Trivariate Splines Research Articles

Efficient Nonparametric Estimation of 3D Point Cloud Signals through Distributed Learning

Advancements in technology have elevated the prominence of 3D point cloud data, making its analysis increasingly vital across various applications. This need drives the demand for advanced statistical analytic approaches to handle challenges such as size, sparsity, and irregularity in 3D point clouds while ensuring accurate and efficient information extraction. This paper introduces a novel nonparametric distributed (NPD) learning framework that utilizes trivariate spline smoothing over a triangulation of domain. The proposed NPD algorithm features a straightforward, scalable, and communication-efficient implementation scheme that can achieve near-linear speedup. In addition, we provide rigorous theoretical support for the NPD estimation framework and demonstrate that the NPD spline estimators attain the same convergence rate as the global spline estimators obtained using the entire dataset and achieve the optimal nonparametric convergence rate established by Stone (1982) under some regularity conditions. To evaluate the efficacy of the proposed NPD method, we conduct simulation studies comparing it with several global nonparametric estimation methods used to smooth the 3D data. The results demonstrate the superior performance of the NPD method in accurately and efficiently processing and learning from 3D point clouds, highlighting its potential to advance large and complex data analysis. Supplementary material, which contains related technical details, proofs of the theoretical results, and additional results in simulation studies, is available online.

A lower bound for the dimension of tetrahedral splines in large degree

Splines are piecewise polynomial functions which are continuously differentiable to some order r. For a fixed integer d the space of splines of degree at most d is a finite dimensional vector space, and a largely open problem in numerical analysis is to determine its dimension. While considerable attention has been given to this problem in the bivariate setting, the literature on trivariate splines is less conclusive. In particular, the dimension of generic trivariate splines is not known even in large degree when r>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r>1$$\\end{document}. In this paper we use a bound we previously derived for splines on vertex stars to compute a new lower bound on the dimension of trivariate splines in large enough degree. We illustrate in several examples that our formula gives the exact dimension of the spline space in large enough degree if vertex positions are generic. In contrast, for splines continuously differentiable of order r>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r>1$$\\end{document}, every lower bound in the literature diverges (often significantly) in large degree from the dimension of the spline space in these examples. We derive the bound using commutative and homological algebra.

Trivariate Spline Collocation Methods for Numerical Solution to 3D Monge-Ampère Equation

Trivariate Spline Collocation Methods for Numerical Solution to 3D Monge-Ampère Equation

A Lower Bound for Splines on Tetrahedral Vertex Stars

A tetrahedral complex all of whose tetrahedra meet at a common vertex is called a vertex star. Vertex stars are a natural generalization of planar triangulations, and understanding splines on vertex stars is a crucial step to analyzing trivariate splines. It is particularly difficult to compute the dimension of splines on vertex stars in which the vertex is completely surrounded by tetrahedra---we call these closed vertex stars. A formula due to Alfeld, Neamtu, and Schumaker gives the dimension of $C^r$ splines on closed vertex stars of degree at least $3r+2$. We show that this formula is a lower bound on the dimension of $C^r$ splines of degree at least $(3r+2)/2$. Our proof uses apolarity and the so-called Waldschmidt constant of the set of points dual to the interior faces of the vertex star. We furthermore observe that arguments of Alfeld, Schumaker, and Whiteley imply that the only splines of degree at most $(3r+1)/2$ on a generic closed vertex star are global polynomials.

A trivariate near-best blending quadratic quasi-interpolant

In this paper, we construct a new trivariate spline quasi-interpolation operator. It is expressed as blending sum of univariate and bivariate C1 quadratic spline quasi-interpolants and it is of near-best type, i.e. it has a small infinity norm and the coefficients functionals defining it are determined by minimizing an upper bound of the operator infinity norm, derived from the Bernstein-Bézier coefficients of its Lebesgue function.

