In this paper, vector-valued multiresolution analysis and orthogonal vector-valued wavelets are introduced. The definition for orthogonal vector-valued wavelet packets is proposed. A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is derived by means of paraunitary vector filter bank theory. An algorithm for constructing a class of compactly supported orthogonal vector-valued wavelets is presented. The properties of the vector-valued wavelet packets are investigated by using operator theory and algebra theory. In particular, it is shown how to construct various orthonormal bases of L2(R, Cs) from the orthogonal vector-valued wavelet packets.

A study on compactly supported orthogonal vector-valued wavelets and wavelet packets

In this paper, we introduce the biorthogonal vector-valued wavelets. We prove that, like in the scalar wavelet case, the existence of a pair of biorthogonal compactly supported vector-valued scaling functions guarantees the existence of a pair of biorthogonal compactly supported vector-valued wavelet functions. An algorithm for constructing a pair of biorthogonal compactly supported vector-valued wavelet functions is presented by means of vector-valued multiresolution analysis and matrix theory. The notion of biorthogonal vector-valued wavelet packets is introduced, and their properties are investigated by virtue of time–frequency analysis and algebra theory. Three biorthogonality formulas concerning the wavelet packets are established. Relation to some physical theories such as E -infinity Cantorian space–time theory is also discussed.

The research of a class of biorthogonal compactly supported vector-valued wavelets

The problem of signal denoising with an orthogonal basis is considered. The existing approaches convert the considered problem into one of finding a threshold for estimates of basis coefficients. In this paper, a new solution to the denoising problem is proposed. The method is based on the description length of the noiseless data in subspaces of the bases. For each subspace, we estimate the desired description length and suggest choosing the subspace for which this quantity is minimized. We provide a method of probabilistically estimating the reconstruction error. This estimate is used for probabilistic validation of the desired description length. In existing thresholding methods, the optimum threshold is obtained as a function of the additive noise variance. In practical problems, where the noise variance is unknown, the first step is to estimate the noise variance. The estimated noise variance is then used in calculating the optimum threshold. Unlike such approaches, in the proposed method, the noise variance estimation and the signal denoising are done simultaneously.

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A new information-theoretic approach to signal denoising and best basis selection

NsrR is a nitric oxide-sensitive regulator of transcription. In Escherichia coli, NsrR is a repressor of the hmp gene encoding the flavohemoglobin that detoxifies nitric oxide. Three other transcription units (ytfE, ygbA, and hcp-hcr) are known to be subject to regulation by NsrR. This chapter describes experimental and statistical protocols used to identify NsrR-binding sites in the E. coli chromosome using chromatin immunoprecipitation and microarray analysis. The methods are applicable, with suitable modifications, to any regulatory protein and any organism.

Genome-wide identification of binding sites for the nitric oxide-sensitive transcriptional regulator NsrR.

In this paper, we introduce matrix-valued multiresolution analysis and orthogonal matrix-valued wavelets with arbitrary integer dilation factor m. A necessary and sufficient condition on the existence of orthogonal matrix-valued wavelets is derived by virtue of paraunitary vector filter bank theory. An algorithm for constructing compactly supported m-scale orthogonal matrix-valued wavelets is presented. The notion of orthogonal matrix-valued wavelet packets is proposed. Their properties are investigated by means of time–frequency method, operator theory and matrix theory. In particular, it is shown how to construct various orthonormal bases of space L2(R, Cr×r) from these wavelet packets, and the orthogonal decomposition relation is also given.

Construction and properties of orthogonal matrix-valued wavelets and wavelet packets☆

Multiwavelets are relative newcomers into the world of wavelets. Thus, it has not been a surprise that the used methods of denoising are modified universal thresholding procedures developed for uniwavelets. On the other hand, the specific of a multiwavelet discrete transform is that typical errors are not identically distributed and correlated, whereas the theory of the universal thresholding is based on the assumption of identically distributed and independent normal errors. Thus, we suggest an alternative denoising procedure based on the Efromovich-Pinsker algorithm. We show that this procedure is optimal over a wide class of noise distributions. Moreover, together with a new cristina class of biorthogonal multiwavelets, which is introduced in this paper, the procedure implies an optimal method for recovering the derivative of a noisy signal. A Monte Carlo study supports these conclusions.

S. Efromovich

Papers

Data-driven and optimal denoising of a signal and recovery of its derivative using multiwavelets