Cubic Spline Interpolation Research Articles

Modeling the Wavelength Dependence of Pixel Response Nonuniformity of a CCD Sensor

Precision measurements in astronomy require stringent control of systematics such as those arising from imperfect correction of sensor effects. In this work, we develop a parametric method to model the wavelength dependence of pixel response nonuniformity (PRNU) for a laser-annealed backside-illuminated charge-coupled device. The model accurately reproduces the PRNU patterns of flat-field images taken at nine wavelengths from 290 to 950 nm, leaving the rms residuals no more than 0.2% in most cases. By removing the large-scale nonuniformity in the flat fields, the rms residuals are further reduced. This model fitting approach gives more accurate predictions of the PRNU than cubic-spline interpolation does with fewer free parameters. It can be applied to make PRNU corrections for individual objects according to their spectral energy distribution to reduce the photometry errors caused by the wavelength-dependent PRNU, if sub-percent level precision is required.

Predicting removal of arsenic from groundwater by iron based filters using deep neural network models

Arsenic (As) contamination in drinking water has been highlighted for its environmental significance and potential health implications. Iron-based filters are cost-effective and sustainable solutions for As removal from contaminated water. Applying Machine Learning (ML) models to investigate and optimize As removal using iron-based filters is limited. The present study developed Deep Learning Neural Network (DLNN) models for predicting the removal of As and other contaminants by iron-based filters from groundwater. A small Original Dataset (ODS) consisting of 20 data points and 13 groundwater parameters was obtained from the field performances of 20 individual iron-amended ceramic filters. Cubic-spline interpolation (CSI) expanded the ODS, generating 1600 interpolated data points (IDPs) without duplication. The Bayesian optimization algorithm tuned the model hyper-parameters and IDPs in a Stratified fivefold Cross-Validation (CV) setup trained all the models. The models demonstrated reliable performances with the coefficient of determination (R2) 0.990–0.999 for As, 0.774–0.976 for Iron (Fe), 0.934–0.954 for Phosphorus (P), and 0.878–0.998 for predicting manganese (Mn) in the effluent. Sobol sensitivity analysis revealed that As (total order index (ST) = 0.563), P (ST = 0.441), Eh (ST = 0.712), and Temp (ST = 0.371) are the most sensitive parameters for the removal of As, Fe, P, and Mn. The comprehensive approach, from data expansion through DLNN model development, provides a valuable tool for estimating optimal As removal conditions from groundwater.

In-situ measurement of particle length distribution by FBRM with application to needle- or rod-shape crystallization processes

In this paper, an in-situ measurement method is proposed for detecting needle- or rod-shape particle length distribution (PLD) during a crystallization process, by using the instrument named focused beam reflectance measurement (FBRM). A data pretreatment strategy is firstly presented to modify the measured chord length distribution (CLD) of particles by FBRM for proper analysis, which could guarantee measurement accuracy when the particle length are smaller than 150 μm. Based on the pretreated CLD, a moment conversion algorithm is established for estimating the moments of particle population density, so as to further improve the measurement accuracy for particles possessing larger sizes. Correspondingly, the particle growth-nucleation model parameters are efficiently estimated by using the interior point method (IPM) in combination with the homotopy analysis method (HAM). Then the PLD is reconstructed based on these estimated low-order moments, through a modified cubic-spline interpolation algorithm with variable interpolation grids. Experimental results on the seeded cooling crystallization process of β form L-glutamic acid (LGA) well demonstrate the effectiveness of the proposed method for in-situ measurement.

Precision forecasting of fertilizer components’ concentrations in mixed variable-rate fertigation through machine learning

