Geometric Interpolation Research Articles

GEOMETRIC MODELING OF MULTI-FACTOR PROCESSES BASED ON VARIABLE POINT ALGORITHMS

In this paper, the geometric theory of multidimensional interpolation was further developed. It has been established that the geometric models of multivariate processes obtained using multidimensional interpolation are characterized by variability, which is a consequence of the multiplicity of choice of reference lines in the process of developing a geometric modeling scheme. At the same time, all possible variations of geometric interpolants fully satisfy the initial experimental and statistical data, but have different curvature between the node points of the interpolation. As the dimension of the space increases, the number of variations increases significantly. The variable approach to geometric modeling of multifactorial processes generates a number of scientific problems that require further research, such as: comparison of geometric objects of multidimensional space, development of criteria for choosing the best solutions, construction of averaged geometric objects as one of the tools for optimizing the results of modeling, etc. The article also presents the results of a computational experiment on geometric modeling of the dependence of the physical and mechanical properties of fine-grained concrete on the composition of the combined aggregate based on variable point algorithms with the subsequent construction of an averaged response surface, the current point of which is the center of gravity of a multidimensional tetrahedron, for which the dimension of space depends on the amount possible interpolation options.

Geometric interpolation of ER frames with G2 Pythagorean-hodograph curves of degree 7

The problem of constructing a curve that interpolates given initial/final positions along with orientational frames is addressed. In more detail, the resulting interpolating curve is a PH curve of degree 7 and among the adaptive frames that can be associated to a spatial PH curve, we consider the Euler-Rodrigues (ER) frame. Moreover G1 continuity between frames is imposed and conditions for achieving general geometric continuity are investigated. It is also shown that our construction of Gk continuity of ER frames implies Gk+1 continuity of the corresponding PH curves, and hence this approach can be useful to define spline motions. Exploiting the relation between rotational matrices and quaternions on the unit sphere, geometric continuity conditions on the frames are expressed through conditions on the corresponding quaternion polynomials. This leads to a nonlinear system of equations whose solvability is investigated, and asymptotic analysis of the solutions in the case of data sampled from a smooth parametric curve and its general adapted frame is derived. It is shown that there exist PH interpolants with optimal approximation order 6, except for the case of the Frenet frame, where the approximation order is at most 4. Several numerical examples are presented, which confirm the theoretical results.

On optimal polynomial geometric interpolation of circular arcs according to the Hausdorff distance

The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the Hausdorff distance and has not been studied yet in the literature. Parametric polynomial curves of low degree are used and a geometric continuity is prescribed at the boundary points of the circular arc. A general theory about the existence and the uniqueness of the optimal approximant is presented and a rigorous analysis is done for some special cases for which the degree of the polynomial curve and the order of the geometric smoothness differ by two. This includes practically interesting cases of parabolic G0, cubic G1, quartic G2 and quintic G3 interpolation. Several numerical examples are presented which confirm theoretical results.

Simulating fullerene polyhedral formation from planar precursors.

The synthesis path of the C60-Buckyball fullerene from a planar precursor developed by Scott et al. [Science, 2002, 295, 5559] is investigated with density functional theory (DFT) methods. Various theoretically possible closing paths are analysed with respect to structural and energetic properties. The initial geometries were obtained by geometric interpolation of a cardboard-like model comprising rigid rings connected by hinges, which were then fully optimized with a selection of DFT-functionals. Analysis of the fully optimised geometries shows remarkable stability of face planarity, bond lengths and bond angles for all studied geometries, indicating soundness of the "cardboard with hinges"-model for approximating reaction paths for molecules of this type. This raises hope for development of a force field description of fullerene precursor molecules that can aid in discovery and analysis of good precursor candidates for rational synthesis of new fullerenes.

