Fourier Harmonics Research Articles

Beyond Numbers: Investigating the Impact of COVID-19 on Suicide Rates in Austria

Abstract During the COVID-19 pandemic, individuals’ mental health faced substantial challenges. This study aims to provide a comprehensive examination of the pandemic’s impact on suicide rates in Austria, with a particular focus on variations by gender, region, and age. The study employs a regression discontinuity in time design and an interrupted time series analysis, utilising monthly death records from January 2015 to December 2021. To account for seasonal variations and potential underreporting of suicides, harmonic Fourier terms are incorporated into the analysis, and both suicides with intended self-harm and self-harm with undetermined intent are included. While existing research offers valuable insights into the impact of the pandemic on suicidal behaviour, it does not yet allow for specific conclusions to be drawn for Austria. In contrast to the methods typically employed in previous research on other countries, this approach enables a comprehensive examination of the temporal effects surrounding the implementation and lifting of pandemic-related measures. The study reveals a significant negative trend in Austria’s suicide rates before the start of the pandemic and more suicides than expected after its onset. In particular, the number of suicides among men aged 64-85 and women in the age groups 0-19, 20-34, and 65-84 was significantly higher than expected. Conversely, the data indicated a positive impact among women aged 35-64. This study contributes to understanding the impact of the pandemic on suicide rates in Austria. It demonstrates that, even within the same country, the impact can differ by gender, age, and region, thereby supporting the need for subgroup analysis. It also highlights that potential underreporting of suicides may result in an underestimation of the impact of the pandemic on suicides in previously conducted studies. Understanding these nuances is crucial for suicide prevention and informing policy decisions to improve health system resilience. Key messages • The study found higher than expected suicide rates in Austria during the COVID-19 pandemic, especially among men aged 64-85 and women aged 0-19, 20-34, and 65-84. • The underreporting of suicides may result in an underestimation of the impact of COVID-19 on suicides in previously conducted studies.

Geometric Algebra Framework Applied to Single-Phase Linear Circuits with Nonsinusoidal Voltages and Currents

We apply a well known technique of theoretical physics, known as geometric algebra or Clifford algebra, to linear electrical circuits with nonsinusoidal voltages and currents. We rederive from the first principles the geometric algebra approach to the apparent power decomposition. The important new point consists of endowing the space of Fourier harmonics with a structure of a geometric algebra (it is enough to define the Clifford product of two periodic functions). We construct a set of commuting invariant imaginary units which are used to define impedance and admittance for any frequency.

Rotation Correction Algorithm Based on Polar Harmonic Fourier Moments and Optimization of Color Image Security Quantization Watermarking Scheme

Rotation Correction Algorithm Based on Polar Harmonic Fourier Moments and Optimization of Color Image Security Quantization Watermarking Scheme

Observation of Enhanced Long-Range Elliptic Anisotropies Inside High-Multiplicity Jets in pp Collisions at sqrt[s]=13 TeV.

A search for collective effects inside jets produced in proton-proton collisions is performed via correlation measurements of charged particles using the CMS detector at the CERN LHC. The analysis uses data collected at a center-of-mass energy of sqrt[s]=13 TeV, corresponding to an integrated luminosity of 138 fb^{-1}. Jets are reconstructed with the anti-k_{T} algorithm with a distance parameter of 0.8 and are required to have transverse momentum greater than 550GeV and pseudorapidity |η^{jet}|<1.6. Two-particle correlations among the charged particles within the jets are studied as functions of the particles' azimuthal angle and pseudorapidity separations (Δϕ^{*} and Δη^{*}) in a jet coordinate basis, where particles' η^{*}, ϕ^{*} are defined relative to the direction of the jet. The correlation functions are studied in classes of in-jet charged-particle multiplicity up to N_{ch}^{j}≈100. Fourier harmonics are extracted from long-range azimuthal correlation functions to characterize azimuthal anisotropy for |Δη^{*}|>2. For low-N_{ch}^{j} jets, the long-range elliptic anisotropic harmonic, v_{2}^{*}, is observed to decrease with N_{ch}^{j}. This trend is well described by MonteCarlo event generators. However, a rising trend for v_{2}^{*} emerges at N_{ch}^{j}≳80, hinting at a possible onset of collective behavior, which is not reproduced by the models tested. This observation yields new insights into the dynamics of jet evolution in the vacuum.

