arXiv subjects

Tomer Michaeli

Publications and source records attributed to Tomer Michaeli.

76 records · Page 5Linked to original sources

arXiv2011

Partially Linear Estimation with Application to Sparse Signal Recovery From Measurement Pairs

We address the problem of estimating a random vector X from two sets of measurements Y and Z, such that the estimator is linear in Y. We show that the partially linear minimum mean squared error (PLMMSE) estimator does not require knowing the joint distribution of X and Y in full, but rather only its second-order moments. This renders it of potential interest in various applications. We further show that the PLMMSE method is minimax-optimal among all estimators that solely depend on the second-order statistics of X and Y. We demonstrate our approach in the context of recovering a signal, which is sparse in a unitary dictionary, from noisy observations of it and of a filtered version of it. We show that in this setting PLMMSE estimation has a clear computational advantage, while its performance is comparable to state-of-the-art algorithms. We apply our approach both in static and dynamic estimation applications. In the former category, we treat the problem of image enhancement from blurred/noisy image pairs, where we show that PLMMSE estimation performs only slightly worse than state-of-the art algorithms, while running an order of magnitude faster. In the dynamic setting, we provide a recursive implementation of the estimator and demonstrate its utility in the context of tracking maneuvering targets from position and acceleration measurements.

cs.IT↗

arXiv2011

Xampling at the Rate of Innovation

We address the problem of recovering signals from samples taken at their rate of innovation. Our only assumption is that the sampling system is such that the parameters defining the signal can be stably determined from the samples, a condition that lies at the heart of every sampling theorem. Consequently, our analysis subsumes previously studied nonlinear acquisition devices and nonlinear signal classes. In particular, we do not restrict attention to memoryless nonlinear distortions or to union-of-subspace models. This allows treatment of various finite-rate-of-innovation (FRI) signals that were not previously studied, including, for example, continuous phase modulation transmissions. Our strategy relies on minimizing the error between the measured samples and those corresponding to our signal estimate. This least-squares (LS) objective is generally non-convex and might possess many local minima. Nevertheless, we prove that under the stability hypothesis, any optimization method designed to trap a stationary point of the LS criterion necessarily converges to the true solution. We demonstrate our approach in the context of recovering pulse streams in settings that were not previously treated. Furthermore, in situations for which other algorithms are applicable, we show that our method is often preferable in terms of noise robustness.

cs.IT↗

arXiv2010

Performance Bounds and Design Criteria for Estimating Finite Rate of Innovation Signals

In this paper, we consider the problem of estimating finite rate of innovation (FRI) signals from noisy measurements, and specifically analyze the interaction between FRI techniques and the underlying sampling methods. We first obtain a fundamental limit on the estimation accuracy attainable regardless of the sampling method. Next, we provide a bound on the performance achievable using any specific sampling approach. Essential differences between the noisy and noise-free cases arise from this analysis. In particular, we identify settings in which noise-free recovery techniques deteriorate substantially under slight noise levels, thus quantifying the numerical instability inherent in such methods. This instability, which is only present in some families of FRI signals, is shown to be related to a specific type of structure, which can be characterized by viewing the signal model as a union of subspaces. Finally, we develop a methodology for choosing the optimal sampling kernels based on a generalization of the Karhunen--Loève transform. The results are illustrated for several types of time-delay estimation problems.

cs.IT↗

arXiv2009

Non-Invertible Gabor Transforms

Time-frequency analysis, such as the Gabor transform, plays an important role in many signal processing applications. The redundancy of such representations is often directly related to the computational load of any algorithm operating in the transform domain. To reduce complexity, it may be desirable to increase the time and frequency sampling intervals beyond the point where the transform is invertible, at the cost of an inevitable recovery error. In this paper we initiate the study of recovery procedures for non-invertible Gabor representations. We propose using fixed analysis and synthesis windows, chosen e.g. according to implementation constraints, and to process the Gabor coefficients prior to synthesis in order to shape the reconstructed signal. We develop three methods to tackle this problem. The first follows from the consistency requirement, namely that the recovered signal has the same Gabor representation as the input signal. The second, is based on the minimization of a worst-case error criterion. Last, we develop a recovery technique based on the assumption that the input signal lies in some subspace of $L_2$. We show that for each of the criteria, the manipulation of the transform coefficients amounts to a 2D twisted convolution operation, which we show how to perform using a filter-bank. When the under-sampling factor is an integer, the processing reduces to standard 2D convolution. We provide simulation results to demonstrate the advantages and weaknesses of each of the algorithms.

math.CA↗