Harmonic Moments Research Articles

Local quaternion polar harmonic Fourier moments-based multiple zero-watermarking scheme for color medical images

Zero-watermarking schemes are currently dedicated to resisting general desynchronization attacks, such as rotation and scaling, but cannot effectively resist complex desynchronization attacks, such as translation, cropping, length–width ratio changing, UnZign, and zero-padding. To this end, a robust zero-watermarking scheme resistant to complex desynchronization attacks was proposed in this paper based on local feature regions (LFRs) and quaternion polar harmonic Fourier moments (QPHFMs). First, the stable feature points of original color medical images were extracted with a speeded-up robust feature (SURF) operator; then, the stable LFRs were obtained through a feature scale-adaptive process and scale-standardized tests. Finally, the QPHFMs of the LFRs were calculated, and multiple feature sequences were constructed using their magnitudes and binarized to generate multiple zero-watermarks. The experimental results indicate that the proposed scheme can not only resist common image processing attacks and general desynchronization attacks but also robustly resist complex desynchronization attacks, such as translation, cropping, length–width ratio changing, UnZign, and zero-padding. The proposed scheme is superior to the state-of-the-art LFR-based multiple embedded watermarking schemes and zero-watermarking schemes.

A second-order maximum-entropy inspired interpolative closure for radiative heat transfer in gray participating media

A new interpolative-based approximation to the second-order maximum-entropy, M2, moment closure for predicting radiative heat transfer in gray participating media is proposed and described. In addition to preserving many of the desirable mathematical properties of the original M2 closure, the proposed interpolative approximation provides significant reductions in computational costs compared to the costs of the original M2 closure by avoiding repeated numerical solution of the corresponding optimization problem for entropy maximization. Theoretical details of the proposed interpolative-based closure, along with a description of an efficient Godunov-type finite-volume scheme that has been developed for the numerical solution of the resulting system of hyperbolic moment equations, are presented. The finite-volume method makes use of limited linear solution reconstruction, multi-block body-fitted quadrilateral meshes with anisotropic adaptive mesh refinement (AMR), and an efficient Newton-Krylov-Schwarz (NKS) iterative method for solution of the resulting non-linear algebraic equations arising from the spatial discretization procedure. The predictive capabilities of the proposed interpolative M2 closure are assessed by considering a number of model problems involving radiative heat transfer within one- and two-dimensional enclosures, the results for which are compared to solutions of the first-order maximum entropy, M1, moment closure, as well as those of the more commonly adopted spherical harmonic moment closure techniques (first-order P1 and third-order P3) and the popular discrete ordinates method (DOM). The latter is used as a benchmark for comparisons, whenever exact solutions are not available. The numerical results illustrate the promise of the proposed M2 closure, with the closure outperforming the M1, P1 and P3 closures for virtually all cases considered.

Experimental investigation of higher harmonic wave loads and moments on a vertical cylinder by a phase-manipulation method

This paper considers wave loads on bottom mounted vertical cylinders in the regime relevant to offshore wind turbine monopiles, examining the higher harmonics of the loading from non-breaking waves which are important in engineering design. Although we consider measurements of total force, our focus is on the moments applied about the seabed which have received much less attention in the literature. We consider an empirical ‘Stokes-like’ model whereby the magnitude of the higher harmonics of force or moment are dependent on the linear force or moment time-series raised to an appropriate power and a fixed coefficient. We compare this representation to high quality wave flume measurements and fully non-linear potential flow simulations. In general we find good agreement for all of the harmonic structure except for the triple frequency component where there are significant discrepancies. We also extend the Stokes-like model and show that harmonics of moments can be robustly predicted from the force harmonic and an appropriate constant moment arm for each harmonic, so a more elaborate calculation is generally not necessary.

Effective quaternion radial harmonic Fourier moments for color image representation

Image moments based on orthogonal manner play an important role in image processing and image analysis tasks. Although many variants have been proposed, there is still big room to reduce the geometrical errors and numerical errors. To address this issue, we propose an effective quaternion radial harmonic Fourier moments (Q-RHFMs) for color image representation. We compare the Q-RHFMs with other image moments and deep data-driven features, on multiple tasks including image reconstruction, watermarking and retrieval. Experimental results show the priorities of Q-RHFMs over other moments and deep features. The codes are available at https://github.com/ZlyaoNjust/Q-RHFMs .

