Continuous Fourier Research Articles

Processing of Idiopathic Pulmonary Fibrosis images based on spatial interpolation using DFT, FFT, DCT and LPF

The work focuses on the study of existing issues with some medical images obtained from patients with Idiopathic Pulmonary Fibrosis (IPF) who have survived this COVID-19 pandemic. This study analyzes potential causes of incorrect medical diagnoses with this disease. In this regard, we employ numerical algorithms such as DFT (Discrete Fourier Transform), FFT (Fast Fourier Transform), DCT (Discrete Cosine Transform), and LPF (Low-pass filter). The main objective of this work is to demonstrate that with these numerical algorithms based on Continuous and Discrete Fourier Theory, it is possible to filter existing noise in IPF radiography images. Furthermore, it is possible to compress, enhance, and improve their resolution so that better decisions can be made in a medical protocol. In this way, it contributes to the understanding of the true images that would be used for an optimal medical diagnosis.

Chebyshev polynomial-based Fourier transformation and its use in low pass filter of gravity data

In this paper, we introduce the novel Chebyshev Polynomials Least-Squares Fourier Transformation (C-LSQ-FT) and its robust variant with the Iteratively Reweighted Least-Squares technique (C-IRLS-FT). These innovative techniques for Fourier transformation are predicated on the concept of inversion, and the C-LSQ-FT method establishes an overdetermined inverse problem within the realm of Fourier transformation. However, given the LSQ approach’s vulnerability to data outliers, we note the potential for considerable errors and potentially unrepresentative model estimations. To circumvent these shortcomings, we incorporate Steiner’s Most Frequent Value method into our framework, thereby providing a more reliable alternative. The fusion of the Iteratively Reweighted Least-Squares (IRLS) algorithm with Cauchy-Steiner weights enhances the robustness of our Fourier transformation process, culminating in the C-IRLS-FT method. We use Chebyshev polynomials as the basis functions in both methods, leading to the approximation of continuous Fourier spectra through a finite series of Chebyshev polynomials and their corresponding coefficients. The coefficients were obtained by solving an overdetermined non-linear inverse problem. We validated the performance of both the traditional Discrete Fourier Transform (DFT) and the newly developed C-IRLS-FT through numerical tests on synthetic datasets. The results distinctly exhibited the reduced sensitivity of the C-IRLS-FT method to outliers and dispersed noise, in comparison with the traditional DFT. We leveraged the newly proposed (C-IRLS-FT) technique in the application of low-pass filtering in the context of gravity data. The results corroborate the technique’s robustness and adaptability, making it a promising method for future applications in geophysical data processing.

Eigenvalue estimates for Fourier concentration operators on two domains

We study concentration operators associated with either the discrete or the continuous Fourier transform, that is, operators that incorporate a spatial cut-off and a subsequent frequency cut-off to the Fourier inversion formula. The spectral profiles of these operators describe the number of prominent degrees of freedom in problems where functions are assumed to be supported on a certain domain and their Fourier transforms are known or measured on a second domain. We derive eigenvalue estimates that quantify the extent to which Fourier concentration operators deviate from orthogonal projectors, by bounding the number of eigenvalues that are away from 0 and 1 in terms of the geometry of the spatial and frequency domains, and a factor that grows at most poly-logarithmically on the inverse of the spectral margin. The estimates are non-asymptotic in the sense that they are applicable to concrete domains and spectral thresholds, and almost match asymptotic benchmarks. Our work covers, for the first time, non-convex and non-symmetric spatial and frequency concentration domains, as demanded by numerous applications that exploit the expected approximate low dimensionality of the modeled phenomena. The proofs build on Israel’s work on one dimensional intervals arXiv:1502.04404v1. The new ingredients are the use of redundant wave-packet expansions and a dyadic decomposition argument to obtain Schatten norm estimates for Hankel operators.

A neural network for forward and inverse nonlinear Fourier transforms for fibre optic communication

We propose a neural network for both forward and inverse continuous nonlinear Fourier transforms, NFT and INFT respectively. We demonstrate the network’s capability to perform NFT and INFT for a random mix of NFDM-QAM signals. The network transformations (NFT and INFT) exhibit true characteristics of these transformations; they are significantly different for low and high-power input pulses. The network shows adequate accuracy with an RMSE of 5×10−3 for forward and 3×10−2 for inverse transforms. We further show that the trained network can be used to perform general nonlinear Fourier transforms on arbitrary pulses beyond the training pulse types.

