5,997 research outputs found
Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization
We establish wavelet characterizations of homogeneous Besov spaces on
stratified Lie groups, both in terms of continuous and discrete wavelet
systems.
We first introduce a notion of homogeneous Besov space in
terms of a Littlewood-Paley-type decomposition, in analogy to the well-known
characterization of the Euclidean case. Such decompositions can be defined via
the spectral measure of a suitably chosen sub-Laplacian. We prove that the
scale of Besov spaces is independent of the precise choice of Littlewood-Paley
decomposition. In particular, different sub-Laplacians yield the same Besov
spaces.
We then turn to wavelet characterizations, first via continuous wavelet
transforms (which can be viewed as continuous-scale Littlewood-Paley
decompositions), then via discretely indexed systems. We prove the existence of
wavelet frames and associated atomic decomposition formulas for all homogeneous
Besov spaces , with and .Comment: 39 pages. This paper is to appear in Journal of Function Spaces and
Applications. arXiv admin note: substantial text overlap with arXiv:1008.451
Orthonormal and biorthonormal filter banks as convolvers, and convolutional coding gain
Convolution theorems for filter bank transformers are introduced. Both uniform and nonuniform decimation ratios are considered, and orthonormal as well as biorthonormal cases are addressed. All the theorems are such that the original convolution reduces to a sum of shorter, decoupled convolutions in the subbands. That is, there is no need to have cross convolution between subbands. For the orthonormal case, expressions for optimal bit allocation and the optimized coding gain are derived. The contribution to coding gain comes partly from the nonuniformity of the signal spectrum and partly from nonuniformity of the filter spectrum. With one of the convolved sequences taken to be the unit pulse function,,e coding gain expressions reduce to those for traditional subband and transform coding. The filter-bank convolver has about the same computational complexity as a traditional convolver, if the analysis bank has small complexity compared to the convolution itself
Multiscale Representations for Manifold-Valued Data
We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere , the special orthogonal group , the positive definite matrices , and the Grassmann manifolds . The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the and maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as , , , where the and maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper
Structure-Aware Sampling: Flexible and Accurate Summarization
In processing large quantities of data, a fundamental problem is to obtain a
summary which supports approximate query answering. Random sampling yields
flexible summaries which naturally support subset-sum queries with unbiased
estimators and well-understood confidence bounds.
Classic sample-based summaries, however, are designed for arbitrary subset
queries and are oblivious to the structure in the set of keys. The particular
structure, such as hierarchy, order, or product space (multi-dimensional),
makes range queries much more relevant for most analysis of the data.
Dedicated summarization algorithms for range-sum queries have also been
extensively studied. They can outperform existing sampling schemes in terms of
accuracy on range queries per summary size. Their accuracy, however, rapidly
degrades when, as is often the case, the query spans multiple ranges. They are
also less flexible - being targeted for range sum queries alone - and are often
quite costly to build and use.
In this paper we propose and evaluate variance optimal sampling schemes that
are structure-aware. These summaries improve over the accuracy of existing
structure-oblivious sampling schemes on range queries while retaining the
benefits of sample-based summaries: flexible summaries, with high accuracy on
both range queries and arbitrary subset queries
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