Smooth Shape‐Aware Functions with Controlled Extrema

Functions that optimize Laplacian‐based energies have become popular in geometry processing, e.g. for shape deformation, smoothing, multiscale kernel construction and interpolation. Minimizers of Dirichlet energies, or solutions of Laplace equations, are harmonic functions that enjoy the maximum principle, ensuring no spurious local extrema in the interior of the solved domain occur. However, these functions are only C⁰ at the constrained points, which often causes smoothness problems. For this reason, many applications optimize higher‐order Laplacian energies such as biharmonic or triharmonic. Their minimizers exhibit increasing orders of continuity but lose the maximum principle and show oscillations. In this work, we identify characteristic artifacts caused by spurious local extrema, and provide a framework for minimizing quadratic energies on manifolds while constraining the solution to obey the maximum principle in the solved region. Our framework allows the user to specify locations and values of desired local maxima and minima, while preventing any other local extrema. We demonstrate our method on the smoothness energies corresponding to popular polyharmonic functions and show its usefulness for fast handle‐based shape deformation, controllable color diffusion, and topologically‐constrained data smoothing.     

STAR ‐ Laplacian Spectral Kernels and Distances for Geometry Processing and Shape Analysis

In geometry processing and shape analysis, several applications have been addressed through the properties of the spectral kernels and distances, such as commute‐time, biharmonic, diffusion, and wave distances. Our survey is intended to provide a background on the properties, discretization, computation, and main applications of the Laplace‐Beltrami operator, the associated differential equations (e.g., harmonic equation, Laplacian eigenproblem, diffusion and wave equations), Laplacian spectral kernels and distances (e.g., commute‐time, biharmonic, wave, diffusion distances). While previous work has been focused mainly on specific applications of the aforementioned topics on surface meshes, we propose a general approach that allows us to review Laplacian kernels and distances on surfaces and volumes, and for any choice of the Laplacian weights. All the reviewed numerical schemes for the computation of the Laplacian spectral kernels and distances are discussed in terms of robustness, approximation accuracy, and computational cost, thus supporting the reader in the selection of the most appropriate method with respect to shape representation, computational resources, and target application.     

Accurate and Efficient Computation of Laplacian Spectral Distances and Kernels

This paper introduces the Laplacian spectral distances, as a function that resembles the usual distance map, but exhibits properties (e.g. smoothness, locality, invariance to shape transformations) that make them useful to processing and analysing geometric data. Spectral distances are easily defined through a filtering of the Laplacian eigenpairs and reduce to the heat diffusion, wave, biharmonic and commute‐time distances for specific filters. In particular, the smoothness of the spectral distances and the encoding of local and global shape properties depend on the convergence of the filtered eigenvalues to zero. Instead of applying a truncated spectral approximation or prolongation operators, we propose a computation of Laplacian distances and kernels through the solution of sparse linear systems. Our approach is free of user‐defined parameters, overcomes the evaluation of the Laplacian spectrum and guarantees a higher approximation accuracy than previous work.     

Multi‐Scale Kernels Using Random Walks

We introduce novel multi‐scale kernels using the random walk framework and derive corresponding embeddings and pairwise distances. The fractional moments of the rate of continuous time random walk (equivalently diffusion rate) are used to discover higher order kernels (or similarities) between pair of points. The formulated kernels are isometry, scale and tessellation invariant, can be made globally or locally shape aware and are insensitive to partial objects and noise based on the moment and influence parameters. In addition, the corresponding kernel distances and embeddings are convergent and efficiently computable. We introduce dual Green's mean signatures based on the kernels and discuss the applicability of the multi‐scale distance and embedding. Collectively, we present a unified view of popular embeddings and distance metrics while recovering intuitive probabilistic interpretations on discrete surface meshes.     

Computing Surface PolyCube‐Maps by Constrained Voxelization

We present a novel method to compute bijective PolyCube‐maps with low isometric distortion. Given a surface and its pre‐axis‐aligned shape that is not an exact PolyCube shape, the algorithm contains two steps: (i) construct a PolyCube shape to approximate the pre‐axis‐aligned shape; and (ii) generate a bijective, low isometric distortion mapping between the constructed PolyCube shape and the input surface. The PolyCube construction is formulated as a constrained optimization problem, where the objective is the number of corners in the constructed PolyCube, and the constraint is to bound the approximation error between the constructed PolyCube and the input pre‐axis‐aligned shape while ensuring topological validity. A novel erasing‐and‐filling solver is proposed to solve this challenging problem. Centeral to the algorithm for computing bijective PolyCube‐maps is a quad mesh optimization process that projects the constructed PolyCube onto the input surface with high‐quality quads. We demonstrate the efficacy of our algorithm on a data set containing 300 closed meshes. Compared to state‐of‐the‐art methods, our method achieves higher practical robustness and lower mapping distortion.     

