Distributed wavelet processing within sensor networks holds promise for reducing communication energy and wireless bandwidth usage at sensor nodes. Local collaboration among nodes de-correlates measurements, yielding a sparser data set with significant values at far fewer nodes. Sparsity can then be leveraged for subsequent processing such as measurement compression, de-noising, and query routing. A number of factors complicate realizing such a transform in real-world deployments, including irregular spatial placement of nodes and a potentially prohibitive energy cost associated with calculating the transform in-network. In this paper, we address these concerns head-on; our contributions are fourfold. First, we propose a simple interpolatory wavelet transform for irregular sampling grids. Second, using ns-2 simulations of network traffic generated by the transform, we establish for a variety of network configurations break-even points in network size beyond which multiscale data processing provides energy savings. Distributed lossy compression of network measurements provides a representative application for this study. Third, we develop a new protocol for extracting approximations given only a vague notion of source statistics and analyze its energy savings over a more intuitive but naïve approach. Finally, we extend the 2-dimensional (2-D) spatial irregular grid transform to a 3-D spatio-temporal transform, demonstrating the substantial gain of distributed 3-D compression over repeated 2-D compression.

This paper is concerned with the construction and analysis of a universal estimator for the regression problem in supervised learning. Universal means that the estimator does not depend on any a priori assumptions about the regression function to be estimated. The universal estimator studied in this paper consists of a least-square fitting procedure using piecewise constant functions on a partition which depends adaptively on the data. The partition is generated by a splitting procedure which differs from those used in CART algorithms. It is proven that this estimator performs at the optimal convergence rate for a wide class of priors on the regression function. Namely, as will be made precise in the text, if the regression function is in any one of a certain class of approximation spaces (or smoothness spaces of order not exceeding one -- a limitation resulting because the estimator uses piecewise constants) measured relative to the marginal measure, then the estimator converges to the regression function (in the least squares sense) with an optimal rate of convergence in terms of the number of samples. The estimator is also numerically feasible and can be implemented on-line.

We introduce nonlinear edge-adapted multiresolution transforms for image processing. These transforms have the same structure as wavelet basis decompositions, but incorporate a specific geometric treatment of edges, which results in sparser representations of piecewise smooth images and in turn better compression properties. We also observe visual improvements over classical wavelets for the compression of real images.

This paper provides a mathematical analysis of transform compression in its relationship to linear and nonlinear approximation theory. Contrasting linear and nonlinear approximation spaces, we show that there are interesting classes of functions/random processes which are much more compactly represented by wavelet-based nonlinear approximation. These classes include locally smooth signals that have singularities, and provide a model for many signals encountered in practice, in particular for images. However, we also show that nonlinear approximation results do not always translate to efficient compress on strategies in a rate-distortion sense. Based on this observation, we construct compression techniques and formulate the family of functions/stochastic processes for which they provide efficient descriptions in a rate-distortion sense. We show that this family invariably leads to Besov spaces, yielding a natural relationship among Besov smoothness, linear/nonlinear approximation order, and compression performance in a rate-distortion sense. The designed compression techniques show similarities to modern high-performance transform codecs, allowing us to establish relevant rate-distortion estimates and identify performance limits.

We introduce new multiscale representations for images which incorporate a specific geometric treatment of edges. The associated transforms are inherently nonlinear and nontensor product, in contrast to classical wavelet basis decompositions over which they exhibit visual improvement in terms of compression. This approach can be viewed as a bridge between edge detection and the nonlinear multiresolution representations of Ami Harten (see Journal of Applied Numerical Mathematics, vol.12, p.153-93, 1993).

The authors give an overview of the continuous and oversampled wavelet transform. They discuss how, by sampling the continuous wavelet transform, orthonormal wavelet bases can be obtained. Multiresolution analysis, as a framework for studying wavelet bases, is also presented. Finally, the authors deal with discrete-time wavelet representations, filter banks, and fast algorithms.

We study the non-orthogonality of perfect-reconstruction (PR) biorthogonal filter banks by measuring the energy preservation between the spatial and transform domains. The mathematical formulation of that issue leads to the computation of the Riesz constants, and a more relevant modelization leads to a measure of near-orthogonality which is well suited for image compression systems based on filter banks. This provides a criterion for the validity of the energy preservation approximation: we can compare the latter approximation with the one that is made when estimating the perceptual quality of an image by the mean square error.