Theory of optimal orthonormal subband coders

The theory of the orthogonal transform coder and methods for its optimal design have been known for a long time. We derive a set of necessary and sufficient conditions for the coding-gain optimality of an orthonormal subband coder for given input statistics. We also show how these conditions can be satisfied by the construction of a sequence of optimal compaction filters one at a time. Several theoretical properties of optimal compaction filters and optimal subband coders are then derived, especially pertaining to behavior as the number of subbands increases. Significant theoretical differences between optimum subband coders, transform coders, and predictive coders are summarized. Finally, conditions are presented under which optimal orthonormal subband coders yield as much coding gain as biorthogonal ones for a fixed number of subbands

Non-separable 2D wavelets with two-row filters

In the literature 2D (or bivariate) wavelets are usually constructed as a tensor product of 1D wavelets. Such wavelets are called separable. However, there are various applications, e.g. in image processing, for which non-separable 2D wavelets are preferable. In this paper, we investigate the class of compactly supported orthonormal 2D wavelets that was introduced by Belogay and Wang [2]. A characteristic feature of this class of wavelets is that the support of the corresponding filter comprises only two rows. We are concerned with the biorthogonal extension of this kind of wavelets. It turns out that the 2D wavelets in this class are intimately related to some underlying 1D wavelet. We explore this relation in detail, and we explain how the 2D wavelet transforms can be realized by means of a lifting scheme, thus allowing an efficient implementation. We also describe an easy way to construct wavelets with more rows and shorter columns

Perfect Reconstruction Two-Channel Filter Banks on Arbitrary Graphs

This paper extends the existing theory of perfect reconstruction two-channel filter banks from bipartite graphs to non-bipartite graphs. By generalizing the concept of downsampling/upsampling we establish the frame of two-channel filter bank on arbitrary connected, undirected and weighted graphs. Then the equations for perfect reconstruction of the filter banks are presented and solved under proper conditions. Algorithms for designing orthogonal and biorthogonal banks are given and two typical orthogonal two-channel filter banks are calculated. The locality and approximation properties of such filter banks are discussed theoretically and experimentally.Comment: 33 pages,11 figures. This manuscript has been submitted to ScienceDirect Applied and Computational Harmonic Analysis (ACHA) on Jan 27,202

Gröbner bases and wavelet design

AbstractIn this paper, we detail the use of symbolic methods in order to solve some advanced design problems arising in signal processing. Our interest lies especially in the construction of wavelet filters for which the usual spectral factorization approach (used for example to construct the well-known Daubechies filters) is not applicable. In these problems, we show how the design equations can be written as multivariate polynomial systems of equations and accordingly how Gröbner algorithms offer an effective way to obtain solutions in some of these cases

Results on optimal biorthogonal filter banks

Optimization of filter banks for specific input statistics has been of interest in the theory and practice of subband coding. For the case of orthonormal filter banks with infinite order and uniform decimation, the problem has been completely solved in recent years. For the case of biorthogonal filter banks, significant progress has been made recently, although a number of issues still remain to be addressed. In this paper we briefly review the orthonormal case, and then present several new results for the biorthogonal case. All discussions pertain to the infinite order (ideal filter) case. The current status of research as well as some of the unsolved problems are described

M-Band Burt–Adelson Biorthogonal Wavelets

AbstractFor every integer M>2 we introduce a new family of biorthogonal MRAs with dilation factor M, generated by symmetric scaling functions with small support. This construction generalizes Burt–Adelson biorthogonal 2-band wavelets. For M∈{3,4} we are able to find simple explicit expressions for two different families of wavelets associated with these MRAs: one with better localization and the other with interesting symmetry–antisymmetry properties. We study the regularity of our scaling functions by determining their Sobolev exponent, for every value of the parameter and every M. We also study the critical exponent when M=3

Biorthogonal multivariate filter banks from centrally symmetric matrices

AbstractWe provide a practical characterization of block centrally symmetric and anti-symmetric matrices which arise in the construction of multivariate filter banks and use these matrices for the construction of biorthogonal filter banks with linear phase