Partial differential equations can be solved on general polygonal and polyhedral meshes, through Polytopal Element Methods (PEMs). Unfortunately, the relation between geometry and analysis is still unknown and subject to ongoing research in order to identify weaker shape-regularity criteria under which PEMs can reliably work. We propose PEMesh, a graphical framework to support the analysis of the relation between the geometric properties of polygonal meshes and the numerical performances of PEM solvers. PEMesh allows the design of polygonal meshes that increasingly stress some geometric properties, by exploiting any external PEM solver, and supports the study of the correlation between the performances of such a solver and geometric properties of the input mesh. Furthermore, it is highly modular, customisable, easy to use, and provides the possibility to export analysis results both as numerical values and graphical plots. PEMesh has a potential practical impact on ongoing and future research activities related to PEM methods, polygonal mesh generation and processing.
Estimating the differential properties of a signal sampled on a surface is of paramount importance in many fields of applied sciences. In the common practice, the surface is discretized with a polygonal mesh, the signal is sampled at its vertices and extended linearly over the triangles. This means that the polyhedral metric is assumed over the surface; the first derivatives of the signal become discontinuous across edges; and the second derivatives vanish. We present a new method based on surface fitting, which efficiently estimates the metric tensor, and the first and second order Riemannian differential operators at any point on the surface. All our differential operators are smooth within each triangle and continuous across the edges, providing a much better estimate of differential quantities on the - yet unknown - underlying smooth manifold.
The following paper proposes the study, the design and the preliminary development of a solution for supporting users without programming experience in creating stories in a Mixed Reality environment. We focus on a Mixed Reality interface split into two parts: the creation and observation phases. During the creation phase, the end user can build his/her own story in the immersive mode of the Mixed Reality experience. The user can also enjoy the stories that other users have designed by seeing the characters appear in their surrounding environment.
In this paper, we discuss the problem of converting a 3D mesh into an assembly of LEGO blocks. The major challenge of this task is how to aggregate the voxels derived by the shape discretization into a set of standard bricks guaranteeing global connectivity. Weproposean outside-in priority-based heuristic method based on the analysis of the critical regions that are more likely to cause the creation of a legal assembly to fail. We show that our graph-building heuristic provides relevant advantages, making it easier to obtain a connected graph with good properties with respect to the layer-based or random aggregation strategies applied in most of the optimization approaches. We also propose BRICKS, a novel dataset for the evaluation of aggregation strategies. It includes voxelizations at 3 different resolutions of 33 shapes and allows the easy comparison of different voxel aggregation strategies independently of the shape discretization step and also considering their scalability. We use it to evaluate our approach with respect to graph-based connectivity measures, showing the advantages of the proposed strategy.
Shape matching is a central problem in geometry processing applications, ranging from texture transfer to statistical shape analysis. The functional maps framework provides a compact representation of correspondences between discrete surfaces, which is then converted into point-wise maps required by real-world applications. The vast majority of methods based on functional maps involve the eigenfunctions of the Laplace-Beltrami Operator (LB) as the functional basis. A primary drawback of the LB basis is that its energy does not uniformly cover the surface. This fact gives rise to regions where the estimated correspondences are inaccurate, typically at tiny parts and protrusions. For this reason, state-of-the-art procedures to convert the functional maps (represented in the LB basis) into point-wise correspondences are often error-prone. We propose PC-G AU , a new functional basis whose energy spreads on the whole shape more evenly than LB. As such, PC-G AU can replace the LB basis in existing shape matching pipelines. PC-G AU consists of the principal vectors obtained by applying Principal Component Analysis (PCA) to a dictionary of sparse Gaussian functions scattered on the surfaces. Through experimental evaluation of established benchmarks, we show that our basis produces more accurate point-wise maps — compared to LB — when employed in the same shape-matching pipeline