Accurate monitoring of fertilizer concentration and variable irrigation components is crucial for achieving precision irrigation through variable-rate fertigation. However, technological and cost constraints pose challenges in effectively managing mixed variable-rate fertigation. This study addresses these challenges by integrating machine learning (ML) with easily monitored physical parameters, such as electrical conductivity (EC), pH, and temperature, to predict fertilizer solution components and concentrations in mixed variable-rate fertigation. Cubic spline interpolation (CSI) was used to enhance the training dataset. Six ML algorithms—Multivariate Linear Regression (MLR), Support Vector Machines (SVM), K-Nearest Neighbors (KNN), Extremely Randomized Trees (ETs), Multilayer Perceptron (MLP), and Extreme Gradient Boosting (XGB)—were used to develop prediction models, with their performance evaluated using the coefficient of determination (R2), root mean square error (RMSE), and Akaike information criterion (AIC). The Extended Fourier Amplitude Sensitivity Test (EFAST) assessed the sensitivity of the ML models to physical parameters. All of the ML models, except for MLR, particularly SVM, demonstrated superior performance in predicting fertilizer solution components’ concentrations, with R2 values between 0.989 and 0.997 and RMSE values between 0.089 and 0.210. The CSI significantly enhanced model performance, resulting in larger R2 values and smaller RMSE values. The AIC results and sensitivity analysis confirmed the exceptional performance of SVM, emphasizing its suitability for predicting fertilizer components using easily measured physical parameters. The developed ML models offer valuable insights for decision-makers managing irrigation and fertilization in mixed variable-rate fertigation.

Reliability enhancement of hybrid microgrid protection against communication data loss and converter faults using cubic-spline interpolation, Savitzky Golay filtering and GRU network

A primary challenge towards adoption of hybrid AC-DC microgrid pertains to the complexity of the protection scheme. Among other challenges, the high resemblance of voltage and current dynamics for converter faults and line faults, hinders use of threshold-based protection schemes. Further the high dependence of the protection mechanism on the communication network results in reduced reliability during data packet loss in the communication channel. In this regard, a protection scheme is proposed based on the combination of cubic spline interpolation, Savitzky-Golay (SG) filtering, maximum overlap discrete wavelet packet transform (MODWPT) and gated recurrent unit (GRU)-neural network. The impact of data packet loss (modelled using Markov chain model) in the communication channel has been minimized by reconstructing the original signal from lossy data using spline interpolation and SG filter. With the reconstructed signal, significant attributes are extracted using MODWPT, which are further utilized to perform fault detection, fault classification and faulty section identification using a set of GRU classifiers. Unlike the reported techniques whose fault detection and classification accuracy are severely compromised during packet loss, the signal reconstruction module in the proposed framework makes it immune to packet loss. The inclusion of a module for distinguishing between converter faults and line faults allows for avoiding relay maloperation during switching failures. The proposed scheme has been extensively validated for a wide range of operating and fault scenarios under converter faults, PV array faults, varying degrees of packet loss, burst length and dynamic loading conditions. For packet loss rate ranging from 5 % to 30 % and burst length from 0.0167 s to 0.1 s, the proposed scheme achieves dependability and security in excess of 99.85 % and 99.46 % respectively.

Classifying marine mammals signal using cubic splines interpolation combining with triple loss variational auto-encoder

In practical applications of passive sonar principles for extracting characteristic frequencies of acoustic signals, scientists typically employ traditional time-frequency domain transformation methods such as Mel-frequency, Short time Fourier transform (STFT), and Wavelet transform (WT). However, these solutions still face limitations in resolution and information loss when transforming data collected over extended periods. In this paper, we present a study using a two-stage approach that combines pre-processing by Cubic-splines interpolation (CSI) with a probability distribution in the hidden space with Siamese triple loss network model for classifying marine mammal (MM) communication signals. The Cubic-splines interpolation technique is tested with the STFT transformation to generate STFT-CSI spectrograms, which enforce stronger relationships between characteristic frequencies, enhancing the connectivity of spectrograms and highlighting frequency-based features. Additionally, stacking spectrograms generated by three consecutive methods, Mel, STFT-CSI, and Wavelet, into a feature spectrogram optimizes the advantages of each method across different frequency bands, resulting in a more effective classification process. The proposed solution using an Siamese Neural Network-Variational Auto Encoder (SNN-VAE) model also overcomes the drawbacks of the Auto-Encoder (AE) structure, including loss of discontinuity and loss of completeness during decoding. The classification accuracy of marine mammal signals using the SNN-VAE model increases by 11% and 20% compared to using the AE model (2013), and by 6% compared to using the Resnet model (2022) on the same actual dataset NOAA from the National Oceanic and Atmospheric Administration - United State of America.

