10,514 research outputs found
Exact Local Reconstruction Algorithms for Signals with Finite Rate of Innovation
Consider the problem of sampling signals which are not bandlimited, but still have a finite number of degrees of freedom per unit of time, such as, for example, piecewise polynomial or piecewise sinusoidal signals, and call the number of degrees of freedom per unit of time the rate of innovation. Classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited. In this paper, we show that many signals with finite rate of innovation can be sampled and perfectly reconstructed using kernels of compact support and a local reconstruction algorithm. The class of kernels that we can use is very rich and includes functions satisfying strang-fix conditions, exponential splines and functions with rational Fourier transforms. Extension of such results to the 2-dimensional case are also discussed and an application to image super-resolution is presente
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
On the accuracy of solving confluent Prony systems
In this paper we consider several nonlinear systems of algebraic equations
which can be called "Prony-type". These systems arise in various reconstruction
problems in several branches of theoretical and applied mathematics, such as
frequency estimation and nonlinear Fourier inversion. Consequently, the
question of stability of solution with respect to errors in the right-hand side
becomes critical for the success of any particular application. We investigate
the question of "maximal possible accuracy" of solving Prony-type systems,
putting stress on the "local" behavior which approximates situations with low
absolute measurement error. The accuracy estimates are formulated in very
simple geometric terms, shedding some light on the structure of the problem.
Numerical tests suggest that "global" solution techniques such as Prony's
algorithm and ESPRIT method are suboptimal when compared to this theoretical
"best local" behavior
Sampling and Reconstruction of Signals in a Reproducing Kernel Subspace of
In this paper, we consider sampling and reconstruction of signals in a
reproducing kernel subspace of L^p(\Rd), 1\le p\le \infty, associated with an
idempotent integral operator whose kernel has certain off-diagonal decay and
regularity. The space of -integrable non-uniform splines and the
shift-invariant spaces generated by finitely many localized functions are our
model examples of such reproducing kernel subspaces of L^p(\Rd). We show that
a signal in such reproducing kernel subspaces can be reconstructed in a stable
way from its samples taken on a relatively-separated set with sufficiently
small gap. We also study the exponential convergence, consistency, and the
asymptotic pointwise error estimate of the iterative approximation-projection
algorithm and the iterative frame algorithm for reconstructing a signal in
those reproducing kernel spaces from its samples with sufficiently small gap
Sampling and Reconstruction of Shapes with Algebraic Boundaries
We present a sampling theory for a class of binary images with finite rate of
innovation (FRI). Every image in our model is the restriction of
\mathds{1}_{\{p\leq0\}} to the image plane, where \mathds{1} denotes the
indicator function and is some real bivariate polynomial. This particularly
means that the boundaries in the image form a subset of an algebraic curve with
the implicit polynomial . We show that the image parameters --i.e., the
polynomial coefficients-- satisfy a set of linear annihilation equations with
the coefficients being the image moments. The inherent sensitivity of the
moments to noise makes the reconstruction process numerically unstable and
narrows the choice of the sampling kernels to polynomial reproducing kernels.
As a remedy to these problems, we replace conventional moments with more stable
\emph{generalized moments} that are adjusted to the given sampling kernel. The
benefits are threefold: (1) it relaxes the requirements on the sampling
kernels, (2) produces annihilation equations that are robust at numerical
precision, and (3) extends the results to images with unbounded boundaries. We
further reduce the sensitivity of the reconstruction process to noise by taking
into account the sign of the polynomial at certain points, and sequentially
enforcing measurement consistency. We consider various numerical experiments to
demonstrate the performance of our algorithm in reconstructing binary images,
including low to moderate noise levels and a range of realistic sampling
kernels.Comment: 12 pages, 14 figure
FRESH – FRI-based single-image super-resolution algorithm
In this paper, we consider the problem of single image super-resolution and propose a novel algorithm that outperforms state-of-the-art methods without the need of learning patches pairs from external data sets. We achieve this by modeling images and, more precisely, lines of images as piecewise smooth functions and propose a resolution enhancement method for this type of functions. The method makes use of the theory of sampling signals with finite rate of innovation (FRI) and combines it with traditional linear reconstruction methods. We combine the two reconstructions by leveraging from the multi-resolution analysis in wavelet theory and show how an FRI reconstruction and a linear reconstruction can be fused using filter banks. We then apply this method along vertical, horizontal, and diagonal directions in an image to obtain a single-image super-resolution algorithm. We also propose a further improvement of the method based on learning from the errors of our super-resolution result at lower resolution levels. Simulation results show that our method outperforms state-of-the-art algorithms under different blurring kernels
- …