A mixed Fourier-variational approach to solve differential or integro-differential wave equations for magnetised plasmas

The All ORders Spectral Algorithm (AORSA) wave equation solver by Jaeger (Jaeger et al 2001 Phys. Plasmas 8 1573) solves the integro-differential wave equation relevant for the radio frequency (RF) domain and for fusion-relevant conditions in tokamaks or stellarators, retaining all finite Larmor radius corrections by substituting the continuous Fourier integrals by a sum over a discrete set of modes. Its strength is also its weakness: the simplicity of the method results in significant computational effort, a full matrix needing to be inverted to solve the associated linear system. Based on the notion that modes are gradually more independent if their eigenvalues differ, the present paper proposes a straightforward numerical method to partly alleviate this need, allowing to substitute the full system matrix by a banded one. The adopted method can be applied to a wide variety of equations. A few 1D examples—of relevance for solving the wave equation in the RF domain of frequencies—are provided: the tunneling equation is used to illustrate the potential of the method, and the all-FLR wave equation (retaining all Finite Larmor Radius corrections in the dielectric response) adopted by Jaeger is solved comparing the solutions found to those based on simpler models (a cold plasma and a ‘tepid plasma’ - i.e. a kinetic model truncated at zero order in Larmor radius—description).

Seasonality as a risk factor for deaths in Parkinson's disease

BackgroundAccording to growing evidence, sleep disruption harms biological processes and circadian homeostasis. Diurnal motor symptom volatility in Parkinson's Disease (PD) has been extensively studied. Few studies examined seasonal variability in PD symptoms, some showing it and others not. ObjectiveTo investigate whether PD patients' deaths follow a rhythmic pattern due to circadian rhythm alterations. Methods: This study used only unidentified patient databases. People with PD, ICD10 code G20, in at least one death certificate field were selected. The Continuous Wavelet Transform and Fourier Transform were checked for oscillation and its duration. ResultsThe 18-year analysis found 43,072 PD deaths. The Continuous Wavelet transform revealed a 351.87-day annual component (p < 0.05). Winter in the southern hemisphere saw more deaths, mainly in July. The Continuous Wavelet transform identified a significant daily component (p < 0.05) of 22.81 hours. Fatalities peaked around 9 a.m. Pneumonia is the leading cause of death in PD, and women and men have the same rhythm pattern. ConclusionParkinson's disease mortality in Brazil follows a pattern. Using over 40.000 death certificates from 18 years, the authors found that Parkinson's patient fatalities rise in winter and peak in July at about 9 a.m. Sunlight reduction increases mortality risk in the long term. Low sunshine lowers temperatures, increasing short-term death risk. This is crucial because it prioritizes the sun, seasons, and circadian rhythm over low temperatures.

Digital reference objects for evaluating algorithm performance in MR image formation

PurposeDigital Reference Objects (DROs) are mathematical phantoms that can serve as a basis for evaluating MR image quality (IQ) in an objective way. Their main purpose is to facilitate the establishment of fully automated and perfectly reproducible IQ metrics to objectively compare different algorithms in MR image formation in a standardized manner. They also allow to re-build parts of standard phantoms. MethodsWe sample DROs directly in k-space, using analytical formulas for the continuous Fourier transform of primitive shapes. We demonstrate this DRO approach by applying a state-of-the-art CNN-based denoising algorithm that is robust to varying noise levels to noisy images of the resolution section of the well-known ACR phantom for IQ assessment, reconstructed from both measured and simulated k-space data. ResultsApplying the CNN-based denoising algorithm to the measured and simulated version of the ACR phantom resolution section produced virtually identical results, as confirmed by visual and quantitative comparison. ConclusionsDROs can help guide technology selection during the development of new algorithms in MR image formation, e.g., via deep learning. This could be an important step towards reproducible MR image formation.