From Simple to Complex Scenes: Learning Robust Feature Representations for Accurate Human Parsing.

Human parsing has attracted considerable research interest due to its broad potential applications in the computer vision community. In this paper, we explore several useful properties, including high-resolution representation, auxiliary guidance, and model robustness, which collectively contribute to a novel method for accurate human parsing in both simple and complex scenes. Starting from simple scenes: we propose the boundary-aware hybrid resolution network (BHRN), an advanced human parsing network. BHRN utilizes deconvolutional layers and multi-scale supervision to generate rich high-resolution representations. Additionally, it includes an edge perceiving branch designed to enhance the fineness of part boundaries. Building on BHRN, we construct a dual-task mutual learning (DTML) framework. It not only provides implicit guidance to assist the parser by incorporating boundary features, but also explicitly maintains the high-order consistency between the parsing prediction and the ground truth. Toward complex scenes: we develop a domain transform method to enhance the model robustness. By transforming the input space from the spatial domain to the polar harmonic Fourier moment domain, the mapping relationship to the output semantic space is highly stable. This transformation yields robust representations for both clean and corrupted data. When evaluated on standard benchmark datasets, our method achieves superior performance compared to state-of-the-art human parsing methods. Furthermore, our domain transform strategy significantly improves the robustness of DTML dramatically in most complex scenes.

Nonlinear modeling of ELM mitigation with RMP on HL-2A

Nonlinear modeling of mitigation of the edge localized mode (ELM) with resonant magnetic perturbation (RMP) is performed for the HL-2A tokamak, utilizing the three-dimensional (3D) magnetohydrodynamic code JOREK. Based on the 3D equilibrium established after application of the n=1 ( n is the toroidal mode number) RMP at 4.9kAt coil current with odd parity, ELM mitigation is successfully simulated consistent with the experimental result. Nonlinear simulations show strong mode coupling among toroidal Fourier harmonics, allowing redistribution of the magnetic energy such that the most unstable toroidal mode saturates at a lower level. This magnetic energy cascade offers an explanation of the RMP-induced ELM mitigation achieved in HL-2A. Detailed examination of the simulation results shows persistent resonant field screening even during the ELM mitigation phase. Finite plasma resistivity however does enable partial penetration of the resonant field thus modifying the edge magnetic topology and characteristics of the edge transport. Plasma radial profiles undergo pronounced changes around the pedestal region, when the magnetic energy of the most unstable toroidal mode reaches the maximum value. Systematic scans of the applied RMP coil current with the JOREK simulations find a threshold value of around 4.5kAt required for achieving the ELM mitigation on HL-2A.

Binary system modes of matrix-coupled multidimensional oscillators

The standard Kuramoto model has been instrumental in explaining synchronization and desynchronization, two emergent phenomena often observed in biological, neuronal, and physical systems. While the Kuramoto model has turned out effective with one-dimensional oscillators, real-world systems often involve high-dimensional interacting units, such as biological swarms, necessitating a model of multidimensional oscillators. However, existing high-dimensional generalizations of the Kuramoto model commonly rely on a scalar-valued coupling strength, which limits their ability to capture the full complexity of high-dimensional interactions. This work introduces a matrix, A, to couple the interconnected components of the oscillators in a d-dimensional space, leading to a matrix-coupled multidimensional Kuramoto model that approximates a prototypical swarm dynamics by its first-order Fourier harmonics. Moreover, the matrix A introduces an inter-dimensional higher-order interaction that partly accounts for the emergence of 2 d system modes in a d-dimensional population, where each dimension can either be synchronized or desynchronized, represented by a set of almost binary order parameters. The binary system modes capture characteristic swarm behaviors such as fish milling or polarized schooling. Additionally, our findings provides a theoretical analogy to cerebral activity, where the resting state and the activated state coexist unihemispherically. It also suggests a new possibility for information storage in oscillatory neural networks.

