Large mesh deformation using the volumetric graph Laplacian

Abstract

We present a novel technique for large deformations on 3D meshes using the volumetric graph Laplacian. We first construct a graph representing the volume inside the input mesh. The graph need not form a solid meshing of the input mesh's interior; its edges simply connect nearby points in the volume. This graph's Laplacian encodes volumetric details as the difference between each point in the graph and the average of its neighbors. Preserving these volumetric details during deformation imposes a volumetric constraint that prevents unnatural changes in volume. We also include in the graph points a short distance outside the mesh to avoid local self-intersections. Volumetric detail preservation is represented by a quadric energy function. Minimizing it preserves details in a least-squares sense, distributing error uniformly over the whole deformed mesh. It can also be combined with conventional constraints involving surface positions, details or smoothness, and efficiently minimized by solving a sparse linear system.We apply this technique in a 2D curve-based deformation system allowing novice users to create pleasing deformations with little effort. A novel application of this system is to apply nonrigid and exaggerated deformations of 2D cartoon characters to 3D meshes. We demonstrate our system's potential with several examples.

Quality encoding for tetrahedral mesh optimization

“…In detail-preserving shape editing, differential representations have gained significant popularity over the past few years [1,5,33,38]. Given a surface mesh, the differential coordinates of a vertex v i are defined by a displacement vector between v i and the weighted average of its 1-ring neighborhood:…”

“…In [38], the volume differential coordinates which encode the so-called volumetric details were introduced for volumepreserving mesh deformation. For a tetrahedral mesh, volume differential coordinates are defined through extending the 1-ring neighbor to the interior, where the 1-ring set is given by…”

“…Our approach is inspired by existing works on Laplacian mesh editing [1,5,33,38], where the main idea is to use a representation that captures the local differential properties of the surface and Figure 1: Given an input tetrahedral mesh (left), our method can well improve the mesh quality (middle). With harmonic field guidance, the quality of boundary tetrahedra (highlighted in red) can be better improved (right).…”

Interactive mesh deformation using equality-constrained least squares

“…Linear PDE-based approaches encode Laplacian vectors and positional constraints in a linear system and obtain the deformed shapes by solving the linear system [1][2][3][4][5]. A discrete Laplacian operator is defined at each vertex by the weighted sum of difference vectors between the vertex and its adjacent neighbors.…”

“…In recent years many discrete deformation techniques based on differential properties have been published [1][2][3][4][5]. They represent the differential properties of a given surface as a linear system and deform the surface so that the differential properties are preserved.…”