Unitary Filter Banks Research Articles

Preconditioning Filter Bank Decomposition Using Structured Normalized Tight Frames

We turn a given filter bank into a filtering scheme that provides perfect reconstruction, synthesis is the adjoint of the analysis part (so-called unitary filter banks), all filters have equal norm, and the essential features of the original filter bank are preserved. Unitary filter banks providing perfect reconstruction are induced by tight generalized frames, which enable signal decomposition using a set of linear operators. If, in addition, frame elements have equal norm, then the signal energy is spread through the various filter bank channels in some uniform fashion, which is often more suitable for further signal processing. We start with a given generalized frame whose elements allow for fast matrix vector multiplication, as, for instance, convolution operators, and compute a normalized tight frame, for which signal analysis and synthesis still preserve those fast algorithmic schemes.

A new design method of CDMA spreading codes based on multi-rate unitary filter bank

It is well-known that the multi-valued CDMA spreading codes can be designed by means of a pair of mirror multi-rate filter banks based on some optimizing criterion. This paper indicates that there exists a theoretical bound in the performance of its circulating correlation property, which is given by an explicit expression. Based on this analysis, a criterion of maximizing entropy is proposed to design such codes. Computer simulation result suggests that the resulted codes outperform the conventional binary balanced Gold codes for an asynchronous CDMA system.

On the optimality of block orthogonal transforms for multiple description coding of Gaussian vector sources

In this work, we consider the application of unitary filter banks for multiple description coding of independent, identically distributed Gaussian vector sources. We derive the redundancy-rate-distortion bound for this class of signals. It is shown that for two-dimensional (2-D) Gaussian vector sources (resulting in a two-description system), 2/spl times/2 block transforms give all the achievable optimal points in the redundancy-rate-distortion function among the class of all unitary filter banks. Similar results can be obtained for higher-dimensional transforms.

Sub-Riemannian geometry and clinical magnetic resonance tomography

Mensch, streckh deine Vernunfft hieher, diese dinge zu begreiffen! Johann Keppler (1571–1630) There is nothing that nuclear spins will not do for you, as long as you treat them as human beings. Erwin Louis Hahn (1949) Ein maschinelles ‘agencement’ ist den Schichten zugewandt, reinen Intensitaten, die sie zirkulieren last um die Selektion der ‘Konsistenzebene’ zu sichern, und der sich die Subjekte zuordnen, welchen sie einen Namen nur als Spur einer Intensitat last. Gilles Deleuze und Felix Guattari (1988) The sense we make of the world is governed by our conceptual means. More specifically, sense is made of nature by projecting a conceptual structure onto observed events. This is accomplished by means of a semantic filter, which transforms raw data into semantically significant events. George L. Farre (1997) Phase coherent wavelets form a unified basis of the multichannel reconstruction analysis–synthesis filter bank, implemented by high-resolution synthetic aperture radar (SAR) imaging and clinical MRI. The construction of unitary bank filters is performed by the Kepplerian stratification technique. It allows for the cross-sectional quadrature filtering of phase histories in contiguous local frequency encoding subband multichannels relative to the rotating coordinate frame of quadrature reference. The Kepplerian quadrature detection strategy, which is based on the unitary filter bank construction takes place in symplectic affine planes of incidence immersed into the three-dimensional compact super-encoding projective space. The planes are implemented in the quadrature format of a phase-splitting network of Fourier analysis of the stratified Heisenberg nilpotent Lie group G. The action of the basic Lie group G of quantum physics under its sub-Riemannian geometry admits a matrix coordinatization by transvections. In terms of projective geometry, the longitudinal dilations act as a spectral transform on the transvections of the G-action. The tomographic slices, frequency selected by the MRI scanner system, are identified with the projectively immersed, symplectic affine leaves ν (ν=0) of the canonical coadjoint orbit foliation of G, on which the continuous affine wavelet transform lifts to a transversal spectral transform. The paper points out that the key point of ensemble quantum computation by Fourier transform MRI is the synergy of the various wavelets, and the coexistence of their geometric substrates recorded by the bi-infinite stratigraphic time line. The paper leads from Keppler to Heisenberg and emphasizes the tight control which Lie groups exercise over the non-invasive MRI modality via geometric quantization. The fascinating aspects of electronic engineering concerned with the calculation of the Weyl symbol are also considered. Copyright © 1999 John Wiley & Sons, Ltd.

