Simultaneous Interpolation Research Articles

Data-driven constitutive meta-modeling of nonlinear rheology via multifidelity neural networks

Predicting the response of complex fluids to different flow conditions has been the focal point of rheology and is generally done via constitutive relations. There are, nonetheless, scenarios in which not much is known from the material mathematically, while data collection from samples is elusive, resource-intensive, or both. In such cases, meta-modeling of observables using a parametric surrogate model called multi-fidelity neural networks (MFNNs) may obviate the constitutive equation development step by leveraging only a handful of high-fidelity (Hi-Fi) data collected from experiments (or high-resolution simulations) and an abundance of low-fidelity (Lo-Fi) data generated synthetically to compensate for Hi-Fi data scarcity. To this end, MFNNs are employed to meta-model the material responses of a thermo-viscoelastic (TVE) fluid, consumer product Johnson’s® Baby Shampoo, under four flow protocols: steady shear, step growth, oscillatory, and small/large amplitude oscillatory shear (S/LAOS). In addition, the time–temperature superposition (TTS) of the material response and MFNN predictions are explored. By applying simple linear regression (without induction of any constitutive equation) on log-spaced Hi-Fi data, a series of Lo-Fi data were generated and found sufficient to obtain accurate material response recovery in terms of either interpolation or extrapolation for all flow protocols except for S/LAOS. This insufficiency is resolved by informing the MFNN platform with a linear constitutive model (Maxwell viscoelastic) resulting in simultaneous interpolation and extrapolation capabilities in S/LAOS material response recovery. The roles of data volume, flow type, and deformation range are discussed in detail, providing a practical pathway to multifidelity meta-modeling of different complex fluids.

Full Hermite Interpolation and Approximation in Topological Fields

By using generalized divided differences, we study the simultaneous interpolation of an m times continuously differentiable function and its derivatives up to a fixed order in a topological field K. If K is a valued field, then simultaneous Hermite interpolation and approximation are considered. Newton interpolating series are used in the case of an infinite number of conditions of interpolation. Applications to the numerical approximation of variational problems, the solution of a functional equation and, in the case of p-adic fields, the representation of solutions of a boundary value problem for an equation of the Fuchsian type illustrate the efficiency of the theoretical results.

Simultaneous Seismic Data Interpolation and Denoising Based on Nonsubsampled Contourlet Transform Integrating With Two-Step Iterative Log Thresholding Algorithm

Seismic data interpolation and denoising play vital roles in obtaining complete and clean data in seismic data processing. Seismic data usually misses along various spatial axes and always mix with random noise. In order to obtain complete and clean seismic data, reconstruction technology can interpolate missing data and attenuate random noise. Nonsubsampled contourlet transform is an effective transform to obtain multi-scale and multi-direction sparse domain data for compression sensing interpolation and denoising. However, conventional iterative shrinkage/thresholding cannot handle ill-posed and ill-conditioned equations for solving linear inverse problem. We present a two-step iterative log thresholding method to overcome ill-posed and ill-conditioned problems and improve the convergence rate and solution accuracy, which can interpolate and denoise seismic data simultaneously in the nonsubsampled contourlet transform framework. First, we use nonsubsampled contourlet transform to convert the seismic missing data with random noise to sparse domain. Then, we apply two-step iterative log thresholding algorithm to interpolate and denoise data in sparse domain. The result of each iteration is based on the results of the previous two iterations, which can accelerate convergence rate. In addition, log thresholding can further improve convergence rate and solution accuracy. Finally, we use inverse nonsubsampled contourlet transform to obtain the interpolated and denoised seismic data. The new method can reconstruct the irregularly missing data and attenuate random noise to obtain complete and clean seismic data with high accuracy, which is crucial for seismic imaging and inversion. We demonstrate the applicability and effectiveness of this simultaneous interpolation and denoising technique with successful applications to both synthetic and field data examples.

