17,816 research outputs found
PRISM: Sparse Recovery of the Primordial Power Spectrum
The primordial power spectrum describes the initial perturbations in the
Universe which eventually grew into the large-scale structure we observe today,
and thereby provides an indirect probe of inflation or other
structure-formation mechanisms. Here, we introduce a new method to estimate
this spectrum from the empirical power spectrum of cosmic microwave background
(CMB) maps.
A sparsity-based linear inversion method, coined \textbf{PRISM}, is
presented. This technique leverages a sparsity prior on features in the
primordial power spectrum in a wavelet basis to regularise the inverse problem.
This non-parametric approach does not assume a strong prior on the shape of the
primordial power spectrum, yet is able to correctly reconstruct its global
shape as well as localised features. These advantages make this method robust
for detecting deviations from the currently favoured scale-invariant spectrum.
We investigate the strength of this method on a set of WMAP 9-year simulated
data for three types of primordial power spectra: a nearly scale-invariant
spectrum, a spectrum with a small running of the spectral index, and a spectrum
with a localised feature. This technique proves to easily detect deviations
from a pure scale-invariant power spectrum and is suitable for distinguishing
between simple models of the inflation. We process the WMAP 9-year data and
find no significant departure from a nearly scale-invariant power spectrum with
the spectral index .
A high resolution primordial power spectrum can be reconstructed with this
technique, where any strong local deviations or small global deviations from a
pure scale-invariant spectrum can easily be detected
Compressive Parameter Estimation for Sparse Translation-Invariant Signals Using Polar Interpolation
We propose new compressive parameter estimation algorithms that make use of
polar interpolation to improve the estimator precision. Our work extends
previous approaches involving polar interpolation for compressive parameter
estimation in two aspects: (i) we extend the formulation from real non-negative
amplitude parameters to arbitrary complex ones, and (ii) we allow for mismatch
between the manifold described by the parameters and its polar approximation.
To quantify the improvements afforded by the proposed extensions, we evaluate
six algorithms for estimation of parameters in sparse translation-invariant
signals, exemplified with the time delay estimation problem. The evaluation is
based on three performance metrics: estimator precision, sampling rate and
computational complexity. We use compressive sensing with all the algorithms to
lower the necessary sampling rate and show that it is still possible to attain
good estimation precision and keep the computational complexity low. Our
numerical experiments show that the proposed algorithms outperform existing
approaches that either leverage polynomial interpolation or are based on a
conversion to a frequency-estimation problem followed by a super-resolution
algorithm. The algorithms studied here provide various tradeoffs between
computational complexity, estimation precision, and necessary sampling rate.
The work shows that compressive sensing for the class of sparse
translation-invariant signals allows for a decrease in sampling rate and that
the use of polar interpolation increases the estimation precision.Comment: 13 pages, 5 figures, to appear in IEEE Transactions on Signal
Processing; minor edits and correction
Sub-Nyquist Sampling: Bridging Theory and Practice
Sampling theory encompasses all aspects related to the conversion of
continuous-time signals to discrete streams of numbers. The famous
Shannon-Nyquist theorem has become a landmark in the development of digital
signal processing. In modern applications, an increasingly number of functions
is being pushed forward to sophisticated software algorithms, leaving only
those delicate finely-tuned tasks for the circuit level.
In this paper, we review sampling strategies which target reduction of the
ADC rate below Nyquist. Our survey covers classic works from the early 50's of
the previous century through recent publications from the past several years.
The prime focus is bridging theory and practice, that is to pinpoint the
potential of sub-Nyquist strategies to emerge from the math to the hardware. In
that spirit, we integrate contemporary theoretical viewpoints, which study
signal modeling in a union of subspaces, together with a taste of practical
aspects, namely how the avant-garde modalities boil down to concrete signal
processing systems. Our hope is that this presentation style will attract the
interest of both researchers and engineers in the hope of promoting the
sub-Nyquist premise into practical applications, and encouraging further
research into this exciting new frontier.Comment: 48 pages, 18 figures, to appear in IEEE Signal Processing Magazin
Compressive Sensing Using Iterative Hard Thresholding with Low Precision Data Representation: Theory and Applications
Modern scientific instruments produce vast amounts of data, which can
overwhelm the processing ability of computer systems. Lossy compression of data
is an intriguing solution, but comes with its own drawbacks, such as potential
signal loss, and the need for careful optimization of the compression ratio. In
this work, we focus on a setting where this problem is especially acute:
compressive sensing frameworks for interferometry and medical imaging. We ask
the following question: can the precision of the data representation be lowered
for all inputs, with recovery guarantees and practical performance? Our first
contribution is a theoretical analysis of the normalized Iterative Hard
Thresholding (IHT) algorithm when all input data, meaning both the measurement
matrix and the observation vector are quantized aggressively. We present a
variant of low precision normalized {IHT} that, under mild conditions, can
still provide recovery guarantees. The second contribution is the application
of our quantization framework to radio astronomy and magnetic resonance
imaging. We show that lowering the precision of the data can significantly
accelerate image recovery. We evaluate our approach on telescope data and
samples of brain images using CPU and FPGA implementations achieving up to a 9x
speed-up with negligible loss of recovery quality.Comment: 19 pages, 5 figures, 1 table, in IEEE Transactions on Signal
Processin
Image registration with sparse approximations in parametric dictionaries
We examine in this paper the problem of image registration from the new
perspective where images are given by sparse approximations in parametric
dictionaries of geometric functions. We propose a registration algorithm that
looks for an estimate of the global transformation between sparse images by
examining the set of relative geometrical transformations between the
respective features. We propose a theoretical analysis of our registration
algorithm and we derive performance guarantees based on two novel important
properties of redundant dictionaries, namely the robust linear independence and
the transformation inconsistency. We propose several illustrations and insights
about the importance of these dictionary properties and show that common
properties such as coherence or restricted isometry property fail to provide
sufficient information in registration problems. We finally show with
illustrative experiments on simple visual objects and handwritten digits images
that our algorithm outperforms baseline competitor methods in terms of
transformation-invariant distance computation and classification
CoSaMP: Iterative signal recovery from incomplete and inaccurate samples
Compressive sampling offers a new paradigm for acquiring signals that are
compressible with respect to an orthonormal basis. The major algorithmic
challenge in compressive sampling is to approximate a compressible signal from
noisy samples. This paper describes a new iterative recovery algorithm called
CoSaMP that delivers the same guarantees as the best optimization-based
approaches. Moreover, this algorithm offers rigorous bounds on computational
cost and storage. It is likely to be extremely efficient for practical problems
because it requires only matrix-vector multiplies with the sampling matrix. For
many cases of interest, the running time is just O(N*log^2(N)), where N is the
length of the signal.Comment: 30 pages. Revised. Presented at Information Theory and Applications,
31 January 2008, San Dieg
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