Tensor Spline Research Articles

Zenith tropospheric delay model in mainland China based on geographically convolutional neural network weighted regression and tensor spline function

Zenith tropospheric delay model in mainland China based on geographically convolutional neural network weighted regression and tensor spline function

Martingale convergence theorems for tensor splines

Martingale convergence theorems for tensor splines

Spatial interpolation of regional PM2.5 concentrations in China during COVID-19 incorporating multivariate data

During specific periods when the PM2.5 variation pattern is unusual, such as during the coronavirus disease 2019 (COVID-19) outbreak, epidemic PM2.5 regional interpolation models have been relatively little investigated, and little consideration has been given to the residuals of optimized models and changes in model interpolation accuracy for the PM2.5 concentration under the influence of epidemic phenomena. Therefore, this paper mainly introduces four interpolation methods (kriging, empirical Bayesian kriging, tensor spline function and complete regular spline function), constructs geographically weighted regression (GWR) models of the PM2.5 concentration in Chinese regions for the periods from January–June 2019 and January–June 2020 by considering multiple factors, and optimizes the GWR regression residuals using these four interpolation methods, thus achieving the purpose of enhancing the model accuracy. The PM2.5 concentrations in many regions of China showed a downward trend during the same period before and after the COVID-19 outbreak. Atmospheric pollutants, meteorological factors, elevation, zenith wet delay (ZWD), normalized difference vegetation index (NDVI) and population maintained a certain relationship with the PM2.5 concentration in terms of linear spatial relationships, which could explain why the PM2.5 concentration changed to a certain extent. By evaluating the model accuracy from two perspectives, i.e., the overall interpolation effect and the validation set interpolation effect, the results showed that all four interpolation methods could improve the numerical accuracy of GWR to different degrees, among which the tensor spline function and the fully regular spline function achieved the most stable effect on the correction of GWR residuals, followed by kriging and empirical Bayesian kriging.

Spatial interpolation of PM2.5 concentrations during holidays in south-central China considering multiple factors

Investigations on PM2.5 regional interpolation models during specific periods with abnormal PM2.5 variation patterns (e.g., during holidays) remain relatively rare. In this paper, we first explored PM2.5 concentration patterns in major provincial capitals in south-central China during the Spring Festival and National Day periods. Then, we analysed the correlation between atmospheric pollutants, meteorological conditions, Zenith Tropospheric Delay (ZTD), elevation and Normalized Difference Vegetation Index (NDVI) with PM2.5 in south-central China during the Spring Festival and National Day periods from 2016 to 2018, followed by GeoDetector application to explore the spatial stratified heterogeneity. In addition, we developed Geographically Weighted Regression (GWR) model, Multiscale Geographically Weighted Regression (MGWR) model, and Geographically and Temporally Weighted Regression (GTWR) model. Then, we applied three interpolation methods (Kriging, Empirical Bayesian Kriging, and Tensor Spline Function) to further optimize the residuals of each regression class model. The results indicated that the model created by the Tensor Spline Function (TSF) interpolation method correcting for the residuals of the GWR model (GWR-TSF) attained the best fit, the GTWR model achieved the worst fit during the Spring Festival period, and the MGWR model attained the worst fit during the National Day period. In terms of the interpolation accuracy, the GWR-TSF model also achieved the highest interpolation accuracy, while the GTWR model attained the lowest interpolation accuracy.

Bayesian Estimation and Comparison of Conditional Moment Models

AbstractWe consider the Bayesian analysis of models in which the unknown distribution of the outcomes is specified up to a set of conditional moment restrictions. The non-parametric exponentially tilted empirical likelihood function is constructed to satisfy a sequence of unconditional moments based on an increasing (in sample size) vector of approximating functions (such as tensor splines based on the splines of each conditioning variable). For any given sample size, results are robust to the number of expanded moments. We derive Bernstein–von Mises theorems for the behaviour of the posterior distribution under both correct and incorrect specification of the conditional moments, subject to growth rate conditions (slower under misspecification) on the number of approximating functions. A large-sample theory for comparing different conditional moment models is also developed. The central result is that the marginal likelihood criterion selects the model that is less misspecified. We also introduce sparsity-based model search for high-dimensional conditioning variables, and provide efficient Markov chain Monte Carlo computations for high-dimensional parameters. Along with clarifying examples, the framework is illustrated with real data applications to risk-factor determination in finance, and causal inference under conditional ignorability.