Simultaneous Interpolation and Deblending of 3-D Seismic Data by Iterative Thresholding

Blended acquisition can improve the efficiency of seismic data acquisition sharply, thus decreasing the acquisition costs. However, there may exist irregularity in blended data, which poses challenges for traditional seismic data processing. Thus, the deblending or interpolation should be done as a prerequisite. Because the deblending and interpolation can affect each other, we propose a unified 3-D joint deblending and interpolation method with a sparsifying transform and a sparsity promotion strategy in the blending fold of 4. To overcome the huge computational cost of 3-D curvelet transform (CT), we implement iterative thresholding in each principal frequency slice efficiently with 2-D CT. With decades of iterations, we can obtain an estimate of the regularized and deblended data. The core idea is that unblended signal can be characterized sparsely, and the blending noise and sampling irregularity have low-amplitude values in the CT domain. If blending was not used during acquisition, the proposed method becomes a pure interpolation algorithm. Similarly, if no sampling irregularity is present in the recorded data, the proposed method turns into a traditional deblending method. Numerical examples on the 3-D synthetic and field artificially blended data demonstrate the validity of the proposed method in simultaneously removing the blending noise and sampling irregularity. The regularized, deblended data can be beneficial for subsequent seismic data inversion and migration procedures.

Simultaneous interpolation and denoising based on a modified thresholding method

Seismic interpolation can provide complete data for some multichannel processing techniques such as time lapse imaging and wave equation migration. However, field seismic data often contains random noise and noisy data interpolation is a challenging task. A traditional method applies interpolation and denoising separately, but this needs two workflows. Simultaneous interpolation and denoising combines interpolation and denoising in one workflow and can also get acceptable results. Most existing interpolation methods can only recover missing traces but fail to attenuate noise in sampled traces. In this study, a novel thresholding strategy is proposed to remove the noise in the sampled traces and meanwhile recover missing traces during interpolation. For each iteration, the residual is multiplied by a weighting factor and then added to the iterative solution, after which the sum in the transformed domain is calculated using the thresholding operation to update the iterative solution. To ensure that the interpolation and denoising results are robust, the exponential method was chosen to reduce the threshold values in small quantities. The curvelet transform was used as sparse representation and three interpolation methods were chosen as benchmarks. Three numerical tests results proved the effectiveness of the proposed method on removing noise in the sampled traces when the minimum threshold values are correctly chosen.

Settings Interpolation and approximation from sublattices of C₀(X; R)

In this paper, we give a proof of a result concerning simultaneous interpolation and approximation from sublattices of the space of real continuous functions vanishing at infinity.

Interpolation and Approximation of Continuous Functions with Values in the Unit Interval

We give applications of a Stone-Weierstrass type theorem concerning uniform density of certain subsets with property V in and establish a simultaneous interpolation and approximation result in when X is a compact Hausdorff space.

Interpolation and denoising of seismic signals using orthogonal matching pursuit algorithm: An aplication in VSP and refraction data

An implementation of the Orthogonal Matching Pursuit (OMP) algorithm was used and the results obtained therefrom are presented for simultaneous interpolation and denoising from seismic signals in the framework of sparse signal representation. OMP is an algorithm for sparse signal representation based on orthogonal projections underlying the signal over an over-complete dictionary. This over-complete dictionary was designed using K-times Singular Values Decomposition (K-SVD). In each iteration, OMP calculates a new signal approximation and the approximation error is used in the next iteration to determine the new element. The new element corresponds to the largest magnitude of the inner products between the current residual and the original elements in the dictionary. The implemented algorithm was applied to VSP seismic data and refraction seismic data; results for the application in restored missing traces and denoise signals are presented.

