The generalization of barycentric coordinates to arbitrary simple polygons with more than three vertices has been a subject of study for a long time. Among the different constructions proposed, mean ...value coordinates have emerged as a popular choice, particularly due to their suitability for the non-convex setting. Since their introduction, they have found applications in numerous fields, and several equivalent formulas for their evaluation have been presented in the literature. However, so far, there has been no study regarding their numerical stability. In this paper, we aim to investigate the numerical stability of the algorithms that compute mean value coordinates. We show that all the known methods exhibit instability in some regions of the domain. To address this problem, we introduce a new formula for computing mean value coordinates, explain how to implement it, and formally prove that our new algorithm provides a stable evaluation of mean value coordinates. We validate our results through numerical experiments.
•A new algorithm for evaluating mean value coordinates.•A formal proof that this algorithm is numerically stable.•A comparison of the numerical stability of all known algorithms for mean value coordinates.
In this paper we consider the problem of creating and controlling volume deformations used to articulate characters for use in high-end applications such as computer generated feature films. We ...introduce a method we call harmonic coordinates that significantly improves upon existing volume deformation techniques. Our deformations are controlled using a topologically flexible structure, called a cage, that consists of a closed three dimensional mesh. The cage can optionally be augmented with additional interior vertices, edges, and faces to more precisely control the interior behavior of the deformation. We show that harmonic coordinates are generalized barycentric coordinates that can be extended to any dimension. Moreover, they are the first system of generalized barycentric coordinates that are non-negative even in strongly concave situations, and their magnitude falls off with distance as measured within the cage.
Constructing a function that interpolates a set of values defined at vertices of a mesh is a fundamental operation in computer graphics. Such an interpolant has many uses in applications such as ...shading, parameterization and deformation. For closed polygons, mean value coordinates have been proven to be an excellent method for constructing such an interpolant. In this paper, we generalize mean value coordinates from closed 2D polygons to closed triangular meshes. Given such a mesh
P
, we show that these coordinates are continuous everywhere and smooth on the interior of
P
. The coordinates are linear on the triangles of
P
and can reproduce linear functions on the interior of
P
. To illustrate their usefulness, we conclude by considering several interesting applications including constructing volumetric textures and surface deformation.
•A coupled S-FEM and Lagrangian particle tracking model is presented for the solution of 3D dilute particle-laden flows.•The unified solution of particle-fluid systems on unstructured meshes with ...multi-type elements based on S-FEM.•A completely new application of S-FEM on multiphase flows.
In this study, a coupled solution algorithm for three-dimensional dilute particle-laden flows is proposed by integrating the Lagrangian particle tracking model into the smoothed finite element method (S-FEM). Initially, an unstructured mesh fluid solver with multi-type elements is developed using the cell-based S-FEM (CS-FEM) in the Eulerian framework. Subsequently, a fluid force-driven strategy is employed to trace the particle trajectories based on the Lagrangian approach. Moreover, a one-way coupling strategy is designed to perform the solution of the particle-fluid system. To ensure accurate computation of fluid information at the particle positions, we introduce the spherical mean value interpolation algorithm that is compatible with polyhedral elements, enabling uniform interpolation across different types of elements. The correctness of both the Eulerian and Lagrangian solvers is validated independently using benchmarks. Numerical results, validated by references and the finite volume method (FVM) software Fluent, demonstrate the effective prediction of particle trajectories and distributions by the proposed algorithm. Overall, this algorithm expands the application of CS-FEM to multiphase flows and exhibits its capability to handle practical particle-laden flow problems.
Seamless cloning of a source image patch into a target image is an important and useful image editing operation, which has received considerable research attention in recent years. This operation is ...typically carried out by solving a Poisson equation with Dirichlet boundary conditions, which smoothly interpolates the discrepancies between the boundary of the source patch and the target across the entire cloned area. In this paper we introduce an alternative,
coordinate-based
approach, where rather than solving a large linear system to perform the aforementioned interpolation, the value of the interpolant at each interior pixel is given by a weighted combination of values along the boundary. More specifically, our approach is based on Mean-Value Coordinates (MVC). The use of coordinates is advantageous in terms of speed, ease of implementation, small memory footprint, and parallelizability, enabling real-time cloning of large regions, and interactive cloning of video streams. We demonstrate a number of applications and extensions of the coordinate-based framework.
In this paper, we introduce a new parametric spline curve, named as P-spline curves. Given a control point set associated with a set of knots and parameters, we first define three sets of points ...using this set of knots and parameters. One set of points lie on a line segment spaced by knots, and two other sets of points lie on two sides of the line segment symmetrically according to the parameters and knots. Then, we compute the mean value coordinates of points on the middle line segment with respect to the two involved quadrilaterals, whose vertices are selected from the three defined point sets. Last, we use these coordinates and blending functions to construct basis functions, which are used to define P-spline curves together with given control point set. There are several desirable features for the P-spline curves. The continuous orders of the resulting curves are determined by the basis functions, and we can adjust the distance between the curves and control points by changing the parameters. Moreover, the construction of P-spline curves is simple, and the relations between the P-spline curves and knots/parameters are intuitive. More importantly, the influences of parameters and control points are local because the four vertices of each quadrilateral only depend on one parameter and three knots. Some numerical examples are used to show that P-spline curves are more local than NURBS (non-uniform rational B-spline) curves, P-curves and P-Bspline curves.