Modeling Geometric Varieties with Given Differential Characteristics and Its Application

The article describes an approach to modeling geometric manifolds with specified differential properties, which is based on the use of geometric interpolants of a multidimensional space. A geometric interpolant is understood as a geometric object of a multidimensional space passing through predetermined points in advance, the coordinates of which correspond to the initial experimental and statistical information. Principles for determining geometric interpolants and an example of an analytical description of a 3-parameter geometric interpolant belonging to a 4-dimensional space in the form of a geometric scheme and a computational algorithm based on a sequence of point equations are given. The main direction of practical use of geometric interpolants is geometric modeling of multifactor processes and phenomena, but they can also be an effective tool for multivariate approximation. Based on this, the article presents a general approach to modeling geometric manifolds with given differential properties and its application in the form of a numerical solution of differential equations by approximating it using geometric interpolants of a multidimensional space. To implement an approach to modeling geometric manifolds with specified differential properties, it is proposed to use a computational algorithm consisting of 10 points. The advantages of using geometric interpolants for the numerical solution of differential equations are highlighted.

Geometric Hermite interpolation by a family of spatial algebraic–trigonometric PH curves

This paper proposes to define a family of spatial algebraic–trigonometric Pythagorean Hodograph (ATPH) curves by integrals of scaled unit tangent vector fields which are originally defined as sphere curves. The obtained ATPH curves have only polynomial parametric speeds but the curves can be employed to represent several typical non-polynomial curves without rational form. A simple algorithm for geometric Hermite interpolation by the proposed spatial ATPH curves without or with arc length constraint has been given. Given two boundary points and two unit tangents at the points, possibly with a prescribed arc length, a unique interpolating ATPH curve can be obtained by solving a simple linear system.

Time series interpolation and application of its methods in financial analysis

In the modern market economy conditions, financial information is very significant when evaluating the results of enterprises’ financial status and activity results. General economic, mathematical, heuristic methods are used for this type of analysis. The article analyses application opportunities for one of the mathematical research methods–time series–as well as its research instruments, discusses the place and role of interpolation methods in a financial analysis. The article reveals that interpolation methods which may be used to determined former values of an enterprise’s financial indicators or identify reasons which determined the enterprise’s financial status and activity results are the following: 1) graphic and arithmetic interpolation in a linear equation; 2) geometric interpolation based on the compound percentage formula and 3) interpolation by averaging.

The Asymptotically Sharp Geometric Rigidity Interpolation Estimate in Thin Bi-Lipschitz Domains

This work is part of a program of development of asymptotically sharp geometric rigidity estimates for thin domains. A thin domain in three dimensional Euclidean space is roughly a small neighborhood of regular enough two dimensional compact surface. We prove an asymptotically sharp geometric rigidity interpolation inequality for thin domains with little regularity. In contrast to that celebrated Friesecke et al. rigidity estimate [Commun. Pure Appl. Math. 55(11):1461–1506, 2002] for plates, our estimate holds for any proper rotations $\boldsymbol {R}\in SO(3)$ . Namely, the estimate bounds the $L^{p}$ distance of the gradient of any $\boldsymbol {y}\in W^{1,p}(\Omega ,\mathbb{R}^{3})$ field from any constant proper rotation $\boldsymbol {R}\in SO(3)$ , in terms of the average $L^{p}$ distance (nonlinear strain) of the gradient $\nabla \boldsymbol {y}$ from the rotation group $SO(3)$ , and the average $L^{p}$ distance of the field itself from the set of rigid motions corresponding to the rotation $\boldsymbol {R}$ . There are several remarkable facts about the estimate: 1. The constants in the estimate are sharp in terms of the domain thickness scaling for any thin domains with the required regularity. 2. In the special cases when the domain has positive or negative Gaussian curvature, the inequality reduces the problem of estimating the gradient $\nabla \boldsymbol {y}$ in terms of the prototypical nonlinear strain $\int _{\Omega }\mathrm{dist}^{p}(\nabla \boldsymbol {y}(x),SO(3))dx$ to the easier problem of estimating only the vector field $\boldsymbol {y}$ in terms of the nonlinear strain without any loss in the constant scalings as the Ansatze suggest. The later will be a geometric rigidity Korn-Poincare type estimate. This passage is major progress in the thin domain rigidity problem. 3. For the borderline energy scaling (bending-to-stretching), the estimate implies improved strong compactness on the vector fields for free. Finally, this being said, our new interpolation inequality reduces the problem of proving “any” geometric one well rigidity problem in thin domains to estimating the vector field itself instead of the gradient, thus reducing the complexity of the problem.