Enhancing Contrast of Spatial Details in X-ray Phase-Contrast Imaging through Modified Fourier Filtering

In-line X-ray phase contrast imaging, which is simple to experiment with, provides significantly higher sensitivity, compared to conventional X-ray absorption imaging. The inversion of the relationship between recorded Fresnel diffraction intensity and the phase shift induced by the object is called phase retrieval. The transport of intensity equation (TIE), a simple method of phase retrieval, which is solved by the fast Fourier transform algorithm proposed by Paganin et al., has been widely adopted. However, the existing method suffers from excessive suppression of high-frequency information, resulting in loss of image details after phase retrieval, or insufficient detail contrast, leading to blurry images. Here, we present a straightforward extension of the two-distance FFT-TIE method by modifying the Fourier filter through the use of a five-point approximation to calculate the inverse Laplacian in a discrete manner. Additionally, we utilize a combination of continuous Fourier transform and a four-point approximation to compute the gradient operator. The method is evaluated by simulating samples with a shape similar to the resolution test map and by using a photograph of a dog for further evaluation. The algorithm that incorporates the modified gradient operator and the algorithm that solely utilizes the continuous Fourier transform for gradient computation were compared with the results obtained using the two-distance FFT-TIE method. The comparisons were conducted using the results obtained from two distances from the sample to the detector. The results show that this method improves the contrast of spatial details and reduces the suppression of high spatial frequencies compared to the two-distance FFT-TIE method. Furthermore, in the low-frequency domain, our algorithm does not lose much information compared to the original method, yielding consistent results. Furthermore, we conducted our experiments using carbon rods. The results show that both our method and the FFT-TIE method exhibit low-frequency distortion due to the requirement of close proximity between the absorption maps and the detector. However, upon closer inspection, our proposed method demonstrates superior accuracy in reproducing the finer details of the carbon rod fibers.

A CNN-based lamb wave processing model for field monitoring of fatigue cracks in orthotropic steel bridge decks

Orthotropic steel bridge decks (OSDs) are prone to fatigue cracking under vehicle cyclic loads, and hence, fatigue crack monitoring is important, especially for the identification of crack growth. To achieve the intelligent monitoring of fatigue cracks in OSDs, Lamb wave technology equipped with machine learning algorithms was proposed in this study. A convolutional neural network (CNN) model with three functional layers was constructed, where a convolution layer was used to eliminate ambient noise disruption from wave feature data, a feature pooling layer to enhance wave features, and a full connection layer to identify crack dimension. The optimal time and frequency parameters of the feature pooling layer were determined based on numerical simulation data and, therefore, could be applied in general situations. In comparisons through field monitoring with continuous wavelet transform (CWT) and continuous Fourier transform (CFT) algorithms, the wave features obtained by the CNN model are more regularized and less fluctuant. The fatigue crack length identification errors of the CNN model are within 1 mm, which are smaller than those of the other algorithms. The CNN model obtains the highest accuracy rates under various loss thresholds. The proposed model also demonstrates superior identification accuracy in comparison to other existing methods. Consequently, when combined with Lamb wave technology, the model can therefore be suitable to applied in the fatigue crack monitoring of real OSDs.

Midpoint Time Allocation and value estimation of Frequency Component for Time-Limited Signal Using Continuous Fractional Fourier Transform

Midpoint Time Allocation and value estimation of Frequency Component for Time-Limited Signal Using Continuous Fractional Fourier Transform

Channel model for the dual-polarization b-modulated nonlinear frequency-division multiplexing optical transmission systems.

We consider optical transmission systems based on the nonlinear frequency division multiplexing (NFDM) concept, i.e., the systems employing the nonlinear Fourier transform (NFT) for signal processing and data modulation. Our work specifically addresses the double-polarization (DP) NFDM setup that utilizes the so-called b-modulation, the most efficient NFDM method proposed up-to-date. We extend the previously-developed analytical approach based on the adiabatic perturbation theory for the continuous nonlinear Fourier spectrum (b-coefficient) onto the DP case to obtain the leading order of continuous input-output signal relation, i.e., the asymptotic channel model, for an arbitrary b-modulated DP-NFDM optical communication system. Our main result is in deriving the relatively simple analytical expressions for the power spectral density of the components of effective conditionally Gaussian input-dependent noise emerging inside the nonlinear Fourier domain. We also demonstrate that our analytical expressions are in remarkable agreement with direct numerical results if one extracts the "processing noise" arising due to the imprecision of numerical NFT operations.

