Approximation Approaches For Complex Environment Modeling And Rendering

In mathematics, an approximation is an inexact representation of function/that is still close enough to be useful. In modeling and rendering, approximating a complex model into a simplified representation is a good strategy to reduce the computation and run applications faster. Though, there have been many progresses in Approximation Theory and Function Analysis, due to the complexity of objects used in Computer Graphics, these existing solutions from mathematics can not be directly applied to decipher problems in Computer Graphics. Thus, we need to propose new approaches for these practical problems that arise from particular applications.In this paper, aiming to solve the approximation problems of modeling and rendering in complex scenes, we have proposed a uniform framework by optimizing the global error to obtain the best approximation. Under such a uniform optimization framework, different types of approximating functions are used for different applications. Based on the types of approximating functions, we carry on our work as follow.1) We take research on one type of functions that are defined on parametric domain and can be formulated analytically. Due to good properties of these functions, we are able to take smooth functions to approximate them and use methods developed in Approximation Theory and Function Analysis. In practice, the diffusion function in the Dipole approximation for Bidirectional Surface Scattering Reflectance Distribution Function is one of these functions with good properties. We present a new factoring form and decomposition method to approximate it. Under our approximation, the translucency of a translucent object can be edited and rendered real-time.2) We take research on approximating more complicated functions that can not be formulated analytically. The bounding volumes approximation for triangle meshes in 3D space is one of such approximation problems. As the triangle mesh is composed by discrete triangles, the methods proposed for approximating smooth function can not be applied any more. To solve it, we propose a new error metric defined on the outside volume and a new optimization strategy based on Llyod clustering. Compared with previous work, our approach is better not only on visual results but also in some time-critial applications, such as real-time shadow computation and collision detection.3) We take research on approximating spatial structures. These bounding volumes used for approximating triangle meshes are simply organized as a union in the 3D space. However, in the tree modeling, not only the shapes of trees are needed to be approximated, but also the rules to grow these shapes should be taken into consideration. In our approach, we take the intrinsic rules that shape these spatial distributions of the stems, flowers and leaves in one type of trees as approximated function and propose a new stochastic representation for trees to approximate these growth rules. After solving such an approximation problem from tree samples acquired from real world, we are able to use the stochastic tree model to accelerate the tree modeling. A bunch of realistic trees, which are similar but visually different, can be generated fast.