Triple parametric tool path interpolation for five-axis machining with three-dimensional cutter compensation

Parametric interpolation obtains a great success in three-axis surface machining with smooth motion, high accuracy, and high machining efficiency, but does not go well in five-axis surface machining due to lack of appropriate and efficient methods of tool path generation, interpolation, and three-dimensional cutter compensation. This article proposes a triple parametric tool path interpolation method for five-axis machining with three-dimensional cutter compensation, which proposes an appropriate triple parametric tool generation method for realizing the three-dimensional cutter compensation in five-axis parametric interpolation. A triple parametric interpolation algorithm is also proposed to realizing the simultaneous interpolation of the source data, which ensures the primitivity and maintains the accuracy. The proposed three-dimensional cutter compensation can compensate the errors caused by minor changes in cutter size, thus machining accuracy can be improved. Finally, illustrated example verifies the feasibility and applicability of the proposed methods.

Simultaneous seismic interpolation and denoising based on sparse inversion with a 3D low redundancy curvelet transform

The simultaneous seismic interpolation and denoising problem can be solved as a sparse inversion problem by using the sparseness of seismic data in a transformed domain as the a priori information, where the properties of the sparse transform will significantly influence the numerical results and computational efficiency. Curvelet transform has nearly optimal sparse expression for seismic data, thus seismic signal processing based on this transform tends to result in preferable results. However, the redundancy of this transform can be 24–32 for three dimensional data which is not computationally cost efficient. This paper introduces a low redundancy curvelet transform to simultaneously interpolate and denoise. The redundancy of the proposed transform can be reduced to 10 for three dimensional data, and this property will improve the computational efficiency of the curvelet transform-based signal processing. The iterative soft thresholding method was chosen to solve the sparse inversion problem. Some practical principles on how to choose the regularisation parameters are discussed, due to the crucial nature of the regularisation parameter for simultaneous interpolation and denoising. Numerical experiments on synthetic and field data demonstrate that the low redundancy transform can provide reliable results while at the same time improving the computational efficiency.

A Hybrid Parallel Spatial Interpolation Algorithm for Massive LiDAR Point Clouds on Heterogeneous CPU-GPU Systems

Nowadays, heterogeneous CPU-GPU systems have become ubiquitous, but current parallel spatial interpolation (SI) algorithms exploit only one type of processing unit, and thus result in a waste of parallel resources. To address this problem, a hybrid parallel SI algorithm based on a thin plate spline is proposed to integrate both the CPU and GPU to further accelerate the processing of massive LiDAR point clouds. A simple yet powerful parallel framework is designed to enable simultaneous CPU-GPU interpolation, and a fast online training method is then presented to estimate the optimal decomposition granularity so that both types of processing units can run at maximum speed. Based on the optimal granularity, massive point clouds are continuously partitioned into a collection of discrete blocks in a data processing flow. A heterogeneous dynamic scheduler based on the greedy policy is also proposed to achieve better workload balancing. Experimental results demonstrate that the computing power of the CPU and GPU is fully utilized under conditions of optimal granularity, and the hybrid parallel SI algorithm achieves a significant performance boost when compared with the CPU-only and GPU-only algorithms. For example, the hybrid algorithm achieved a speedup of 20.2 on one of the experimental point clouds, while the corresponding speedups of using a CPU or a GPU alone were 8.7 and 12.6, respectively. The interpolation time was reduced by about 12% when using the proposed scheduler, in comparison with other common scheduling strategies.

Joint seismic data denoising and interpolation with double-sparsity dictionary learning

Seismic data quality is vital to geophysical applications, so methods of data recovery, including denoising and interpolation, are common initial steps in the seismic data processing flow. We present a method to perform simultaneous interpolation and denoising, which is based on double-sparsity dictionary learning. This extends previous work that was for denoising only. The original double sparsity dictionary learning algorithm is modified to track the traces with missing data by defining a masking operator that is integrated into the sparse representation of the dictionary. A weighted low-rank approximation algorithm is adopted to handle the dictionary updating as a sparse recovery optimization problem constrained by the masking operator. Compared to traditional sparse transforms with fixed dictionaries that lack the ability to adapt to complex data structures, the double-sparsity dictionary learning method learns the signal adaptively from selected patches of the corrupted seismic data while preserving compact forward and inverse transform operators. Numerical experiments on synthetic seismic data indicate that this new method preserves more subtle features in the dataset without introducing pseudo-Gibbs artifacts when compared to other directional multiscale transform methods such as curvelets.