Forced torsional vibration of nanobeam via nonlocal strain gradient theory and surface energy effects under moving harmonic torque

The forced torsional vibration of a nanobeam by using the nonlocal strain gradient theory and incorporating the surface effects is investigated in the current work. The aim of this paper was to analyze the surface effect on the forced torsional vibration of a nanobeam. The simulated nanobeam is under the distributed external torque and moving external harmonic torque. The governing equations are derived by employing the Hamilton principle. The extracted partial differential equation is converted to ordinary differential equation by deploying the Assumed Modes method. The dynamic torsion of clamped–clamped end supports of nanobeam is determined by using the Convolution integral. The effect of nondimensional material length scale parameter, the nondimensional nonlocal parameter, the velocity parameter and the nondimensional moment, on the maximum nondimensional dynamic torsion of nanobeam are studied. Eventually, the effect of residual surface torsion and surface shear modulus on the maximum nondimensional dynamic torsion of nanobeam, which is obtained from the nonlocal strain gradient theory under the distributed torsional and moving harmonic dynamic moments, is studied.

A Three-Dimensional Ring Model for Uncertainty Quantification in Natural Frequencies and Sound Radiation Characteristics of Tires

Material and geometrical parameters of tires involve some degree of uncertainty mainly related to production processes. Accordingly, the associated structural responses are affected by these uncertainties. In this study, a novel theoretical ring model is presented to describe the in-plane and out-of-plane vibrations as well as the steady-state response of tires, and then to evaluate the influence of the uncertainties in structural parameters on the natural frequencies and the sound radiation characteristics under uncertain excitations. The Hamilton principle is applied here to derive the governing equations. The modal superposition method is used to calculate the steady-state response of the tire. In the sound radiation analysis, the in-plane and out-of-plane bending and torsional vibrations under a set of harmonic unit forces and moments are treated as the source of noise generation. On this basis, the generalized polynomial chaos expansion method is then adopted to evaluate the influence of the uncertainty on the natural frequencies and the sound power. To obtain the unknown coefficients of the expansions, the nonintrusive probabilistic collocation method is employed. Moreover, considering the concept of linear independence of vectors, the number of collocation points is reduced. It is applied to investigate the impacts of the elastic and structural uncertainties on the natural frequencies of the tire. This yields an efficient simulation in terms of computational costs. Finally, the distributions of the sound power due to the forced vibration under the random concentrated line forces are given.

Accurate quaternion radial harmonic Fourier moments for color image reconstruction and object recognition

Orthogonal moments have become a powerful tool for object representation and image analysis. Radial harmonic Fourier moments (RHFMs) are one of such image descriptors based on a set of orthogonal projection bases, which outperform other moments because of their computational efficiency. However, the conventional computational framework of RHFMs produces geometric error and numerical integration error, which will affect the accuracy of RHFMs, thus degrading the image reconstruction performance. To overcome this shortcoming, we propose a new computational framework of RHFMs, namely accurate quaternion radial harmonic Fourier moments (AQRHFMs), for color image processing, and also analyze the properties of AQRHFMs. Firstly, we propose a precise computation method of RHFMs to reduce the geometric and numerical errors. Secondly, by using the algebra of quaternions, we extend the accurate RHFMs to AQRHFMs in order to deal with the color images in a holistic manner. Experimental results show the proposed AQRHFMs achieve promising performance in image reconstruction and object recognition in both noise-free and noisy conditions.

Image description by harmonic Fourier moment

In this paper, Two kinks of orthogonal moments are constructed with Trigonometric functions, named Harmonic-Fourier moments (HFMs), which include Cosine-Fourier moments (CFMs) and Sine-Fourier moments (SFMs). These kinds of moments can be used in image and object analysis. The radial functions of HFMs have more zeros than Zernike polynomials of the same degree so that the property makes HFMs have stronger ability in image description. Furthermore, the formulas of HFMs are extremely simple. This property makes it possible that HFMs can be computed at a higher speed.

Robust image watermarking using invariant accurate polar harmonic Fourier moments and chaotic mapping

Polar harmonic Fourier moments (PHFMs) are very effective global image descriptors and can be used in many areas of digital image processing. However, the numerical integration error of traditional computation methods significantly affects its computation accuracy. This paper proposed an accurate PHFMs computation method based on Gaussian numerical integration (GNI), which effectively mitigated the numerical integration error, and proposed a novel watermarking algorithm resistant to geometric attacks based on accurate PHFMs and chaotic mapping. This algorithm improves its robustness against geometric attacks by relying on the geometric invariance of accurate PHFMs and scrambles the watermark image by relying on the sensitivity of the Tent chaotic system to the initial value, enhancing its security. The experimental results show that the proposed algorithm can effectively resist geometric attacks, such as scaling, rotation, length-width ratio change and flipping, and is superior to other watermarking algorithms.