Spectral Affine‐Kernel Embeddings

In this paper, we propose a controllable embedding method for high‐ and low‐dimensional geometry processing through sparse matrix eigenanalysis. Our approach is equally suitable to perform non‐linear dimensionality reduction on big data, or to offer non‐linear shape editing of 3D meshes and pointsets. At the core of our approach is the construction of a multi‐Laplacian quadratic form that is assembled from local operators whose kernels only contain locally‐affine functions. Minimizing this quadratic form provides an embedding that best preserves all relative coordinates of points within their local neighborhoods. We demonstrate the improvements that our approach brings over existing nonlinear dimensionality reduction methods on a number of datasets, and formulate the first eigen‐based as‐rigid‐as‐possible shape deformation technique by applying our affine‐kernel embedding approach to 3D data augmented with user‐imposed constraints on select vertices.     

On the Optimality of Valence-based Connectivity Coding

We show that the average entropy of the distribution of valences in valence sequences for the class of manifold 3D triangle meshes and the class of manifold 3D polygon meshes is strictly less than the entropy of these classes themselves. This implies that, apart from a valence sequence, another essential piece of information is needed for valence‐based connectivity coding of manifold 3D meshes. Since there is no upper bound on the size of this extra piece of information, the result implies that the question of optimality of valence‐based connectivity coding is still open.     

Stability of nonlinear feedback systems in a Volterra representation

Presented in this paper is a stability condition for a class of nonlinear feedback systems where the plant dynamics can be represented by a finite series of Volterra kernels. The class of Volterra kernels are limited to p‐linear stable operators and may contain pure delays. The stability condition requires that the linear kernel is non‐zero and that the closed loop characteristic equation associated with the linearized system is stable. Next, a sufficient condition is developed to upper bound the infinity‐norm of an external disturbance signal thereby guaranteeing that the internal and output signals of the closed loop nonlinear system are contained in L_∞. These results are then demonstrated on a design example. A frequency domain controller design procedure is also developed using these results where the trade‐off between performance and stability are considered for this class of nonlinear feedback systems. Copyright © 2000 John Wiley & Sons, Ltd.     

Smooth relevance vector machine: a smoothness prior extension of the RVM

Enforcing sparsity constraints has been shown to be an effective and efficient way to obtain state-of-the-art results in regression and classification tasks. Unlike the support vector machine (SVM) the relevance vector machine (RVM) explicitly encodes the criterion of model sparsity as a prior over the model weights. However the lack of an explicit prior structure over the weight variances means that the degree of sparsity is to a large extent controlled by the choice of kernel (and kernel parameters). This can lead to severe overfitting or oversmoothing—possibly even both at the same time (e.g. for the multiscale Doppler data). We detail an efficient scheme to control sparsity in Bayesian regression by incorporating a flexible noise-dependent smoothness prior into the RVM. We present an empirical evaluation of the effects of choice of prior structure on a selection of popular data sets and elucidate the link between Bayesian wavelet shrinkage and RVM regression. Our model encompasses the original RVM as a special case, but our empirical results show that we can surpass RVM performance in terms of goodness of fit and achieved sparsity as well as computational performance in many cases. The code is freely available. Action Editor: Dale Schuurmans.     

Reproducing kernel Hilbert spaces and variable metric algorithms in PDE-constrained shape optimization

In this paper we investigate and compare different gradient algorithms designed for the domain expression of the shape derivative. Our main focus is to examine the usefulness of kernel reproducing Hilbert spaces for PDE-constrained shape optimization problems. We show that radial kernels provide convenient formulas for the shape gradient that can be efficiently used in numerical simulations. The shape gradients associated with radial kernels depend on a so-called smoothing parameter that allows a smoothness adjustment of the shape during the optimization process. Besides, this smoothing parameter can be used to modify the movement of the shape. The theoretical findings are verified in a number of numerical experiments.     

渐进插值的LOOP曲面细分*

目前很多细分方法都存在不能用同一种方法处理封闭网格和开放网格的问题。对此,一种新的基于插值技术的LOOP曲面细分方法,其主要思想就是给定一个初始三角网格M,反复生成新的顶点,新顶点是通过其相邻顶点的约束求解得到的,从而构造一个新的控制网格M,在取极限的情况下,可以证明插值过程是收敛的;因为生成新顶点使用的是与其相连顶点的约束求解得到的,本质上是一种局部方法,所以,该方法很容易定义。它在本地方法和全局方法中都有优势,能处理任意顶点数量和任意拓扑结构的网格,从而产生一个光滑的曲面并忠实于给定曲面的形状,其控制     

Fluid-structure interactions using different mesh motion techniques

In this work, we compare different mesh moving techniques for monolithically-coupled fluid-structure interactions in arbitrary Lagrangian–Eulerian coordinates. The mesh movement is realized by solving an additional partial differential equation of harmonic, linear-elastic, or biharmonic type. We examine an implementation of time discretization that is designed with finite differences. Spatial discretization is based on a Galerkin finite element method. To solve the resulting discrete nonlinear systems, a Newton method with exact Jacobian matrix is used. Our results show that the biharmonic model produces the smoothest meshes but has increased computational cost compared to the other two approaches.     