Barycentric coordinates provide a simple way of expressing the linear interpolant to data given at the vertices of a triangle and have numerous applications in computer graphics and other fields. The ...generalization of barycentric coordinates to polygons with more than three vertices is not unique and many constructions have been proposed. Among them, mean value coordinates stand out by having a simple closed form and being well-defined for arbitrary polygons, but they may take on large negative values in the case of concave polygons, leading to artefacts in applications like shape deformation. We present a modification of mean value coordinates that is based on the observation that the mean value coordinates of some point v inside a polygon can be negative if the central projection of the polygon onto the unit circle around v folds over. By iteratively smoothing the projected polygon and carrying over this smoothing procedure to the barycentric coordinates of v, these fold-overs as well as the negative coordinate values and shape deformation artefacts gradually disappear, and they are guaranteed to completely vanish after a finite number of iterations.
•We introduce a pipeline based on surface parametrization to remesh triangulations.•We give clear geometrical arguments to explain how ensuring one-to-one parametrizations.•Mean Value Coordinates ...(‘MVC’ Floater, 2003) are not a Laplacian on structured triangulations.•Holes in triangulations are virtually filled through MVC to improve the parametrization quality.•We show there is no Lagrange P2 version of MVC.
Triangulations are an ubiquitous input for the finite element community. However, most raw triangulations obtained by imaging techniques are unsuitable as-is for finite element analysis. In this paper, we give a robust pipeline for handling those triangulations, based on the computation of a one-to-one parametrization for automatically selected patches of input triangles, which makes each patch amenable to remeshing by standard finite element meshing algorithms. Using only geometrical arguments, we prove that a discrete parametrization of a patch is one-to-one if (and only if) its image in the parameter space is such that all parametric triangles have a positive area. We then derive a non-standard linear discretization scheme based on mean value coordinates to compute such one-to-one parametrizations, and show that the scheme does not discretize a Laplacian on a structured mesh. The proposed pipeline is implemented in the open source mesh generator Gmsh, where the creation of suitable patches is based on triangulation topology and parametrization quality, combined with feature edge detection. Several examples illustrate the robustness of the resulting implementation.
Summary
The rotation degree of freedom in discontinuous deformation analysis (DDA) may cause false volume expansion when a block undergoes a large rotation. We propose a block displacement function ...to prevent this defect. Specifically, the degrees of freedom in a block are redefined by incremental displacements at its vertices, and displacement is formulated based on the mean value coordinates. In addition, the finite element method with updated Lagrangian formulation is employed to derive the equilibrium equations, while the contact analysis and implicit time integration for dynamics is maintained from the original DDA. After each time step, the block configuration is updated by adding the new degrees of freedom to the previous coordinates of the block vertices. Results from numerical examples confirm the effectiveness of the proposed approach to prevent false volume expansion, ensure correctness of contact analysis, and provide realistic stress results when simulating large rotations.
Mean value coordinates for quad cages in 3D Thiery, Jean-Marc; Memari, Pooran; Boubekeur, Tamy
ACM transactions on graphics,
01/2018, Letnik:
37, Številka:
6
Journal Article
Recenzirano
Space coordinates offer an elegant, scalable and versatile framework to propagate (multi-)scalar functions from the boundary vertices of a 3-manifold, often called a
cage
, within its volume. These ...generalizations of the barycentric coordinate system have progressively expanded the range of eligible cages to triangle and planar polygon surface meshes with arbitrary topology, concave regions and a spatially-varying sampling ratio, while preserving a smooth diffusion of the prescribed on-surface functions. In spite of their potential for major computer graphics applications such as freeform deformation or volume texturing, current space coordinate systems have only found a moderate impact in applications. This follows from the constraint of having only triangles in the cage most of the time, while many application scenarios favor arbitrary (non-planar) quad meshes for their ability to align the surface structure with features and to naturally cope with anisotropic sampling. In order to use space coordinates with arbitrary quad cages currently, one must triangulate them, which results in large propagation distortion. Instead, we propose a generalization of a popular coordinate system - Mean Value Coordinates - to quad and tri-quad cages, bridging the gap between high-quality coarse meshing and volume diffusion through space coordinates. Our method can process non-planar quads, comes with a closed-form solution free from global optimization and reproduces the expected behavior of Mean Value Coordinates, namely smoothness within the cage volume and continuity everywhere. As a result, we show how these coordinates compare favorably to classical space coordinates on triangulated quad cages, in particular for freeform deformation.
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