Mesh Deformation with Penalty-Based Collision Response

Abstract

In this short note I consider the problem of triangular mesh deformation in R3 with simultaneous collision response between the deformed mesh and other fixed objects in the scene. My aim is an interactive mesh modeling system with quasi-static mesh deformation where the user manipulates a set of handles and the system helps him (interactively or in a post-processing step) to avoid or reduce collisions. In recent years a number of physically plausible mesh deformation methods were introduced that are based on finding the deformed geometry as a (local) minimum of a specifically designed deformation energy (see, e.g., [1, 2, 7]). In this note I will focus mainly on As-Rigid-As-Possible (ARAP) mesh deformation [7] due to its simplicity, sufficiently plausible results and good performance for the intended application in anatomical modeling. In ARAP, a mesh deformation energy is defined in terms of the current and deformed mesh geometry. It is a non-linear energy that tries to keep the deformation locally as close as possible to rigid transformation while satisfying user constraints. These are typically given by a set of fixed vertices (that do not change their position during the deformation) and a set of handle vertices (that are manipulated interactively by the user and are not affected by the deformation). Here, I combine ARAP mesh deformation with a standard penalty-based approach to collision response. In penalty methods artificial forces are introduced into the system when a collision is detected. The force is proportional to the penetration depth and its magnitude and direction act to resolve the collision (see, e.g., [8] and references therein). A nice and successful combination of physically plausible energy-based mesh deformation with penalty-based response to collision was presented in [5]. In their approach, the authors introduce a penalty force in the form of a spring with zero rest length and anisotropic Young\u27s modulus. In this note, the ARAP deformation energy is modified by adding a penalty term that penalizes vertex positions that are inside or near other objects in the scene. It consists of a smooth scalar function that takes the distance from a vertex to another mesh as its parameter. The choice of this function gives flexibility in the desired behavior of the response to collision (e.g., response to collision after it occured or repulsion before the collision occurs). Note that this approach does not guarantee that all collisions are avoided or resolved. Also, self-collisions are not considered