Path Following of an Underwater Snake-like Robot Exposed to Ocean Currents and Locomotion Efficiency Optimization Based on Multi-Strategy Improved Sparrow Search Algorithm

In this paper, we propose a cubic spline interpolation method to generate a desired curve path and present an integral line of sight (ILOS) method and a control strategy for course tracking control based on nonsingular terminal sliding mode to enable an underwater snake-like robot (USR) to move towards and follow the path generated by the parametric cubic-spline interpolation (PCSI) path-planning method, while considering the disturbances caused by ocean currents. The efficiency of robot locomotion is an important evaluation criterion for robot design. Thus, we introduce a multi-strategy improved sparrow search algorithm (MISSA) to dynamically choose gait parameters that significantly enhance the efficiency of robot movement. We conduct simulations to demonstrate that the proposed controller enables the USR, subjected to ocean currents, to accurately converge towards and follow the target path. Our results also reveal that MISSA effectively enhances the locomotion efficiency of the robot.

Impact of resampling interpolation FIR filter in the practical Kramers-Kronig receiver.

The practical Kramers-Kronig (KK) receiver has been a competitive receiving technique in the data-center, medium reach, and even long-haul metropolitan networks. Nevertheless, an extra digital resampling operation is required at both ends of the KK field reconstruction algorithm due to the spectrum broadening caused by adopting the nonlinear function. Generally, the digital resampling function can be implemented by using linear interpolation (LI-ITP), the Lagrange cubic interpolation (LC-ITP), the spline cubic interpolation (SC-ITP), time-domain anti-aliasing finite impulse response (FIR) filter method (TD-FRM) scheme, and fast Fourier transform (FFT)-based scheme. However, the performance and the computational complexity analysis of different resampling interpolation schemes in the KK receiver have not been thoroughly investigated yet. Different from the interpolation schemes of conventional coherent detection, the interpolation function of the KK system is followed by the nonlinear operation, which will broaden the spectrum significantly. Due to the frequency-domain transfer function of different interpolation schemes, the broadened spectrum will have a potential spectrum aliasing, which will cause serious inter-symbol interference (ISI) and further impair the KK phase retrieval performance. We experimentally investigate the performance of different interpolation schemes under different digital up-sampling rates (i.e. the computational complexity) as well as the cut-off frequency, the tap number of the anti-aliasing filter, and the shape factor of the TD-FRM scheme in a 112-Gbit/s SSB DD 16-QAM system over 1920-km Raman amplification (RFA)-based standard single-mode fiber (SSMF). The experimental results involve that the TD-FRM scheme outperforms other interpolation schemes and the complexity is reduced by at least 49.6%. In fiber transmission results, take 20% soft decision-forward error correction (SD-FEC) of 2×10-2 as the threshold, the LI-ITP and LC-ITP schemes only reach 720-km while others can reach up to 1440-km.

Comparison of uniform resampling and nonuniform sampling direct-reconstruction methods in k-space for FD-OCT

The nonuniform distribution of interference spectrum in wavenumber k-space is a key issue to limit the imaging quality of Fourier-domain optical coherence tomography (FD-OCT). At present, the reconstruction quality at different depths among a variety of processing methods in k-space is still uncertain. Using simulated and experimental interference spectra at different depths, the effects of common six processing methods including uniform resampling (linear interpolation (LI), cubic spline interpolation (CSI), time-domain interpolation (TDI), and K-B window convolution) and nonuniform sampling direct-reconstruction (Lomb periodogram (LP) and nonuniform discrete Fourier transform (NDFT)) on the reconstruction quality of FD-OCT were quantitatively analyzed and compared in this work. The results obtained by using simulated and experimental data were coincident. From the experimental results, the averaged peak intensity, axial resolution, and signal-to-noise ratio (SNR) of NDFT at depth from 0.5 to 3.0[Formula: see text]mm were improved by about 1.9[Formula: see text]dB, 1.4 times, and 11.8[Formula: see text]dB, respectively, compared to the averaged indices of all the uniform resampling methods at all depths. Similarly, the improvements of the above three indices of LP were 2.0[Formula: see text]dB, 1.4 times, and 11.7[Formula: see text]dB, respectively. The analysis method and the results obtained in this work are helpful to select an appropriate processing method in k-space, so as to improve the imaging quality of FD-OCT.

