Multidimensional (MD) multirate systems, which find applications in the coding and compression of image and video data, have recently attracted much attention. The basic building blocks in an MD multirate system are the decimation matrix M, the expansion matrix L, and MD digital filters. With D denoting the number of dimensions, M and L are D×D nonsingular integer matrices. When these matrices are diagonal, most of the one-dimensional (1-D) multirate results can be extended automatically, using separable approaches (i.e., separable operations in each dimension). Separable approaches are commonly used in practice due to their low complexity in implementation. However, nonseparable operations, with respect to nondiagonal decimation and expansion matrices, often provide more flexibility and better performance. Several applications, such as the conversion between progressive and interlaced video signals, actually require the use of nonseparable operations. For the nonseparable case, extensions of 1-D results to the MD case are nontrivial. Some of these extensions, e.g., polyphase decomposition and maximally decimated perfect reconstruction systems, have already been successfully accomplished by some authors. However, there exist several 1-D results in multirate processing for which the MD extensions are even more difficult. In this paper, we will introduce some recent developments in these extensions. Some important results are: the design of nonseparable MD decimation / interpolation filters derived from 1-D filters, the generalized pseudocirculant property of alias-free maximally decimated filter banks, the commutativity of MD decimators and expanders, and applications in the efficient polyphase implementation of MD rational decimation systems. We will also introduce several other results of theoretical importance.
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