Theory adn design of two-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 cascading one-dimensional (1D) filter banks in the form of a tree structure. In this case, the individual analysis and synthesis filters are separable and the filter bank is called a separable filter bank. MD filter banks with nonseparable filters offer more flexibility and usually provide 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 techniques for MD filter banks, mostly two-dimensional (2D) filter banks. In 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-channel 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, transformations 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 cosine modulated filter bank (CMFB) is well-known for its design and implementation 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. We will construct the 2D CMFB by following a very close analogy of 1D case. It is well-known that the 1D lossless systems can be characterized by state space description. In 1D, the connection between the losslessness of a transfer matrix and the unitariness of the realization matrix is well-established. We will present the developments on the study of 2D lossless systems. As in ID case, the 2D FIR lossless systems can be characterized in terms of state space realizations. We will review this, and then address the factorizability of 2D FIR lossless systems by using the properties of state space realizations.