Linear Combination Of Trigonometric Functions Research Articles

Experimental and Theoretical Analysis of Bispectrum Characteristics of Phase Coupled Signals

Fourier transform points out that a certain function that meets certain conditions can be decomposed into a linear combination of trigonometric functions (sine or cosine functions). The functions of high-order cumulant include suppressing the Gaussian noise, eliminating independent signal components, and identifying the phase coupling phenomenon of the signal. To prove this hypothesis, this study constructs a cosine signal with the phase coupling phenomenon based on Fourier transform theory which substitutes it into the third-order cumulant expression and performs detailed reasoning. The constructed signal is extended to the complex signal domain and the same conclusion is obtained. The number of coupled signals is expanded from three to a higher value. The results of the study give definite mathematical and physical meaning to the bispectral peaks. The collected mechanical vibration signals are given to demonstrate this conclusion. The demonstrated characteristics of high-order cumulants have made them widely used in many fields.

Learning Orientation-Aware Distances for Oriented Object Detection

Oriented object detectors have suffered severely from the discontinuous boundary problem for a long time. In this work, we ingeniously avoid this problem by relating regression outputs to regression target orientations. The core idea of our method is to build a contour function which imports orientations and outputs the corresponding distance predictions. Inspired by Fourier transformations, we assume this function can be represented as a linear combination of trigonometric functions and Fourier series. We replace the final 4D layer in the regression branch of fully convolutional one-stage object detector (FCOS) with a Fourier Series Transformation (FST) module and term this new network FCOSF. By this unique design, the regression outputs in FCOSF can adaptively vary according to the regression target orientations. Thus, the discontinuous boundary has no impact on our FCOSF. More importantly, FCOSF avoids building complicated oriented box representations, which usually cause extra computations and ambiguities. With only flipping augmentation and single-scale training and testing, FCOSF with ResNet-50 achieves 73.64% mAP on the DOTA-v1.0 dataset with up to 23.6 FPS speed, surpassing all one-stage oriented object detectors. On the more challenging DOTA-v2.0 dataset, FCOSF also achieves the highest results of 51.75% mAP among one-stage detectors. More experiments on DIOR-R and HRSC2016 are also conducted to verify the robustness of FCOSF. Code and models will be available at https://github.com/DDGRCF/FCOSF.

A novel use of time separation technique to improve flat detector CT perfusion imaging in stroke patients.

CT perfusion imaging (CTP) is used in the diagnostic workup of acute ischemic stroke (AIS). CTP may be performed within the angio suite using flat detector CT (FDCT) to help reduce patient management time. In order to significantly improve FDCT perfusion (FDCTP) imaging, data-processing algorithms need to be able to compensate for the higher levels of noise, slow rotation speed, and a lower frame rate of current FDCT devices. We performed a realistic simulation of FDCTP acquisition based on CTP data from seven subjects. We used the time separation technique (TST) as a model-based approach for FDCTP data processing. We propose a novel dimension reduction in which we approximate the time attenuation curves by a linear combination of trigonometric functions. Our goal was to show that the TST can be used even without prior assumptions on the shape of the attenuation profiles. We first demonstrated that a trigonometric basis is suitable for dimension reduction of perfusion data. Using simulated FDCTP data, we have shown that a trigonometric basis in the TST provided better results than the classical straightforward processing even with additional noise. Average correlation coefficients of perfusion maps were improved for cerebral blood flow (CBF), cerebral blood volume, mean transit time (MTT) maps. In a moderate noise scenario, the average Pearson's coefficient for the CBF map was improved using the TST from 0.76 to 0.81. For the MTT map, it was improved from 0.37 to 0.45. Furthermore, we achieved a total processing time from the reconstruction of FDCTP data to the generation of perfusion maps of under 5 min. In our study cohort, perfusion maps created from FDCTP data using the TST with a trigonometric basis showed equivalent perfusion deficits to classic CT perfusion maps. It follows, that this novel FDCTP technique has potential to provide fast and accurate FDCTP imaging for AIS patients.

Analysis and trajectory planning for anti-swing on a one-stage cart pendulum

In order to achieve smooth and anti-swing positioning process, a smooth trajectory planning scheme, which is mainly the linear combination of trigonometric functions, is proposed for a one-stage cart pendulum. Based on the internal dynamics of the cart pendulum system, the planning trajectory of the cart is derived from the anticipated swing angle of the pendulum bar, which satisfies the initial and terminal state constraints. Experiments are conducted to demonstrate the performance of anti-swing with the introduced planned trajectory.

