Geometric Interpolation Research Articles

Improving Realism of Facial Interpolation and Blendshapes with Analytical Partial Differential Equation-Represented Physics

How to create realistic shapes by interpolating two known shapes for facial blendshapes has not been investigated in the existing literature. In this paper, we propose a physics-based mathematical model and its analytical solutions to obtain more realistic facial shape changes. To this end, we first introduce the internal force of elastic beam bending into the equation of motion and integrate it with the constraints of two known shapes to develop the physics-based mathematical model represented with dynamic partial differential equations (PDEs). Second, we propose a unified mathematical expression of the external force represented with linear and various nonlinear time-dependent Fourier series, introduce it into the mathematical model to create linear and various nonlinear dynamic deformations of the curves defining a human face model, and derive analytical solutions of the mathematical model. Third, we evaluate the realism of the obtained analytical solutions in interpolating two known shapes to create new shape changes by comparing the shape changes calculated with the obtained analytical solutions and geometric linear interpolation to the ground-truth shape changes and conclude that among linear and various nonlinear PDE-based analytical solutions named as linear, quadratic, and cubic PDE-based interpolation, quadratic PDE-based interpolation creates the most realistic shape changes, which are more realistic than those obtained with the geometric linear interpolation. Finally, we use the quadratic PDE-based interpolation to develop a facial blendshape method and demonstrate that the proposed approach is more efficient than numerical physics-based facial blendshapes.

Existence and uniqueness of s-curve segments of tensioned elastica satisfying geometric Hermite interpolation conditions

It has been recently proved that every proper restricted elastic spline is a stable nonlinear spline, and this yields a broad existence proof for stable nonlinear splines. When tension is included in the setup, stable nonlinear splines under tension always exist, but they do not always have the property that each piece (connecting one interpolation point to the next) is an s-curve. Being correlated with the fairness of an interpolating curve, this property is desirable and we conjecture that the framework employed successfully with restricted elastic splines will also work well with nonlinear splines under tension. Our purpose is to prove the following foundational result: Given points P1≠P2, in the plane, along with corresponding unit directions d1,d2 that satisfy d1⋅(P2−P1)≥0 and d2⋅(P2−P1)≥0, there exists a unique s-curve segment of Euler–Bernoulli elastica under tension λ>0 that connects P1 to P2 with initial direction d1 and terminal direction d2.

GeoLatent: A Geometric Approach to Latent Space Design for Deformable Shape Generators

We study how to optimize the latent space of neural shape generators that map latent codes to 3D deformable shapes. The key focus is to look at a deformable shape generator from a differential geometry perspective. We define a Riemannian metric based on as-rigid-as-possible and as-conformal-as-possible deformation energies. Under this metric, we study two desired properties of the latent space: 1) straight-line interpolations in latent codes follow geodesic curves; 2) latent codes disentangle pose and shape variations at different scales. Strictly enforcing the geometric interpolation property, however, only applies if the metric matrix is a constant. We show how to achieve this property approximately by enforcing that geodesic interpolations are axis-aligned, i.e., interpolations along coordinate axis follow geodesic curves. In addition, we introduce a novel approach that decouples pose and shape variations via generalized eigendecomposition. We also study efficient regularization terms for learning deformable shape generators, e.g., that promote smooth interpolations. Experimental results on benchmark datasets show that our approach leads to interpretable latent codes, improves the generalizability of synthetic shapes, and enhances performance in geodesic interpolation and geodesic shooting.

A lattice Boltzmann model for the interface tracking of immiscible ternary fluids based on the conservative Allen-Cahn equation

