10,171 research outputs found
Optimal recovery of integral operators and its applications
In this paper we present the solution to the problem of recovering rather
arbitrary integral operator based on incomplete information with error. We
apply the main result to obtain optimal methods of recovery and compute the
optimal error for the solutions to certain integral equations as well as
boundary and initial value problems for various PDE's
Diffusion Maps Kalman Filter for a Class of Systems with Gradient Flows
In this paper, we propose a non-parametric method for state estimation of
high-dimensional nonlinear stochastic dynamical systems, which evolve according
to gradient flows with isotropic diffusion. We combine diffusion maps, a
manifold learning technique, with a linear Kalman filter and with concepts from
Koopman operator theory. More concretely, using diffusion maps, we construct
data-driven virtual state coordinates, which linearize the system model. Based
on these coordinates, we devise a data-driven framework for state estimation
using the Kalman filter. We demonstrate the strengths of our method with
respect to both parametric and non-parametric algorithms in three tracking
problems. In particular, applying the approach to actual recordings of
hippocampal neural activity in rodents directly yields a representation of the
position of the animals. We show that the proposed method outperforms competing
non-parametric algorithms in the examined stochastic problem formulations.
Additionally, we obtain results comparable to classical parametric algorithms,
which, in contrast to our method, are equipped with model knowledge.Comment: 15 pages, 12 figures, submitted to IEEE TS
Sampling in the Analysis Transform Domain
Many signal and image processing applications have benefited remarkably from
the fact that the underlying signals reside in a low dimensional subspace. One
of the main models for such a low dimensionality is the sparsity one. Within
this framework there are two main options for the sparse modeling: the
synthesis and the analysis ones, where the first is considered the standard
paradigm for which much more research has been dedicated. In it the signals are
assumed to have a sparse representation under a given dictionary. On the other
hand, in the analysis approach the sparsity is measured in the coefficients of
the signal after applying a certain transformation, the analysis dictionary, on
it. Though several algorithms with some theory have been developed for this
framework, they are outnumbered by the ones proposed for the synthesis
methodology.
Given that the analysis dictionary is either a frame or the two dimensional
finite difference operator, we propose a new sampling scheme for signals from
the analysis model that allows to recover them from their samples using any
existing algorithm from the synthesis model. The advantage of this new sampling
strategy is that it makes the existing synthesis methods with their theory also
available for signals from the analysis framework.Comment: 13 Pages, 2 figure
A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal
Unveiling meaningful geophysical information from seismic data requires to
deal with both random and structured "noises". As their amplitude may be
greater than signals of interest (primaries), additional prior information is
especially important in performing efficient signal separation. We address here
the problem of multiple reflections, caused by wave-field bouncing between
layers. Since only approximate models of these phenomena are available, we
propose a flexible framework for time-varying adaptive filtering of seismic
signals, using sparse representations, based on inaccurate templates. We recast
the joint estimation of adaptive filters and primaries in a new convex
variational formulation. This approach allows us to incorporate plausible
knowledge about noise statistics, data sparsity and slow filter variation in
parsimony-promoting wavelet frames. The designed primal-dual algorithm solves a
constrained minimization problem that alleviates standard regularization issues
in finding hyperparameters. The approach demonstrates significantly good
performance in low signal-to-noise ratio conditions, both for simulated and
real field seismic data
A Tensor-Based Dictionary Learning Approach to Tomographic Image Reconstruction
We consider tomographic reconstruction using priors in the form of a
dictionary learned from training images. The reconstruction has two stages:
first we construct a tensor dictionary prior from our training data, and then
we pose the reconstruction problem in terms of recovering the expansion
coefficients in that dictionary. Our approach differs from past approaches in
that a) we use a third-order tensor representation for our images and b) we
recast the reconstruction problem using the tensor formulation. The dictionary
learning problem is presented as a non-negative tensor factorization problem
with sparsity constraints. The reconstruction problem is formulated in a convex
optimization framework by looking for a solution with a sparse representation
in the tensor dictionary. Numerical results show that our tensor formulation
leads to very sparse representations of both the training images and the
reconstructions due to the ability of representing repeated features compactly
in the dictionary.Comment: 29 page
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