Computation of greatest common divisor for the blind deconvolution of transient impulsive signals

Comunicación presentada al First Internacional Workshop on Marine technology. Villanova i La Geltrú (Barcelona), 2005.We propose a new blind deconvolution method for transient impulsive signals in a single input – multiple output (SIMO) system. The method exploits the data redundancy inherent to SIMO multichannel systems to obtain an estimation of the input signal. The method is built upon the assumptions of finite-length signals and channel diversity

The L1-Potts functional for robust jump-sparse reconstruction

We investigate the non-smooth and non-convex $L^1$-Potts functional in discrete and continuous time. We show $\Gamma$-convergence of discrete $L^1$-Potts functionals towards their continuous counterpart and obtain a convergence statement for the corresponding minimizers as the discretization gets finer. For the discrete $L^1$-Potts problem, we introduce an $O(n^2)$ time and $O(n)$ space algorithm to compute an exact minimizer. We apply $L^1$-Potts minimization to the problem of recovering piecewise constant signals from noisy measurements $f.$ It turns out that the $L^1$-Potts functional has a quite interesting blind deconvolution property. In fact, we show that mildly blurred jump-sparse signals are reconstructed by minimizing the $L^1$-Potts functional. Furthermore, for strongly blurred signals and known blurring operator, we derive an iterative reconstruction algorithm

Generic Feasibility of Perfect Reconstruction with Short FIR Filters in Multi-channel Systems

We study the feasibility of short finite impulse response (FIR) synthesis for perfect reconstruction (PR) in generic FIR filter banks. Among all PR synthesis banks, we focus on the one with the minimum filter length. For filter banks with oversampling factors of at least two, we provide prescriptions for the shortest filter length of the synthesis bank that would guarantee PR almost surely. The prescribed length is as short or shorter than the analysis filters and has an approximate inverse relationship with the oversampling factor. Our results are in form of necessary and sufficient statements that hold generically, hence only fail for elaborately-designed nongeneric examples. We provide extensive numerical verification of the theoretical results and demonstrate that the gap between the derived filter length prescriptions and the true minimum is small. The results have potential applications in synthesis FB design problems, where the analysis bank is given, and for analysis of fundamental limitations in blind signals reconstruction from data collected by unknown subsampled multi-channel systems.Comment: Manuscript submitted to IEEE Transactions on Signal Processin

Hierarchical Bayesian sparse image reconstruction with application to MRFM

This paper presents a hierarchical Bayesian model to reconstruct sparse images when the observations are obtained from linear transformations and corrupted by an additive white Gaussian noise. Our hierarchical Bayes model is well suited to such naturally sparse image applications as it seamlessly accounts for properties such as sparsity and positivity of the image via appropriate Bayes priors. We propose a prior that is based on a weighted mixture of a positive exponential distribution and a mass at zero. The prior has hyperparameters that are tuned automatically by marginalization over the hierarchical Bayesian model. To overcome the complexity of the posterior distribution, a Gibbs sampling strategy is proposed. The Gibbs samples can be used to estimate the image to be recovered, e.g. by maximizing the estimated posterior distribution. In our fully Bayesian approach the posteriors of all the parameters are available. Thus our algorithm provides more information than other previously proposed sparse reconstruction methods that only give a point estimate. The performance of our hierarchical Bayesian sparse reconstruction method is illustrated on synthetic and real data collected from a tobacco virus sample using a prototype MRFM instrument.Comment: v2: final version; IEEE Trans. Image Processing, 200

Learning earthquake sources using symmetric autoencoders

We introduce Symmetric Autoencoder (SymAE), a neural-network architecture designed to automatically extract earthquake information from far-field seismic waves. SymAE represents the measured displacement field using a code that is partitioned into two interpretable components: source and path-scattering information. We achieve this source-path representation using the scale separation principle and stochastic regularization, which traditional autoencoding methods lack. According to the scale separation principle, the variations in far-field band-limited seismic measurements resulting from finite faulting occur across two spatial scales: a slower scale associated with the source processes and a faster scale corresponding to path effects. Once trained, SymAE facilitates the generation of virtual seismograms, engineered to not contain subsurface scattering effects. We present time-reversal imaging of virtual seismograms to accurately infer the kinematic rupture parameters without knowledge of empirical Green's function. SymAE is an unsupervised learning method that can efficiently scale with large amounts of seismic data and does not require labeled seismograms, making it the first framework that can learn from all available previous earthquakes to accurately characterize a given earthquake. The paper presents the results of an analysis of nearly thirty complex earthquake events, revealing differences between earthquakes in energy rise times, stopping phases, and providing insights into their rupture complexity