A lattice Boltzmann (LB) model for the interface tracking of immiscible ternary fluids is proposed based on the conservative Allen-Cahn (A-C) equation. A nonlocal equilibrium distribution function source term is adopted in the LB equation to eliminate the unwanted numerical items. With the new LB equation and the carefully designed equilibrium distribution function, Chapman-Enskog analysis shows that the present model can correctly recover the conservative A-C equation with second-order accuracy. A series of benchmark cases have been used to validate the accuracy and stability of the present model, including the classical diagonal translation of two concentric circular interfaces, the rigid-body rotation of Zalesak's disk, and the periodic shear deformation of two tangent circular interfaces. By coupling the dynamic LB model for Navier-Stokes equations, the stationary droplets, the spreading of a liquid lens, the spinodal decomposition, and the Raleigh-Taylor instability have also been simulated in the framework of ternary fluids. Utilizing the geometric interpolation method, the wettability effect of the horizontal boundary is considered and numerical tests with different wall contact angles have been performed to verify the applicability of the present model. All the numerical tests show that the present model can capture the interfaces among the immiscible ternary fluids accurately. Furthermore, using fewer discrete velocity directions, the present model shows higher numerical accuracy and stability compared with other existing numerical models. Therefore, it is promising to improve computational efficiency.

Geometric Hermite interpolation by rational curves of constant width

A constructive characterization of the support function for a rationally parameterized curve of constant width is given. In addition, a Hermite interpolation problem for such kind of curves is solved, which yields a method to determine a rational curve of constant width that passes through a set of free points with the corresponding tangent directions. Finally, the case of piecewise rational support functions is considered, which increases the design freedom. The procedure is presented in the general case of hedgehogs of constant width taking the advantage of projective hedgehogs, so that some constraints must be taken to ensure convexity of the desired curve.

Deep Spatial Prediction via Heterogeneous Multi-source Self-supervision

Spatial prediction is to predict the values of the targeted variable, such as PM2.5 values and temperature, at arbitrary locations based on the collected geospatial data. It greatly affects the key research topics in geoscience in terms of obtaining heterogeneous spatial information (e.g., soil conditions, precipitation rates, wheat yields) for geographic modeling and decision-making at local, regional, and global scales. In situ data, collected by ground-level in situ sensors, and remote sensing data, collected by satellite or aircraft, are two important data sources for this task. In situ data are relatively accurate while sparse and unevenly distributed. Remote sensing data cover large spatial areas, but are coarse with low spatiotemporal resolution and prone to interference. How to synergize the complementary strength of these two data types is still a grand challenge. Moreover, it is difficult to model the unknown spatial predictive mapping while handling the tradeoff between spatial autocorrelation and heterogeneity. Third, representing spatial relations without substantial information loss is also a critical issue. To address these challenges, we propose a novel Heterogeneous Self-supervised Spatial Prediction (HSSP) framework that synergizes multi-source data by minimizing the inconsistency between in situ and remote sensing observations. We propose a new deep geometric spatial interpolation model as the prediction backbone that automatically interpolates the values of the targeted variable at unknown locations based on existing observations by taking into account both distance and orientation information. Our proposed interpolator is proven to both be the general form of popular interpolation methods and preserve spatial information. The spatial prediction is enhanced by a novel error-compensation framework to capture the prediction inconsistency due to spatial heterogeneity. Extensive experiments have been conducted on real-world datasets and demonstrated our model’s superiority in performance over state-of-the-art models.

Geometric Hermite interpolation in $$\mathbb {R}^{n}$$ by refinements

Geometric Hermite interpolation in $$\mathbb {R}^{n}$$ by refinements

Exact Probability Distribution for the ROC Area under Curve.

The Receiver Operating Characteristic (ROC) is a de facto standard for determining the accuracy of in vitro diagnostic (IVD) medical devices, and thus the exactness in its probability distribution is crucial toward accurate statistical inference. We show the exact probability distribution of the ROC AUC-value, hence exact critical values and p-values are readily obtained. Because the exact calculations are computationally intense, we demonstrate a method of geometric interpolation, which is exact in a special case but generally an approximation, vastly increasing computational speeds. The method is illustrated through open access data, demonstrating superiority of 26 composite biomarkers relative to a predicate device. Especially under correction for testing of multiple hypotheses, traditional asymptotic approximations are encumbered by considerable imprecision, adversely affecting IVD device development. The ability to obtain exact p-values will allow more efficient IVD device development.

