27 research outputs found
Solving physics-driven inverse problems via structured least squares
Numerous physical phenomena are well modeled by partial differential equations (PDEs); they describe a wide range of phenomena across many application domains, from model- ing EEG signals in electroencephalography to, modeling the release and propagation of toxic substances in environmental monitoring. In these applications it is often of interest to find the sources of the resulting phenomena, given some sparse sensor measurements of it. This will be the main task of this work. Specifically, we will show that finding the sources of such PDE-driven fields can be turned into solving a class of well-known multi-dimensional structured least squares prob- lems. This link is achieved by leveraging from recent results in modern sampling theory – in particular, the approximate Strang-Fix theory. Subsequently, numerical simulation re- sults are provided in order to demonstrate the validity and robustness of the proposed framework
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
Exact Feature Extraction Using Finite Rate of Innovation Principles With an Application to Image Super-Resolution
The accurate registration of multiview images is of central importance in many advanced image processing applications. Image super-resolution, for example, is a typical application where the quality of the super-resolved image is degrading as registration errors increase. Popular registration methods are often based on features extracted from the acquired images. The accuracy of the registration is in this case directly related to the number of extracted features and to the precision at which the features are located: images are best registered when many features are found with a good precision. However, in low-resolution images, only a few features can be extracted and often with a poor precision. By taking a sampling perspective, we propose in this paper new methods for extracting features in low-resolution images in order to develop efficient registration techniques. We consider, in particular, the sampling theory of signals with finite rate of innovation and show that some features of interest for registration can be retrieved perfectly in this framework, thus allowing an exact registration. We also demonstrate through simulations that the sampling model which enables the use of finite rate of innovation principles is well suited for modeling the acquisition of images by a camera. Simulations of image registration and image super-resolution of artificially sampled images are first presented, analyzed and compared to traditional techniques. We finally present favorable experimental results of super-resolution of real images acquired by a digital camera available on the market
Multichannel sampling of finite rate of innovation signals
Recently there has been a surge of interest in sampling theory in signal processing
community. New efficient sampling techniques have been developed that allow
sampling and perfectly reconstructing some classes of non-bandlimited signals at
sub-Nyquist rates. Depending on the setup used and reconstruction method involved,
these schemes go under different names such as compressed sensing (CS),
compressive sampling or sampling signals with finite rate of innovation (FRI).
In this thesis we focus on the theory of sampling non-bandlimited signals
with parametric structure or specifically signals with finite rate of innovation. Most
of the theory on sampling FRI signals is based on a single acquisition device with
one-dimensional (1-D) signals. In this thesis, we extend these results to the case of
2-D signals and multichannel acquisition systems. The essential issue in multichannel
systems is that while each channel receives the input signal, it may introduce
different unknown delays, gains or affine transformations which need to be estimated
from the samples together with the signal itself. We pose both the calibration of
the channels and the signal reconstruction stage as a parametric estimation problem
and demonstrate that a simultaneous exact synchronization of the channels and reconstruction
of the FRI signal is possible. Furthermore, because in practice perfect
noise-free channels do not exist, we consider the case of noisy measurements and
show that by considering Cramer-Rao bounds as well as numerical simulations, the
multichannel systems are more resilient to noise than the single-channel ones.
Finally, we consider the problem of system identification based on the multichannel and finite rate of innovation sampling techniques. First, by employing our
multichannel sampling setup, we propose a novel algorithm for system identification
problem with known input signal, that is for the case when both the input signal and
the samples are known. Then we consider the problem of blind system identification
and propose a novel algorithm for simultaneously estimating the input FRI signal
and also the unknown system using an iterative algorithm
Multichannel Sampling of Signals With Finite Rate of Innovation
In this letter, we present a possible extension of the theory of sampling signals with finite rate of innovation (FRI) to the case of multichannel acquisition systems. The essential issue of a multichannel system is that each channel introduces different unknown delays and gains that need to be estimated for the calibration of the channels. We pose both the synchronization stage and the signal reconstruction stage as a parametric estimation problem and demonstrate that a simultaneous exact synchronization of the channels and reconstruction of the FRI signal is possible. We also consider the case of noisy measurements and evaluate the Cramér-Rao bounds (CRB) of the proposed system. Numerical results as well as the CRB show clearly that multichannel systems are more resilient to noise than the single-channel ones
Sampling Sparse Signals on the Sphere: Algorithms and Applications
We propose a sampling scheme that can perfectly reconstruct a collection of
spikes on the sphere from samples of their lowpass-filtered observations.
Central to our algorithm is a generalization of the annihilating filter method,
a tool widely used in array signal processing and finite-rate-of-innovation
(FRI) sampling. The proposed algorithm can reconstruct spikes from
spatial samples. This sampling requirement improves over
previously known FRI sampling schemes on the sphere by a factor of four for
large . We showcase the versatility of the proposed algorithm by applying it
to three different problems: 1) sampling diffusion processes induced by
localized sources on the sphere, 2) shot noise removal, and 3) sound source
localization (SSL) by a spherical microphone array. In particular, we show how
SSL can be reformulated as a spherical sparse sampling problem.Comment: 14 pages, 8 figures, submitted to IEEE Transactions on Signal
Processin
Sampling signals with finite rate of innovation
Consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation. Examples of signals with a finite rate of innovation include streams of Diracs (e.g., the Poisson process), nonuniform splines, and piecewise polynomials. Even though these signals are not bandlimited, we showthat they can be sampled uniformly at (or above) the rate of innovation using an appropriate kernel and then be perfectly reconstructed. Thus, we prove sampling theorems for classes of signals and kernels that generalize the classic "bandlimited and sinc kernel" case. In particular, we show how to sample and reconstruct periodic and finite-length streams of Diracs, nonuniform splines, and piecewise polynomials using sinc and Gaussian kernels. For infinite-length signals with finite local rate of innovation, we show local sampling and reconstruction based on spline kernels. The key in all constructions is to identify the innovative part of a signal (e.g., time instants and weights of Diracs) using an annihilating or locator filter: a device well known in spectral analysis and error-correction coding. This leads to standard computational procedures for solving the sampling problem, which we show through experimental results. Applications of these new sampling results can be found in signal processing, communications systems, and biological systems
Annihilation-driven Localised Image Edge Models
We propose a novel edge detection algorithm with sub-pixel accuracy based on annihilation of signals with finite rate of innovation. We show that the Fourier domain annihilation equations can be interpreted as spatial domain multiplications. From this new perspective, we obtain an accurate estimation of the edge model by assuming a simple parametric form within each localised block. Further, we build a locally adaptive global mask function (i.e, our edge model) for the whole image. The mask function is then used as an edge- preserving constraint in further processing. Numerical experiments on both edge localisations and image up-sampling show the effectiveness of the proposed approach, which out- performs state-of-the-art method