Approximation of monotone clothoid segments by degree 7 Pythagorean–hodograph curves

The clothoid is a planar curve with the intuitive geometrical property of a linear variation of the curvature with arc length, a feature that is important in many geometric design applications. However, the exact parameterization of the clothoid is defined in terms of the irreducible Fresnel integrals, which are computationally expensive to evaluate and incompatible with the polynomial/rational representations employed in computer aided geometric design. Consequently, applications that seek to exploit the simple curvature variation of the clothoid must rely on approximations that satisfy a prescribed tolerance. In the present study, we investigate the use of planar Pythagorean-hodograph (PH) curves as polynomial approximants to monotone clothoid segments, based on geometric Hermite interpolation of end points, tangents, and curvatures, and precise matching of the clothoid segment arc length. The construction, employing PH curves of degree 7, involves iterative solution of a system of five algebraic equations in five real unknowns. This is achieved by exploiting a closed-form solution to the problem of interpolating the specified data (except the curvatures) using quintic PH curves, to determine starting values that ensure rapid and accurate convergence to the desired solution.

Application of mixed geometric interpolants for modeling the strength characteristics of steel fiber concrete

The paper proposes an approach to modeling geometric multifactorial processes and phenomena using mixed geometrical interpolants, for example of the steel fiber concrete strength characteristics’ simulation A key role is played by the formation of the geometric model, as developed in detail in the article. Analytical description of the geometric model obtained by means of algebraic curves (that pass through the predetermined point in advance) is realized by means of the BN-calculus mathematical formalism.

Cortical thickness computation by solving tetrahedron-based harmonic field

Cortical thickness computation in magnetic resonance imaging (MRI) is an important method to study the brain morphological changes induced by neurodegenerative diseases. This paper presents an algorithm of thickness measurement based on a volumetric Laplacian operator (VLO), which is able to capture accurately the geometric information of brain images. The proposed algorithm is a novel three-step method: 1) The rule of parity and the shrinkage strategy are combined to detect and fix the intersection error regions between the cortical surface meshes separated by FreeSurfer software and the tetrahedral mesh is constructed which reflects the original morphological features of the cerebral cortex, 2) VLO and finite element method are combined to compute the temperature distribution in the cerebral cortex under the Dirichlet boundary conditions, and 3) the thermal gradient line is determined based on the constructed local isothermal surfaces and linear geometric interpolation results. Combined with half-face data storage structure, the cortical thickness can be computed accurately and effectively from the length of each gradient line. With the obtained thickness, we set experiments to study the group differences among groups of Alzheimer's disease (AD, N = 110), mild cognitive impairment (MCI, N = 101) and healthy control people (CTL, N = 128) by statistical analysis. The results show that the q-value associated with the group differences is 0.0458 between AD and CTL, 0.0371 between MCI and CTL, and 0.0044 between AD and MCI. Practical tests demonstrate that the algorithm of thickness measurement has high efficiency and is generic to be applied to various biological structures that have internal and external surfaces.

ГЕОМЕТРИЧЕСКАЯ ТЕОРИЯ МНОГОМЕРНОЙ ИНТЕРПОЛЯЦИИ

The paper presents a geometric theory of multidimensional interpolation based on invariants of affine geometry. The analytical description of geometric interpolants is performed within the framework of the mathematical apparatus BN-calculation using algebraic curves that pass through preset points. A geometric interpretation of the interaction of parameters, factors, and the response function is presented, which makes it possible to generalize the geometric theory of multidimensional interpolation in the direction of increasing the dimension of space. The conceptual principles of forming the tree of the geometric interpolant model as a geometric basis for modeling multi-factor processes and phenomena are described.