Determine the Solution of Delay Differential Equations using Runge-Kutta Methods with Cubic-Spline Interpolation

This paper describes some iterations for term delay in Delay Differential Equation (DDE), which is causing a huge number of iteration calculations. Time-delay was approximated using Cubic-Spline Interpolation, so DDE can rewrite as Differential Equations. Then, Runge-Kutta methods have been used to determine the solution of Differential equations from DDE.

Process Modelling and Simulation of Key Volatile Compounds of Maillard Reaction Products Derived from Beef Tallow Residue Hydrolysate Based on Proxy Models.

The hydrolysis time is directly related to the flavor of the Maillard reaction, but existing proxy models cannot simulate and model the variation curves of vital volatile components. This study developed a predictive model for modelling and simulating key volatile compounds of Maillard reaction products (MRPs) derived from beef tallow residue hydrolysate. Results showed the degree of hydrolysis increased with hydrolysis time, and the most significant improvement in the roast flavor and overall acceptance was when hydrolyzing 4 h. Based on flavor dilution value and the relative odor activity value, nine key volatile components were identified, and 2-ethyl-3,5-dimethylpyrazine with roast flavor was the highest. Compared with Polynomial Curve Fitting (PCF) and Cubic Spline Interpolation (CSI), key volatile compounds of MRPs could be better modeled and simulated by the Curve Prediction Model (CPM). All results suggested that CPM could predict the changes in key volatile components produced by MRPs.

An improved compressed sensing method for dynamic impact signals based on cubic spline interpolation.

During the dynamic acquisition of impact signals, a high sampling frequency brings significant challenges to the analog-to-digital converter and other test systems. To address this issue, in this study, an improved compressed sensing (CS) method is proposed for the measurement of impact signals based on cubic spline interpolation (CSI). According to the characteristics of the dynamic impact signal, a random non-uniform sampling strategy combining CS and CSI is presented. The CSI obviously reduces the number of observation points required by the traditional CS. To resolve the problem that the traditional orthogonal matching pursuit (OMP) algorithm can only guarantee the local optimal solution but cannot obtain the global optimal solution, an improved orthogonal matching pursuit (IOMP) algorithm is proposed. First, n atoms related to residuals are selected to build a local atomic dictionary. Subsequently, the atom most relevant to the signal observation result is selected from the local atomic dictionary. The iteration process is repeated until enough atoms are selected. The IOMP algorithm effectively improves the success rate of reconstruction. Finally, an impact signals test platform based on the Machete hammer is established. The results of theoretical simulations and several experiments indicate that the data reconstruction error of the proposed improved CS method for impact signals is approximately 5.0%.

Application of the Trigonometric Polynomial Interpolation for the Estimation of the Vertical Eddy Viscosity Coefficient Based on the Ekman Adjoint Assimilation Model

In this study, a triangular polynomial interpolation (TPI) scheme was developed to estimate the vertical eddy viscosity coefficient (VEVC) on the basis of the Ekman model with adjoint assimilation. In the twin experiments, the advantages and disadvantages of estimating the VEVC using the TPI scheme under different factors are discussed. The results indicated that (1) the TPI scheme proves to be better than the cubic spline interpolation (CSI) and Cressman interpolation (CI) schemes; (2) the inversion results are more sensitive to observations from upper ocean layers than those from lower layers, and the TPI scheme is less likely to be influenced by missing data; (3) for various boundary layer depths, the inversion results of the TPI scheme remain consistent with the given distributions; (4) the inversion results can be influenced considerably by observational errors, and the TPI scheme is more resistant to noise than the CSI and CI schemes; and (5) the inversion accuracy of the TPI scheme can be improved by selecting the temporal wind stress drag coefficients. In practical experiments, the adjoint method with the TPI scheme was developed to estimate the Ekman currents by assimilating the observations from a buoy stationed in the Yellow Sea. The results showed the successful estimation of the VEVC and demonstrated that more precise current velocities can be obtained with this estimation scheme. In summary, this study provides a useful approach for the effective estimation of the VEVC.

