High-order Polynomial Fitting Research Articles

Bivariate polynomial for time-dependent dielectric properties of ex-vivo porcine and human liver tissue at frequencies below 100 MHz

Time-dependent dielectric properties of ex vivo tissue are important for biological viability assessment, successful organ transplantation and finding the optimal substitute for the in vivo tissue. However, there are relatively few studies on the time dependence of dielectric properties and especially the mathematical models related to it. This study mathematically describes the time-dependent dielectric properties of excised liver tissues at frequencies below 100 MHz. A bivariate high-order polynomial was used to reflect the mathematical correlation of dielectric properties (relative permittivity and conductivity ) with frequency and ex vivo time under controlled temperature and humidity. Statistical analysis, cross validation and error analysis were used to assess the fitting performance and prediction ability of the bivariate high-order polynomial fitting model. The results showed that the dielectric properties of excised porcine liver tissues and human liver tissues could be well fitted and predicted by the bivariate high-order polynomial fitting model at frequencies below 100 MHz. Encouragingly, the dielectric properties can be calculated at a given frequency and ex vivo time at controlled temperature and humidity. This presents a new way of obtaining accurate dielectric properties of ex vivo biological tissues in a specific outer environment. It is also possible to extrapolate back the ex vivo time after excision from the body using the dielectric properties.

Theoretical Feasibility Analysis of Fast Back-Projection Algorithm for Moon-Based SAR in Time Domain

Nowadays, the Earth observation based on the Moon has attracted attention from many researchers and relevant departments. There also exists a considerable amount of interest in monitoring large-scale and long-term geoscience phenomena using the Moon-based SAR (MBS). However, the Earth’s observation from MBS has long transmission time, and the relative motion of MBS with its Earth ground target (EGT) is much different to the space-borne SAR, the above reasons indicate that the traditional stop-and-go model is no longer suitable for MBS in frequency domain imaging. Here a dual-path separate calculation method for single pulse is presented in this paper for a better match of a real scenario, and then the slant range is fitted to a high-order polynomial series. The MBS’s location, the synthetic aperture time and other factors have effects on length of the dual- path and fit bias. Without thoroughly investigated phase de-correlation processing in frequency domain, and to avoid computational costs in traditional back-projection (BP) algorithm, the paper first proposes a fast back-projection (FBP) algorithm in time domain for MBS, a platform that has long transmission time and long synthetic aperture time. In the FBP algorithm, the original method, that projected echo on all pixels in the imaging area, is changed to projected echo on a centerline instead. A suitable interpolation for points on the centerline is adopted to reduce the projected error; the synthetic aperture length and imaging area are also divided into subsections to reduce computation cost. The formula indicates that the range error could be control once the product of sub-imaging area’s length and sub-aperture’s length stay constant. Through the theoretical analysis, the detailed range difference mainly at apogee, perigee, ascending, and descending nodes indicate the necessity to separately calculate the dual-path for MBS’s single pulse transmission in Earth-Moon motion, with real ephemeris been adopted; then, the high-order polynomial fitting will better describe the motion trajectory. Lastly, the FBP algorithm proposed is simulated in a specific scenario under acceptable resolution, and the result shows its feasibility for image compression.

Human gait modelling and tracking based on motion functionalisation

Abstract This paper proposes a mathematical function movement model based on the gait movement of the human body and, in particular, on the trajectory of the limbs during human movement. The article systematically measures and experimentally deals with the trajectory of the limbs of 40 students in the walking movement. The linear high-order polynomial fitting method eliminates the motion error. Simultaneously, the linear relationship least square method is used to obtain the expression of the limb motion function. Finally, the mathematical model of the limb motion trajectory is obtained. It is verified through experiments that the model proposed in the thesis can calculate the law of limb movement and movement parameters of any person under normal walking movement. This research has high research value for human movement rehabilitation and the design of wearable equipment.