The microlocal irregularity of Gaussian noise

The study of random Fourier series, linear combinations of trigonometric functions whose coefficients are independent (in our case Gaussian) random variables with polynomially bounded means and standard deviations, dates back to Norbert Wiener in one of the original constructions of Brownian motion. A geometric generalization -- relevant e.g.\ to Euclidean quantum field theory with an infrared cutoff -- is the study of random Gaussian linear combinations of the eigenfunctions of the Laplace-Beltrami operator on an arbitrary compact Riemannian manifold $(M,g)$, Gaussian noise $\Phi$. I will prove that, when our random coefficients are independent Gaussians whose standard deviations obey polynomial asymptotics and whose means obey a corresponding polynomial upper bound, the resultant random $\mathscr{H}^s$-wavefront set $\operatorname{WF}^s(\Phi)$ (defined as a subset of the cosphere bundle $\mathbb{S}^*M$) is either almost surely empty or almost surely the entirety of $\mathbb{S}^*M$, depending on $s \in \mathbb{R}$, and we will compute the threshold $s$ and the behavior of the wavefront set at it. Consequently, the random $C^\infty$-wavefront set $\operatorname{WF}(\Phi)$ is almost surely the entirety of the cosphere bundle. The method of proof is as follows: using Sazonov's theorem and its converse, it suffices to understand which compositions of microlocal cutoffs and inclusions of $L^2$-based fractional order Sobolev spaces are Hilbert-Schmidt (HS), and the answer follows from general facts about the HS-norms of the elements of the pseudodifferential calculus of Kohn and Nirenberg.

Image iterative method for handwritten Chinese character recognition

Although accuracy and calculation time of handwritten Chinese character recognition have been greatly improved, it has not yet reached the practical application standard. Recently literatures, iterative method is used to recognize face images and good results are obtained. Therefore, image recognition is improved based on this idea. In this paper, linear combination of trigonometric functions is used as auxiliary function and image to construct discrete dynamic system. By adding bevel and shift to construct font surface and so on, the system structure is improved to solve problem of system convergence caused by large area flatness of handwritten Chinese character image, and good recognition effect is achieved. Preliminary experiments(for example, extracting 20 Chinese characters written by 9 people in the data set) are carried out, and recognition rate can reach 100% when all of them are trained, and recognition rate can reach 76.7% when each Chinese character is trained to 3 pieces; in addition, experiments were conducted on handwritten Chinese character sets changing from different angles, and the recognition rate reached 80%, which was compared with other algorithms to verify the feasibility of handwritten Chinese character recognition algorithm based on chaotic iterative trajectory characteristics of images.

Cancellable biometric system based on linear combination of trigonometric functions with special application to forensic dental biometrics

Biometric templates in its original form are susceptible to attacks and irrevocable if compromised, and thus need to be protected. The proposed template protection approach based on a linear combination of trigonometric functions secures the dental templates stored in forensic records and also maintains the recognition performance of the identification system simultaneously. Furthermore, the security analysis of this non invertible transformation has been done by making many attempts to revert the transformed templates back to the original domain which proved effectual in maintaining the non-invertibility. The experimental results indicate that this approach is practical with a low EER of 2%. Moreover, this system has a high recognition rate of almost 98% and rank 1 identification rate of 93.33% for original as well as transformed dental templates and it is worth to mention that varying the key value does not degrade the system performance in case of template renewability.

Cancellable biometric system based on linear combination of trigonometric functions with special application to forensic dental biometrics

Biometric templates in its original form are susceptible to attacks and irrevocable if compromised, and thus need to be protected. The proposed template protection approach based on a linear combination of trigonometric functions secures the dental templates stored in forensic records and also maintains the recognition performance of the identification system simultaneously. Furthermore, the security analysis of this non invertible transformation has been done by making many attempts to revert the transformed templates back to the original domain which proved effectual in maintaining the non-invertibility. The experimental results indicate that this approach is practical with a low EER of 2%. Moreover, this system has a high recognition rate of almost 98% and rank 1 identification rate of 93.33% for original as well as transformed dental templates and it is worth to mention that varying the key value does not degrade the system performance in case of template renewability.

Application of iterative algorithms for gas-turbine blades cooling optimization

This paper presents two iterative algorithms used to solve the inverse problem of gas-turbine blade cooling. First of them was obtained by variational calculus methods, and the second one is reduced to solving the least-square approximation problem. To determine the temperature distribution on the inner boundary of the blade (of the multiply-connected region), a combination of trigonometric functions defined on the boundary of a unit circle was used. Both algorithms are presented in two variants: the first one is about reducing the number of trigonometric functions, and in the second one, coefficients of trigonometric functions are regularized in the process of iteration. Results of calculations indicate that both algorithms operate more effectively in the variant where coefficients of linear combination of trigonometric functions are regularized.