Elementary proof of Funahashi's theorem

Funahashi established that the space of two-layer feedforward neural networks is dense in the space of all continuous functions defined over compact sets in $n$-dimensional Euclidean space. The purpose of this short survey is to reexamine the proof of Theorem 1 in Funahashi \cite{Funahashi}. The Tietze extension theorem, whose proof is contained in the appendix, will be used. This paper is based on harmonic analysis, real analysis, and Fourier analysis. However, the audience in this paper is supposed to be researchers who do not specialize in these fields of mathematics. Some fundamental facts that are used in this paper without proofs will be collected after we present some notation in this paper.

Quaternion fast and accurate polar harmonic Fourier moments for color image analysis and object recognition

Image moments, as a kind of global feature descriptor of images, have become a valuable tool for pattern recognition and image analysis. However, traditional methods are mainly used to deal with grayscale images. In this paper, we apply quaternions to fast and accurate polar harmonic Fourier moments, proposing a kind of quaternion fast and accurate polar harmonic Fourier moment (QFAPHFM) capable of handling color images. Furthermore, this paper provides a detailed analysis of the invariance of QFAPHFMs under rotation, scaling, and translation transformations. The experimental results show that QFAPHFMs exhibit excellent performance in both image reconstruction and object recognition tasks. QFAPHFMs achieve accurate image reconstruction under noiseless and noisy conditions, and demonstrate excellent recognition performance in the color-based object recognition tasks.

Blind watermarking algorithm with 2D-SCCM chaotic system encryption based on SURF and RHFMs

In this paper, a digital watermarking algorithm was proposed based on local feature regions and radial harmonic Fourier moments. It used a quantized modulation algorithm to embed the watermark information into the frequency domain information of the image. Compared with traditional watermarks directly embedded into the pixel values, the algorithm enhanced the invisibility of the watermark. Many existing digital watermarking algorithms neglected the security of the embedding process, because of more attention on the effect of post-embedding and extraction. To overcome this problem, we proposed a two-dimensional hyperchaotic system named 2D-SCCM chaotic system, which used classical sine mapping and cubic mapping for coupling. Before embedding, the correlation among the pixels of the watermarked image was disrupted, resulting in a full process security enhancement. Combining the clustering algorithm with feature point extraction, the obtained local feature region was more stable and suitable for watermark embedding. Results showed that the improved watermark embedding algorithm was more effective for resisting geometric attacks such as cropping attacks. Moreover, the PSNR values of the final watermark-embedded images were all greater than 50 dB, indicating a good balance between imperceptibility and robustness.

Formation of small-scale modes via ECCD injection into KSTAR plasma core

In KSTAR experiments exhibiting sawtooth instability, the formation of multiple flux tubes (MFTs) has been frequently observed when electron cyclotron resonance heating or a current drive is applied near the inversion radius of the sawtooth. On the global scale, these MFTs evolve into a single flux tube mode or dual modes. The modes are observed as multiple Fourier harmonics in the spectrogram. A comprehensive correlation analysis of 2D imaging diagnostic data reveals a notable energy transfer within structures of varying sizes during the global mode transition. Broadband fluctuations are enhanced, and energy transfer between Fourier harmonics occurs in the presence of MFTs. Cross-power spectrum in the presence of multiple Fourier harmonics aligns with power law of inverse cascade. This suggests that energy inverse cascade process can contribute to formation of MFTs.