Application of the Residue Theorem to Euler Integral, Gaussian Integral, and Beyond

The paper is divided into three different parts, which use the residue theorem to solve several different integrals, namely, the Euler integral, the Gaussian integral, the Fresnel integral, and so forth. The process of using the resiude theorem to determine these integrals is to first turn the integrals into convenient forms of complex integrals, and then find integral perimeters so that any integral on one of the curves is the required integral, through the drawing observation of the contour to write the original integral into the form of multiple integral. By studying the resiude theorem to solve the problem of complex integrals, it is demonstrated that the resiude theorem is actually a process that makes the calculation easier. These solved integrals have a wide range of applications including the study of the refraction of light, analytics, probability theory, combinatorial mathematics, and unification of the continuous Fourier transform.

SOME DISTINGUISHING CHARACTERISTICS OF THE LINEAR IMMUTABILITY OF CONTINUOUS TIME SERIES VIA BIVARIATE VECTOR VALUED STOCHASTIC PROCESSES

During the process of analyzing the continuous expanded finite Fourier transforms of strictly stable $(i+j)$ vector-valued time series, it is presumed that some of the observations have been misplaced. This is done on the basis of an assumption. This is due to the fact that the method entails looking at continuous extended finite Fourier transforms. This is done so that the findings can be interpreted in a manner that is as precise as is practical given the information that is available. The reason for this is so that the findings can be used to make better decisions. Consequently of this additional data, the continuous Fourier transform will become the focal point of the researchers' achievements. At the present time, the concept of asymptotic moments is garnering a large amount of interest from researchers all around the world. In this investigation we will apply our theoretical study in case study on the subject of Electricity Energy,

A continuous approach to Floquet theory for pulse-sequence optimization in solid-state NMR.

We present a framework that uses a continuous frequency space to describe and design solid-state nuclear magnetic resonance (NMR) experiments. The approach is similar to the well-established Floquet treatment for NMR, but it is not restricted to periodic Hamiltonians and allows the design of experiments in a reverse fashion. The framework is based on perturbation theory on a continuous Fourier space, which leads to effective, i.e., time-independent, Hamiltonians. It allows the back-calculation of the pulse scheme from the desired effective Hamiltonian as a function of spin-system parameters. We show as an example how to back-calculate the rf irradiation in the MIRROR experiment from the desired chemical-shift offset behavior of the sequence.

Comparison of simulation methods for dynamic internal air distribution in naturally ventilated livestock buildings

Accurate internal air distribution prediction facilitates the analysis of occupant thermal comfort and the control of gaseous contaminants within naturally ventilated livestock buildings (NVLB). However, the dynamic and random change in external wind conditions makes it difficult to predict the air distribution. This study compares three scenarios based on real-time monitored external wind conditions: the quasi-static simulation method with moving average data, the transient simulation method with raw data, the transient simulation method with Kalman filtering, and the continuous fast Fourier transform. The quasi-static simulation was verified for computational efficiency and reliability within different time spans. The simulation results were validated with measurement data in a low-rise stand-alone NVLB surrounding a low-density building complex. The quasi-static simulation results of internal air distribution in oblique wind directions partially agreed with the hourly averaged wind speed measurements and turbulent kinetic energy. Additionally, the quasi-static simulation agrees with the direct transient method for predicting the air distribution within a long-time span. The quasi-static simulation shows promise for the future simulation of internal air distribution in naturally ventilated livestock buildings under dynamic boundary conditions.

INVERTIBILITY OF FOURIER CONVOLUTION OPERATORS WITH PC SYMBOLS ON VARIABLE LEBESGUE SPACES WITH KHVEDELIDZE WEIGHTS

Let \(p(\cdot ):\mathbb {R}\rightarrow (1,\infty )\) be a sufficiently regular variable exponent and \(\varrho\) be a Khvedelidze weight on \(\mathbb {R}\). Suppose that a function a belongs to the algebra \(PC_{p(\cdot ),\varrho }\) of piecewise continuous Fourier multipliers on the weighted variable Lebesgue space \(L^{p(\cdot )}(\mathbb {R},\varrho )\). We show that the Fourier convolution operator \(W^0(a)=F^{-1}aF\) is invertible on the space \(L^{p(\cdot )}(\mathbb {R},\varrho )\) if and only if its symbol a is invertible in \(L^\infty (\mathbb {R})\).