WAVELETS IN HIGH-RESOLUTION RADAR IMAGING AND CLINICAL MAGNETIC RESONANCE TOMOGRAPHY

Coherent wavelets form a unified basis of the multichannel perfect reconstruction analysis-synthesis filter bank of high-resolution synthetic aperture radar SAR imaging and clinical magnetic resonance imaging MRI . The unitary filter bank construction is performed by the Kepplerian temporospatial Hilbert bundle strategy of physical astronomy, which allows stroboscopic and synchronous cross-sectional quadrature filtering of phase histories in contiguous local frequency encoding subband multichannels with respect to the rotating coordinate frame of quadrature reference. The Kepplerian temporospatial quadrature strategy and the associated unitary filter bank construction take place in symplectic affine planes and are implemented in the quadrature format of a phase-splitting network by Fourier analysis of the Heisenberg nilpotent Lie group G. The tomographic slices, frequency selected by the MRI scanner, are identified with the nontrivial coadjoint orbits of G. Thus the pathway of this paper leads from Keppler to Heisenberg and to the fascinating aspects of electronic engineering concerned with the implementation of the symmetries inherent to the semiclassical approach to clinical magnetic resonance tomography by large-scale integrated LSI circuit technology.

On the optimality of nonunitary filter banks in subband coders

This paper investigates the energy compaction capabilities of nonunitary filter banks in subband coding. It is shown that nonunitary filter banks have larger coding gain than unitary filter banks because of the possibility of performing half-whitening in each channel. For long filter unit pulse responses, optimization of subband coding gain for stationary input signals results in a filter bank decomposition, where each channel works as an optimal open-loop DPCM system. We derive a formula giving the optimal filter response for each channel as a function of the input power spectral density (PSD). For shorter filter bank responses, good gain is obtained by suboptimal half-whitening responses, but the impact on the theoretical coding gain is still highly significant. Image coding examples demonstrate that better performance is achieved using nonunitary filter banks when the input images are in correspondence with the signal model.

Factorization approach to unitary time-varying filter bank trees and wavelets

A complete factorization of all optimal (in terms of quick transition) time-varying FIR unitary filter bank tree topologies is obtained. This has applications in adaptive subband coding, tiling of the time-frequency plane and the construction of orthonormal wavelet and wavelet packet bases for the half-line and interval. For an M-channel filter bank the factorization allows one to construct entry/exit filters that allow the filter bank to be used on finite signals without distortion at the boundaries. One of the advantages of the approach is that an efficient implementation algorithm comes with the factorization. The factorization can be used to generate filter bank tree-structures where the tree topology changes over time. Explicit formulas for the transition filters are obtained for arbitrary tree transitions. The results hold for tree structures where filter banks with any number of channels or filters of any length are used. Time-varying wavelet and wavelet packet bases are also constructed using these filter bank structures. the present construction of wavelets is unique in several ways: 1) the number of entry/exit functions is equal to the number of entry/exit filters of the corresponding filter bank; 2) these functions are defined as linear combinations of the scaling functions-other methods involve infinite product constructions; 3) the functions are trivially as regular as the wavelet bases from which they are constructed. >

On the optimality of nonunitary filter banks in subband coders

This paper investigates the energy compaction capabilities of nonunitary filter banks in subband coding. It is shown that nonunitary filter banks have larger coding gain than unitary filter banks because of the possibility of performing half-whitening in each channel. For long filter unit pulse responses, optimization of subband coding gain for stationary input signals results in a filter bank decomposition, where each channel works as an optimal open-loop DPCM system. We derive a formula giving the optimal filter response for each channel as a function of the input power spectral density (PSD). For shorter filter bank responses, good gain is obtained by suboptimal half-whitening responses, but the impact on the theoretical coding gain is still highly significant. Image coding examples demonstrate that better performance is achieved using nonunitary filter banks when the input images are in correspondence with the signal model.

Unitary FIR filter banks and symmetry

Recently, unitary FIR filter banks with linear-phase have been completely parameterized by exploiting the eigenstructure of the exchange matrix. This correspondence gives new characterizations of polyphase matrices of filter banks with various types of symmetry on the filters. Using matrix extensions of the well-known hyperbolic and orthogonal lattices, we give a alternative proof for the parameterization of linear-phase unitary filter banks. A complete parameterization of FIR unitary filter banks with each of the different types of symmetries considered (not just linear-phase) is also given. These results can also be used to generate non-unitary filter banks with symmetries, though no completeness results can be obtained. In some cases implicit, and in others explicit parameterization of wavelet tight frames associated with these filter banks are also given. This paper only considers filter banks with an even number of channels. A similar theory can be developed if the number of channels is an odd integer. >