0.7 Generalizations of the basic multiresolution wavelet system  (Page 16/28)

There are at least two descriptions of the problem. We may want a single expansion system to handle several different classes of signals, each ofwhich are well-represented by a particular basis system or we may have a single class of signals, but the elements of that class are linearcombinations of members of the well-represented classes. In either case, there are several criteria that have been identified as important [link] , [link] :

• Sparsity: The expansion should have most of the important information in the smallest number of coefficients so that the others aresmall enough to be neglected or set equal to zero. This is important for compression and denoising.
• Separation: If the measurement consists of a linear combination of signals with different characteristics, the expansion coefficientsshould clearly separate those signals. If a single signal has several features of interest, the expansion should clearly separate thosefeatures. This is important for filtering and detection.
• Superresolution: The resolution of signals or characteristics of a signal should be much better than with a traditional basis system. Thisis likewise important for linear filtering, detection, and estimation.
• Stability: The expansions in terms of our new overcomplete systems should not be significantly changed by perturbations or noise.This is important in implementation and data measurement.
• Speed: The numerical calculation of the expansion coefficients in the new overcomplete system should be of order $O\left(N\right)$ or $O(Nlog(N\left)\right)$ .

These criteria are often in conflict with each other, and various compromises will be made in the algorithms and problem formulations for anacceptable balance.

Overcomplete representations

This section uses the material in  Chapter: Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Right Frames, and unconditional Bases on bases and frames. One goal is to represent a signal using a “dictionary" ofexpansion functions that could include the Fourier basis, wavelet basis, Gabor basis, etc. We formulate a finite dimensional version of thisproblem as

for $n=0,1,2,\cdots ,N-1$ and $k=0,1,2,\cdots ,K-1$ . This can be written in matrix form as

where $\mathbf{y}$ is a $N\times 1$ vector with elements being the signal values $y\left(n\right)$ , the matrix $\mathbf{X}$ is $N\times K$ the columns of which are made up of all the functions in the dictionaryand $\alpha$ is a $K\times 1$ vector of the expansion coefficients ${\alpha }_k$ . The matrix operator has the basis signals ${\mathbf{x}}_{\mathbf{k}}$ as its columns so that the matrix multiplication [link] is simply the signal expansion [link] .

For a given signal representation problem, one has two decisions: what dictionary to use (i.e., choice of the $\mathbf{X}$ ) and how to represent the signal in terms of this dictionary (i.e., choice of $\alpha$ ). Since the dictionary is overcomplete, there are several possible choices of $\alpha$ and typically one uses prior knowledge or one or more of the desired properties we saw earlier to calculatethe $\alpha$ .

A matrix example

Consider a simple two-dimensional system with orthogonal basis vectors

which gives the matrix operator with ${\mathbf{x}}_{\mathbf{1}}$ and ${\mathbf{x}}_{\mathbf{2}}$ as columns