NONLINEAR ROLLING STABILITY AND CHAOS RESEARCH OF TRIMARAN VESSEL WITH VARIABLE LAY-OUTS IN REGULAR AND IRREGULAR WAVES UNDER WIND LOAD

The trimaran vessel rolls strongly at low forward speed and may capsize in high sea conditions due to chaos and loss of stability, which is not usually considered in conventional limit-based criteria. In order to perfect the method of measuring roll performance of trimaran, a set of nonlinear roll motion stability analysis method based on Lyapunov and Melnikov theory was established. The nonlinear roll motion equation was constructed by CFD and high-order polynomial fitting method. The wave force threshold of rolling chaos in regular waves is calculated by Gauss-Legendre numerical integration method. The limited significant wave height of rolling chaos in random sea conditions is deduced by the phase space transfer rate, and the complex effect of wind load is superposed in the calculation. The influence of trimaran configuration on the roll system is analyzed through the state differentiation of homoclinic and heteroclinic orbit in phase portrait. The calculation of the maximum Lyapunov exponent further verified the applicability of Melnikov method, and the topological structure change of gradual failure of the rolling system is analyzed by the erosion of safe basin. The complex changes of the nonlinear damping coefficient and the nonlinear restoring moment coefficient caused by the change of the transverse lay-outs between the main hull and side hull have a significant influence on chaos and stability, and the existence of wind load has a certain weakening effect on the stability and symmetry of the system. The conclusion also further indicates the importance of the lay-outs to the dynamic stability of the trimaran vessel, which is significant for its seakeeping design.

Focusing Translational-Variant Bistatic Forward- Looking SAR Data Using the Modified Omega-K Algorithm

Accurate 2-D frequency spectrum (2-D FS) with acceleration, two-way range coupling terms, and spatial-variant Doppler parameters are the main problems to be solved in translational-variant bistatic forward-looking synthetic aperture radar (SAR) (TV BFSAR) with curved trajectory. For these issues, a modified omega-K imaging algorithm is derived in this article. The maximum usage of 2-D FS based on the method of series reversion (MSR) is achieved by linear range cell migration correction, and 2-D FS is linearized in bistatic range by using high-order polynomial fitting. Then, a method of azimuth resampling is introduced to implement compensation of spatial-variant Doppler parameters. Different from other bistatic omega-K methods, our newly proposed method focuses on the small-aperture data in the azimuth frequency domain to avoid azimuth aliasing without padding zeros and uses the frequency focusing position to study the model of spatial-variant phase. Simulation results and real data verify the effectiveness of the proposed method.

Seismic time–frequency spectrum analysis based on local polynomial Fourier transform

Time–frequency analysis technology is widely used in non-stationary seismic data analysis. The energy concentration of the spectrum depends on the consistency of the kernel function of the time–frequency analysis method and the instantaneous frequency variation of the signals. The conventional time–frequency analysis methods usually require that the local instantaneous frequency of the signals remains unchanged or linearly changed. So it is difficult to accurately characterize the instantaneous frequency nonlinear variation of the non-stationary signal. The local polynomial Fourier transform (LPFT) method can effectively describe the instantaneous frequency variation by local high-order polynomial fitting and obtain the results with high spectral and energy concentration. The numerical simulations and field seismic data applications show that the time–frequency spectrum results obtained by LPFT can reflect the instantaneous frequency variation characteristics of the seismic data, while ensuring the concentration of time–frequency energy.

Gauss polynomials based modelling of grid-connected PV system output power in high-altitude area

Complicated ordering management and improper providing electricity are a problem encountered during the PV system generate electricity. Accurately predict the output power is an effective approach for minimizing these problems. In this paper, electric power exchange relationship of each link is analysed, according to the topology of the photovoltaic (PV) power stations. We propose the under prediction model of output power, based on the theory model of output PV current in the high-altitude environment. The composite error could be calculated by the difference between the under prediction power and the measure power. Comprehensive error expression is obtained by using the high-order polynomial fitting method. An improved prediction model, which adapts to the high-altitude environment, is established, while compensate the original model with the composite error. Compared with the actual output power of the PV power station, the results show that the curves of the two are highly consistent, the correlation between the two is more than 0.8 and the relative error between the predicted and measured power is less than that of the under-predicted power model. The results show that the improved predictive power model has a good tracking performance.