Gravity–capillary wave-making resistance on deep water

For supercritical cases (forcing speed &amp;gt; the minimum phase speed, 0.23 m/s), the problem of two-dimensional linear, inviscid gravity–capillary waves generated by a moving delta-function type pressure source is well known. Using harmonic functions or Fourier transform, Lamb [Hydrodynamics, 6th ed. (Cambridge University Press, 1993)] and Rayleigh [Proc. London Math. Soc. s1-15(1), 69–78 (1883)] detailed the steady-state full-space wave-profile solution using an artificial viscosity. Whitham [Linear and Nonlinear Waves (Wiley-Interscience, 1974)] presented the same solution for the region that is far-from-the-forcing using a slowly varying exponential function. For the same problem, but, considering not only supercritical but also subcritical cases, and, without using the artificial viscosity, the present work provides a detailed solution procedure to find full-space wave-profile solutions based on Fourier transform where complex integration is needed; different analytical expressions of the same wave profile will be provided depending on different paths. The associated wave-making resistance is shown to be equal to the integral of the product of a moving pressure source, and the resultant wave slope and is calculated in two ways. One is a direct calculation in the physical domain, which requires the wave-profile solution, and the other is an indirect calculation in the wavenumber domain, which does not require the wave-profile solution. For supercritical cases, short and long sinusoidal waves are calculated ahead of and behind the pressure source. The associated wave-making resistance decreases toward a certain constant as the forcing speed increases and the associated required power features a minimum at the forcing speed of 0.3027 m/s. For subcritical cases, a simple symmetric dimple is calculated and the wave-making resistance becomes zero due to its symmetry.

ШЛЯХИ ЗМЕНШЕННЯ ПОХИБОК ВИМІРЮВАННЯ ДІЕЛЕКТРИЧНОЇ ПРОНИКНОСТІ СЛАБОПОГЛИНАЮЧИХ ДІЕЛЕКТРИКІВ У МІЛІМЕТРОВОМУ ТА СУБМІЛІМЕТРОВОМУ (ТЕРАГЕРЦОВОМУ) ДІАПАЗОНАХ ДОВЖИН ХВИЛЬ МЕТОДОМ ПОВЕРХНЕВОГО ПЛАЗМОННОГО РЕЗОНАНСУ

Subject and Purpose. The sources of errors are identified that may arise in the course of terahertz-range measurements of the dielec- tric constant of weakly absorbent dielectrics, if performed within the surface plasmon resonance (SPR) technique. Possible ways are analyzed for reducing or fully eliminating such errors. Methods and Methodology. Specific details of applying the SPR method for dielectric constant measurements have been analyzed, with the aim of identifying the major factors that particularly affect the measurement accuracy. Results. It has been noted that in order to reduce the level of backlight interference (which may lead to blinding the receiver), it is expedient to make surface resonance records via frequency scanning. In that case the impact of the interference signal nonstationa- rity arising from the partial conversion of the surface wave energy into that of the volume wave, which occurs at the grating edges, is markedly lower than in the case of angular scanning. A mathematical expression has been derived which suggests a relation between scanning step sizes in angle and in frequency (for the angular and frequency scanning, respectively). As has been shown, a better mea- surement accuracy is achievable if the SPR is recorded as a function of frequency. Indeed, the frequency can be varied, with the use of familiar technologies, in steps of a much smaller size than such adopted for angular scanning. Errors in the above measurements can also arise if the resonance is excited on a grating whose Fourier spectrum contains many high-frequency components which carry a noticeable portion of the diffracted radiation energy. These energy losses can be greatly reduced if the SPR is excited on a grating whose profile involves the lowest number of spatial Fourier harmonics. Conclusions. The method suggested allows a significant reduction in the level of errors of the dielectric constant measurements in weakly absorbing dielectrics if the surface plasmon resonance effects are registered in dependence on the incident frequency, while the SPR is excited at a diffraction grating whose troughs-and-peaks profile is close to harmonical.

