3,468 research outputs found
Superposition frames for adaptive time-frequency analysis and fast reconstruction
In this article we introduce a broad family of adaptive, linear
time-frequency representations termed superposition frames, and show that they
admit desirable fast overlap-add reconstruction properties akin to standard
short-time Fourier techniques. This approach stands in contrast to many
adaptive time-frequency representations in the extant literature, which, while
more flexible than standard fixed-resolution approaches, typically fail to
provide efficient reconstruction and often lack the regular structure necessary
for precise frame-theoretic analysis. Our main technical contributions come
through the development of properties which ensure that this construction
provides for a numerically stable, invertible signal representation. Our
primary algorithmic contributions come via the introduction and discussion of
specific signal adaptation criteria in deterministic and stochastic settings,
based respectively on time-frequency concentration and nonstationarity
detection. We conclude with a short speech enhancement example that serves to
highlight potential applications of our approach.Comment: 16 pages, 6 figures; revised versio
Weighted Thresholding and Nonlinear Approximation
We present a new method for performing nonlinear approximation with redundant
dictionaries. The method constructs an term approximation of the signal by
thresholding with respect to a weighted version of its canonical expansion
coefficients, thereby accounting for dependency between the coefficients. The
main result is an associated strong Jackson embedding, which provides an upper
bound on the corresponding reconstruction error. To complement the theoretical
results, we compare the proposed method to the pure greedy method and the
Windowed-Group Lasso by denoising music signals with elements from a Gabor
dictionary.Comment: 22 pages, 3 figure
Compressed Sensing with Coherent and Redundant Dictionaries
This article presents novel results concerning the recovery of signals from
undersampled data in the common situation where such signals are not sparse in
an orthonormal basis or incoherent dictionary, but in a truly redundant
dictionary. This work thus bridges a gap in the literature and shows not only
that compressed sensing is viable in this context, but also that accurate
recovery is possible via an L1-analysis optimization problem. We introduce a
condition on the measurement/sensing matrix, which is a natural generalization
of the now well-known restricted isometry property, and which guarantees
accurate recovery of signals that are nearly sparse in (possibly) highly
overcomplete and coherent dictionaries. This condition imposes no incoherence
restriction on the dictionary and our results may be the first of this kind. We
discuss practical examples and the implications of our results on those
applications, and complement our study by demonstrating the potential of
L1-analysis for such problems
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