Efficient slicing of Catmull–Clark solids for 3D printed objects with functionally graded material

In the competition for the volumetric representation most suitable for functionally graded materials in additively manufactured (AM) objects, volumetric subdivision schemes, such as Catmull–Clark (CC) solids, are widely neglected. Although they show appealing properties, efficient implementations of some fundamental algorithms are still missing. In this paper, we present a fast algorithm for direct slicing of CC-solids generating bitmaps printable by multi-material AM machines. Our method optimizes runtime by exploiting constant time limit evaluation and other structural characteristics of CC-solids. We compare our algorithm with the state of the art in trivariate trimmed spline representations and show that our algorithm has similar runtime behavior as slicing trivariate splines, fully supporting the benefits of CC-solids.

Flexible identification procedure for thermodynamic constitutive models for magnetostrictive materials.

We present a novel approach for identifying a multiaxial thermodynamic magneto-mechanical constitutive law by direct bi- or trivariate spline interpolation from available magnetization and magnetostriction data. Reference data are first produced with a multiscale model in the case of a magnetic field and uniaxial and shear stresses. The thermodynamic model fits well to the results of the multiscale model, after which the models are compared under complex multiaxial loadings. A surprisingly good agreement between the two models is found, but some differences in the magnetostrictive behaviour are also pointed out. Finally, the model is fitted to measurement results from an electrical steel sheet. The spline-based constitutive law overcomes several drawbacks of analytical approaches used earlier. The presented models and measurement results are openly available.

Trivariate spline representations for computer aided design and additive manufacturing

Digital representations targeting design and simulation for Additive Manufacturing (AM) are addressed from the perspective of Computer Aided Geometric Design. We discuss the feasibility for multi-material AM for B-rep based CAD, STL, sculptured triangles as well as trimmed and block-structured trivariate locally refined spline representations. The trivariate spline representations support Isogeometric Analysis (IGA), and topology structures supporting these for CAD, IGA and AM are outlined. The ideas of (Truncated) Hierarchical B-splines, T-splines and LR B-splines are outlined and the approaches are compared. An example from the EC H2020 Factories of the Future Research and Innovation Actions CAxMan illustrates both trimmed and block-structured spline representations for IGA and AM.

Exact conversion from Bézier tetrahedra to Bézier hexahedra

Modeling and computing of trivariate parametric volumes is an important research topic in the field of three-dimensional isogeometric analysis. In this paper, we propose two kinds of exact conversion approaches from Bézier tetrahedra to Bézier hexahedra with the same degree by reparametrization technique. In the first method, a Bézier tetrahedron is converted into a degenerate Bézier hexahedron, and in the second approach, a non-degenerate Bézier tetrahedron is converted into four non-degenerate Bézier hexahedra. For the proposed methods, explicit formulas are given to compute the control points of the resulting tensor–product Bézier hexahedra. Furthermore, in the second method, we prove that tetrahedral spline solids with Ck-continuity can be converted into a set of tensor–product Bézier volumes with Gk-continuity. The proposed methods can be used for the volumetric data exchange problems between different trivariate spline representations in CAD/CAE. Several experimental results are presented to show the effectiveness of the proposed methods.

On the dimension of trivariate spline spaces with the highest order smoothness on 3D T-meshes

T-meshes are a type of rectangular partitions of planar domains which allow hanging vertices. Because of the special structure of T-meshes, adaptive local refinement is possible for splines defined...

Trivariate near-best blending spline quasi-interpolation operators

A method to define trivariate spline quasi-interpolation operators (QIOs) is developed by blending univariate and bivariate operators whose linear functionals allow oversampling. In this paper, we construct new operators based on univariate B-splines and bivariate box splines, exact on appropriate spaces of polynomials and having small infinity norms. An upper bound of the infinity norm for a general blending trivariate spline QIO is derived from the Bernstein-Bezier coefficients of the fundamental functions associated with the operators involved in the construction. The minimization of the resulting upper bound is then proposed and the existence of a solution is proved. The quadratic and quartic cases are completely worked out and their exact solutions are explicitly calculated.

