Dyadic Decomposition Research Articles

On wavelet denoising and its applications to time delay estimation

The application of dyadic wavelet decomposition in the context of time delay estimation is investigated. We consider a model in which the source signal is deterministic and the received sensor outputs are corrupted by additive noises. Wavelet denoising is exploited to provide an effective solution for the problem. Denoising is first applied to preprocess the received signals from two spatially separated sensors with an attempt to remove the contamination, and the peak of their cross correlation function is then located from which the time delay between the two signals can be derived. A novel wavelet shrinkage/thresholding technique for denoising is introduced, and the performance of the algorithm is analyzed rigorously. It is proved that the proposed method achieves global convergence with a high probability. Simulation results also corroborate that the technique is efficient and performs significantly better than both the generalized cross correlator (GCC) and the direct cross correlator (CC).

Global regularity of 3D rotating Navier-Stokes equations for resonant domains

Abstract We prove existence on infinite time intervals of regular solutions to the 3D rotating Navier-Stokes equations in the limit of strong rotation (large Coriolis parameter Ω). This uniform existence is proven for periodic or stress-free boundary conditions for all domain aspect ratios, including the case of three wave resonances which yield nonlinear “ 21 2 - dimensional ” limit equations; smoothness assumptions are the same as for local existence theorems. The global existence is proven using techniques of the Littlewood-Paley dyadic decomposition. Infinite time regularity for solutions of the 3D rotating Navier-Stokes equations is obtained by bootstrapping from global regularity of the limit equations and convergence theorems.

Discrete quadratic estimates and holomorphic functional calculi in Banach spaces

We develop a discrete version of the weak quadratic estimates for operators of type w explained by Cowling, Doust, McIntosh and Yagi, and show that analogous theorems hold. The method is direct and can be generalised to the case of finding necessary and sufficient conditions for an operator T to have a bounded functional calculus on a domain which touches σ(T) nontangentially at several points. For operators on Lp, 1 < p < ∞, it follows that T has a bounded functional calculus if and only if T satisfies discrete quadratic estimates. Using this, one easily obtains Albrecht's extension to a joint functional calculus for several commuting operators. In Hilbert space the methods show that an operator with a bounded functional calculus has a uniformly bounded matricial functional calculus.The basic idea is to take a dyadic decomposition of the boundary of a sector Sv. Then on the kth ingerval consider an orthonormal sequence of polynomials . For h ∈ H∞(Sν), estimates for the uniform norm of h on a smaller sector Sμ are obtained from the coefficients akj = (h, ek, j). These estimates are then used to prove the theorems.

Detail Preserving Wavelet-based Compression with Adaptive Context-based Quantisation

A novel compression method called MBWT (modified basic wavelet technique) for efficient medical image compression is presented. MBWT is based on classic wavelet compression scheme with dyadic decomposition, uniform scalar quantisation and entropy coding of wavelet coefficients. By exploiting of space-frequency data characteristics in wavelet domain, an algorithm of adaptive quantisation and separable context-based binary coding with two (bitewise and wordwise) arithmetic coders are proposed. Also the most suitable filter banks selection is included. Quantisation step size evaluation is done for each subband on the basis of two items: established image quality level and wavelet coefficients variance estimation. Additionally adaptive threshold data selection is performed to reduce of unimportant coefficients in noisy areas. 9-order noncausal context model depicts the importance of the data from spatial correlations. The significance of parent node is also taken into account as a frequency aspect of data dependence. The idea of entropy coding includes three main elements: a) the construction of a decomposition tree with significant and insignificant nodes, b) pruning the tree - modified zerotree construction with four symbols of alphabet, c) separate arithmetic coding of two distinct data streams - a set of significant coefficients and a significant root node map with the signs of significant data. The MBWT algorithm seems to be simpler than the majority of efficient wavelet compression techniques and the effectiveness of our method is competitive with the best algorithms in the literature across diverse classes of medical images. Objective and subjective image quality evaluation supports this conlusion. Significant efficiency improvement over SPIHT algorithm is irrefutable for all images tested.

The shape characteristic detection of tool breakage in milling operations

Detection of tool failure is very important in automated manufacturing. All previously developed tool breakage detection approaches in milling operations have adopted the strategy of parameter detection in which the detection of tool breakage was carried out according to values of specific parameters selected to reflect tool state (with or without tool breakage). In this paper the new concept of shape characteristic detection of tool breakage in milling operations is proposed. The detection of tool breakage is conducted according to the shape characteristics of discrete dyadic wavelet decomposition of cutting force. By means of the proposed method, the influence caused by the variation of cutting parameters and transients is eliminated. The proposed method is conducted in two steps. In the first step, cutting force signals are decomposed by discrete dyadic wavelet, with the shape characteristic vectors then being generated by the proposed shape characteristic vector-generating algorithm. In the second step, the shape characteristic vectors are fast classified by the ART2 neural networks. The accuracy and effectiveness of the proposed method are verified by numerous experiments.

