The wavelet transform is a simple and elegant tool that can be used for many digital signal processing applications. It overcomes some of the limitations of the Fourier transform with its ability to represent a function simultaneously in the frequency and time domains using a single prototype function (or wavelet) and its scales and shifts. This webpage focuses specifically on two recently developed discrete wavelet transforms (DWTs), namely the double-density DWT and the double-density complex DWT.

The double-density DWT is an improvement upon the critically sampled DWT with important additional properties: (1) It employs one scaling function and two distinct wavelets, which are designed to be offset from one another by one half, (2) The double-density DWT is overcomplete by a factor of two, and (3) It is nearly shift-invariant. In two dimensions, this transform outperforms the standard DWT in terms of denoising; however, there is room for improvement because not all of the wavelets are directional. That is, although the double-density DWT utilizes more wavelets, some lack a dominant spatial orientation, which prevents them from being able to isolate those directions.

A solution to this problem is provided by the double-density complex DWT, which combines the characteristics of the double-density DWT and the dual-tree DWT. The double-density complex DWT is based on two scaling functions and four distinct wavelets, each of which is specifically designed such that the two wavelets of the first pair are offset from one other by one half, and the other pair of wavelets form an approximate Hilbert transform pair. By ensuring these two properties, the double-density complex DWT possesses improved directional selectivity and can be used to implement complex and directional wavelet transforms in multiple dimensions.

To understand each transform's behavior, this project utilizes the MATLAB programming environment to implement the execution of both DWTs on 1-D signals and 2-D images. We construct the filter bank structures for both the double-density DWT and the double-density complex DWT using finite impulse response (FIR) perfect reconstruction filter banks, which are discussed in detail at the beginning of each section. These filter banks are then applied recursively to the lowpass subband, using the analysis filters for the forward transform and the synthesis filters for the inverse transform. By doing this, it is then possible to evaluate each transform's performance in several applications including signal denoising, image enhancement.