Simultaneous Calibration Method for Doppler Velocity Log Errors Based on a Genetic Algorithm

Doppler velocity log (DVL) errors affect the accuracy of strapdown inertial-navigation system (SINS)/DVL integrated navigation. Most existing calibration methods do not consider the problem of DVL sampling frequencies that are not fixed, and they cannot cope with large misalignment angles. This study proposes a method to calibrate internal and external DVL errors simultaneously, based on a genetic algorithm. To solve the problem of asynchronous SINS and DVL sampling, a method for unifying the time reference using cubic-spline interpolation is proposed. An error model for the DVL is established; the scale factor and misalignment angles are used as optimization genes for a genetic algorithm. The method can achieve high accuracy without requiring the misalignment angles to be small, significantly reducing the practical difficulty of installation. The results of simulation and shipboard experiments indicate that the calibration accuracy of the proposed algorithm is significantly better than that of the traditional method. For example, when the misalignment angle reaches 15°, the proposed algorithm has only a small error. If the carrier pitch angle changes to 5°, the root-mean-square error is reduced by 31.15% in the x-direction and 69.15% in the z-direction under highly dynamic conditions. The method thus seems likely to be useful in real-world engineering settings.

Deep learning based missing data recovery of non-stationary wind velocity

Wireless sensor technology and wireless sensor network are the developing trend of structural health monitoring (SHM) system in the future. Due to working principles, environment and other various factors, random data missing may occur easily during data transmission by mobile wireless sensors. The randomness and fluctuation of non-stationary wind velocity is strong and complex, affected by turbulence effect. Regarding random missing of non-stationary wind velocity in data transmission, a missing data recovery algorithm based on secondary decomposition of signals, convolutional neural network (CNN) and Kriging based sequence interpolation (KSI) is proposed in this paper. Before data transmission, the non-stationary wind velocity is decomposed by wavelet transform (WT) and empirical wavelet transform (EWT) to obtain a set of sub-signals with uniform frequency. Then low-frequency sub-signals are supplemented by cubic spline interpolation (CSI). Due to the randomness and complexity of high-frequency sub-signals, CNN is used to establish the recovery model to achieve preliminary recovery results. Generally, there exists lots of extreme points in high-frequency sub-signals. The recovery error may be large due to strong randomness of extreme points. Therefore, KSI based extreme value estimation is proposed for high-frequency sub-signals to estimate extreme curves, which are subsequently used to correct the preliminary recovery results. Numerical examples of field monitoring downburst data testify the superiority of the proposed algorithm. In addition, structural dynamic analysis fully verifies the effectiveness of extreme value correction.

Information Retrieval from Photoplethysmographic Sensors: A Comprehensive Comparison of Practical Interpolation and Breath-Extraction Techniques at Different Sampling Rates.

The increasingly widespread diffusion of wearable devices makes possible the continuous monitoring of vital signs, such as heart rate (HR), heart rate variability (HRV), and breath signal. However, these devices usually do not record the “gold-standard” signals, namely the electrocardiography (ECG) and respiratory activity, but a single photoplethysmographic (PPG) signal, which can be exploited to estimate HR and respiratory activity. In addition, these devices employ low sampling rates to limit power consumption. Hence, proper methods should be adopted to compensate for the resulting increased discretization error, while diverse breath-extraction algorithms may be differently sensitive to PPG sampling rate. Here, we assessed the efficacy of parabola interpolation, cubic-spline, and linear regression methods to improve the accuracy of the inter-beat intervals (IBIs) extracted from PPG sampled at decreasing rates from 64 to 8 Hz. PPG-derived IBIs and HRV indices were compared with those extracted from a standard ECG. In addition, breath signals extracted from PPG using three different techniques were compared with the gold-standard signal from a thoracic belt. Signals were recorded from eight healthy volunteers during an experimental protocol comprising sitting and standing postures and a controlled respiration task. Parabola and cubic-spline interpolation significantly increased IBIs accuracy at 32, 16, and 8 Hz sampling rates. Concerning breath signal extraction, the method holding higher accuracy was based on PPG bandpass filtering. Our results support the efficacy of parabola and spline interpolations to improve the accuracy of the IBIs obtained from low-sampling rate PPG signals, and also indicate a robust method for breath signal extraction.

