Multidimensional Multirate Systems: Characterization, Design, and Applications

Multidimensional multirate systems have been used widely in signal processing, communications, and computer vision. Traditional multidimensional multirate systems are tensor products of one-dimensional systems. While these systems are easy to implement and design, they are inadequate to represent multidimensional signals since they cannot capture the geometric structure. Therefore, “true” multidimensional systems are more suited to multidimensional signals, such as images and videos. This thesis focuses on the characterization, design, and applications of “true” multidimensional multirate systems. One key property of multidimensional multirate systems is perfect reconstruction, which guarantees the original input can be perfectly reconstructed from the outputs. The most popular multidimensional multirate systems are multidimensional filter banks, including critically sampled and oversampled ones. Characterizing and designing multidimensional perfect reconstruction filter banks have been challenging tasks. For critically sampled filter banks, previous one-dimensional theory cannot be extended to the multidimensional case due to the lack of a multidimensional factorization theorem. For oversampled filter banks, existing one-dimensional theory does not work in the multidimensional case. We derive complete characterizations of multidimensional critically sampled and oversampled filter banks and propose novel design methods for multidimensional filter banks. We illustrate our multidimensional multirate system theory by several image processing applications.