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:

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

where f ( x ), f j 0 ,k (X) , and ? 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.

j 0 ,k (X) is a member of the set of expansion functions derived from a scaling function ?(X) , by translation and scaling using:

j, k ( x ) is a member of the set of wavelets derived from a wavelet function ?( x ), by translation and scaling using:

The DWT can be formulated as a filtering operation with two related FIR filters, low-pass filter h f and high-pass filter h ? . Both W f ( j,k ) and W ( j,k ), the scale j approximation and the detail coefficients, can be computed by convolving W f ( j + 1, k ), the scale j +1 approximation coefficients, with the time-reversed scaling and wavelet vectors, h (- n ) and h (- n ), and sub-sampling the results by 2, as expressed in Equations (6) and (7) and illustrated in Figure 1.

The filter bank in Figure 1 can be iterated to implement multi-resolution analysis. The IDWT can be implemented by up-sampling and synthesis filtering. The one-dimensional DWT and IDWT can be extended to two-dimensional.