Modelling temporal genetic and spatio‐temporal residual effects for high‐throughput phenotyping data*

SummaryHigh‐throughput phenomics data are being collected in both the laboratory and the field. The data are often collected at many time points and there may be spatial variation in the laboratory or field that impacts on the growth of the plants, and that may influence the traits of interest. Modelling the genetic effects is of primary interest in such studies, but these effects might be biased if non‐genetic effects present in the experiment are ignored. With data that are collected both in time and space, there may be a need to jointly model these multi‐dimensional non‐genetic effects. Thus both modelling of genetic effects over time and non‐genetic effects over time and space in a one‐stage analysis is considered. An experiment that involves field phenomics data with four dimensions, two in space and two in time, provides the vehicle to examine the models. Factor analytic (FA) models are often used for genetic effects for different environments to provide reliable estimates of genetic variances and correlations. As the time dimension defines the environments, FA models are examined for the phenomics data. Reduced rank tensor smoothing splines are presented as a possible approach for modelling the spatio‐temporal effects, although an additional term is included for heterogeneity over the two time dimensions. This approach is feasible, although very time‐consuming. The process of model selection for the genetic effects is presented including tests, information criteria and diagnostics. Comparisons of more simplistic models are made with the reduced rank tensor spline. This also shows the interplay between the genetic and residual models in model selection.

Influence of Drone Altitude, Image Overlap, and Optical Sensor Resolution on Multi-View Reconstruction of Forest Images

Recent technical advances in drones make them increasingly relevant and important tools for forest measurements. However, information on how to optimally set flight parameters and choose sensor resolution is lagging behind the technical developments. Our study aims to address this gap, exploring the effects of drone flight parameters (altitude, image overlap, and sensor resolution) on image reconstruction and successful 3D point extraction. This study was conducted using video footage obtained from flights at several altitudes, sampled for images at varying frequencies to obtain forward overlap ratios ranging between 91 and 99%. Artificial reduction of image resolution was used to simulate sensor resolutions between 0.3 and 8.3 Megapixels (Mpx). The resulting data matrix was analysed using commercial multi-view reconstruction (MVG) software to understand the effects of drone variables on (1) reconstruction detail and precision, (2) flight times of the drone, and (3) reconstruction times during data processing. The correlations between variables were statistically analysed with a multivariate generalised additive model (GAM), based on a tensor spline smoother to construct response surfaces. Flight time was linearly related to altitude, while processing time was mainly influenced by altitude and forward overlap, which in turn changed the number of images processed. Low flight altitudes yielded the highest reconstruction details and best precision, particularly in combination with high image overlaps. Interestingly, this effect was nonlinear and not directly related to increased sensor resolution at higher altitudes. We suggest that image geometry and high image frequency enable the MVG algorithm to identify more points on the silhouettes of tree crowns. Our results are some of the first estimates of reasonable value ranges for flight parameter selection for forestry applications.

Sparse transfer for facial shape-from-shading

We present an image-based 3D face shape reconstruction method which transfers shape cues inferred from source face images to guide the reconstruction of the target face. Specifically, a sparse face shape adaption mechanism is used to generate a target-specific reference shape by adaptively and selectively combining source face shapes. This reference shape can also facilitate the reconstruction optimization for the target shape. As an off-line process, each source shape has been derived from a set of given sufficient source images (more than 9) based on a non-Lambertian reflectance model. Such a process allows for the existence of cast shadow and specularity, and more accurately infers the source shape. Guided by the target-specific reference shape, the shape of a target face can be estimated using a small number of images (even only one). The proposed reconstruction method refers to a lighting estimation and an albedo estimation for the target face. No standard 3D shape (such as the high-precision scanned 3D face) is required in the reconstruction process. Compared to the state-of-the-arts including the Photometric Stereo, Tensor Spline, the single reference based method, and the GEM algorithm, the proposed sparse transfer model can produce visually better facial details and obtain smaller reconstruction errors.

Tensor Body: Real-Time Reconstruction of the Human Body and Avatar Synthesis From RGB-D

Real-time 3-D reconstruction of the human body has many applications in anthropometry, telecommunications, gaming, fashion, and other areas of human-computer interaction. In this paper, a novel framework is presented for reconstructing the 3-D model of the human body from a sequence of RGB-D frames. The reconstruction is performed in real time while the human subject moves arbitrarily in front of the camera. The method employs a novel parameterization of cylindrical-type objects using Cartesian tensor and b-spline bases along the radial and longitudinal dimension respectively. The proposed model, dubbed tensor body, is fitted to the input data using a multistep framework that involves segmentation of the different body regions, robust filtering of the data via a dynamic histogram, and energy-based optimization with positive-definite constraints. A Riemannian metric on the space of positive-definite tensor splines is analytically defined and employed in this framework. The efficacy of the presented methods is demonstrated in several real-data experiments using the Microsoft Kinect sensor.

