Physically-based Modeling for Graphics and Vision

The elastic properties of materials constrain the motion and dynamics of objects in the real world, hence modeling and simulating the physical characteristics of these objects is essential to obtain realistic computer modeling for graphics, vision and animation. This type of modeling is referred to as physically-based modeling and is the main focus of this chapter. A major problem for physically-based modeling has been to establish a unified approach for representing geometry that allows efficient calculation of both geometric relations (e.g., surface-to-point distances) and the non-rigid physical response to imposed forces. This problem has been addressed by researchers in both computer graphics [7, 38, 31, 44], and machine vision [40, 12, 25, 41, 30, 28]. In computer graphics, the central goal is efficient detection of collisions and simulation of the resulting non-rigid motion. In machine vision the hope is to develop general purpose models that are capable of accurately representing the free-form shapes and nonrigid motions of natural objects. As part of this chapter, we will present a general overview of this past research and then present our approach to 3-D physical modeling in graphics and vision. Our approach to geometric modeling is based on the use of parameterized implicit functions, particularly polynomial deformations of superquadric functions. The physical model is developed by discretizing the geometric model via the finite element method, making use of Galerkin polynomial approximations for shape interpolation. The geometric and physical models are then combined into a unified model by using the finite element modal shape functions as polynomial deformations applied to an underlying superquadric implicit function. These polynomial deformations are known as the object’s “deformation modes” and are derived directly from the finite element equations. This approach significantly reduces the computational cost of both geometric distance calculations and modeling of physical response. By describing the finite element equations in terms of the object’s free vibration modes, the system of equations is decoupled into a sum of independent, frequency-ordered deformations. This not only allows closed-form solutions of the finite element equations,but also allows us to smoothly trade accuracy against computational cost. Similarly, by using an implicit function representation for geometry we can greatly reduce the computational cost of contact detection and collision response. By combining the vibration mode deformations with an underlying implicit function representation, we obtain both of these advantages.