Investigation on Resonance Response of Mass-Adjustable Float in Wave Energy Conversion System

Abstract In the floating wave power system, when the natural frequency of the float coincides with the wave frequency, the system would theoretically work in the resonance state, which consequently makes the capture efficiency of wave energy conversion device reach the maximum. The natural frequency of the float is related to many parameters, including the float size and shape, float stiffness, float mass, float additional mass, etc. The variation of float mass might cause other parameters, like draught, additional mass and hydrodynamic coefficients, to change. This is one of the important reason why there is still no efficient and accurate method for mass adjustment. In order to solve this problem, a mass-adjustable float is designed, and a new optimization calculation method for the float based on the high-order polynomial fitting theory is proposed. The optimized method combines the advantages of boundary element analysis and numerical calculation. It introduces an additional coefficient K related to the wave frequency and the draught of the float to optimize the Hooft’s approximate equation between the additional mass and the float size. This optimized method can greatly improve the computational efficiency without reducing calculation accuracy. Finally, the heaving response of the mass-adjustable floats at different wave frequencies are calculated in AQWA, which verifies that the mass-adjustable floats have higher wave capture efficiency than ordinary floats.

Uncertainty analysis of linear least square fitting apply to non-linear model

Linear least square is a common method to fit two-dimensional data set with linear relation. In non-linear case we need to do some linearization before fitting. Parameters can be extracted from the fitted line and used to predict new dependent variable values at new independent variable points. In this report we investigate the uncertainty introduced to a nonlinear model prediction from linear least square fitting. As an example, we fit a data set with the relation D = a/h, which is the physical model of measurement we are interested. The fitting goes well, and we get the curve with R-square very closed to one. However, when we use the fitted parameters to actual measurements, the accuracy is poor, especially at large y. At first, we expect that this due to the functional form of the model. When compare with high order polynomial fitting we realize that this is not the case, since, for example, the sixth-degree polynomial gives less than one percent error, about 10 times less than linear fitting prediction. It is because the linearization and the inverse transformation to the original space in the linear least square method that give rise to the uncertainty. Our analysis can be generalized to any non-linear model prediction. We expect our results to be a caution to anyone using linear fitting to their non-linear model prediction.

An effective method for monitoring tissue temperature using low-field MRI system.

This technical note describes an effective method for monitoring temperature based on the minimum curvature surface fitting (MCSF) referenceless proton resonance frequency shift (PRFs) at 0.35T. Experimental tests on phantom, ex vivo porcine livers and human brain were conducted. A comparison study between the proposed method and a fiber optic temperature probe was conducted for microwave (MW) ablation. Experimental results showed that temperature root mean square error (RMSE) calculated by MCSF model were lower than those computed by high-order polynomial fitting. For ablation experiments, the temperature errors between temperature probe and MCSF method were all less than 2∘C.

A parametric interpolation method with minimal feedrate fluctuation by nonuniform rational basis spline

Because the relation between the arc length s and curve parameter u cannot be represented by explicit function for most of the curves, it is difficult to consider the accuracy, robustness, and computational efficiency for most of the parametric interpolation, especially when the curves are complex or extremes. Therefore, an off-line fitting interpolation method by using nonuniform rational basis spline is presented in this paper. As nonuniform rational basis spline has many geometry implementation tools and numerous good properties as compared to the polynomial, the required fitting accuracy can be obtained more easily than with polynomial. After the de Boor method is applied, the computational load of nonuniform rational basis spline is decreased as compared to the Taylor approximation and the higher order polynomial fitting method. In order to obtain the proper s-u fitting nonuniform rational basis spline and reduce the computational load of the fitting process, the sampled s-u data points are divided according to the properties of nonuniform rational basis spline, and in each segment, the knot vectors, control points, and weights are calculated by the iterative-optimization method. Then the s-u nonuniform rational basis spline can be applied in real-time interpolation, and the accuracy, robustness, and computational efficiency are demonstrated by simulations and experiments.

Establishing a direct connection between detrended fluctuation analysis and Fourier analysis

To understand methodological features of the detrended fluctuation analysis (DFA) using a higher-order polynomial fitting, we establish the direct connection between DFA and Fourier analysis. Based on an exact calculation of the single-frequency response of the DFA, the following facts are shown analytically: (1) in the analysis of stochastic processes exhibiting a power-law scaling of the power spectral density (PSD), S(f)∼f(-β), a higher-order detrending in the DFA has no adverse effect in the estimation of the DFA scaling exponent α, which satisfies the scaling relation α=(β+1)/2; (2) the upper limit of the scaling exponents detectable by the DFA depends on the order of polynomial fit used in the DFA, and is bounded by m+1, where m is the order of the polynomial fit; (3) the relation between the time scale in the DFA and the corresponding frequency in the PSD are distorted depending on both the order of the DFA and the frequency dependence of the PSD. We can improve the scale distortion by introducing the corrected time scale in the DFA corresponding to the inverse of the frequency scale in the PSD. In addition, our analytical approach makes it possible to characterize variants of the DFA using different types of detrending. As an application, properties of the detrending moving average algorithm are discussed.

