Wavelet Radiosity: Wavelet Methods for Integral Equations

In this chapter we will explain how wavelets can be used to solve integral equations. The example we use is an important integral equation in graphics, the radiosity equation. The radiosity equation governs the transport of light between surfaces under the assumption that all reflection occurs isotropically. The resulting integral equation is linear and can be analyzed as a linear operator. Since wavelets can be used as bases for function spaces, linear operators can be expressed in them. If these operators satisfy certain smoothness conditions—as radiosity does— the resulting matrices are approximately sparse and can be solved asymptotically faster if only finite precision is required of the answer. We develop this subject by first introducing the Galerkin method which is used to solve integral equations. Applying the method results in a linear system whose solution approximates the solution of the original integral equation. This discussion is kept very general. In a subsequent section the realization of linear operators in wavelet bases is discussed. There we will show why the vanishing moment property of wavelets results in (approximately) sparse matrix systems for integral operators of the Calderon-Zygmund type. After these foundations we change gears and describe some techniques recently introduced in the radiosity literature. A technique, known as Hierarchical Radiosity, is shown to be equivalent to the use of the Haar basis in the context of solving integral equations. Treating this example in more detail allows us to fill many of the mathematical definitions with geometric intuition. Finally we discuss the implementation of a particular wavelet radiosity algorithm and the construction of an oracle function which is crucial for a linear time algorithm. In general we will concentrate on the arguments and intuition behind the use of wavelet methods for integral equations and in particular their application to radiosity. Many of the implementation details will be deliberately abstracted and they can be found by the interested reader in the references ([21, 11, 14]).