawtchouk polynomials. In this paper, we propose a new method to derive the translation and scale invariants of Krawtchouk moments directly from the Krawtchouk polynomials. The performance of the proposed method is verified using binary characters. Experimental results show that the values of the Krawtchouk moments are invariant under image translation and scale. Furthermore, the speed of the proposed method is faster than conventional methods.

Vector field images are a type of new multidimensional data that appear in many engineering areas. Although the vector fields can be visualized as images, they differ from graylevel and color images in several aspects. To analyze them, special methods and algorithms must be originally developed or substantially adapted from the traditional image processing area. In this paper, we propose a method for the description and matching of vector field patterns under an unknown rotation of the field. Rotation of a vector field is so-called total rotation, where the action is applied not only on the spatial coordinates but also on the field values. Invariants of vector fields with respect to total rotation constructed from orthogonal Gaussian–Hermite moments and Zernike moments are introduced. Their numerical stability is shown to be better than that of the invariants published so far. We demonstrate their usefulness in a real world template matching application of rotated vector fields.

Image moments such as zernike, tchebichef and geometric moments have been widely used in image processing applications. They have useful properties to detect edges. In this paper, we present how one of the moment families, in particular geometric moments (GM) can be utilized in estimating the sigma and size of the Gaussian point spread function (PSF) that degrades the images. With the knowledge of how edges vary in the presence of Gaussian blur, a method that uses low order geometric moments is proposed to estimate the PSF parameter. This is achieved by using the difference of the GMs of the original and the reblurred images as feature vectors to train extreme learning machine (ELM) to estimate the PSF parameters respectively. Further, a novel method that uses a cascaded digital filters operating as subtractors is proposed to transform the image from geometric moment domain to spatial domain. The effectiveness of the proposed method of estimating the PSF parameters is examined using cross database validation. The results show that the proposed method in most of the cases performs better than the three existing methods when examined in terms of the visual quality evaluated using structural similarity (SSIM) index.

A quadrature-based moment method for the approximate solution of the generalized population balance equation (GPBE) governing the evolution of the joint size-velocity number density function (NDF) of a particle population is formulated and tested. The proposed method relies on the third-order hyperbolic conditional quadrature method of moments developed for velocity distribution transport. This approach is combined with the conditional quadrature method of moments to incorporate the dependency of the NDF on particle size, leading to an efficient, stable, quadrature method that uses an analytical solution to determine the size-conditioned velocity moments. The incorporation of source terms accounting for aggregation, breakup, and collisions, as well as acceleration terms such as gravity and drag, is performed using a realizable ODE solver. The approach is then demonstrated by considering zero-dimensional cases to verify the correct integration of the source terms. A set of one-dimensional cases involving droplet evaporation and coalescence is used to validate the velocity-dependent source terms. A two-dimensional case of crossing jets of particles with different sizes is used to demonstrate the proposed method in the case of a polydisperse flow with inertial particles. All work has been implemented in the open-source framework OpenQBMM, based on OpenFOAM®.

Compound ion distributions, fi(v), have been measured with high-time resolution by NASA's Magnetospheric Multi-Scale Mission (MMS) and have been found in reconnection simulations. A compound distribution, fi(v), consisting, for example, of essentially disjoint pieces will be called a multi-beam distribution and modeled as a sum of “beams,” fi(v) = f1(v) + ⋯ + fN(v). Velocity moments of fi(v) are taken beam by beam and summed. Such multi-beam moments of fi(v) have advantages over the customary standard velocity moments of fi(v), for which there is only one mean flow velocity. For example, the standard thermal energy moment of a pair of equal and opposite cold particle beams is non-zero even though each beam has zero thermal energy. We therefore call this thermal energy pseudothermal. By contrast, a multi-beam moment of two or more beams has no pseudothermal energy. We develop three different ways of approximating a compound ion velocity distribution, fi(v), as a sum of beams and finding multi-beam moments for both a compound fi(v) measured by MMS in the dayside magnetosphere during reconnection and a compound fi(v) found in a particle-in-cell simulation of magnetotail reconnection. The three methods are (i) a visual method in which the velocity centroid of each beam is estimated and the beam densities are determined self-consistently, (ii) a k-means method in which particles in a particle representation of fi(v) are sorted into a minimum energy configuration of N (= k) clusters, and (iii) a nonlinear least squares method based on a fit to a sum of N kappa functions. Multi-beam energy moments are calculated and compared with standard moments for the thermal energy density, pressure tensor, thermal energy flux (heat plus enthalpy fluxes), bulk kinetic energy density, ram pressure, and bulk kinetic energy flux. Applying this new formalism to real data demonstrates in detail how multi-beam techniques provide new insights into the energetics of observed space plasmas.