Flexible Copula Density Estimation with Penalized Hierarchical B‐splines

ABSTRACTThe paper introduces a new method for flexible spline fitting for copula density estimation. Spline coefficients are penalized to achieve a smooth fit. To weaken the curse of dimensionality, instead of a full tensor spline basis, a reduced tensor product based on so called sparse grids (Notes Numer. Fluid Mech. Multidiscip. Des., 31, 1991, 241‐251) is used. To achieve uniform margins of the copula density, linear constraints are placed on the spline coefficients, and quadratic programming is used to fit the model. Simulations and practical examples accompany the presentation.

Tensor Spline Approximation in Economic Dynamics with Uncertainties

Modern economic theory views the economy as a dynamical system in which rational decisions are made in the face of uncertainties. Optimizing decisions over time on market behavior such as consumption, investment, labor supply, and technology innovation is of practical importance. Interpreting all market behavior in a broad sense, the problem finds further applications in many areas other than economics. Finding the policy function inherent in the associated Euler equation has been an important but challenging task. This note proposes using composite 1-dimensional cubic splines in tensor form to process the Newton iterative scheme on approximating the unknown policy functions. This tensor spline approach has the advantages of freedom in the node collocation, simplicity in the derivative calculation, fast convergence, and high precision over the conventional projection methods. Applications to the neoclassical growth model with leisure choice are used to demonstrate the working of the idea. In particular, tensor products are employed throughout to simplify and effectuate the operations.

Non-Lambertian Reflectance Modeling and Shape Recovery of Faces Using Tensor Splines

Modeling illumination effects and pose variations of a face is of fundamental importance in the field of facial image analysis. Most of the conventional techniques that simultaneously address both of these problems work with the Lambertian assumption and thus fall short of accurately capturing the complex intensity variation that the facial images exhibit or recovering their 3D shape in the presence of specularities and cast shadows. In this paper, we present a novel Tensor-Spline-based framework for facial image analysis. We show that, using this framework, the facial apparent BRDF field can be accurately estimated while seamlessly accounting for cast shadows and specularities. Further, using local neighborhood information, the same framework can be exploited to recover the 3D shape of the face (to handle pose variation). We quantitatively validate the accuracy of the Tensor Spline model using a more general model based on the mixture of single-lobed spherical functions. We demonstrate the effectiveness of our technique by presenting extensive experimental results for face relighting, 3D shape recovery, and face recognition using the Extended Yale B and CMU PIE benchmark data sets.

Partially monotone tensor spline estimation of the joint distribution function with bivariate current status data

The analysis of the joint cumulative distribution function (CDF) with bivariate event time data is a challenging problem both theoretically and numerically. This paper develops a tensor spline-based sieve maximum likelihood estimation method to estimate the joint CDF with bivariate current status data. The I-splines are used to approximate the joint CDF in order to simplify the numerical computation of a constrained maximum likelihood estimation problem. The generalized gradient projection algorithm is used to compute the constrained optimization problem. Based on the properties of B-spline basis functions it is shown that the proposed tensor spline-based nonparametric sieve maximum likelihood estimator is consistent with a rate of convergence potentially better than $n^{1/3}$ under some mild regularity conditions. The simulation studies with moderate sample sizes are carried out to demonstrate that the finite sample performance of the proposed estimator is generally satisfactory.

Optimal dynamic production from a large oil field in Saudi Arabia

We model the economically optimal dynamic oil production decisions for a stylized oilfield resembling the largest developed field in Saudi Arabia, Ghawar, paying particular attention to the engineering aspects of oil production. Specifically, we begin with a fluid dynamics model composed of differential equations describing the dynamics of fluid flow as a function of fluid pressure, formation characteristics, water injection, new wells, and how these parameters change as oil extraction occurs. We then link this physical description of the field to an intertemporal optimizing economic model. The cost and revenue functions are based on data from a number of sources. We use tensor splines to approximate the value function. The optimal solution depends on exogenous variables, such as the discount rate or the timing of breakthroughs in the cost of alternative energy sources, which are uncertain. We examine solutions under a number of scenarios to account for these uncertainties.

