3:30, SPTM-L2.1
RESULTS ON VECTOR BIORTHOGONAL PARTNERS
B. VRCELJ, P. VAIDYANATHAN
The concept of Multiple Input Multiple Output (MIMO) biorthogonal partners arises in many different contexts, one of them being multiwavelet theory. They also play a central role in the theory of MIMO channel equalization, especially with fractionally spaced equalizers. In this paper we will explore some further theoretical properties of MIMO biorthogonal partners. These include the conditions for the existence of MIMO biorthogonal partners and their application in finding the solution for the least squares signal approximation.

3:50, SPTM-L2.2
FRAME RECONSTRUCTION OF THE LAPLACIAN PYRAMID
M. DO, M. VETTERLI
The Laplacian pyramid (LP) is studied as a frame operator and this reveals that the usual reconstruction is suboptimal. With orthogonal filters, the LP is shown to be a tight frame, and thus the optimal linear reconstruction using the dual frame operator has a simple structure as symmetrical with the forward transform. In more general cases, we propose a simple and efficient filter bank for reconstruction in the LP that is shown to performs better than the usual method. Numerical results indicate that gains of more than 1 dB are actually achieved.

4:10, SPTM-L2.3
BIORTHOGONAL BUTTERWORTH WAVELETS
V. ZHELUDEV, A. AVERBUCH, A. PEVNYI
In the paper we present a new family of biorthogonal wavelet transforms and a related library of biorthogonal symmetric wavelets. For the construction we use the interpolatory discrete splines which enable us to design perfect reconstruction filter banks related to the Butterworth filters. The construction is performed in a ``lifting'' manner. The proposed scheme is based on interpolation and, as such, it involves only samples of signals and it does not require any use of quadrature formulas. These filters have linear phase property. The filters yield perfect frequency resolution.

4:30, SPTM-L2.4
ISSUES OF FILTER DESIGN FOR BINARY WAVELET TRANSFORM
N. LAW, W. SIU
Wavelet decomposition has recently been generalized to binary field in which the arithmetic is performed wholly in GF(2). In order to maintain an invertible binary wavelet transform with desirable multiresolution properties, the bandwidth, the perfect reconstruction and the vanishing moment constraints are placed on the binary filters. While they guarantee an invertible transform, the transform becomes non-orthogonal and non-biorthogonal in which the inverse filters could be signal length-dependent. We propose to apply the perpendicular constraint on the binary filters to make them length independent. A filter design strategy is outlined in which a filter design for a length of eight is given. We also propose an efficient implementation structure for the binary filters that saves memory space and reduces the computational complexity.

4:50, SPTM-L2.5
A NEW DIRECTIONAL, LOW-REDUNDANCY, COMPLEX-WAVELET TRANSFORM
F. FERNANDES, R. VAN SPAENDONCK, C. BURRUS
Shift variance and poor directional selectivity, two major disadvantages of the discrete wavelet transform, have previously been circumvented either by using highly redundant, non-separable wavelet transforms or by using restrictive designs to obtain a pair of wavelet trees with a transform-domain redundancy of 4.0 in 2D. In this paper, we demonstrate that excellent shift-invariance properties and directional selectivity may be obtained with a transform-domain redundancy of only 2.67 in 2D. We achieve this by projecting the wavelet coefficients from Selesnick's almost shift-invariant, double-density wavelet transform so as to separate approximately the positive and negative frequencies, thereby increasing directionality. Subsequent decimation and a novel inverse projection maintain the low redundancy while ensuring perfect reconstruction. Although our transform generates complex-valued coefficients allowing processing capabilities that are impossible with real-valued coefficients, it may be implemented with a fast algorithm that uses only real arithmetic. To demonstrate the efficacy of our new transform, we show that it achieves state-of-the-art performance in a seismic image-processing application.

5:10, SPTM-L2.6
ON FILTER BANKS WITH RATIONAL OVERSAMPLING
R. VON BORRIES, R. DE QUEIROZ, C. BURRUS
Despite the great popularity of critically-decimated filter banks, oversampled filter banks are useful in applications where data expansion is not a problem. We studied oversampled filter banks and showed that for some popular classes of filter banks it is not possible to obtain perfect reconstruction with rational (non-integer) oversampling ratios. Nevertheless, it is always possible to oversample the analysis filter bank by an integer factor, i.e. there will be a similar synthesis bank which would provide perfect reconstruction. The analysis is carried within a time-aliasing framework developed to analyze non-critically decimated filter banks.