9 research outputs found
Nonlinear approximation with nonstationary Gabor frames
We consider sparseness properties of adaptive time-frequency representations
obtained using nonstationary Gabor frames (NSGFs). NSGFs generalize classical
Gabor frames by allowing for adaptivity in either time or frequency. It is
known that the concept of painless nonorthogonal expansions generalizes to the
nonstationary case, providing perfect reconstruction and an FFT based
implementation for compactly supported window functions sampled at a certain
density. It is also known that for some signal classes, NSGFs with flexible
time resolution tend to provide sparser expansions than can be obtained with
classical Gabor frames. In this article we show, for the continuous case, that
sparseness of a nonstationary Gabor expansion is equivalent to smoothness in an
associated decomposition space. In this way we characterize signals with sparse
expansions relative to NSGFs with flexible time resolution. Based on this
characterization we prove an upper bound on the approximation error occurring
when thresholding the coefficients of the corresponding frame expansions. We
complement the theoretical results with numerical experiments, estimating the
rate of approximation obtained from thresholding the coefficients of both
stationary and nonstationary Gabor expansions.Comment: 19 pages, 2 figure
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
A Phase Vocoder based on Nonstationary Gabor Frames
We propose a new algorithm for time stretching music signals based on the
theory of nonstationary Gabor frames (NSGFs). The algorithm extends the
techniques of the classical phase vocoder (PV) by incorporating adaptive
time-frequency (TF) representations and adaptive phase locking. The adaptive TF
representations imply good time resolution for the onsets of attack transients
and good frequency resolution for the sinusoidal components. We estimate the
phase values only at peak channels and the remaining phases are then locked to
the values of the peaks in an adaptive manner. During attack transients we keep
the stretch factor equal to one and we propose a new strategy for determining
which channels are relevant for reinitializing the corresponding phase values.
In contrast to previously published algorithms we use a non-uniform NSGF to
obtain a low redundancy of the corresponding TF representation. We show that
with just three times as many TF coefficients as signal samples, artifacts such
as phasiness and transient smearing can be greatly reduced compared to the
classical PV. The proposed algorithm is tested on both synthetic and real world
signals and compared with state of the art algorithms in a reproducible manner.Comment: 10 pages, 6 figure
Uncertainty in the cost-effectiveness of health care interventions
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