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]:

((1))

((2))

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

((3))

where f(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) are usually called approximation and detail coefficients, respectively.

is a member of the set of expansion functions derived from a scaling functionϕ(x), by translation and scaling using:

((4))

Ψ _j,k (x) is a member of the set of wavelets derived from a wavelet functionΨ(x), by translation and scaling using:

((5))

The DWT can be formulated as a filtering operation with two related FIR filters, low-pass filter and high-pass...