Biharmonic Coordinates

Barycentric coordinates are an established mathematical tool in computer graphics and geometry processing, providing a convenient way of interpolating scalar or vector data from the boundary of a planar domain to its interior. Many different recipes for barycentric coordinates exist, some offering the convenience of a closed‐form expression, some providing other desirable properties at the expense of longer computation times. For example, harmonic coordinates, which are solutions to the Laplace equation, provide a long list of desirable properties (making them suitable for a wide range of applications), but lack a closed‐form expression. We derive a new type of barycentric coordinates based on solutions to the biharmonic equation. These coordinates can be considered a natural generalization of harmonic coordinates, with the additional ability to interpolate boundary derivative data. We provide an efficient and accurate way to numerically compute the biharmonic coordinates and demonstrate their advantages over existing schemes. We show that biharmonic coordinates are especially appealing for (but not limited to) 2D shape and image deformation and have clear advantages over existing deformation methods.     

A posteriori error estimation for the heterogeneous Maxwell equations on isotropic and anisotropic meshes

We consider residual-based a posteriori error estimators for the heterogeneous Maxwell equations using isotropic as well as anisotropic meshes. The continuous problem is approximated by using conforming approximated spaces with minimal assumptions. Lower and upper bounds are obtained under standard assumptions on the meshes. The lower bound holds unconditionally, while the upper bound depends on alignment properties of the meshes with respect to the solution. In particular for isotropic meshes the upper bound also holds unconditionally. A numerical test is presented which confirms our theoretical results.     

A Linear Variational System for Modelling From Curves

We present a linear system for modelling 3D surfaces from curves. Our system offers better performance, stability and precision in control than previous non‐linear systems. By exploring the direct relationship between a standard higher‐order Laplacian editing framework and Hermite spline curves, we introduce a new form of Cauchy constraint that makes our system easy to both implement and control. We introduce novel workflows that simplify the construction of 3D models from sketches. We show how to convert existing 3D meshes into our curve‐based representation for subsequent editing and modelling, allowing our technique to be applied to a wide range of existing 3D content.     

Feature-preserving adaptive mesh generation for molecular shape modeling and simulation

We describe a chain of algorithms for molecular surface and volumetric mesh generation. We take as inputs the centers and radii of all atoms of a molecule and the toolchain outputs both triangular and tetrahedral meshes that can be used for molecular shape modeling and simulation. Experiments on a number of molecules are demonstrated, showing that our methods possess several desirable properties: feature-preservation, local adaptivity, high quality, and smoothness (for surface meshes). We also demonstrate an example of molecular simulation using the finite element method and the meshes generated by our method. The approaches presented and their implementations are also applicable to other types of inputs such as 3D scalar volumes and triangular surface meshes with low quality, and hence can be used for generation/improvement of meshes in a broad range of applications.     

Critical parameters of integral delay systems

The critical frequencies of integral delay systems with a class of analytic kernels are determined via an auxiliary delay‐free system, and an upper bound on the number of critical frequencies is obtained. The critical delays are obtained by substituting these frequencies into the characteristic equation of the system. The procedure is validated with a number of nontrivial examples. Copyright © 2013 John Wiley & Sons, Ltd.     

An Evaluation of the Sparsity Degree for Sparse Recovery with Deterministic Measurement Matrices

The paper deals with the estimation of the maximal sparsity degree for which a given measurement matrix allows sparse reconstruction through ? ₁-minimization. This problem is a key issue in different applications featuring particular types of measurement matrices, as for instance in the framework of tomography with low number of views. In this framework, while the exact bound is NP hard to compute, most classical criteria guarantee lower bounds that are numerically too pessimistic. In order to achieve an accurate estimation, we propose an efficient greedy algorithm that provides an upper bound for this maximal sparsity. Based on polytope theory, the algorithm consists in finding sparse vectors that cannot be recovered by ? ₁-minimization. Moreover, in order to deal with noisy measurements, theoretical conditions leading to a more restrictive but reasonable bounds are investigated. Numerical results are presented for discrete versions of tomography measurement matrices, which are stacked Radon transforms corresponding to different tomograph views.     

Local input‐to‐state stabilization and ℓ ∞ ‐induced norm control of discrete‐time quadratic systems

This paper addresses the problems of local stabilization and control of open‐loop unstable discrete‐time quadratic systems subject to persistent magnitude bounded disturbances and actuator saturation. Firstly, for some polytopic region of the state‐space containing the origin, a method is derived to design a static nonlinear state feedback control law that achieves local input‐to‐state stabilization with a guaranteed stability region under nonzero initial conditions and persistent bounded disturbances. Secondly, the stabilization method is extended to deliver an optimized upper bound on the ? _∞ ‐induced norm of the closed‐loop system for a given set of persistent bounded disturbances. Thirdly, the stabilization and ? _∞ designs are adapted to cope with actuator saturation by means of a generalized sector bound constraint. The proposed controller designs are tailored via a finite set of state‐dependent linear matrix inequalities. Numerical examples are presented to illustrate the potentials of the proposed control design methods. Copyright © 2014 John Wiley & Sons, Ltd.