Blunt extension of discrete universal multifractal cascades: development and application to downscaling

ABSTRACT Scale issues are ubiquitous in geosciences. Because of their simplicity and intuitiveness, and despite strong limitations, notably its non-stationarity features, discrete random multiplicative cascade processes are very often used to address these scale issues. A novel approach based on the parsimonious framework of Universal Multifractals (UM) is introduced to tackle this issue while preserving the simple structure of discrete cascades. It basically consists in smoothing at each cascade step the random multiplicative increments with the help of a geometric interpolation over a moving window. The window size enables to introduce non-conservativeness in the simulated fields. It is established theoretically,] and numerically confirmed, that the simulated fields also exhibit a multifractal behaviour with expected features. It is shown that such an approach remains valid over a limited range of UM parameters. Finally, we test downscaling of rainfall fields with the help of this blunt discrete cascade process, and we discuss challenges for future developments.

Spatial modulation with signature constellations for increased robustness to antenna and channel correlations

Spatial modulation (SM) has a strong sensitivity to transmit antenna and channel correlations, because some of the information bits are assigned to active antenna selection, and the correlation limits the detection reliability of these bits. Recent approaches to solve this problem rely on either unequal error protection (UEP) of antenna and symbol bits with the addition of a channel encoder/decoder pair to the transceiver or precoding in the form of antenna-dependent rotation (or joint rotation and amplitude scaling) of the signal constellation. The UEP approaches have been shown to offer only limited efficiency in compensating for the adverse channel effects while increasing the latency and complexity due to the addition of the encoder/decoder. The precoding based approaches achieve good results for small number of transmit antennas and for small constellations such as BPSK and QPSK, but the performance quickly degrades for higher-level signal constellations. Moreover, the complexity of the precoder optimization problem increases with the number of transmit antennas and the modulation order, making this approach not very practical to use for high spectral efficiencies. This paper introduces a novel approach to this problem, which is particularly suited for high-level signal constellations. This is achieved by the use of active-antenna dependent signature constellations which are derived from each other through successive geometric interpolations. This approach preserves the minimum inter-constellation Euclidean distance given the primary constellation. The theoretical analysis and the simulation results show that compared to previous methods it gives significant performance improvements in terms of robustness to transmit antenna correlation, particularly for Rician fading channels.

Description of neutral line of harmonic drive flexspline with geometric property interpolation

Description of neutral line of harmonic drive flexspline with geometric property interpolation

Generation of geometric interpolations of building types with deep variational autoencoders

AbstractThis work presents a methodology for the generation of novel 3D objects resembling wireframes of building types. These result from the reconstruction of interpolated locations within the learnt distribution of variational autoencoders (VAEs), a deep generative machine learning model based on neural networks. The data set used features a scheme for geometry representation based on a ‘connectivity map’ that is especially suited to express the wireframe objects that compose it. Additionally, the input samples are generated through ‘parametric augmentation’, a strategy proposed in this study that creates coherent variations among data by enabling a set of parameters to alter representative features on a given building type. In the experiments that are described in this paper, more than 150 k input samples belonging to two building types have been processed during the training of a VAE model. The main contribution of this paper has been to explore parametric augmentation for the generation of large data sets of 3D geometries, showcasing its problems and limitations in the context of neural networks and VAEs. Results show that the generation of interpolated hybrid geometries is a challenging task. Despite the difficulty of the endeavour, promising advances are presented.

Representation theory in chiral conformal field theory: from fields to observables

This article develops new techniques for understanding the relationship between the three different mathematical formulations of two-dimensional chiral conformal field theory: conformal nets (axiomatizing local observables), vertex operator algebras (axiomatizing fields), and Segal CFTs. It builds upon previous work (Tener in Adv Math 349:488–563, 2019), which introduced a geometric interpolation procedure for constructing conformal nets from VOAs via Segal CFT, simultaneously relating all three frameworks. In this article, we extend this construction to study the relationship between the representation theory of conformal nets and the representation theory of vertex operator algebras. We define a correspondence between representations in the two contexts, and show how to construct representations of conformal nets from VOAs. We also show that this correspondence is rich enough to relate the respective ‘fusion product’ theories for conformal nets and VOAs, by constructing local intertwiners (in the sense of conformal nets) from intertwining operators (in the sense of VOAs). We use these techniques to show that all WZW conformal nets can be constructed using our geometric interpolation procedure.