Viscous Rankine vortices

A generalized version of the Rankine vortex is taken as initial state in an infinite fluid domain in two dimensions. The inner rigid-body motion of the Rankine vortex is surrounded by an outer irrotational vortex, where the inner flow and outer flow are allowed to have different amplitudes. We study the time evolution in a viscous fluid, where the kinematic viscosity scales the unit of dimensionless time. By an inverse Hankel transform, the Rankine vortex appears as an integrated continuous spectrum of axisymmetric Fourier–Bessel modes with radial wave numbers k. Each of these velocity modes is represented by a Bessel function J1(kr) where r is the radial distance from the vortex center. The amplitude of each individual mode will decay exponentially at a rate of νk2, where ν is the kinematic viscosity. The decaying continuous Fourier–Bessel spectrum is calculated analytically, while the evolution of the flow field is found by numerical evaluation of an inverse Hankel transform. We also consider the presence of a free surface in the gravity field, where pressure distribution in the vertical direction is hydrostatic. The far field remains irrotational during the viscous decay.

Spectral Analysis With a Piecewise Separation Occlusion Model for Image-Based Rendering

Occlusion is ubiquitous in the real world, and neglecting it will certainly lead to quality degeneration in a reconstructed light field (LF). In this paper, we derive a piecewise separation occlusion model (PSOM) for LF reconstruction that is optimized for occlusion in the continuous Fourier and integral domains. The PSOM is first obtained by determining the definite integral intervals between the occlusion and nonocclusion boundaries in the spatial domain along the camera direction. We then use these definite integral intervals to perform spectral analysis of the plenoptic function (POF) to derive the occluded and nonoccluded plenoptic spectra. Finally, we show that using the two sampling rates derived from the occluded and nonoccluded spectra is more effective than using a uniform sampling rate for reconstructing a single scene unless the scene is free of occlusion. Through numerical experiments on simulated signals and real-world inward-looking and outward-looking scenes, we demonstrate the effectiveness of our proposed approach.

Stochastically Forced Ensemble Dynamic Mode Decomposition for Forecasting and Analysis of Near-Periodic Systems

Time series forecasting remains a central challenge problem in almost all scientific disciplines. We introduce a novel load forecasting method in which observed dynamics are modeled as a forced linear system using Dynamic Mode Decomposition (DMD) in time delay coordinates. Central to this approach is the insight that grid load, like many observables on complex real-world systems, has an “almost-periodic” character, i.e., a continuous Fourier spectrum punctuated by dominant peaks, which capture regular (e.g., daily or weekly) recurrences in the dynamics. The forecasting method presented takes advantage of this property by (i) regressing to a deterministic linear model whose eigenspectrum maps onto those peaks, and (ii) simultaneously learning a stochastic Gaussian process regression (GPR) process to actuate this system. Our forecasting algorithm is compared against state-of-the-art forecasting techniques not using additional explanatory variables and is shown to produce superior performance. We further show that this joint approach offers marked improvement over existing methods that use purely DMD or GPR. Moreover, its use of linear intrinsic dynamics offers a number of desirable properties in terms of interpretability and parsimony. Results are presented for a test case using load data from an electrical grid. Load forecasting is an essential challenge in power systems engineering, with major implications for real-time control, pricing, maintenance, and security decisions.

Neural networks for computing and denoising the continuous nonlinear Fourier spectrum in focusing nonlinear Schr\xf6dinger equation

We combine the nonlinear Fourier transform (NFT) signal processing with machine learning methods for solving the direct spectral problem associated with the nonlinear Schrödinger equation. The latter is one of the core nonlinear science models emerging in a range of applications. Our focus is on the unexplored problem of computing the continuous nonlinear Fourier spectrum associated with decaying profiles, using a specially-structured deep neural network which we coined NFT-Net. The Bayesian optimisation is utilised to find the optimal neural network architecture. The benefits of using the NFT-Net as compared to the conventional numerical NFT methods becomes evident when we deal with noise-corrupted signals, where the neural networks-based processing results in effective noise suppression. This advantage becomes more pronounced when the noise level is sufficiently high, and we train the neural network on the noise-corrupted field profiles. The maximum restoration quality corresponds to the case where the signal-to-noise ratio of the training data coincides with that of the validation signals. Finally, we also demonstrate that the NFT b-coefficient important for optical communication applications can be recovered with high accuracy and denoised by the neural network with the same architecture.