Improving temperature interpolation usingMODIS LSTand local topography: a comparison of methods in south east Australia

ABSTRACTAvailable climate data for south east Australia is reliant upon elevational lapse rates, which do not account for mesoscale processes that can affect temperatures, such as cold air drainage. Additional predictor variables are available for generating new climate datasets such as topographic indices and Moderate Resolution Imaging Spectroradiometer land surface temperature (MODIS LST); however, these have not been thoroughly tested to date. In this study, the relative benefits of including a localized topographic index and standardizedMODIS LSTvalues for temperature interpolation were assessed using partial bivariate splines, full and partial trivariate splines, and regression kriging. Trivariate splines provided the best interpolation performance in most cases; however, the partial bivariate spline with a fixed dependence upon elevation performed marginally better than the full trivariate spline for minimum temperature. The local topographic index improved theRMSEof minimum temperature climate normals by 17% in comparison to the best performing elevation only model. A further improvement for minimum temperature performance was achieved by including standardized night timeMODIS LSTvalues as covariates (34–39% reduction inRMSE). Standardized day timeMODIS LSTvalues improved maximum temperature interpolation performance; however, the improvement was only marginal in comparison to the full trivariate spline (6% reduction inRMSE). Cross validation of daily maximum and minimum temperature anomalies reflected performance trends shown in the climate normal analysis. Results suggest that the use of alternative approaches to interpolating temperature data may have significant implications for the calculation of bioclimatic variables and provide new opportunities to study extremes at high spatial and temporal resolutions using existing weather station networks. Furthermore, improving minimum temperature surfaces by accounting for temperature inversions driven by cold air drainage regimes may improve our ability to incorporate mesoscale temperature variability into a variety of applications, such as deriving temperature dependent climatic variables, species distribution modelling and assessments of fire risk.

Isogeometric analysis with Bézier tetrahedra

This paper presents an approach for isogeometric analysis of 3D objects using rational Bézier tetrahedral elements. In this approach, both the geometry and the physical field are represented by trivariate splines in Bernstein Bézier form over the tetrahedrangulation of a 3D geometry. Given a NURBS represented geometry, either untrimmed or trimmed, we first convert it to a watertight geometry represented by rational triangular Bézier splines (rTBS). For trimmed geometries, a compatible subdivision scheme is developed to guarantee the watertightness. The rTBS geometry preserves exactly the original NURBS surfaces except for an interface layer between trimmed surfaces where controlled approximation occurs. From the watertight rTBS geometry, a Bézier tetrahedral partition is generated automatically. By imposing continuity constraints on Bézier ordinates of the elements, we obtain a set of global Cr smooth basis functions and use it as the basis for analysis. Numerical examples demonstrate that our method achieves optimal convergence in Cr spaces and can handle complicated geometries.

Isogeometric analysis-suitable trivariate NURBS models from standard B-Rep models

This paper presents an effective method to automatically construct trivariate spline models of complicated geometry and arbitrary topology required for NURBS-based isogeometric analysis. The input is a triangulation of the solid model’s boundary. The boundary surface is decomposed into a set of cuboids in two steps: pants decomposition and cuboid decomposition. The novelty of our method is the pants-to-cuboids decomposition algorithm. The algorithm is completely automatic and very robust even for low-quality and noisy meshes. The set of cuboids composes a generalized polycube approximating very roughly the input model’s geometry while faithfully replicating its topology. Due to its regular structure, the polycube is suitable for serving as the parametric domain for the tensor-product spline representation. Based on a discrete harmonic mapping between the cuboids’ boundary and the polycube’s boundary, surface and volume parameterizations are generated in order to fit the trivariate splines. The efficiency and the robustness of the proposed approach are illustrated by several examples.

Trivariate spline quasi-interpolants based on simplex splines and polar forms

In this work we describe an approximating scheme based on simplex splines on a tetrahedral partition using volumetric data. We use the trivariate simplex B-spline representation of polynomials or piecewise polynomials in terms of their polar forms to construct several differential or discrete quasi interpolants which have an optimal approximation order.

Quantifying the lifetime circadian rhythm of physical activity: a covariate-dependent functional approach.

