Vector graph representation for 3D meshes and algorithms for mesh deformation, enveloping, interpolation and morphing

Abstract

The performance of the tools developed for graphics and animation depends on the underlying representation of the geometric model of the objects. The geometric model in turn describes the geometry of the real world object. In this work, we propose a unified framework for mesh editing based on the Vector Graph (VG) representation of the geometric model. The VG representation is an instance of the barycentric coordinates. It represents various geometric models such as triangular, quad and hybrid meshes as a collection of vectors. For various mesh editing applications, the deformations of the geometric models are considered in terms of the deformation of the VG representation. For this task, the deformation of a vector of the VG is modelled in terms of rotation and scaling which are orientation preserving transformations. Moreover, the composition is commutative and forms a matrix group which is also a smooth manifold. The simplicity of the VG allows for the straight forward formulations of mesh editing approaches, such as Deformation Transfer, Enveloping, Interpolation and Morphing. For the proposed method, the major computation is involved in solving an verdetermined system of linear equations with appropriate constraints. The efficient methods exist for solving this systems at a large scale. This ensures the real-time execution of the algorithm for large geometric models. In this work, the proposed algorithms for Deformation Transfer, enveloping, interpolation and morphing using the VG representations are compared qualitatively and quantitatively with state-of-the-art methods. In addition, we have proposed a face selection algorithm which reduces the computational complexity without compromising on the quality of editing. This can be applied to other face based formulations also. Apart from computational efficiency, based on the experiments, it is encouraging to note that (a) the proposed Deformation Transfer method reserves the shape and the geometric details of the target mesh for a wide range of deformations, (b) the enveloped mesh using proposed enveloping method can be edited in real-time to generate complex poses without artifacts, (c) qualitatively the proposed interpolation scheme performs at par with the existing approaches and is computationally real-time compared to others and (d) proposed morphing is qualitatively similar to the other approaches. In this work, our objective is two fold. We first propose computationally efficient algorithms which ensure smooth and realistic deformations of the mesh for various applications and then implement these algorithms in Blender as prototypes for interactive user experience. The simplicity of the VG is reflected in the implementation of all the proposed algorithms in Blender 2.82. The created Blender add-ons are easy to use. We have included a detailed user manual for the developed add-ons as a part of the thesis.