PRACTICAL METHODS OF THE VALVE SEATS WEAR ASSESSING
 IN THE FLOW DEFECTION PROCESS

The actual issue of obtaining objective results when assessing the state of the valve seats worn surfaces of the valve timing gear of an internal combustion engine has been considered in this article, as well as the imperfections of existing methods for assessing local and integral wear in the problems of flaw detection during overhauls in vehicle repair industry. The need for a mathematical approach in solving the posed problems has been confirmed by assessing the state of worn surfaces with complex-variable topography due to the developed geometric interpolation 3D model that describes the profile to be restored in the valve seats flaw detection process. Practical methods for assessing the degree of valve seats wear have been offered on the basis of a mathematical 3D model through the introduction of a technique for using circular diagrams and profilograms using precise control and measuring equipment.

Improved Least-Squares Progressive Iterative Approximation for Tensor Product Surfaces

Geometric iterative methods, including progressive iterative approximation and geometric interpolation methods, are efficient for fitting a given data set. With the development of big data technology, the number of fitting data points has become massive, and the progressive iterative approximation for least-squares fitting (LSPIA) is generally applied to fit mass data. Combining the Schulz iterative method for calculating the Moore–Penrose generalized inverse matrix with the traditional LSPIA method, this paper presents an accelerated LSPIA method for tensor product surfaces and shows that the corresponding iterative surface sequence converged to the least-squares fitting surface of the given data set. The iterative format is that of a non-stationary iterative method, and the convergence rate increased rapidly as the iteration number increased. Some numerical examples are provided to illustrate that the proposed method has a faster convergence rate.

Anisotropic elastic applications in composite materials using the isogeometric boundary element method

This paper describes an isogeometric analysis of the boundary element method, called IGABEM, applied to anisotropic 2D plane elastic problems. The Lekhnitskii's anisotropic fundamental solution is used and the singular integral terms is regularized. For the weak singularity kernel, the Telles transform is used. On the other hand, in the strong singularity term, the Singularity Subtraction Technique - SST is used. The shape functions used are the Non-Uniform Rational B-Spline - NURBS. Thus, the same mathematical representation of Computer Aided Design - CAD is used in the implemented computational code. This avoids the generation of meshes and provides an exact representation of most complex geometries. As a reflection of this isoparametric concept, errors from geometric interpolation are excluded, improving numerical results. Furthermore, to increase the numerical efficiency of the code, the NURBS are decomposed into Bézier curves. To evaluate the accuracy of the formulation, complex problems using high-order isogeometric boundary elements are analyzed. Their results are compared to analytical solutions showing good agreement.

The Analysis of Some Spherical Mechanism Movements and Joint Designs by The New SLERP Interpolations

The movements of the spherical mechanisms take an important place in the robotic studies. In this paper, the design and the trajectory motion equations of the joints in the spherical mechanisms were presented by the methods of the sequential SLERP, sequential fast SLERP, sequential SLERP with De-Moivre, and modular SLERP interpolation that we firstly defined. Additionally, the Serret Frenet frame and curvatures of the geometrical SLERP interpolation curve for the locations of the joints in the spherical mechanisms were computed, and then, several numerical examples on the interpolation equations that produce the motion equations of the spherical mechanisms with 2 and 3-degree of freedom were given at the end of the paper.

Multivariant spatial and geometric interpolation of field maps through model order reduction

Complex three-dimensional field maps can be efficiently denoised and compressed in data volume by high order single value decomposition. Extension of the method to inter/extrapolation and determination of basis function representation is reported here. Inter/extrapolation of denoised and compressed maps is applied to spatial coordinates and especially to the parameters defining the geometry of devices creating such maps. It significantly increases the efficiency of the provision of compact and noise-free maps combined with high resolution. The method is illustrated by creating corresponding maps for rf cells of radio-frequency quadrupoles.

Generic bond energy formalism within the modified quasichemical model for ternary solutions

The Modified Quasichemical Model in the Pair Approximation (MQMPA) can effectively capture the thermodynamic features of a binary solution with Short-Range Ordering (SRO). If the model is used to treat a ternary solution, a geometric interpolation method must be employed to extend the bond energy expression from binary to ternary formalism. The aim of the present work is to implement such extension by means of a generic geometric interpolation approach. The generic method is unbiased and can be transformed into the widely used Kohler, Toop and Muggianu approaches with special interpolation parameters. The interpolation parameters can be calculated by the integration method as well as be optimized by ternary experimental data. The generic bond energy formalism (GBEF) has thus been derived to provide the MQMPA great flexibility to describe ternary solutions with complex configurations. Moreover, the GBEF is more concise than the formula derived by a combinatorial Kohler-Toop method. The concise GBEF is in the respect more conveniently programmed. Eventually, the Cu-Li-Sn liquid where both SRO and clustering among atoms occur is employed to validate the effectiveness and reliability of the GBEF within the MQMPA.