Objective measurement of physical activity using wearable devices such as accelerometers may provide tantalizing new insights into the association between activity and health outcomes. Accelerometers can record quasi-continuous activity information for many days and for hundreds of individuals. For example, in the Baltimore Longitudinal Study on Aging physical activity was recorded every minute for [Formula: see text] adults for an average of [Formula: see text] days per adult. An important scientific problem is to separate and quantify the systematic and random circadian patterns of physical activity as functions of time of day, age, and gender. To capture the systematic circadian pattern, we introduce a practical bivariate smoother and two crucial innovations: (i) estimating the smoothing parameter using leave-one-subject-out cross validation to account for within-subject correlation and (ii) introducing fast computational techniques that overcome problems both with the size of the data and with the cross-validation approach to smoothing. The age-dependent random patterns are analyzed by a new functional principal component analysis that incorporates both covariate dependence and multilevel structure. For the analysis, we propose a practical and very fast trivariate spline smoother to estimate covariate-dependent covariances and their spectra. Results reveal several interesting, previously unknown, circadian patterns associated with human aging and gender.

Bounds on the Dimension of Trivariate Spline Spaces: A Homological Approach

We consider the vector space of globally differentiable piecewise polynomial functions defined on a three-dimensional polyhedral domain partitioned into tetrahedra. We prove new lower and upper bounds on the dimension of this space by applying homological techniques. We give an insight of different ways of approaching this problem by exploring its connections with the Hilbert series of ideals generated by powers of linear forms, fat points, the so-called Froberg–Iarrobino conjecture, and the weak Lefschetz property.

Trivariate Optimal Smoothing Splines with Dynamic Shape Modeling of Deforming Object

We develop a method of constructing trivariate optimal smoothing splines using normalized uniform B-spline as the basis functions. Such splines are useful particularly for modeling dynamic shape of 3-dimensional deformable object by using two variables for 3D shape and one for time evolution. The trivariate splines are constructed as a tensor product of three B-splines, and an optimal smoothing spline problem is solved together with typical examples of constraints as periodicity. The problem is formulated as convex quadratic programming (QP) problem in such a way that 3D array of control points is vectorized and a MATLAB QP solver is readily applicable for numerical solutions. We demonstrate usefulness of the method by dynamic shape modeling of red blood cell, where we will see that relatively small number of observation data yield satisfactory results.

Trivariate quartic splines on type-6 tetrahedral partitions and their applications

In this paper we develop a 3D quasi-interpolating spline scheme, on a bounded domain, based on trivariate quartic C 2 box splines on type-6 tetrahedral partitions. Then, we describe some applications related both to the reconstruction of gridded volume data and to numerical integration. We also provide some numerical tests.

Component-aware tensor-product trivariate splines of arbitrary topology

The fundamental goal of this paper aims to bridge the large gap between the shape versatility of arbitrary topology and the geometric modeling limitation of conventional tensor-product splines for solid representations. Its contribution lies at a novel shape modeling methodology based on tensor-product trivariate splines for solids with arbitrary topology. Our framework advocates a divide-and-conquer strategy. The model is first decomposed into a set of components as basic building blocks. Each component is naturally modeled as tensor-product trivariate splines with cubic basis functions while supporting local refinement. The key novelty is our powerful merging strategy that can glue tensor-product spline solids together subject to C2 continuity. As a result, this new spline representation has many attractive advantages. At the theoretical level, the integration of the top-down topological decomposition and the bottom-up spline construction enables an elegant modeling approach for arbitrary high-genus solids. Each building block is a regular tensor-product spline, which is CAD-ready and facilitates GPU computing. In addition, our new spline merging method enforces the features of semi-standardness (i.e., ∑iwiBi(u,v,w)≡1 everywhere) and boundary restriction (i.e., all blending functions are confined exactly within parametric domains) in favor of downstream CAE applications. At the computational level, our component-aware spline scheme supports meshless fitting which completely avoids tedious volumetric mapping and remeshing. This divide-and-conquer strategy reduces the time and space complexity drastically. We conduct extensive experiments to demonstrate its shape flexibility and versatility towards solid modeling with complicated geometries and non-trivial genus.