Discrete Wavelet Transform (DWT)

Definition

Discrete 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]:

$$W_\varphi (j_0 ,k) = {1\over\sqrt M }\sum\limits_{x} {f(x)\varphi _{j_0 ,k} } (x)$$

((1))

$$W_\psi (j,k) = {1\over\sqrt M }\sum\limits_k {f(x)\psi _{j,k} (x)}$$

((2))

for j≥j ₀ and the Inverse DWT (IDWT) is defined as:

$$\begin{array}{*{20}l}f(x) = & {1\over{\sqrt {M} }}\sum\limits_{k} {W_\varphi (j_0 ,k)\varphi_{j_0 ,k} (x)} \\ &+ {1\over{\sqrt M }}\sum\limits_{j = j_0 }^\infty{\sum\limits_{k} {W_\psi (\,j,k)\psi_{j,k}} (x).}\end{array}$$

((3))

where f(x), $\varphi _{j_0 ,k} (x)$, and ψ _j,k (x) are functions of the discrete variable x = 0,1,2,…,M−1. Normally we let j ₀ = 0 and select M to be a power of 2 (i.e., M = 2^J) 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)...