We examine the vertical structure of Galactic disk kinematics over a Galactocentric radial distance range of $R=5-15$ $\rm{kpc}$ and up to $3$ $\rm{kpc}$ away from the Galactic plane, using the K-type giants surveyed by LAMOST. Based on robust measurements of three-dimensional velocity moments, a wobbly disk is detected in a phenomenological sense. An outflow dominates the radial motion of the inner disk, while in the outer disk there exist alternate outward and inward flows. The vertical bulk velocities is a combination of breathing and bending modes. A contraction-like breathing mode with amplitudes increasing with the distance to the plane and an upward bending mode dominate the vertical motion outside $R_0$, and there are reversed breathing mode and bending mode at $R, Comment: 17 pages, 23 figures

In the current study, model expressions for fifth-order velocity moments obtained from the truncated Gram-Charlier series expansions model for a turbulent flow field probability density function are validated using data from direct numerical simulation (DNS) of a planar turbulent flow in a strained channel. Simplicity of the model expressions, the lack of unknown coefficients, and their applicability to non-Gaussian turbulent flows make this approach attractive to use for closing turbulent models based on the Reynolds-averaged Navier-Stokes equations. The study confirms validity of the model expressions. It also shows that the imposed flow deformation improves an agreement between the model and DNS profiles for the fifth-order moments in the flow buffer zone including when the flow reverses its direction. The study reveals sensitivity of particularly odd velocity moments to the grid resolution. A new length scale is proposed as a criterion for the grid generation near the wall and in the other flow areas dominated by high mean velocity gradients when higher-order statistics have to be collected from DNS.

In this paper, we propose a new method for the rapid calculation of Meixner’s discrete orthogonal moments and its inverses. In this method, we have used the notion of digital filters based on the Z transform both to accelerate the computation time of Meixner and to reduce the reconstruction error of the images. To guarantee the numerical stability and robustness with respect to the noise, we propose two algorithms that treat the images as a set of blocks where each block will be treated independently. In fact, through the first algorithm, the moments of Meixner are computed from a set of geometric blocks of fixed size. On the other hand, in the second algorithm, the images are represented by a slice set where each slice contains several homogeneous blocks of different sizes. The moments of Meixner, in this case, are calculated from each block in each slice. The application of these two algorithms allowed us to deduce a significant reduction in processed information and image space, that permits the use of Meixner moments of low order for a better description of the fine details of the images. The performances of the proposed method are demonstrated through several simulations on different image bases.

In this paper, we introduce new sets of separable discrete moments for 3D image analysis, named: TKKM (Tchebichef-Krawtchouk-Krawtchouk Moments) and TTKM (Tchebichef-Tchebichef-Krawtchouk Moments). Firstly, we present a detailed comparative study between the proposed separable 3D moments and the classical ones in terms of global feature extraction capability under noisy and noise-free conditions. Also, their local feature extraction ability is examined. Secondly, our study investigates the ability of the proposed separable 3D moments in pattern recognition. For this, new sets of separable 3D discrete moment invariants are introduced. The proposed rotation, scaling and translation 3D moment invariants have been rigorously tested under different sets of mixed transforms. The obtained results show that the representation capability, in comparison with traditional Krawtchouk and Tchebichef moments, has been significantly improved by using the new proposed 3D separable moments and can be highly useful in the field of 3D image analysis.