Exploring Accurate Invariants on Polar Harmonic Fourier Moments in Polar Coordinates for Robust Image Watermarking

Exploring Accurate Invariants on Polar Harmonic Fourier Moments in Polar Coordinates for Robust Image Watermarking

Multi-user communications for line-of-sight large intelligent surface systems

In this paper, we propose a new interference-suppression beamformer design and a Direction-of-Arrival (DoA) estimation method for Line-of-Sight (LoS) multi-user communication systems with circular Large Intelligent Surfaces (LISs). This symmetrical circular-shaped LIS allows signals defined on it to be formed by combining Fourier and Bessel series. The Fourier harmonics are orthogonal along the rotational direction and the Bessel functions constituting the Bessel series are orthogonal along the radial direction. Synthesizing basis functions from these two series results in the proposed interference-suppression beamformer, with their coefficients designed to cancel inter-user interference. Also by exploiting signal structure of a circular LIS communicating with Mobile Stations (MSs), we devise a DoA estimation method to obtain directional information of incoming signals. Under LoS channel, the DoA corresponds to elevation and Azimuth angles of MSs relative to the LIS. This method is numerically demonstrated to have promising performance with high-frequency carriers.

Model based optimal magnetic feedback control of multiple resistive wall modes with discrete coil and sensor arrays

A model-based optimal control method for multiple resistive wall mode (RWM) feedback stabilization has been developed and tested in plasma experiments at the EXTRAP T2R reversed field pinch (RFP) device. The controller is designed to target issues that arise in connection with RWM magnetic feedback stabilization systems based on discrete control coil and sensor arrays in tokamak and reversed field pinch devices. Multi-mode control capabilities in these systems is limited due to coupling of modes induced by the control system. The coupling originates from the generation of side-band control field harmonics and from aliasing of multiple harmonics in the sensor measurements. These couplings naturally leads to a multiple-input, multiple-output (MIMO) control problem. A model based state space controller has been designed based on a relatively simple physics model of the RWM plasma response. The physics model, which is applicable for the high aspect ratio RFP, is the basis for Fourier decoupling of the MIMO control problem into a set of single-input, multiple-output (SIMO) systems using the discrete Fourier transform (DFT). The linear, time-invariant physics model allows for the design of a state space model with states representing physical quantities; in this case the Fourier harmonics of the radial field at the resistive wall. Since the states cannot be directly measured, a Kalman filter is used for estimation of the states from the aliased sensor array measurements. A linear–quadratic (LQ) optimal state controller has been implemented. Design parameters, such as the LQ control cost function state weights and the Kalman filter input error covariances have been used to optimize the control operation in various ways. The controller has enhanced multi-mode control capabilities compared to earlier designs. For example it allows the prioritizing of suppression of one of the multiple magnetic field Fourier harmonics produced by a given control current DFT component. The controller has been tested in plasma experiments at EXTRAP T2R device, utilizing a newly installed extended sensor array, and the enhanced capabilities for multiple RWM feedback stabilization has been demonstrated.

Robust multifunctional watermarking system based zero-watermarking and polar harmonic Fourier moments

The pseudo-randomness exhibited by chaotic systems has proven valuable not only in plaintext information encryption of images with varying dimensions but also as a key signal for protecting image copyright information. Image copyright protection is a crucial part of image information protection. Currently, digital watermarking technology is predominantly employed for copyright protection of digital images. However, most of the existing watermarking technologies primarily focus on the improvement of watermarking performance, neglecting the security of the watermarking system itself. Drawing inspiration from the Latin square rule, this paper proposes a novel hyperchaotic mapping modulation framework, which is employed to construct a new 3D hyperchaotic mapping and enhance the security of the fragile watermark and the zero-watermark generation process. Additionally, it is integrated into the classical image encryption framework of scrambling-diffusion to safeguard meaningful copyright information. Moreover, a functional watermarking system is designed based on the RSA asymmetric encryption algorithm, least significant bit (LSB), and polar harmonic Fourier moments (PHFMs). The proposed system enables identity verification, tampering location, and copyright authentication simultaneously. During the generation of zero watermarks, the binarization threshold of the PHFMs information matrix is determined by the method of maximum variance. The experimental results demonstrate that this scheme achieves a high level of security, better robustness to both conventional and geometric attacks, and precise localization for different types of tampering attacks.

