One- and two- dimensional cosine modulated filter banks

Subband coding as a lossy data compression technique was first introduced for speech coding. It has been demonstrated to be a very competitive coding method for general audio signals as well as images. Subband coding has been incorporated in various popular coding standards. Essential to the implementation of subband coding is an M-channel filter bank that partitions the input signal into M subbands. In the context of 1D (one-dimensional) filter bank design, the CMFB (cosine modulated filter bank) is well-known for design and implementation efficiency. All the filters in the filter bank are cosine modulated versions of a prototype filter. As a result the cost of design as well as complexity is reduced dramatically by a factor of M. In this thesis we study the design of CMFB in 1D case and 2D (two-dimensional) case. In previous works on 1D CMFB, the filters in the filter bank do not have linear phase, which is considered an important feature in image coding applications. The design of cosine modulated filter banks with linear-phase filters is the first topic to be presented in this thesis. Design examples will be given to show that filter banks with filters having good frequency selectivity can be obtained in spite of the linear phase constraint. For the design of 2D cosine modulated filter banks, the simplest approach is to cascade 1D filter banks in the form of a tree. This type of 2D filter banks are referred to as separable. The frequency support of the filters in a separable filter bank are restricted to rectangular shapes. Nonseparable filter banks allow more flexible partitions of the frequency plane and achieve better performance. Almost all the existing design techniques for 2D nonseparable filter banks are developed exclusively for the two-channel case. We will consider two types of 2D M-channel nonseparable filter banks, the two-parallelogram type and the four-parallelogram type. These are respectively the classes of filter banks in which the passbands of the filters consist of two and four parallelograms. In these designs additional cosine modulated constraints will be incorporated for design and implementation economy.