THEORY ADN DESIGN OF 2-DIMENSIONAL FILTER BANKS - A REVIEW

There has been considerable interest in the design of multidimensional (MD) filter banks. MD filter banks find application in subband coding of images and video data. MD filter banks can be designed by cascadin g one-dimensional (1D) filter banks in the form of a tree structure. I n this case, the individual analysis and synthesis filters are separab le and the filter bank is called a separable filter bank. MD filter ba nks with nonseparable filters offer more flexibility and usually provi de better performance. Nonetheless, their design is considerably more difficult than separable filter banks. The purpose of this paper is to provide an overview of developments in this field on the design techn iques for MD filter banks, mostly two-dimensional (2D) filter banks. I n some image coding applications, the 2D two-channel filter banks are of great importance, particularly the filter bank with diamond-shaped filters. We will present several design techniques for the 2D two-chan nel nonseparable filter banks, As the design of MD filters are not as tractable as that of 1D filters, we seek design techniques that do not involve direct optimization of MD filters. To facilitate this, transf ormations that turn a separable MD filter bank into a nonseparable one are developed. Also, transformations of 1D filter banks to MD filter banks are investigated. We will review some designs of MD filter banks using transformations. In the context of 1D filter bank design, the c osine modulated filter bank (CMFB) is well-known for its design and im plementation efficiency. All the analysis filters are cosine modulated versions of a prototype filter. The design cost of the filter bank is equivalent to that of the prototype and the implementation complexity is comparable to that of the prototype plus a low-complexity matrix. The success with 1D CMFB motivate the generalization to the 2D case. W e will construct the 2D CMFB by following a very close analogy of 1D c ase. It is well-known that the 1D lossless systems can be characterize d by state space description. In 1D, the connection between the lossle ssness of a transfer matrix and the unitariness of the realization mat rix is well-established. We will present the developments on the study of 2D lossless systems. As in ID case, the 2D FIR lossless systems ca n be characterized in terms of state space realizations. We will revie w this, and then address the factorizability of 2D FIR lossless system s by using the properties of state space realizations.