Tensor Splines for Interpolation and Approximation of DT-MRI With Applications to Segmentation of Isolated Rat Hippocampi

In this paper, we present novel algorithms for statistically robust interpolation and approximation of diffusion tensors-which are symmetric positive definite (SPD) matrices-and use them in developing a significant extension to an existing probabilistic algorithm for scalar field segmentation, in order to segment diffusion tensor magnetic resonance imaging (DT-MRI) datasets. Using the Riemannian metric on the space of SPD matrices, we present a novel and robust higher order (cubic) continuous tensor product of B-splines algorithm to approximate the SPD diffusion tensor fields. The resulting approximations are appropriately dubbed tensor splines. Next, we segment the diffusion tensor field by jointly estimating the label (assigned to each voxel) field, which is modeled by a Gauss Markov measure field (GMMF) and the parameters of each smooth tensor spline model representing the labeled regions. Results of interpolation, approximation, and segmentation are presented for synthetic data and real diffusion tensor fields from an isolated rat hippocampus, along with validation. We also present comparisons of our algorithms with existing methods and show significantly improved results in the presence of noise as well as outliers.

Robust Tensor Splines for Approximation of Diffusion Tensor MRI Data.

In this paper, we present a novel and robust spline approximation algorithm given a noisy symmetric positive definite (SPD) tensor field. Such tensor fields commonly arise in the field of Medical Imaging in the form of Diffusion Tensor (DT) MRI data sets. We develop a statistically robust algorithm for constructing a tensor product of B-splines - for approximating and interpolating these data - using the Riemannian metric of the manifold of SPD tensors. Our method involves a two step procedure wherein the first step uses Riemannian distances in order to evaluate a tensor spline by computing a weighted intrinsic average of diffusion tensors and the second step involves minimization of the Riemannian distance between the evaluated spline curve and the given data. These two steps are alternated to achieve the desired tensor spline approximation to the given tensor field. We present comparisons of our algorithm with four existing methods of tensor interpolation applied to DT-MRI data from fixed heart slices of a rabbit, and show significantly improved results in the presence of noise and outliers. We also present validation results for our algorithm using synthetically generated noisy tensor field data with outliers. This interpolation work has many applications e.g., in DT-MRI registration, in DT-MRI Atlas construction etc. This research was in part funded by the NIH ROI NS42075 and the Department of Radiology, University of Florida.

A Multiresolution Tensor Spline Method for Fitting Functions on the Sphere

We present the details of a multiresolution method which we proposed at the Taormina Wavelet Conference in 1993 (see L-spline wavelets in Wavelets: Theory, Algorithms, and Applications, C. Chui, L. Montefusco, and L. Puccio, eds., Academic Press, New York, pp. 197--212) which is suitable for fitting functions or data on the sphere. The method is based on tensor products of polynomial splines and trigonometric splines and can be adapted to produce surfaces that are either tangent plane continuous or almost tangent plane continuous. The result is a convenient compression algorithm for dealing with large amounts of data on the sphere. We give full details of a computer implementation that is highly efficient with respect to both storage and computational cost. We also demonstrate the performance of the method on several test examples.

Faster Adaptive Importance Sampling in Low Dimensions

Adaptive importance sampling using kernel density estimation techniques was introduced by West. This technique adapts the importance sampling function to the underlying integrand, thus yielding small-variance estimates. One drawback of this approach is that evaluation of the kernel mixture density is slow. We present a linear tensor spline representation of the adaptive importance function using variable bandwidth kernels that retains the small variance properties of West's approach but executes more quickly.

A philosophy for scientific computing tools

We argue that subroutine libraries are not enough but need to be supplemented by common file formats and directly executable commands. A concrete example, tensor spline primitives, illustrates this style of programming.

Modeling MOS VLSI circuits for transient analysis

Modeling plays a significant role in the efficient simulation of VLSI circuits. By simplifying the models used to analyze these circuits, it is possible to perform transient analyses with reasonable accuracy at speeds of one or two orders of magnitude faster than in conventional circuit simulation programs. The author discusses the models that are used in the second-generation MOTIS timing simulator. The methods used have been applied to a wide variety of MOS digital integrated circuits. All MOS transistors are modeled as voltage-controlled current sources using multidimensional tables. The actual currents are computed by approximation using variation-diminishing tensor splines. Nonlinear device capacitances in the circuit are approximated using linear models which are derived from experimental simulations using a circuit simulator. At the subcircuit level, special structures in the circuit are identified automatically by a preprocessor and are modeled using macro-models. Driver-load MOS transistor gates and bootstrapped circuits are examples of these structures. Their modeling is achieved by an experimental process before implementation in the preprocessor. The simplifications in the device and circuit models presented here have provided a significant improvement in the speed of transient analysis for large MOS digital circuits with relatively little loss in accuracy. This has resulted in a viable design verification environment using MOTIS.