We investigate the dynamical evolution of isolated equal-mass star cluster models by means of direct N-body simulations, primarily focusing on the effects of the presence of primordial anisotropy in the velocity space. We found evidence of the existence of a monotonic relationship between the moment of core collapse and the amount and flavour of anisotropy in the stellar system. Specifically, equilibria characterised by the same initial structural properties (Plummer density profile) and with different degrees of tangentially-biased (radially-biased) anisotropy, reach core collapse earlier (later) than isotropic models. We interpret this result in light of an accelerated (delayed) phase of the early evolution of collisional stellar systems "anisotropic-response"), which we have characterised both in terms of the evolution of the velocity moments and of a fluid model of two-body relaxation. For the case of the most tangentially anisotropic model the initial phase of evolution involves a catastrophic collapse of the inner part of the system which continues until an isotropic velocity distribution is reached. This study represents a first step towards a comprehensive investigation of the role played by kinematic richness in the long-term dynamical evolution of collisional systems., 13 pages, 12 figures, accepted for publication in MNRAS

We describe a subpixel edge detection approach in images. The proposed approach is based on the algebraic moments of the brightness function of halftone images. For an ideal two-dimensional edge, we consider a model with the following four parameters: the edge orientation, the distance from the edge to the center of the mask, and the brightness values from both sides of the edge. To obtain all subpixel parameters of the edge, six algebraic moments are used. To compute the moments rapidly, masks are used. The specificity of the proposed approach is as follows: masks of almost all sizes can be used and they are computed by means of explicit relations provided in the present paper as well. Increasing mask sizes, one can increase the accuracy of the detection of subpixel edge parameters, which is especially important for high-definition images. We present experiments displaying the efficiency of the proposed approach.

Computation procedure presented in Fast computation of JacobiFourier moments for invariant image recognition has been analyzed and it has been demonstrated that the proposed domain of the kernel functions causes the loss of the orthogonality.Some imprecisions in the determination of the particular cases of JacobiFourier kernel, as well as some errata in the recursive computation of the polynomials have been corrected.It has proposed the use of a polar pixel tiling scheme, which allows a more accurate moment computation.It is demonstrated that the image reconstruction error is reduced, and this error continuously decreases as the number of considered moments increases. In the recent work: Fast computation of JacobiFourier moments for invariant image recognition, Pattern Recognition 48 (2015) 18361843, the authors propose a new method for the recursive computation of JacobiFourier moments. This method reduces the computational complexity in radial and angular kernel functions of the moments, improving the numerical stability of the computation procedure. However, they use a rectangular domain for the computation of the JacobiFourier moments. In this work, we demonstrate that the use of this domain involves the loss of kernel orthogonality. Also, errata and inaccuracies which could lead to erroneous results have been corrected and clarified. Furthermore, we propose a more precise procedure of the moments computation by using a circular pixel tiling scheme, which is based on the image interpolation and an adaptive Simpson quadrature method for the numerical integration.

Recently, orthogonal moments have become efficient tools for two-dimensional and three-dimensional (2D and 3D) image not only in pattern recognition, image vision, but also in image processing and applications engineering. Yet, there is still a major difficulty in 3D rotation invariants. In this paper, we propose new sets of invariants for 2D and 3D rotation, scaling and translation based on orthogonal radial Hahn moments. We also present theoretical mathematics to derive them. Thus, this paper introduces in the first case new 2D radial Hahn moments based on polar representation of an object by one-dimensional orthogonal discrete Hahn polynomials, and a circular function. In the second case, we present new 3D radial Hahn moments using a spherical representation of volumetric image by one-dimensional orthogonal discrete Hahn polynomials and a spherical function. Further 2D and 3D invariants are derived from the proposed 2D and 3D radial Hahn moments respectively, which appear as the third case. In order to test the proposed approach, we have resolved three issues:the image reconstruction, the invariance of rotation, scaling and translation, and the pattern recognition. The result of experiments show that the Hahn moments have done better than the Krawtchouk moments, with and without noise. Simultaneously, the mentioned reconstruction converges quickly to the original image using 2D and 3D radial Hahn moments, and the test images are clearly recognized from a set of images that are available in COIL-20 database for 2D image, and Princeton shape benchmark (PSB) database for 3D image.

