A Fast Geometric Multigrid Method for Curved Surfaces

We introduce a geometric multigrid method for solving linear systems arising from variational problems on surfaces in geometry processing, Gravo MG. Our scheme uses point clouds as a reduced representation of the levels of the multigrid hierarchy to achieve a fast hierarchy construction and to extend the applicability of the method from triangle meshes to other surface representations like point clouds, nonmanifold meshes, and polygonal meshes. To build the prolongation operators, we associate each point of the hierarchy to a triangle constructed from points in the next coarser level. We obtain well-shaped candidate triangles by computing graph Voronoi diagrams centered around the coarse points and determining neighboring Voronoi cells. Our selection of triangles ensures that the connections of each point to points at adjacent coarser and finer levels are balanced in the tangential directions. As a result, we obtain sparse prolongation matrices with three entries per row and fast convergence of the solver.Comment: Ruben Wiersma and Ahmad Nasikun contributed equally. To be published in SIGGRAPH 2023. 16 pages total (8 main, 5 supplement), 14 figure

Animation space: a truly linear framework for character animation

Skeletal subspace deformation (SSD), a simple method of character animation used in many applications, has several shortcomings; the best-known being that joints tend to collapse when bent. We present animation space, a generalization of SSD that greatly reduces these effects and effectively eliminates them for joints that do not have an unusually large range of motion.While other, more expensive generalizations exist, ours is unique in expressing the animation process as a simple linear transformation of the input coordinates. We show that linearity can be used to derive a measure of average distance (across the space of poses), and apply this to improving parametrizations.Linearity also makes it possible to fit a model to a set of examples using least-squares methods. The extra generality in animation space allows for a good fit to realistic data, and overfitting can be controlled to allow fitted models to generalize to new poses. Despite the extra vertex attributes, it is possible to render these animation-space models in hardware with no loss of performance relative to SSD

Complex Functional Maps : a Conformal Link Between Tangent Bundles

International audienceIn this paper, we introduce complex functional maps, which extend the functional map framework to conformal maps between tangent vector fields on surfaces. A key property of these maps is their orientation awareness. More specifically, we demonstrate that unlike regular functional maps that link functional spaces of two manifolds, our complex functional maps establish a link between oriented tangent bundles, thus permitting robust and efficient transfer of tangent vector fields. By first endowing and then exploiting the tangent bundle of each shape with a complex structure, the resulting operations become naturally orientationaware, thus favoring orientation and angle preserving correspondence across shapes, without relying on descriptors or extra regularization. Finally, and perhaps more importantly, we demonstrate how these objects enable several practical applications within the functional map framework. We show that functional maps and their complex counterparts can be estimated jointly to promote orientation preservation, regularizing pipelines that previously suffered from orientation-reversing symmetry errors

Grid-based Finite Elements System for Solving Laplace-Beltrami Equations on 2-Manifolds

Solving the Poisson equation has numerous important applications. On a Riemannian 2-manifold, the task is most often formulated in terms of finite elements and two challenges commonly arise: discretizing the space of functions and solving the resulting system of equations. In this work, we describe a finite elements system that simultaneously addresses both aspects. The idea is to define a space of functions in 3D and then restrict the 3D functions to the mesh. Unlike traditional approaches, our method is tessellation-independent and has a direct control over system complexity. More importantly, the resulting function space comes with a multi-resolution structure supporting an efficient multigrid solver, and the regularity of the function space can be leveraged in parallelizing/streaming the computation. We evaluate our framework by conducting several experiments. These include a spectral analysis that reveals the embedding-invariant robustness of our discretization, and a benchmark for solver convergence/performance that reveals the competitiveness of our approach against other state-of-the-art methods. We apply our work to several geometry-processing applications. Using curvature flows, we show that we can support efficient surface evolution where the embedding changes with time. Formulating surface filtering as a solution to the screened-Poisson equation, we demonstrate that we can support an anisotropic surface editing system that processes high resolution meshes in real time

User-appropriate viewer for high resolution interactive engagement with 3D digital cultural artefacts.

The core mission of museums and cultural institutions is the preservation, study and presentation of cultural heritage content. In this technological age, the creation of digital datasets and archives has been widely adopted as one way of seeking to achieve some or all of these goals. However, there are many challenges with the use of these data, and in particular the large numbers of 3D digital artefacts that have been produced using methods such as non- contact laser scanning. As public expectation for more open access to information and innovative digital media increases, there are many issues that need to be rapidly addressed. The novel nature of 3D datasets and their visualisation presenting unique issues that impede use and dissemination. Key questions include the legal issues associated with 3D datasets created from cultural artefacts; the complex needs of users who are interacting with them; a lack of knowledge to texture and assess the visual quality of the datasets; and how the visual quality of the presented dataset relates to the perceptual experience of the user. This engineering doctorate, based on an industrial partnership with the National Museums of Liverpool and Conservation Technologies, investigates these questions and offers new ways of working with 3D cultural heritage datasets. The research outcomes in the thesis provide an improved understanding of the complexity of intellectual property law in relation to 3D cultural heritage datasets and how this impacts dissemination of these types of data. It also provides tools and techniques that can be used to understand the needs of a user when interacting with 3D cultural content. Additionally, the results demonstrate the importance of the relationship between texture and polygonal resolution and how this can affect the perceived visual experience of a visitor. It finds that there is an acceptable cost to texture and polygonal resolution to offer the best perceptual experience with 3D digital cultural heritage. The results also demonstrate that a non-textured mesh may be as highly received as a high resolution textured mesh. The research presented provides methodologies and guidelines to improve upon the dissemination and visualisation of 3D cultural content; enhancing and communicating the significance of their 3D collections to their physical and virtual visitors. Future opportunities and challenges for disseminating and visualising 3D cultural content are also discussed