Gabor Shearlets

In this paper, we introduce Gabor shearlets, a variant of shearlet systems, which are based on a different group representation than previous shearlet constructions: they combine elements from Gabor and wavelet frames in their construction. As a consequence, they can be implemented with standard filters from wavelet theory in combination with standard Gabor windows. Unlike the usual shearlets, the new construction can achieve a redundancy as close to one as desired. Our construction follows the general strategy for shearlets. First we define group-based Gabor shearlets and then modify them to a cone-adapted version. In combination with Meyer filters, the cone-adapted Gabor shearlets constitute a tight frame and provide low-redundancy sparse approximations of the common model class of anisotropic features which are cartoon-like functions.Comment: 24 pages, AMS LaTeX, 4 figure

On power-complementary FIR filters

Conditions are derived, under which two linear-phase FIR filter transfer functions H(z)and G(z) have the power-complementary property, i.e., |H(e^{j omega})|^{2} + |G(e^{jomega})|^{2} = 1. It is shown that, the constraint of linear phase on the transfer functions strongly restricts the class of frequency responses that can be realized by a power-complementary pair

A zero-assignment approach to two-channel filter banks and wavelets

Cataloged from PDF version of article.It is well-known that subband decomposition and perfect reconstruction of an arbitrary input signal is possible by a proper design of four lters. Besides having a wide range of applications in signal processing, perfect reconstruction lter banks have a strong connection with wavelets as pointed out by Mallat. Daubechies managed to design minimal order, maximally at lters and she proposed a cascade algorithm to construct compactly supported orthogonal wavelets from the orthogonal perfect reconstruction lter banks. The convergence of the cascade algorithm requires at least one zero at z = 1 and z = 1 for the lowpass and the highpass lters, respectively. This thesis focuses on the design of two-channel lter banks with assigned zeros. The fact that causal, stable and rational transfer functions form a Euclidean domain is used to pose the problem in an abstract setup. A polynomial algorithm is proposed to design lter banks with lters having assigned zeros and a characterization of all solutions having the same zeros in terms of a free, even, causal, stable and rational transfer function is obtained. A generalization of Daubechies design of orthogonal lter banks is given. The free parameter can be used to improve the lter bank design and the design of corresponding orthogonal or biorthogonal wavelets. The results also nd an application in examining the robustness of regularity of minimal length compactly supported wavelets with respect to perturbation of lter zeros at 1 and -1.Akbaş, MustafaM.S

Analysis of transonic flow about lifting wing-body configurations

An analytical solution was obtained for the perturbation velocity potential for transonic flow about lifting wing-body configurations with order-one span-length ratios and small reduced-span-length ratios and equivalent-thickness-length ratios. The analysis is performed with the method of matched asymptotic expansions. The angles of attack which are considered are small but are large enough to insure that the effects of lift in the region far from the configuration are either dominant or comparable with the effects of thickness. The modification to the equivalence rule which accounts for these lift effects is determined. An analysis of transonic flow about lifting wings with large aspect ratios is also presented

Passive cascaded-lattice structures for low-sensitivity FIR filter design, with applications to filter banks

A class of nonrecursive cascaded-lattice structures is derived, for the implementation of finite-impulse response (FIR) digital filters. The building blocks are lossless and the transfer function can be implemented as a sequence of planar rotations. The structures can be used for the synthesis of any scalar FIR transfer function H(z) with no restriction on the location of zeros; at the same time, all the lattice coefficients have magnitude bounded above by unity. The structures have excellent passband sensitivity because of inherent passivity, and are automatically internally scaled, in an L_2 sense. The ideas are also extended for the realization of a bank of MFIR transfer functions as a cascaded lattice. Applications of these structures in subband coding and in multirate signal processing are outlined. Numerical design examples are included

Bandwidth Efficient Transmultiplexers, Part 2: Subband Complements and Performance Aspects

Abstract-This paper examines the performance issues relating to the quadrature amplitude modulation (QAM) and vestigial sideband (VSB) transmultiplexers synthesized in [l]. First, an analysis of the limitations of the configured systems regarding intersymbol interference and crosstalk suppression arising from the use of practical filters is made. Based on these observations, a new design technique for an FIR low-pass prototype that takes the practical degradations into account is formulated. The procedure involves the unconstrained optimization of an error function. A performance evaluation reveals that for four of the five systems, the new method is superior to a minimax approach in that lower intersymbol interference and crosstalk distortions are achieved with a smaller number of filter taps. For the other transmultiplexer, the advantage of the optimized design over the minimax design is in the added flexibility of taking crosstalk into account thereby diminishing the crosstalk distortion. The five transmultiplexers can be converted into new subband systems. We show how the optimized design approach formulated for the transmultiplexers carries over to the new subband systems. I