This study aims to explore a novel approach to reconstruct multi-gray-level images based on circular blocks reconstruction method using two exact and fast moments: Zernike (CBR-EZM) and pseudo-Zernike (CBR-EPZM): An image is first divided into a set of sub-images which are then reconstructed independently. We also introduced Chamfer distance (CD) to capitalize on the use of discrete distance instead of Euclidean one. The combination of our methods and CD leads to CBR-EZM-CD and CBR-EPZM-CD methods. Obviously, image partitioning offers significant advantages, but an undesirable circular blocking effect can occur. To mitigate this effect, we have implemented overlapping feature to our new methods leading to OCBR-EZM-CD and OCBR-EPZM-CD, by exploiting neighborhood information of the circular blocks. The main motivation of this novel approach is to explore new applications of Zernike and pseudo-Zernike moments. One of the fields is feature extraction for pattern recognition: Zernike and pseudo-Zernike moments are well known to capture only the global features, but thanks to the circular block reconstruction, we can now use those moments to extract also local features.

In this paper, we introduce new sets of 2D and 3D rotation, scaling and translation invariants based on orthogonal radial Racah moments. We also provide theoretical mathematics to derive them. Thus, this work proposes in the first case a new 2D radial Racah moments based on polar representation of an object by one-dimensional orthogonal discrete Racah polynomials on non-uniform lattice, and a circular function. In the second case, we present new 3D radial Racah moments using a spherical representation of volumetric image by one-dimensional orthogonal discrete Racah polynomials and a spherical function. Further 2D and 3D invariants are extracted from the proposed 2D and 3D radial Racah moments respectively will appear in the third case. To validate the proposed approach, we have resolved three problems. The 2D/ 3D image reconstruction, the invariance of 2D/3D rotation, scaling and translation, and the pattern recognition. The result of experiments show that the Racah moments have done better than the Krawtchouk moments, with and without noise. Simultaneously, the mentioned reconstruction converges rapidly to the original image using 2D and 3D radial Racah moments, and the test 2D/3D images are clearly recognized from a set of images that are available in COIL-20 database for 2D image, and PSB database for 3D image.

Exponent Fourier moments (EFMs) are suitable for image representation and invariant pattern recognition. EFMs posses more number of uniformly distributed zeros as compared to Zernike Moments. However, these moments tend to be unstable near the center of image and also show a rise in reconstruction error for higher order of moments. In this paper, we propose a new computational framework for calculating the traditional EFM by partitioning the radial and angular part into equally spaced sectors. The proposed approach is simple and results in better image representation capability, numerical stability, and computational speed. Moreover, the proposed approach is completely stable near the center of image.

Orthogonal moments play an important role in image analysis and other similar applications. However, existing orthogonal moments are restricted to integer order, and little investigation of non-integer order orthogonal moments has been conducted to date. In this paper, a general framework of real-order orthogonal moments, also known as fractional-order orthogonal moments, is proposed. In this general framework, fractional-order orthogonal moments can be defined in Cartesian and polar coordinate systems. Shifted Legendre polynomials are implemented in this paper to investigate the properties of fractional-order orthogonal moments. A series of experiments are performed, which demonstrate that fractional-order orthogonal moments are not only capable of region-of-interest (ROI) feature extraction but also have potential for image reconstruction and face recognition and have high noise robustness in invariant image recognition.

Invariants to image scaling composed of Gaussian–Hermite moments are introduced in this paper for the first time. To achieve the invariance, we propose to modulate the Gaussian–Hermite polynomial basis using variable parameter σ , the value of which depends on the input image. The scaling invariance property can be coupled with the rotation invariance presented earlier. This approach is applicable in 2D as well as in 3D and provides very good numerical stability, as demonstrated by several experiments on real data.