An Improved Weighted Projection Onto Convex Sets Method for Seismic Data Interpolation and Denoising

Due to the environment effects, economy restrictions, and acquisition equipment limitations, observed seismic data always have several traces missing and contain some random noise, affecting the performance of surface-related multiple elimination (SRME), wave-equation-based imaging, and inversion. Projection onto convex sets (POCS) is an effective interpolation algorithm, while the performance is unsatisfactory in noisy situations. Weighted POCS (WPOCS) method can weaken the random noise effects to some extent, but the performance is still unsatisfactory. Thus, an improved WPOCS (IWPOCS) method is proposed in this paper, for seismic data interpolation and denoising simultaneously based on Curvelet transform. First, the POCS formula is derived from the iterative hard threshold (IHT) view. Then, its shortcoming is analyzed because there is an implicit assumption that the observed seismic data should have a high signal-to-noise ratio (SNR). Finally, a novel method named IWPOCS is proposed based on WPOCS method, which can achieve simultaneous interpolation and denoising. Among the above three methods, the IWPOCS method is the most effective to interpolate and denoise seismic data in terms of recovered SNR and visual view. Numerical experiments on the synthetic data and the real seismic data from the marine acquisition with towed streamers confirm the validity of the proposed IWPOCS method.

A highly-efficient technique for evaluating bond-orientational order parameters

We propose a novel, highly-efficient approach for the evaluation of bond-orientational order parameters (BOPs). Our approach exploits the properties of spherical harmonics and Wigner 3j-symbols to reduce the number of terms in the expressions for BOPs, and employs simultaneous interpolation of normalised associated Legendre polynomials and trigonometric functions to dramatically reduce the total number of arithmetic operations. Using realistic test cases, we show how the above, combined with a CPU-cache-friendly data structure, leads to a 10 to 50-fold performance increase over approaches currently in use, depending on the size of the interpolation grids and the machine used. As the proposed approach is an approximation, we demonstrate that the errors it introduces are well-behaved, controllable and essentially negligible for practical grid sizes. We benchmark our approach against other structure identification methods (centro-symmetry parameter (CSP), common neighbour analysis (CNA), common neighbourhood parameter (CNP) and Voronoi analysis), generally regarded as much faster than BOPs, and demonstrate that our formulation is able to outperform them for all studied systems.

AN IMPROVED PROJECTION ONTO CONVEX SETS METHOD FOR SIMULTANEOUS INTERPOLATION AND DENOISING

AbstractSparsity‐promoted seismic interpolation is an important interpolation strategy; however, most existing interpolation methods merely focused on high signal‐to‐noise ratio data. Actually, noise is always contained in sampled data; the existence of noise will deteriorate interpolation results and makes interpolation challenging. Projection onto convex sets (POCS) method is an efficient method; however, it cannot get acceptable results for noisy data interpolation. A strategy for simultaneous interpolation and denoising is the weighted POCS method which uses an additional weighting factor to achieve denoising. In this paper, we demonstrate that the weighted projection onto convex sets (POCS) method is a method for unconstrained optimization model, and it can be deduced from the iterative thresholding method. In addition, an improved POCS method is proposed for interpolation and denoising without extra computation effort. Numerical examples on synthetic and field data demonstrate that the improved POCS method can get reliable results for noiseless and noisy data interpolation. First interpolation and then denoising examples proved that simultaneous interpolation and denoising is robust when the least thresholds are properly chosen.