基于相位同步的分布式全相参雷达参数估计及相参性能

For distributed aperture coherent radar with a general architecture, a signal model for coherent parameter estimation and coherent processing based on system phase synchronization is proposed in this paper, and the sensitivity of parameter estimation and coherent processing performance to synchronization errors is analyzed in detail. A closed-form hybrid Cramer-Rao bound (HCRB) of the estimation of time delay difference is first derived in multiple-input and multiple output (MIMO) work mode. Following this, in fully coherent work mode, the analytical formula of output signal-to-noise ratio gain (oSNRg) is addressed, and a numerical solution of the analytical formula is obtained by using high-order polynomial fitting. The principle of antenna configuration is then studied. It is concluded that the estimation accuracy of time delay difference or coherent processing performance is determined by both synchronization error and input SNR. Compared with existing methods, the method based on phase synchronization can yield higher estimation accuracy of time delay differences and better coherent processing performance, given the assumption that the phase synchronization error is small and the input SNR is low.

Estimation of temperature coefficient of resistance for microfabricated platinum thermometers in thermal conductivity measurements of one-dimensional nanostructures

The suspended-microdevice method is widely used in thermal conductivity measurements of one-dimensional nanostructures, in which microfabricated platinum coils are used as thermometers to determine temperature rises of both the heat source and heat sink. The error in the estimated temperature coefficient of resistance (TCR) of thermometers usually dominates the total error in this measurement. In this paper, we compare different methods to calculate TCR of platinum resistance thermometers (PRT) in thermal conductivity measurement of an individual boron nanoribbon performed on the suspended-microdevice setup. Our results show that electrical resistances of PRTs deviate from the Bloch–Grüneisen formula due to the high defect and impurity concentration in the resistance thermometer. This deviation cannot be fully compensated by considering the interference between electron–phonon scattering and electron–impurity scattering. It is also shown that simple numerical differentiation can give an accurate TCR since the random error in the measured electrical resistance is small enough. High-order polynomial fitting can also work well except at the boundary of temperature range.

Streaming BP for Non-Linear Motion Compensation SAR Imaging Based on GPU

As a high-quality imaging method for non-linear motion SAR, back projection (BP) algorithm's huge computational cost hinders its applications. In recent years, the emergence of GPU as a high-performance computing platform counteracts this disadvantage greatly, and brings the BP algorithm into people's perspective. In accord with the SAR's working procedures, the streaming structure is suitable for massive SAR data processing. Thus, a streaming BP algorithm and a piecewise high-order polynomial fitting method for IMU data preprocessing that is compatible to the streaming structure are proposed in this paper. The validity and performances of the fitting method are verified and discussed using actual IMU data. We find that the 9th-order polynomial fitting can meet the requirement of typical actual IMU data processing. The validity of the streaming BP for non-linear motion compensation is verified via actual SAR data, and its computing performance is analyzed and optimized on NVidia's GTX 590. The total runtime of the streaming BP on the GPU platform for an image with size 8192x8192 pixels is about 160 s, which can meet the requirements of many off-line SAR imaging applications.

Dynamic Recrystallization Behavior and Microstructure Evolution of Bridge Weathering Steel in Austenite Region

AbstractThe austenite dynamic recrystallization (DRX) behavior and microstructure evolution of a bridge weathering steel was systematically investigated at a deformation temperature range of 800–1100°C and strain rate of 0.1–10 s−1 by using hot compression test and optical microscopy. The stress exponent and hot deformation energy were obtained by regression method to determine thermal deformation constitutive equation. The curve of stress versus strain is used, combined with high order polynomial fitting, to accurately determine the critical value of DRX. The relationships between critical strain, critical stress, and Z parameter of the bridge weathering steel were obtained by regression method. Moreover, the influence factors of DRX kinetics of the bridge weathering steel were studied in the light of the experimental results. It is shown that the strain rate has a more significant effect on the rate of DRX than that of the deformation temperature, and there is almost 0.85 orders of magnitude increment in the rate of DRX as the strain rate increases an order of magnitude. The dynamically recrystallized grain size can be decreased with decreasing the deformation temperature and increasing the strain rate during the austenite deformation.