Zernike moments (ZMs) are very useful image descriptors which belong to a family of orthogonal rotation invariant moments. Due to their many attractive characteristics, they have been used in many pattern recognition, image processing and computer vision applications. However, they suffer from very high computation complexity which prohibits their use in many practical problems. The ZMs are computed as a convolution process between the image data and the Zernike kernel functions. In the past, various attempts have been made for the efficient computation of ZMs and considerable success has been achieved using recursive relations and 8-way symmetry/anti-symmetry of Zernike function. In this paper, we propose a new computational flow model for the convolution of the image data with the Zernike kernel functions. The proposed model also takes advantage of the 8-way symmetry/anti-symmetry property of the kernel function and builds up the convolution process which reduces the number of additions/subtractions from 56 to 24 and the number of multiplications from 12 to 8 (refer Table 6 in the text) for each location in an octant of a circular disk on which the moments are computed. Detailed experimental results show that the speed of the ZMs computation increases by a factor varying from 15% to 41% (depending upon the order of moments) for multiple images as compared to the existing fast algorithms available in the literature. When ZMs are computed at each pixel of an image on overlapping blocks, the improvement in computation time varies from 10% to 33%.

Image moments have been used in many research fields of the engineering. However, the related computation of invariant moments mostly adopted the polar coordinate system, which not only increase the computational load, but also cause large quantized error. To solve this problem, an improved algorithm to compute Pseudo-Jacobi-Fourier moments in the Cartesian coordinate system is proposed in this paper. The experimental results show that the reconstructed image with improved PJFM’s has more advantages than polar coordinate system, such as more information, fewer moments, less time consuming. And the recognition rate of the microscopic images of 8 helminth eggs was also higher than in polar coordinate system.

Image recognition is widely used in different application areas such as shape recognition, gesture recognition and eye recognition. In this research, we introduced image recognition using efficient invariant moments and Principle Component Analysis (PCA) for gray and color images using different number of invariant moments. We used twelve moments for each image of gray images and Hu’s seven moments for color images to decrease dimensionality of the problem to 6 PCA’s for gray and 5 PCA’s for color images and hence the recognition time. PCA is then employed to decrease dimensionality of the problem and hence the recognition time and this is our main objective. The PCA is derived from Karhunen-Loeve’s transformation. Given an N-dimensional vector representation of each image, PCA tends to find a K-dimensional subspace whose basis vectors correspond to the maximum variance direction in the original image space. This new subspace is normally lower dimensional (K N). Three known datasets are used. The first set is the known Flower dataset. The second is the Africans dataset, and the third is the Shapes dataset. All these datasets were used by many researchers.

In this work, we propose new sets of 2D and 3D rotation invariants based on orthogonal radial dual Hahn moments, which are orthogonal on a non-uniform lattice. We also present theoretical mathematics to derive them. Thus, this paper presents in the first case new 2D radial dual Hahn moments based on polar representation of an image by one-dimensional orthogonal discrete dual Hahn polynomials and a circular function. The dual Hahn polynomials are general case of Tchebichef and Krawtchouk polynomials. In the second case, we introduce new 3D radial dual Hahn moments employing a spherical representation of volumetric image by one-dimensional orthogonal discrete dual Hahn polynomials and a spherical function, which are orthogonal on a non-uniform lattice. The 2D and 3D rotational invariants are extracts from the proposed 2D and 3D radial dual Hahn moments respectively. In order to test the proposed approach, three problems namely image reconstruction, rotational invariance and pattern recognition are attempted using the proposed moments. The result of experiments shows that the radial dual Hahn moments have performed better than the radial Tchebichef and Krawtchouk moments, with and without noise. Simultaneously, the mentioned reconstruction converges quickly to the original image using 2D and 3D radial dual Hahn moments, and the test images are clearly recognized from a set of images that are available in COIL-20 database for 2D image and PSB database for 3D image.