Approximation of 3D objects by piecewise linear geometric interpolants of their 1D cross-sections

In this paper we introduce a method for reconstruction of 3D objects from their 1D parallel cross-sections by set-valued interpolation. We regard a 3D object as the graph of a set-valued function defined on a planar domain, with the given 1D cross-sections as its samples. The method is based on a triangulation T of the sampling points in the planar domain and it is modular: for each triangle in T the corresponding 1D cross-sections are geometrically interpolated by a union of polyhedrons. The union of these interpolants over all triangles of T constitutes the approximating 3D object. Properties of this approximation are studied, in particular we derive the approximation order of the error, measured by the symmetric difference metric.

Highly efficient facial blendshape animation with analytical dynamic deformations

Adding physics to facial blendshape animation is an active research topic. Existing physics-based approaches of facial blendshape animation are numerical, so they require special knowledge and skills, additional preprocess, large computer capacity, and expensive calculations leading to low animation frame rates, and are not easy to learn, implement and use. To tackle these problems, we propose an analytical approach and develop a blending force-based framework for physics-based facial animation. The proposed approach introduces the equation of motion to consider inertial effects, damping effects and the resistance against deformations, combines them with source and target facial shapes to formulate the mathematical model of dynamic deformations, and develops a simple and efficient closed-form solution. The blending force-based framework incorporates the new proposed slider force-based, exponentiation force-based and random force-based methods built on the obtained closed form solution to achieve highly efficient facial animation. Compared with facial blendshape animation using geometric linear interpolation, the proposed approach is physics-based. It not only creates all the blended shapes generated by linear interpolation, but also a much larger superset of blended shapes. Unlike linear interpolation which can only generate blended shapes with a same deformation rate, the proposed approach can generate blended shapes with different deformation rates, resulting in special effects of acceleration and deceleration. Compared to existing physics-based approaches of facial blendshape animation which are numerical, the proposed approach is the first time to develop an analytical approach of physics-based facial blendshapes. It does not require any special knowledge and skills and is easy to learn, implement and use. More importantly, it can avoid the additional preprocess of numerical methods and create various physics-based facial blendshape animations highly efficiently. Moreover, it can be used to estimate physical parameters from real shapes and developed into an interactive and real-time physics-based shape manipulation tool.

TOPOLOGY OPTIMIZATION ANALYSIS OF ADHESIVE SOUND ABSORBING MATERIALS STRUCTURE WITH SUBDIVISION SURFACE BOUNDARY ELEMENT METHOD

Isogeometric analysis (IGA) realizes the seamless integration of CAD and CAE by using spline basis functions to represent geometric models and implement the numerical analysis, and is widely used in elastic mechanics, electromagnetic field, potential problem and other fields. However, it is difficult to construct directly a complex structure by using IGA. The subdivision surface method can be used to subdivide and fit the discrete mesh of a traditional model in order to construct smooth surfaces. Such a method is suitable for complicated problems. This method has the following advantages: (1) It is suitable for any topological structure; (2) The numerical calculation is stable; (3) It is simple to implement; (4) Local refinement and continuity control. Because of its flexibility and convenience in the construction of complex structural models, this method has been widely used in aerospace, automobile, animation, game making and other modeling fields. The subdivision surface method (SSM) and the boundary element method (BEM) were integrated to perform structural acoustic analysis. The box-spline interpolation of SSM was used for both geometric interpolation and physical interpolation, achieving high-order approximation of the structural surface and physical field. The acoustic scattering of the structure of adhesive sound absorption materials was taken as an example to test the effectiveness of the algorithm. The above analysis was combined with the method of moving asymptotes (MMA) algorithm to conduct a topological optimization of the distribution of sound absorption materials. In this study, the adjoint variable method and the acoustic BEM were used to analyze the sensitivity of the distribution of sound absorption materials on the surface of the structure. Each update of design variables brings small changes in the layout of sound absorbing materials, and ultimately the optimal solution is obtained. The resulting solvers provide an efficient computational tool for topology optimization design. The proposed algorithm is then applied to some numerical examples to illustrate the potential for engineering optimization design.