Knee angle-specific EMG normalization: The use of polynomial based EMG-angle relationships

The normalization of EMG signals to those recorded during a maximal voluntary contraction provides a valid construct for comparisons of relative muscle activity. However, the length dependence of muscle activation and purported, substantial, muscle translocation and changes in muscle architecture during dynamic movements presents a need for joint angle-dependent normalization processes. The purposes of the present study were to: (1) quantify variations in muscle activity across a large ROM, (2) determine the accuracy with which fitted EMG-joint angle curves accurately characterized these variations, and (3) compare peak (EMG-P) and average (EMG-A) EMG amplitudes obtained during a countermovement leg extension when normalized to both absolute peak and joint angle-specific muscle activity. Fifteen subjects performed a large ROM (110°) isokinetic (30°s−1) leg extension from which EMG-joint angle relationships were derived using polynomial fitting of different complexities. Ten subjects also performed loaded countermovement leg extensions from which EMG signals were normalized using peak muscle activity and EMG-angle curves. EMG amplitude varied significantly over the ROM and the use of EMG-angle curves for signal normalization resulted in significantly greater EMG-P and EMG-A than those normalized using the absolute peak EMG. Higher-order polynomial fitting better matched the filtered EMG amplitudes. Thus, there is a strong rationale for using EMG-angle polynomial fits to normalize EMG signals for large ROM movements.

Polynomial Moving Fitting Method for Edge Identification

A moving fitting method for edge detection is proposed in this work. Polynomial function is used for the curve fitting of the column of pixels near the edge. Proposed method is compared with polynomial fitting method without sub-segment. The comparison shows that even with low order polynomial, the effects of moving fitting are significantly better than that with high order polynomial fitting without sub-segment.

An approach for on-line signature authentication using Zernike moments

In this work, shape analysis of the acceleration plot, using lower order Zernike moments is performed for authentication of on-line signature. The on-line signature uses time functions of the signing process. The lower order Zernike moments represent the global shape of a pattern. The derived feature, acceleration vector is computed for the sample signature which comprises on-line pixels. The Zernike moment represent the shape of the acceleration plot. The summation value of a Zernike moment for a signature sample is obtained on normalized acceleration values. This type of substantiation decreases the influence of primary features with respect to translation, scaling and rotation at preprocessing stage. Zernike moments provide rotation invariance. In this investigation it was evident that the summation of magnitude of a Zernike moment for a genuine sample was less as compared to the summation of magnitude of a imposter sample. The number of derivatives of acceleration feature depends on the structural complexity of the signature sample. The computation of best order by polynomial fitting and reference template of a subject is discussed. The higher order derivatives of acceleration feature are considered. Signatures with higher order polynomial fitting and complex structure require higher order derivatives of acceleration. Each derivative better represents a portion of signature. The best result obtained is 4% of False Rejection Rate [FRR] and 2% of False Acceptance Rate [FAR].

DYNAMIC RECRYSTALLIZATION AND PRECIPITATION BEHAVIORS OF A KIND OF LOW CARBON V–MICROALLYED STEEL

By using thermo-mechanical simulator,OM and TEM,the dynamic recrystallization (DRX) and precipitation behaviors of a kind of low carbon V-microalloyed steel have been investigated at temperatures ranging from 900 to 1050℃and strain rates from 0.01 to 10 s~(-1).The activation energy (Qdef) for hot deformation of this kind of V-microalloyed steel was calculated to be 341.97 kJ/mol by regression analysis,while the apparent stress exponent(n) was calculated to be 4.24.The equation describing the hot working process was also obtained.The critical strain for DRX was accurately determined based on the P-J method and high order polynomial fitting between strain hardening rate and true stress,and mathematical models of critical strain and peak strain versus Z parameter were deduced.The dynamic precipitation behavior of V(C,N) particles at low strain rate was further investigated.The results show that with increasing the strain,the average size of V(C,N) particles increases and the size distribution of the precipitates become wide.The calculations of the driving force for recrystallization and pinning force show that once the dynamic recrystallization take place, the dynamic precipitation could not prevent dynamic recrystallization from occurring.