5D seismic data completion and denoising using a novel class of tensor decompositions

We have developed a novel strategy for simultaneous interpolation and denoising of prestack seismic data. Most seismic surveys fail to cover all possible source-receiver combinations, leading to missing data especially in the midpoint-offset domain. This undersampling can complicate certain data processing steps such as amplitude-variation-with-offset analysis and migration. Data interpolation can mitigate the impact of missing traces. We considered the prestack data as a 5D multidimensional array or otherwise referred to as a 5D tensor. Using synthetic data sets, we first found that prestack data can be well approximated by a low-rank tensor under a recently proposed framework for tensor singular value decomposition (tSVD). Under this low-rank assumption, we proposed a complexity-penalized algorithm for the recovery of missing traces and data denoising. In this algorithm, the complexity regularization was controlled by tuning a single regularization parameter using a statistical test. We tested the performance of the proposed algorithm on synthetic and real data to show that missing data can be reliably recovered under heavy downsampling. In addition, we demonstrated that compressibility, i.e., approximation of the data by a low-rank tensor, of seismic data under tSVD depended on the velocity model complexity and shot and receiver spacing. We further found that compressibility correlated with the recovery of missing data because high compressibility implied good recovery and vice versa.

The Simultaneous Interpolation of Target Radar Cross Section in Both the Spatial and Frequency Domains by Means of Legendre Wavelets Model-Based Parameter Estimation

The understanding of the target radar cross section (RCS) is significant for target identification and for radar designing and optimization. In this paper, a numerical algorithm for calculating target RCS is presented which is based on Legendre wavelet model-based parameter estimation (LW-MBPE). The Padé rational function fitting model applied for MBPE in the frequency domain is enhanced to include spatial dependence on the numerator and denominator coefficients. This allows the function to interpolate target RCS in both the frequency and spatial domains simultaneously. The combination of Legendre wavelets guarantees the convergence of the algorithm. The method is convergent by increasing the sampling frequency and spatial points. Numerical results are provided to demonstrate the validity and applicability of the new technique.

S1310203 R-testを用いた複合加工機の補間運動精度に関する研究

Multi-tasking machine tools are rapidly expanding because they can do both turning and milling processes. There is no accuracy test standard because multi-tasking machine tool is not defined exactly. However, it can be tested by referring to the accuracy test standard for turning centers and the newly revised one for 5-axis machining centers. In this research, three axes simultaneous interpolation motion measurement with R-test was performed and the motion errors for the multi-tasking machine were evaluated. Moreover, measurement with two-dimensional scale was also performed. The result was revealed much more information at once by performing data processing similar to the R-test.

Stark broadening of the He I 492.2 nm line with forbidden components in dense low-temperature plasma

We present the results of an experimental study of the He I 492.2nm line with forbidden components emitted by low-electron temperature and high electron number density helium plasma with or without addition of hydrogen. The shapes of the He I 447.1nm and 501.6nm lines and hydrogen Hβ line were used to measure electron number density in the range (1–5)×1023m−3, while electron temperatures in the range from 25,000K to 35,000K were determined from the line to underlying continuum ratio of He I lines at 388.8, 667.8, 706.5 and 728.1nm. The experimental profiles of the He I lines are compared with the results of a computer simulation method. The comparison was performed by use of a program for simultaneous interpolation of data tables obtained by computer simulation for 492.2 and 501.6nm He I and hydrogen Hβ lines. The electron number densities obtained from this comparison agree within 10% with corresponding values determined from the 447.1nm He I line which is well within theoretical and experimental uncertainties for all lines included in comparison.

Simultaneous image interpolation for stereo images

Simultaneous interpolation of stereo images aims to use a pair of stereo images to reconstruct another pair of images with higher resolutions. To tackle this problem, a simultaneous approach is proposed in this paper. Since the prior image model is important for solving the ill-posed numerical issues encountered in the image interpolation computation, the proposed approach exploits a prior image model that considers both the spatial local smoothness constraint within each reconstructed high-resolution image and the disparity-compensated local smoothness constraint between the pair of reconstructed high-resolution images. The proposed approach requires the disparity to be known in advance or has been accurately estimated by the existing stereo matching algorithm. Experiments are conducted using both artificially generated images and real-life images to demonstrate the superior performance of the proposed approach.