The generalized lapped biorthogonal transform

A lattice structure based on the singular value decomposition (SVD) is introduced. The lattice can also be proven to use a minimal number of delay elements and to completely span a large class of M-channel linear phase perfect reconstruction filter bank (LPPRFB): all analysis and synthesis filters have the same FIR length of L = K M , sharing the same center of symmetry. The lattice also structurally enforces both linear phase and perfect reconstruction properties, is capable of providing fast and efficient implementation, and avoids the costly matrix inversion problem in the optimization process. From a block transform perspective, the new lattice represents a family of generalized lapped biorthogonal transform (GLBT) with arbitrary integer overlapping factor K . The relaxation of the orthogonal constraint allows the GLBT to have significantly different analysis and synthesis basis functions which can then be tailored appropriately to fit a particular application. Several design examples are presented along with a high-performance GLBT-based progressive image coder to demonstrate the superiority of the new lapped transforms. 1. I N T R O D U C T I O N Linear phase perfect reconstruction filter banks have been used extensively in numerous applications, especially image processing [l]. In the two-channel case, all solutions have been found whereas there are still many open problems in M-channel cases. An attractive approach to the design and implementation of LPPRFB is the parameterization by lattice structures based on the factorization of the polyphase matrices E(z) and R(z) shown in Figure 1. The lattice structure offers fast and efficient implementation, retains both LP and PR properties regardless of coefficient quantization, and (if it is general enough) guarantees that no optimal solution will be excluded in the optimization process. Complete and minimal two-channel P R lattice structure has been reported in [2]. M-channel lattices have been presented for the more restricted paraunitary case [3], resulting in the generalized lapped orthogonal transform (GenLOT) [4]. No general lattice has been reported for the biorthogonal case (defined as R(z)E(z) = z-IiI). Only several particular solutions were proposed so far: Chan replaced some orthogonal matrices in [4] by cascades of invertible block diagonal matrices [5]; Malvar suggested a simple scaling of the first antisymmetric basis function of the initial block (which was chosen to be the DCT) [6]. XIftl