Path Tracking of an Underwater Snake Robot and Locomotion Efficiency Optimization Based on Improved Pigeon-Inspired Algorithm

This paper considers the tracking control of curved paths for an underwater snake robot, and investigates the methods used to improve energy efficiency. Combined with the path-planning method based on PCSI (parametric cubic-spline interpolation), an improved LOS (light of sight) method is proposed to design the controller and guide the robot to move along the desired path. The evaluation of the energy efficiency of robot locomotion is discussed. In particular, a pigeon-inspired optimization algorithm improved by quantum rules (QPIO) is proposed for dynamically selecting the gait parameters that maximize energy efficiency. Simulation results show that the proposed controller enables the robot to accurately follow the curved path and that the QPIO algorithm is effective in improving robot energy efficiency.

A Novel Classification Method with Cubic Spline Interpolation

Classification is the last, and usually the most time-consuming step in recognition. Most recently proposed classification algorithms have adopted machine learning (ML) as the main classification approach, regardless of time consumption. This study proposes a statistical feature classification cubic spline interpolation (FC-CSI) algorithm to classify emotions in speech using a curve fitting technique. FC-CSI is utilized in a speech emotion recognition system (SERS). The idea is to sketch the cubic spline interpolation (CSI) for each audio file in a dataset and the mean cubic spline interpolations (MCSIs) representing each emotion in the dataset. CSI interpolation is generated by connecting the features extracted from each file in the feature extraction phase. The MCSI is generated by connecting the mean features of 70% of the files of each emotion in the dataset. Points on the CSI are considered the new generated features. To classify each audio file according to emotion, the Euclidian distance (ED) is found between each CSI and all MCSIs of all emotions in the dataset. Each audio file is classified according to the nearest MCSI to the CSI representing it. The three datasets used in this work are Ryerson Audio-Visual Database of Emotional Speech and Song (RAVDESS), Berlin (Emo-DB), and Surrey Audio-Visual Expressed Emotion (SAVEE). The proposed work shows fast classification and high accuracy of results. The classification accuracy, i.e., the proportion of samples assigned to the correct class, using FC-CSI without feature selection (FS), was 69.08%, 92.52%, and 89.1% with RAVDESS, Emo-DB, and SAVEE, respectively. The results of the proposed method were compared to those of a designed neural network called SER-NN. Comparisons were made with and without FS. FC-CSI outperformed SER-NN on Emo-DB and SAVEE, and underperformed on RAVDESS, without using an FS algorithm. It was noticed from experiments that FC-CSI operated faster than the same system utilizing SER-NN.

A Cylindrical Equivalent Source-Based Physical Optics Method for Rapid Analysis of Airborne Radomes

The physical optics (PO) method treats the outer surface of airborne radomes in front of antennas as a secondary source region when analyzing the electromagnetic performance. The effective reduction of the secondary source region can significantly improve the computational efficiency, which is helpful for the rapid analysis and design of airborne radomes. In this paper, we present a cylindrical equivalent source-based PO method by introducing an antenna radiation cylinder that can cover the main near-field radiation characteristics of the antenna. The cubic-spline interpolation technique is used to standardize the solution method for the boundary of the cylindrical equivalent source of different antenna-radome systems. The results of a tangent-ogival radome verify the validity of the proposed method. Compared with the conical equivalent source-based PO method, the proposed method improves the efficiency by 71.83%. It can be applied to airborne antenna-radome systems with antenna diameters of 12.14 times of wavelength and above, using 10% as the error threshold.

Fault location of hybrid three-terminal HVDC transmission line based on improved LMD

To reduce the impact of traveling wave velocity, wave head recognition, and LMD (Local Mean Decomposition) end effect and mode mixing on location accuracy, this paper presents a fault location method based on improved LMD for hybrid three-terminal HVDC transmission lines, which is not affected by wave velocity. Firstly, the improved LMD is used to get the time when the fault traveling wave first arrives at the four measuring terminals. Then the fault branch is identified according to the ratio of the time of the fault traveling wave arriving at the measuring terminal. Finally, the equation is listed according to the principle of traveling wave location and the measurement equation which is not affected by the wave velocity is deduced. A hybrid three-terminal HVDC transmission system model is built by PSCAD to simulate various faults and output fault data under various fault conditions. MATLAB is used to process the fault data and verify the location algorithm. The simulation results show that the location method proposed in this paper is not affected by fault distances, fault types, infeed levels and transition resistances. In addition, it has higher accuracy than LMD based on CSI (Cubic Spline Interpolation) algorithm, wavelet packet location algorithm and double-ended location method of using wave velocity calculation.