The principal investigator's research is focused on the study of alternatives to the tensor product for constructing multi-dimensional wavelet functions from one-dimensional wavelet functions. These alternative methods may overcome some limitations of the tensor product approach while also complementing the tensor product formulation. The tensor product concept is also prevalent in many other application areas, and thus new developments have the potential for broader impact. From the mathematical perspective, the research includes extending current efforts in developing the coset sum methodology, searching for alternative mechanisms to construct wavelet bases while assessing their distinguishing properties for certain applications, and studying systematic ways to construct multi-dimensional wavelet systems where the tensor product does not work.
Wavelets have been used in a wide range of applications including Image Compression. Examples where wavelets are a key tool include the JPEG 2000 digital image standard and fingerprint compression for data storage. This work concerns improvements in the construction of multi-dimensional wavelet systems focused toward specific application areas, and provides an opportunity for mathematics graduate students to study mathematics from an application perspective.
Wavelets have been used in a wide range of applications including Image Compression. Examples where wavelets are a key tool include the JPEG 2000 digital image standard and fingerprint compression for data storage. The principal investigator's research was focused on finding and studying alternatives to the tensor product in constructing multi-dimensional wavelet functions from one-dimensional wavelet functions, which can provide improvements over the current use of multi-dimensional wavelet systems. The research provided totally new, yet simple, ways of constructing multi-dimensional wavelet functions and gave an opportunity to look for other methods outside the tensor product construction. For example, a new alternative--called coset sum methodology--to the tensor product method was introduced by using the sum rather than the product of one-dimensional wavelet functions, and it was shown that the coset sum wavelet systems provide much more effective representations for the two-dimensional images with directional contents than the current tensor product wavelet representations. The research also provided an opportunity for mathematics graduate students to study mathematics from an application perspective. The research results and discoveries were reported in peer-reviewed journals, and presented at mathematical, scientific and engineering conferences and meetings. The grant generated research topics suitable for a doctoral dissertation and provided partial support for one doctoral student.