A Study of the Solvability of the Position Problem for Multi-Circuit Mechanisms by Way of Example of the Double Butterfly Linkage

It is frequently forgotten that the common methods of position analysis of planar linkages, which are based on dyadic decomposition, are not general. This is true regardless of whether a graphical or analytical approach is used. It is demonstrated in this paper that relatively simple linkages can give rise to problems that cannot be addressed by dyadic decomposition methods, and that solution of these problems can be far more complex.

Dyadic Affine Decompositions and Functional Wavelet Transforms

Decomposition of continuous functions can be accomplished by considering the difference of consecutive interpolation operators. When such a difference is expressed as an infinite series of some “wavelets” basis, the coefficient sequence becomes Donoho’s “interpolating wavelet transform.” Here, in contrast to the usual $L^2 $-setting, no analyzing wavelet is used to describe the wavelet transform. The objective of this paper is to study the structure of such decomposition spaces, including the formulation of bases and their duals, which leads to the notion of functional wavelet transforms (FnWT) using the duals as analyzing wavelets. Such a transform retains some of the most important properties of the integral wavelet transform of Grossmann and Morlet, such as the property of vanishing moments, which has significant applications to engineering problems.

Generalized Strichartz Inequalities for the Wave Equation

We make a synthetic exposition of the generalized Strichartz inequalities for the wave equation obtained in [6] together with the limiting cases recently obtained in [13] with as simple proofs as possible. The proofs combine stationary phase estimates, dyadic decompositions, the Hardy-Littlewood inequality or the Hölder inequality in time, and abstract duality and interpolation arguments.

Displacement Structures of Covariance Matrices, Lossless Systems, and Numerical Algorithm Design

Low displacement rank theory underlies many fast algorithms designed for structured covariance matrices. Some of these have gained notoriety for their numerical instability problems, particularly fast least-squares algorithms. Recent studies have shown that instability is not inherent to fast algorithms, but rather comes from the violation of backward consistency constraints. This paper thus details the connection between covariance matrices of a given displacement inertia and lossless rational matrices, as well as the role of this connection in numerically consistent algorithms. This basic connection allows displacement structures to be parametrized via a sequence of rotation angles obtained from a lossless system. The utility of this approach is that, irrespective of errors in the rotation parameter set, they remain consistent with a positive definite matrix of a prescribed displacement inertia. This property in turn may be rephrased as meaningful forms of backward consistency in numerical algorithms. The rotation parameters then take the form of Givens or Jacobi angles applied to data, in contrast to classical approaches which directly manipulate dyadic decompositions of the displacement structure. The concepts are illustrated in popular signal processing applications. In particular, these connections lend clear insight into the stable computation of reflection coefficients of Toeplitz systems, and also serve to resolve the numerical instability problem of fast least-squares algorithms.

An image distance measure insensitive to amplitude and mean value variations: Application to data reduction through SVD

The traditional m.s.e. distance evaluation, used commonly as a measure of degradation of images (or signals in general) through a given process, is very much affected by any linear amplification or mean value variation in the image. The significative contents of an image doesn't change with any one of the mentioned operations, so a fair distance measure should be independent from their effects. This paper describes a distance measure that fulfill this condition, and uses it to evaluate the evolution of the degradation during the Singular Values Dyadic Decomposition of some given discrete images represented by matrices. Results are plotted and discussed.

Applications of the theory of Hardy spaces to harmonic analysis on the Heisenberg group

The aim of this paper is to prove some results about the (non-commutative) Fourier transform on the Heisenberg groups H n. These results are all consequences ofa"containing Banach space" embedding result for H p spaces (Theorems 2.1 and 2.2). Theorems 4.4 and 4.5 are of interest because they give estimates for the size of Fourier transforms of elements of L p and H p. These estimates differ from Hausdorff-Young phenomena in that they do not involve fractional powers of operators. Theorem 4.3 gives sufficient conditions for an operator valued function M(2) (not necessarily diagonal) to be a "Fourier multiplier" of H q to L 2. More generally, Theorems 3.2 and 3.3 give sufficient conditions for a tempered distribution to be a convolution operator from H q to L p where 0 < q < 1 and l <_p<_ oo(p4=q). We mention that some important ingredients of this paper are the atomic definition of H p, the work of Geller [5], and the dyadic decomposition of the identity introduced by de Michele and Mauceri [2]. Related material can be found in [9].

Banach convolution algebras of functions II

LetG be a noncompact, locally compact group. By means of “generalized dyadic decompositions” ofG, translation invariant Banach spacesF(B, B, X) of (classes of) measurable functions onG are constructed, e. g. certain weighted amalgams ofLp-spaces. Basic properties of these spaces are derived and connections with spaces considered in the literature are indicated. As a main result, sufficient conditions are given which imply that a space of this type is a Banach algebra with respect to convolution.

Improved algorithm for optimal stabilisation of linear systems

An algorithm for the optimal stabilisation of linear systems is proposed, based on the dyadic decomposition of the plant matrix. The algorithm is superior to others that have been given.