Discrete Wavelet Transform (DWT)

Abstract

DefinitionDiscrete Wavelet Transform is a technique to transform image pixels into wavelets, which are then used for wavelet-based compression and coding.The DWT is defined as [1]:Wφ(j0,k)=1M‾‾√∑xf(x)φj0,k(x)Wφ(j0,k)=1M∑xf(x)φj0,k(x)W_\varphi (j_0 ,k) = {1\over\sqrt M }\sum\limits_{x} {f(x)\varphi _{j_0 ,k} } (x) (1)Wψ(j,k)=1M‾‾√∑kf(x)ψj,k(x)Wψ(j,k)=1M∑kf(x)ψj,k(x)W_\psi (j,k) = {1\over\sqrt M }\sum\limits_k {f(x)\psi _{j,k} (x)} (2)for j≥j0 and the Inverse DWT (IDWT) is defined as:f(x)=1M√∑kWφ(j0,k)φj0,k(x)+1M√∑j=j0∞∑kWψ(j,k)ψj,k(x).f(x)=1M∑kWφ(j0,k)φj0,k(x)+1M∑j=j0∞∑kWψ(j,k)ψj,k(x).\begin{array}{*{20}l}f(x) = & {1\over{\sqrt {M} }}\sum\limits_{k} {W_\varphi (j_0 ,k)\varphi_{j_0 ,k} (x)} \\ &+ {1\over{\sqrt M }}\sum\limits_{j = j_0 }^\infty{\sum\limits_{k} {W_\psi (\,j,k)\psi_{j,k}} (x).}\end{array} (3)where f(x), φj0,k(x)φj0,k(x)\varphi _{j_0 ,k} (x), and ψj,k (x) are functions of the discrete variable x = 0,1,2,…,M−1. Normally we let j0 = 0 and select M to be a power of 2 (i.e., M = 2J) so that the summations in Equations (1), (2) and (3) are performed over x = 0,1,2,…,M−1, j = 0,1,2,…, J−1, and k = 0,1,2,…,2 j − 1. The coefficients defined in Equations (1) and (2) are usually ...