Geometric Transformation by Moment Method with Wavelet Matrix

Abstract

A novel approach to geometric transformation, which can be modeled by a partial differential equation with boundary conditions, is presented in this chapter. Typical numerical methods, which are available for partial differential equation such as finite element method, finite difference method and moment method, are discussed. Based on the discussion, we propose a novel approach which consists of three stages: (i) Laplace equation (LE) is converted into a boundary integral equation (BIE) by the potential theory, (ii) based on the moment method, the boundary integral equation (BIE) is changed to a dense matrix equation (DME), and (iii) the dense matrix equation (DME) is transformed to a sparse matrix equation (SME) according to the wavelet matrix transform. It can considerably reduce the computation time in solving the linear equations. The method of construction of the wavelet matrix, especially for the multi-wavelet one, is presented. In this case, a preprocessor is needed. The choice of the threshold which can balance the sparsity and accuracy, is discussed. Our approach has two main advantages, namely, the pixels to be transformed are arbitrary and the computational cost is very low. The performance is evaluated through experiments.