The wavelet transform is by definition a linear transform/ operator. It is also orthonormal operator. Therefore it lends itself nicely to other numerical computations. In these other numerical computation methods, the objective is simplify the complexity of computation by transforming the original matrix into a new sparse matrix.
One concept of the wavelet transform is its filtering aspect. Every time a wavelet operator is applied, it divides a signal in two. For one-dimensional arrays, a wavelet transform has the same effect as a high and low pass filter. With multi-resolution wavelets, the array is divided into segments that frequency specific. These segments are considered to be neighborhoods, provide feature extraction of the two dimensional structure, which constitutes spatial features. The image analogy are edge extraction.
Another concept with wavelets is the distribution and significance of energy. Each time a wavelet transform is applied, the energy distribution changed. These changes represent contributions by the higher and lower frequency components. In the case of multi-resolution wavelets, some of the segments carry very little energy in comparison to the original. The significance of such energy can used as a filtering mechanism to eliminate noise or insignificant elements from the array making it more sparse. This sparseness allows the array to be represented with fewer elements.
These two qualities make the wavelet transform a reasonably good preconditioner. This preconditioning is useful for feature extraction, data compression, and scientific computation. The frequency components reveal properties of domain. However, the sparesness offers opportunities for the use of faster numerical algorithms on the same data. Thus wavelets offer a computational "catalysist" form these numerical methods.
Attached is a copy of "Applications of Wavelets to Image Processing and Matrix Multiplication" by Daniel Beatty, MSCS Master's Thesis Chairman Eric Sinzinger Ph.D., and thesis committee members Alan Sill Ph.D., Philip Smith, Ph.D., and Noe Lopez-Benitez, Ph.D..