Algebraic Relations between Harmonic and Anti-harmonic Moments of Plane Polygons

Abstract In this paper we describe the algebraic relations satisfied by the harmonic and anti-harmonic moments of simply connected, but not necessarily convex planar polygons with a given number of vertices.

Multiparticle correlation studies in pPb collisions at sNN=8.16 TeV

The second- and third-order azimuthal anisotropy Fourier harmonics of charged particles produced in pPb collisions, at $\sqrt{s_\mathrm{NN}} =$ 8.16 TeV, are studied over a wide range of event multiplicities. Multiparticle correlations are used to isolate global properties stemming from the collision overlap geometry. The second-order "elliptic" harmonic moment is obtained with high precision through four-, six-, and eight-particle correlations and, for the first time, the third-order "triangular" harmonic moment is studied using four-particle correlations. A sample of peripheral PbPb collisions at $\sqrt{s_\mathrm{NN}} =$ 5.02 TeV that covers a similar range of event multiplicities as the pPb results is also analyzed. Model calculations of initial-state fluctuations in pPb and PbPb collisions can be directly compared to the high precision experimental results. This work provides new insight into the fluctuation-driven origin of the $v_3$ coefficients in pPb and PbPb collisions, and into the dominating overall collision geometry in PbPb collisions at the earliest stages of heavy ion interactions.

Large deviation rates for Markov branching processes

Large deviation rates are determined for quantities associated with a Markov branching process [Formula: see text] having offspring mean [Formula: see text] and split rate [Formula: see text]. The principal quantities examined are [Formula: see text] and [Formula: see text], where [Formula: see text] is the almost sure limit of an appropriately normed version of [Formula: see text]. Modifications and conditional versions are examined. Some of this requires determination of the asymptotic behavior of harmonic moments [Formula: see text].

Novel Multi-Channel Fractional-Order Radial Harmonic Fourier Moments for Color Image Analysis

The classical radial harmonic Fourier moments (RHFMs) and the quaternion radial harmonic Fourier moments (QRHFMs) are gray-scale and color image descriptors. The radial harmonic functions with integer orders are not able to extract fine features from the input images. In this paper, the authors derived novel fractional-order radial harmonic functions in polar coordinates. The obtained functions are used to defined novel multi-channel fractional-order radial harmonic moments (FrMRHFMs) for color image description and analysis. The invariants to geometric transformations for these new moments are derived. A theoretical comparison between FrMRHFMs and QRHFMs is performed from the aspects of kernel function and the spectrum analysis. Numerical simulation is carried out to test these new moments in terms of image reconstruction capabilities, invariance to the similarity transformations, color image recognition and the CPU computational times. The obtained theoretical and numerical results clearly show that the proposed FrMRHFMs is superior to the QRHFMs and the existing fractional-order orthogonal moments.

Deviations for Martingale Convergence of a Branching Process with Random Index

The purpose of this paper is to obtain large deviations, moderate deviations and normal deviations for the convergence of martingale generated by a supercritical Galton–Watson process with a Poisson process as its time index. Harmonic moments and Shroder index play an important role in the proofs.

Image Description With Polar Harmonic Fourier Moments

Due to their good rotational invariance and stability, image continuous orthogonal moments are intensively applied in rotationally invariant recognition and image processing. However, most moments produce numerical instability, which impacts the image reconstruction and recognition performance. In this paper, a new set of invariant continuous orthogonal moments, polar harmonic Fourier moments (PHFMs), free of numerical instability is designed. The radial basis functions (RBFs) of the PHFMs are much simpler than those of the Chebyshev-Fourier moments (CHFMs), orthogonal Fourier-Mellin moments (OFMMs), Zernike moments (ZMs), and pseudo-Zernike moments (PZMs). For the same degree, the RBFs of the PHFMs have more zeros and are more evenly distributed than those of the ZMs and PZMs. Therefore, PHFMs do not suffer from information suppression problem; hence, the image description ability of the PHFMs is superior to that of the ZMs and PZMs. Moreover, the RBFs of the PHFMs are always less than or equal to 1.0 near the unit disk center, whereas those of the OFMMs, PZMs, CHFMs, and radial harmonic Fourier moments (RHFMs) are infinite (implying numerical instability). This indicates that PHFMs can outperform these moments in image reconstruction tasks. We theoretically and experimentally demonstrate that PHFMs outperform the above moments in reconstructing images and recognizing rotationally invariant objects considering noise and various attacks. This paper also details the significance of the PHFM phase in image reconstruction, angle estimation using PHFMs, and the accurate moment selection of the PHFMs.

