Randomized algorithms play a central role in low rank approximations of large matrices. In this paper, the scheme of the randomized SVD is extended to a randomized LU algorithm. Several error bounds are introduced, that are based on recent results from random matrix theory related to subgassian matrices. The bounds also improve the existing bounds of already known randomized algorithm for low rank approximation. The algorithm is fully parallelized and thus can utilize efficiently GPUs without any CPU-GPU data transfer. Numerical examples, which illustrate the performance of the algorithm and compare it to other decomposition methods, are presented.Lemma 3.7 ([46]). For any m × n matrix A, let R be the n × l SRFT matrix. Then, Y = AR can be computed in O(mn log l) floating point operations. Interpolative decomposition (ID)Let A be an m × n of rank r. A ≈ A (:,J) X is the ID of rank r of A if:1. J is a subset of r indices from 1, . . . , n. 2. The r × n matrix A (:,J) is a subset of J columns from A.

In many applications, sampled data are collected in irregular fashion or are partly lost or unavailable. In these cases, it is necessary to convert irregularly sampled signals to regularly sampled ones or to restore missing data. We address this problem in the framework of a discrete sampling theorem for band-limited discrete signals that have a limited number of nonzero transform coefficients in a certain transform domain. Conditions for the image unique recovery, from sparse samples, are formulated and then analyzed for various transforms. Applications are demonstrated on examples of image superresolution and image reconstruction from sparse projections.

We present an algorithm for low rank approximation of matrices where only some of the entries in the matrix are taken into consideration. This algorithm appears in recent literature under different names, where it is described as an EM based algorithm that maximizes the likelihood for the missing entries without any relation for the mean square error minimization. When the algorithm is minimized from a mean-square-error point of view, we prove that the error produced by the algorithm is monotonically decreasing. It guarantees to converge to a local MSE minimum. We also show that an extension of the EM based algorithm for weighted low rank approximation, which appeared in recent literature, claiming that it converges to a local minimum of the MSE is wrong. Finally, we show the use of the algorithm in different applications for physics, electrical engineering and data interpolation.

It is shown that one can make use of local instabilities in turbulent video frames to enhance image resolution beyond the limit defined by the image sampling rate. The paper outlines the processing algorithm, presents its experimental verification on simulated and real-life videos and discusses its potentials and limitations.

A fast algorithm for the approximation of a low rank LU decomposition is presented. In order to achieve a low complexity, the algorithm uses sparse random projections combined with FFTbased random projections. The asymptotic approximation error of the algorithm is analyzed and a theoretical error bound is presented. Finally, numerical examples illustrate that for a similar approximation error, the sparse LU algorithm is faster than recent state-of-the-art methods. The algorithm is completely parallelizable that enables to run on a GPU. The performance is tested on a GPU card, showing a significant improvement in the running time in comparison to sequential execution.

Color videos acquired with a single CCD through turbulent media can be enhanced in their resolution beyond the limit defined by the image sampling rate. We provide a mathematical justification for this claim, present an efficient superresolution algorithm and its experimental verification on a real-life video, and finally, discuss its potentials and limitations.

The pseudo-polar Fourier transform is a specialized non-equally spaced Fourier transform, which evaluates the Fourier transform on a near-polar grid known as the pseudo-polar grid.The advantage of the pseudo-polar grid over other non-uniform sampling geometries is that the transformation, which samples the Fourier transform on the pseudo-polar grid, can be inverted using a fast and stable algorithm. For other sampling geometries, even if the nonequally spaced Fourier transform can be inverted, the only known algorithms are iterative.The convergence speed of these algorithms as well as their accuracy are difficult to control, as they depend both on the sampling geometry as well as on the unknown reconstructed object. In this paper, a direct inversion algorithm for the three-dimensional pseudo-polar Fourier transform is presented. The algorithm is based only on one-dimensional resampling operations, and is shown to be significantly faster than existing iterative inversion algorithms.

Abstract-Particle filter is a powerful method that tracks the state of a target based on non-linear observations. We present a multiscale based method that accelerates the computation of particle filters. Unlike the conventional way, which calculates weights over all particles in each cycle of the algorithm, we sample a small subset from the source particles using matrix decomposition methods. Then, we apply a function extension algorithm that uses a particle subset to recover the density function for all the rest of the particles not included in the chosen subset. The computational effort is substantial especially when multiple objects are tracked concurrently. The proposed algorithm reduces significantly the computational load. By using the Fast Gaussian Transform, the complexity of the particle selection step is reduced to a linear time in n and k , where n is the number of particles and k is the number of particles in the selected subset. We demonstrate our method on both simulated and on real data such as objects tracking in videos sequences.