To the issue of forecasting price dynamics in the property valuation

The problems of short-, medium- and long-term forecasting price dynamics are considered. The improvement of short-term forecasting techniques based on exponential smoothing is proposed. A modified autoregressive model of the first-order differences for the medium-term forecasting is developed. A method of constructing the approximating function of the first-order differences as a linear combination of trigonometric functions, which can be used for long-term forecasting, is proposed.

The estimation of heavy-tailed probability density functions, their mixtures and quantiles

This paper is devoted to the estimation of heavy-tailed probability density functions (p.d.f.s), their mixtures and high quantiles. We discuss the relevance of this issue in teletraffic engineering and propose a new combined estimation technique for such p.d.f.s. The “tail” of the p.d.f. is estimated by a parametric tail model and its “body” by a non-parametric method in terms of a finite linear combination of trigonometric functions. To provide the minimum of the mean-squared error of the estimation, the parameters of the parametric and non-parametric parts are estimated by means of the bootstrap method and the structural risk minimization method. The latter parameters are determined by the number of extreme-valued data that are used in Hill’s estimate of the tail index and the number of terms and coefficients of the linear combination. The new method is illustrated using some relevant mixtures of heavy-tailed p.d.f.s and applied to construct a high-quantile estimate. Furthermore, its effectiveness is shown by an application to real data arising from Web-traffic characteristics.

Mixed collocation methods for y″= f( x, y)

The collocation methods introduced here are based on linear combinations of trigonometric functions and powers. The motivation is to provide better approximations for oscillatory solutions of initial-value problems for differential equations of the special form y″= f( x, y). The resulting methods, for two or more collocation points, are implicit Runge–Kutta–Nyström methods with coefficients which depend on both the fitted angular frequency and the steplength. Algebraic and trigonometric order conditions are considered and the stability properties of some methods are examined. Particular mixed collocation methods, and other methods for the same class of problems, are compared by applying them to a variety of test problems.

Natural frequencies of finite circular cylinders in axially symmetric longitudinal vibration

A method for computing natural frequencies of finite circular cylinders in axially symmetric longitudinal vibration is developed which is applicable to both solid and hollow cylinders with any type of homogeneous boundary conditions. The procedure involves representation of the components of displacement in series of the corresponding pure radial and axial modes of the infinite cylinder. The Rayleigh-Ritz method is used to derive a set of differential equations for the coefficients of the expansion functions. The solution of these equations is given as a linear combination of trigonometric functions with coefficients which are determined by satisfying the boundary conditions at the ends of the cylinder. When the end conditions are homogeneous, a set of homogeneous algebraic equations is obtained, and the resulting determinantal equation can be used to compute natural frequencies. Numerical results are given for a solid cylinder with free lateral surface and free ends and a solid cylinder with fixed lateral surface and free ends. For the former case, experimental results were found in the literature with which the present results are in very good agreement.

Investigation of an efficient representation of speech spectra for segmentation and classification of speech sounds

A functional representation of speech sounds in orthogonal polynomial space is described and preliminary results are presented. Speech spectra are approximated by a linear combination of orthogonal polynomials which are found to be more efficient than a linear combination of trigonometric functions. The original spectra (100 samples in frequency) and the polynomial approximations are represented by points in their respective Hilbert spaces, the distance between successive points being a measure of the dissimilarity of successive spectra. Segment boundaries are indicated where the distance between successive spectra exceeds a threshold. The effectiveness in segmentation of connected utterances using these spectral forms is compared. Also, representing speech in orthogonal polynomial space appears to be applicable to clustering and separating transformations which yield simple decision boundaries for phoneme classification. Although only one polynomial class is investigated, the procedure is valid for other functional representations of speech data.

Approach to More Efficient Segmentation and Classification of Speech Sounds

The success of lumped-parameter analogs of the vocal tract suggests that speech spectra may be optimally approximated by a rational function. As a step in this direction, speech spectra have been approximated by a linear combination of Gram polynomials and have been found to be more efficient than a linear combination of trigonometric functions. The original spectra (100 samples in frequency) and the polynomial and trigonometric approximations have been represented by points in their respective Hilbert spaces, the distance between successive points being a measure of the dissimilarity of successive spectra. Segment boundaries are indicated where the distance between successive spectra exceeds a threshold. The effectiveness in segmentation of connected utterances using these spectral forms will be compared. Results will be presented and implications made for real-time machine recognition of speech. [This research was supported by NASA Fellowships and the Electrical Engineering Department at Northeastern University.]