Efficient copyright protection for three CT images based on quaternion polar harmonic Fourier moments

The current zero-watermarking scheme mainly offers the copyright protection to one image. Although the scheme can be used for many images repeatedly, the repeat operation requires a lot of time and large storage space. Using the quaternion polar harmonic Fourier moments (QPHFM), the paper considers three CT images as a whole to construct the key image and proposes a robust zero-watermarking algorithm offering the copyright protection to three CT images simultaneously, improving the efficiency of watermarking system and saving the storage space. The algorithm first considers the three CT images as three imaginary parts of an array of pure quaternion and calculates their QPHFM and then constructs feature image using the amplitude of QPHFM, and finally makes the chaotic scrambling to the feature image and then the XOR operation to the feature image and watermark image to generate key image. Experiment results prove the proposed scheme can resist common image processing attacks and geometric attacks effectively and thus can be well applied to the copyright protection of three images.

Color image segmentation using proximal classifier and quaternion radial harmonic Fourier moments

Segmentation involves separating an object from the background in a given image. Image segmentation has a variety of applications and has received considerable attention in multimedia application and computer vision. Although numerous approaches have been introduced, image segmentation is still far from being solved due to most of image segmentation algorithms are often so complicated and some unsatisfactory results appear frequently. Therefore, developing a suitable technique of image segmentation is still a challenging problem. In this article, a novel color image segmentation will be introduced based on quaternion radial harmonic Fourier moments (QRHFMs) and proximal classifier. Firstly, the image feature of pixel-level is represented by the accurate and invariant QRHFMs holistically as a vector field, which can describe sufficiently the image pixel information due to take into account the relationship among different color channels. Secondly, the image feature from pixel-level is utilized as the input of the proximal classifier with consistency (PCC), which not only has lower computation time but also has better generalization compared to traditional support vector machines classifiers. Then, we choose the training samples by Tsallis entropy thresholding to train PCC classification model. Finally, the color image is classified by the trained PCC classification model. Our algorithm can make full use of the accurate and robust local image feature, as well the quickness and generalization ability of PCC classifier. A series of experimental results shows that this algorithm has better segmentation performance than the state-of-the-art method from the literature.

Rocking vibrations of rigid foundations on multi-layered poroelastic media

This paper presents rocking vibrations of a rigid foundation resting on a multi-layered poroelastic half-space. The foundation is assumed to be rigid and massless, and subjected to a time–harmonic moment. In addition, each layer of the multi-layered half-space is governed by Biot’s theory of poroelastodynamics. The contact surface between the foundation and the layered half-space is smooth, and either fully permeable or impermeable. This dynamic interaction problem is studied by employing a discretization technique and an exact stiffness matrix scheme. Comparisons with existing solutions on rocking vibrations of rigid foundations on elastic and poroelastic media are shown to verify the accuracy of the present scheme. Selected numerical results on rocking compliances of rigid foundations of various shapes and mudmat foundations are presented. In addition, a dynamic interaction problem involving closely spaced foundations under rocking vibrations is also presented to demonstrate the applicability of the present solution scheme.

Quaternion Harmonic moments and extreme learning machine for color object recognition

The quaternary orthogonal moments have been widely used as color image descriptors owe to their remarkable color and shape information encapsulation capability. Their computation, however, depends on finding the optimal value of a unit pure quaternion parameter, which is done empirically and with no warranty of optimality. We propose a 2D color object recognition method that relies on the quaternion-valued parameter-free disc-harmonic moment invariants (QHMs) fed into the quaternion extreme learning machine (QELM). The role of this latter is to maintain the correlation between the four parts, real and imaginary, of the quaternary descriptor coefficients. Several datasets are used for recognition experiments. We draw the conclusion that: (1) our quaternion-valued QHMs invariants outperform other quaternary moments, (2) the quaternion-valued moment invariants give results better than the modulus-based moment invariants and (3) the QELM yields results better than the state-of-the-art classifiers.

On the measurement of the helicity of intergalactic magnetic fields using ultra-high-energy cosmic rays

The origin of the first magnetic fields in the Universe is a standing problem in cosmology. Intergalactic magnetic fields (IGMFs) may be an untapped window to the primeval Universe, providing further constrains on magnetogenesis. We demonstrate the feasibility of using ultra-high-energy cosmic rays (UHECRs) to constrain the helicity of IGMFs by performing simulations of cosmic-ray propagation in simple magnetic field configurations. We show that the first harmonic moments of the arrival distribution of UHECRs may be used to measure the absolute value of the helicity and its sign.