Raster image processing using 2D Padé-type approximations

We have developed a method called the two-dimensional Padé-type approximants method, which can be used to reduce the Gibbs phenomenon in the harmonic two-dimensional Fourier series. This method can be applied to both monochrome and color raster images. To do this, we implement the generalized two-dimensional Padé approximation proposed by Chisholm. In this approach, we select the range of frequency values on the integer grid according to the Vavilov method. We propose a definition of a Padé-type functional and provide examples of its application to simple discontinuous templates represented as raster images. Through this study, we are able to draw conclusions about the practical usage and advantages of the Padé-type approximation. We demonstrate that the Padé-type approximant effectively eliminates distortions associated with the Gibbs phenomenon, and it is visually more appropriate than the Fourier approximant. Additionally, the application of the Padé-type approximation reduces the number of parameters without sacrificing precision.

Statistical Image Watermark Algorithm for FAPHFMs Domain Based on BKF–Rayleigh Distribution

In the field of image watermarking, imperceptibility, robustness, and watermarking capacity are key indicators for evaluating the performance of watermarking techniques. However, these three factors are often mutually constrained, posing a challenge in achieving a balance among them. To address this issue, this paper presents a novel image watermark detection algorithm based on local fast and accurate polar harmonic Fourier moments (FAPHFMs) and the BKF–Rayleigh distribution model. Firstly, the original image is chunked without overlapping, the entropy value is calculated, the high-entropy chunks are selected in descending order, and the local FAPHFM magnitudes are calculated. Secondly, the watermarking signals are embedded into the robust local FAPHFM magnitudes by the multiplication function, and then MMLE based on the RSS method is utilized to estimate the statistical parameters of the BKF–Rayleigh distribution model. Finally, a blind image watermarking detector is designed using BKF–Rayleigh distribution and LO decision criteria. In addition, we derive the closed expression of the watermark detector using the BKF–Rayleigh model. The experiments proved that the algorithm in this paper outperforms the existing methods in terms of performance, maintains robustness well under a large watermarking capacity, and has excellent imperceptibility at the same time. The algorithm maintains a well-balanced relationship between robustness, imperceptibility, and watermarking capacity.

Robust dense-field based copy-move forgery localization using generic radial harmonic Fourier moment invariants.

As the importance of reliable multimedia content increases in today's society, image forensics is a growing field of research. The act of copying and pasting specific parts of an image, known as copy-move forgery, may be utilized for illegal or unethical purposes. Just as with other vision-related technologies, the accuracy of forensic analysis depends on having an appropriate image representation. Most existing feature extraction techniques do not accurately reflect the underlying image content leading to reduced performance. In this article, to detect the copy-move forgery attack, the Generic Radial Harmonic Fourier Moment (GRHFM) is proposed for reliable and distinctive image representation. The algorithm has the ability to effectively manipulate the distribution of zeros to emphasize certain image regions. Additionally, the relationships between complex exponentials and trigonometric functions are exploited to efficiently compute and easily implement the transform kernels. The efficacy of the algorithm is illustrated through experiments on dense-domain-based matching patterns. Experimental results on five benchmarking databases prove the effectiveness of the proposed approach compared with the state-of-the-art methods. According to the average scores, the proposed method demonstrates superior accuracy in overall localization performance. The F1 score, precision, and recall percentage values obtained are 92.5, 95.44, and 91.96, respectively. Robustness experiments on more challenging attacks are also conducted on FAU dataset. Results show that the proposed framework satisfies invariance to the various image variations, and thus an enhanced robustness compared to the previous methods. Moreover, the advantage of reasonable computational cost implies its potential use in real-world forensic applications.