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A simple construction of an orthonormal basis starting with a so called mother wavelet, together with an efficient implementation gained the wavelet decomposition easy acceptance and generated a great research interest in its applications. An orthonormal basis may not, however, always be a suitable representation of a signal, particularly when time (or space) invariance is a required property. The conventional way around this problem is to use a redundant decomposition. In this paper, we address the time invariance problem for orthonormal wavelet transforms and propose an extension to wavelet packet decompositions. We show that it is possible to achieve time invariance and preserve the orthonormality. We subsequently propose an efficient approach to obtain such a decomposition. We demonstrate the importance of our method by considering some application examples in signal reconstruction and time delay estimation.

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We propose a best basis algorithm for signal enhancement in white Gaussian noise.The best basis search is performed in families of orthonormal bases constructed with wavelet packets or local cosine bases. We base our search for the "best" basis on a criterion of minimal reconstruction error of the underlying signal. This approach is intuitively appealing because the enhanced or estimated signal has an associated measure of performance, namely the resulting mean-square error. Previous approaches in this framework have focused on obtaining the most "compact" signal representations, which consequently contribute to effective denoising. These approaches, however, do not possess the inherent measure of performance which our algorithm provides.We first propose an estimator of the mean-square error, based on a heuristic argument and subsequently compare our simple error criterion to the Stein unbiased risk estimator. We compare the two proposed estimators by providing both qualitative and quantitative analyses of the bias term. Having two estimators of the mean-square error, we incorporate these cost functions into the search for the "best" basis, and subsequently provide a substantiating example to demonstrate their performance.

We introduce a family of first-order multidimensional ordinary differential equations (ODEs) with discontinuous right-hand sides and demonstrate their applicability in image processing. An equation belonging to this family is an inverse diffusion everywhere except at local extrema, where some stabilization is introduced. For this reason, we call these equations "stabilized inverse diffusion equations" (SIDEs). Existence and uniqueness of solutions, as well as stability, are proven for SIDEs. A SIDE in one spatial dimension may be interpreted as a limiting case of a semi-discretized Perona-Malik equation. In an experiment, SIDE's are shown to suppress noise while sharpening edges present in the input signal. Their application to image segmentation is also demonstrated.

In this paper we consider the problem of estimating the eigenvectors of the sample covariance matrix of decentralized measurements in a distributed fashion. The need for a distributed scheme is motivated by the many moment based methods that resort to the covariance of the data to extract information from the measurements. For large sensor network, gathering the data at a central processor generates a communication bottleneck. Our algorithm is based on a combination of the so called power method, that is used to compute the eigenvectors, and the average consensus protocol, that is utilized to structure the information exchange into a gossiping protocol. Our work shows how a completely distributed scheme based on near neighbors communications is feasible, and applies the proposed method to the estimation of the direction of arrival of a signal source.

Nonlinear filtering techniques based on the theory of robust estimation are introduced. Some deterministic and asymptotic properties are derived. The proposed denoising methods are optimal over the Huber-contaminated normal neighborhood and are highly resistant to outliers. Experimental results showing a much improved performance of the proposed filters in the presence of Gaussian and heavy-tailed noise are analyzed and illustrated.

We propose a new approach to detect and quantify the periodic structure of dynamical systems using topological methods. We propose to use delay-coordinate embedding as a tool to detect the presence of harmonic structures by using persistent homology for robust analysis of point clouds of delay-coordinate embeddings. To discover the proper delay, we propose an autocorrelation like (ACL) function of the signals, and apply the introduced topological approach to analyze breathing sound signals for wheeze detection. Experiments have been carried out to substantiate the capabilities of the proposed method.Index Terms-Algebraic topology algorithms, audio analysis, biomedical signal processing, topological signal analysis.