On planar polynomial geometric interpolation

In the paper, the planar polynomial geometric interpolation of data points is revisited. Simple sufficient geometric conditions that imply the existence of the interpolant are derived in general. They require data points to be convex in a certain discrete sense. Since the geometric interpolation is based precisely on the known data only, one may consider it as the parametric counterpart to the polynomial function interpolation. The established result confirms the Höllig–Koch conjecture on the existence and the approximation order in the planar case for parametric polynomial curves of any degree stated quite a while ago.

A Look Inside the Black Box of Machine Learning Photodynamics Simulations.

Photochemical reactions are of great importance in chemistry, biology, and materials science because they take advantage of a renewable energy source, mild reaction conditions, and high atom economy. Light absorption can excite molecules to a higher energy electronic state of the same spin multiplicity. The following nonadiabatic processes induce molecular transformations that afford exotic molecular architectures and high-energy-isomers that are inaccessible by thermal means. Computational simulations now complement time-resolved instrumentation to reveal ultrafast excited-state mechanistic information for photochemical reactions that is essential in disentangling elusive spectroscopic features, excited-state lifetimes, and excited-state mechanistic critical points. Nonadiabatic molecular dynamics (NAMD), powered by surface hopping techniques, is among the most widely applied techniques to model the photochemical reactions of medium-sized molecules. However, the computational efficiency is limited because of the requisite thousands of multiconfigurational quantum-chemical calculations multiplied by hundreds of trajectories. Machine learning (ML) has emerged as a revolutionary force in computational chemistry to predict the outcome of the resource-intensive multiconfigurational calculations on the fly. An ML potential trained with a substantial set of quantum-chemical calculations can predict the energies and forces with errors under chemical accuracy at a negligible cost. The integration of ML potentials in NAMD dramatically extends the maximum simulation time scale by ∼10 000-fold to the nanosecond regime.In this Account, we present a comprehensive demonstration of ML photodynamics simulations and summarize our most recent applications in resolving complex photochemical reactions. First, we address three fundamental components of ML techniques for photodynamics simulations: the quantum-chemical data set, the ML potential, and NAMD. Second, we describe best practices in building training data and our procedure toward training the ML photodynamics model with our recent literature contributions. We introduce a convenient training data generation scheme combining Wigner sampling and geometrical interpolation. It trains reliable and effective ML potentials suitable for subsequent active learning to detect undersampled data. We demonstrate how active learning automatically discovers new mechanistic pathways and reproduces experimental results. We point out that atomic permutation is an essential data augmentation approach to improve the learnability of distance-based molecular descriptors for highly symmetric molecules. Third, we demonstrate the utility of ML-photodynamics by showing the results of ML photodynamics simulations of (1) photo-torquoselective 4π disrotatory electrocyclic ring closing of norbornyl cyclohexadiene, which reveals a thermal conversion from experimentally unobserved intermediates to the reactant in 1 ns; (2) [2 + 2] photocycloaddition of substituted [3]-syn-ladderdienes in competition with 4π and 6π electrocyclic ring-opening reactions, uncovering substituent effects to explain the reported increased quantum yield of substituted cubane precursors; and (3) photochemical 4π disrotatory electrocyclic reactions of fluorobenzenes in nanoseconds with XMS-CASPT2-level training data. We expect this Account to broaden understanding of ML photodynamics and inspire future developments and applications to increasingly large molecules within complex environments on long time scales.

ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ НАПРЯЖЁННО-ДЕФОРМИРОВАННОГО СОСТОЯНИЯ МЕТАЛЛОКОНСТРУКЦИЙ С ПОМОЩЬЮ ГЕОМЕТРИЧЕСКИХ ИНТЕРПОЛЯНТОВ

The work is devoted to carrying out multidimensional interpolation and approximation methods for the numerical solution of differential equations and computer model development of the stress-strain state of metal structures. The core of the work is a fundamental computational algorithm for the numerical solution of differential equations using geometric interpolants on regular and irregular networks. On its basis, computational experiments are carried out on numerical simulation of the stress-strain state of operated reservoirs for storing petroleum products, which form a software package implemented in the Maple interpreter. At the same time, the differential equation for modelling the stress-strain state of an elastic cylindrical shell under axisymmetric loading is improved for the numerical analysis of the stress-strain state of a cylindrical reservoir with geometric imperfections. Also a new approach is proposed to take into consideration the initial conditions of the differential equation, which consists of parallel transfer of the numerical solution to the point, its coordinates correspond to the initial conditions. The advantage of the proposed approach for the numerical solution of differential equations using geometric interpolants is that it eliminates the need to coordinate geometric information in the process of interaction between CAD and FEA systems, by analogy with the isogeometric method.

An n-order hold approach for fractional-delay interpolation in auralization

During the signal processing chain of an auralization simulating the propagation of a sound from a moving source to a stationary receiver, it is often necessary to interpolate between the samples of the source signal in order to arrive at uniformly-spaced samples at the receiver. In some cases, this interpolation is done in the receiver time frame—where the “input” samples of the source have become irregularly spaced due to time dilation effects. Canonical band-limited interpolation methods (i.e., sinc and sinc-derived approaches) cannot be applied in this case as they rely on having a uniformly-spaced input. The use of geometric interpolation methods that can handle irregularly-spaced input may not be grounded in signal processing principles and may produce unwanted artifacts and noise. This presentation outlines the possibility of embedding an irregularly spaced n-order hold signal within a highly over-sampled uniformly-spaced signal and then processing down to the desired sampling rate through successive decimations. Initial distortion and noise characteristics of the approach are shown for some basic propagation geometries. The possible benefits of using such an approach in an auralization scheme with a time-varying Doppler shift are discussed including: the prevention of aliasing, processing time advantages, and real-time processing.

Large yacht resistance reduction by hydrodynamic multi-objective shape optimization

ABSTRACT This paper presents the results of the hydrodynamic optimization of a large yacht concept design that is meant to be used as a reference for future buildings. The optimization framework is used to improve the calm water resistance performance of a generic baseline design while satisfying a set of geometrical constraints. The shape of the hull is modified according to a geometrical interpolation approach where two shapes are combined to obtain a third design. The search towards the optimum is driven by a global convergence, multi-objective, genetic algorithm where the resistance of each candidate is evaluated by a Boundary Element Method. The final optimal design is then verified by means of a high-fidelity Reynolds Averaged Navier–Stokes technique, preliminary validated against available experimental data. The obtained results show the strong influence of the stern shape of the hull on the final resistance performance, highlighting the effects of the pressure recovery due to a so-called S-shape bottom. The bow shape is modified consistently to fulfil the geometric constraints and to further improve the resistance performance by inducing a positive interference effect on the generated wave pattern.

The use of interpolation methods for modelling multifactor processes based on an experiment planning matrix

This article is dedicated to the development of experimental data processing tools based on the geometric theory of multidimensional interpolation. At the same time, a scientific problem was solved, which is dedicated to the analytical determination of models of multifactor processes while maintaining the existing approach to experiment planning This makes it possible not only to use interpolation methods for the mathematical description of new experiments, but also to use the experimental data obtained earlier to improve their mathematical interpretation. The paper provides two examples that confirm the effectiveness of the proposed approach to constructing models of multivariate processes using multivariate interpolation methods. The first of the above examples contains two geometric models of the stress-strain state of metal polyhedral bent struts, which are presented in the form of ruled response hypersurfaces passing through 8 predetermined points in the 4-dimensional space. The second one is dedicated to the construction of a 2-factor process process using a 2- parameter parabolic geometric interpolant with subsequent optimization by methods of mathematical analysis of a function of two variables. As a result, the optimization of the aerated concrete manufacturing process was carried out to achieve the maximum values of the ultimate strength in compression after heat and moisture treatment.