The Projection Method for Multidimensional Framelet and Wavelet Analysis

Abstract

The projection method is a simple way of constructing functions and filters by integrating multidimensional functions and filters along parallel superplanes in the space domain. Equivalently expressed in the frequency domain, the projection method constructs a new function by simply taking a cross-section of the Fourier transform of a multidimensional function. The projection method is linked to several areas such as box splines in approximation theory and the projection-slice theorem in image processing. In this paper, we shall systematically study and discuss the projection method in the area of multidimensional framelet and wavelet analysis. We shall see that the projection method not only provides a painless way for constructing new wavelets and framelets but also is a useful analysis tool for studying various optimal properties of multidimensional refinable functions and filters. Using the projection method, we shall explicitly and easily construct a tight framelet filter bank from every box spline filter having at least order one sum rule. As we shall see in this paper, the projection method is particularly suitable to be applied to frequency-based nonhomogeneous framelets and wavelets in any dimensions, and the periodization technique is a special case of the projection method for obtaining periodic wavelets and framelets from wavelets and framelets on Euclidean spaces.