Development of a 2-D Wavelet Transform based on Kronecker Product

Abstract

Advances in wavelet transform have produces algorithms capable of surpassing the existing digital signal processing. This paper presents a new wavelet transform computation method that verifies the potential benefits of the kronecker product and gains much improvement in terms of low computational complexity. A fast algorithm for computing 2-D wavelet transform based on a modified orthogonal matrix is developed using kronecker product. The algorithm has several promising features which make it suitable for 2-D signal processing applications. Introduction The discrete wavelet transform (DWT) is found to be an efficient and useful tool for signal processing applications [1, 2]. It has become a powerful tool in various digital processing such as audio, image and multimedia [1-3]. Fast computation methods introduced recently applied in several approaches for directional representations of image data, each one with the intent of efficiently representing image features. Among these examples include Ridglet [4], Curvelets [5], Contourlets [6], and Shearlets [7]. As the choice of transform used depends in particular, on computational complexity which is measured in terms of the number of multiplications and additions required for the implementation of the transform. This paper facilitates the computation of discrete 2-D wavelet transform involving a computation procedure consisting mainly of basic arithmetic operations like matrix multiplication, permutations, shuffling and other easy to verify operations. A simple way to perform wavelet decomposition on 2-D signal is to alternate between operations on the rows and columns. First, wavelet decomposition is performed on each row of the 2-D signal matrix. Then, wavelet decomposition is performed to each column of the previous result. The process is repeated to perform the complete wavelet decomposition. The basic idea developed in this paper is that of solving the problem of cascaded two steps multiplication of 2-D DWT computation into only one step of multiplication. A particular set of wavelets is specified by a particular set of numbers, called wavelet filter coefficients. For simplicity we will restrict to wavelet filters in a class discovered by Daubechies. The most localized embers often used are Haar and Daubechies 4 (Db4) coefficients. For easy of notation we will use the notation h(0) and h(1) for Haar coefficients and h(0), h(1), h(2), and h(3) for Daubechies 4 coefficients. In the 2-D Haar bases wavelet, the matrix form will be as