Sampling systems matched to input processes and image classes

This dissertation investigates sampling and reconstruction of wide sense stationary (WSS) random processes from their sample random variables . In this context, two types of sampling systems are studied, namely, interpolation and approximation sampling systems. We aim to determine the properties of the filters in these systems that minimize the mean squared error between the input process and the process reconstructed from its samples. More specifically, for the interpolation sampling system we seek and obtain a closed form expression for an interpolation filter that is optimal in this sense. Likewise, for the approximation sampling system we derive a closed form expression for an optimal reconstruction filter given the statistics of the input process and the antialiasing filter. Using these expressions we show that Meyer-type scaling functions and wavelets arise naturally in the context of subsampled bandlimited processes. We also derive closed form expressions for the mean squared error incurred by both the sampling systems. Using the expression for mean squared error we show that for an approximation sampling system, minimum mean squared error is obtained when the antialiasing filter and the reconstruction filter are spectral factors of an ideal brickwall-type filter. Similar results are derived for the discrete-time equivalents of these sampling systems. Finally, we give examples of interpolation and approximation sampling filters and compare their performance with that of some standard filters. The implementation of these systems is based on a novel framework called the perfect reconstruction circular convolution (PRCC) filter bank framework. The results obtained for the one dimensional case are extended to the multidimensional case. Sampling a multidimensional random field or image class has a greater degree of freedom and the sampling lattice can be defined by a nonsingular matrix D. The aim is to find optimal filters in multidimensional sampling systems to reconstruct the input image class from its samples on a lattice defined by D. Closed form expressions for filters in multidimensional interpolation and approximation sampling systems are obtained as are expressions for the mean squared error incurred by each system. For the approximation sampling system it is proved that the antialiasing and reconstruction filters that minimize the mean squared error are spectral factors of an ideal brickwall-type filter whose support depends on the sampling matrix D. Finally. we give examples of filters in the interpolation and approximation sampling systems for an image class derived from a LANDSAT image and a quincunx sampling lattice. The performance of these filters is compared with that of some standard filters in the presence of a quantizer

An optimally well-localized multi-channel parallel perfect reconstruction filter bank.

Joint uncertainty for the overall L-channel, one-dimensional, parallel filter bank is quantified by a metric which is a weighted sum of the time and frequency localizations of the individual filters. Evidence is presented to show that a filter bank possessing a lower joint filter bank uncertainty with respect to this metric results in a computed multicomponent AM-FM image model that yields lower reconstruction errors. This strongly supports the theory that there is a direct relationship between joint uncertainty as quantified by the measures developed and the degree of local smoothness or "local coherency" that may be expected in the filter bank channel responses. Thus, as demonstrated by the examples, these new measures may be used to construct new filter banks that offer excellent localization properties on par with those of Gabor filter banks.This dissertation defines a measure of uncertainty for finite length discrete-time signals. Using this uncertainty measure, a relationship analogous to the well known continuous-time Heisenberg-Weyl inequality is developed. This uncertainty measure is applied to quantify the joint discrete time-discrete frequency localization of finite impulse response filters, which are used in a quadrature mirror filter bank (QMF). A formulation of a biorthogonal QMF where the low pass analysis filter minimizes the newly defined measure of uncertainty is presented. The search algorithm used in the design of the length-N linear phase low pass analysis FIR filter is given for N = 6 and 8. In each case, the other three filters, which constitute a perfect reconstruction QMF, are determined by adapting a method due to Vetterli and Le Gall. From a set of well known QMFs comprised of length six filters, L-channel perfect reconstruction parallel filter banks (PRPFB) are constructed. The Noble identities are used to show that the L-channel PRPFB is equivalent to a L - 1 level discrete wavelet filter bank. Several five-channel PRPFBs are implemented. A separable implementation of a five-channel, one-dimensional filter bank produces twenty-five channel, two-dimensional filter bank. Each non-low pass, two-dimensional filter is decomposed in a novel, nonseparable way to obtain equivalent channel filters that possess orientation selectivity. This results in a forty-one channel, two-dimensional, orientation selective, PRPFB