On Sampling Methods for Linear Scale-Invariant Systems

4 pagesInternational audienceWe study a class of self-similar processes that are not stationary, nor have stationary increments. They are called Euler-Cauchy (EC) processes and are built as output of linear scale-invariant parametric systems. This article study several discretization methods of EC processes which are not bandlimited processes: direct sampling, bilinear transformation and approximation on fractional B-splines. For the three different methods, we obtain theoretical formulae and compute numerical realizations and properties

Accelerated Spectral Clustering Using Graph Filtering Of Random Signals

We build upon recent advances in graph signal processing to propose a faster spectral clustering algorithm. Indeed, classical spectral clustering is based on the computation of the first k eigenvectors of the similarity matrix' Laplacian, whose computation cost, even for sparse matrices, becomes prohibitive for large datasets. We show that we can estimate the spectral clustering distance matrix without computing these eigenvectors: by graph filtering random signals. Also, we take advantage of the stochasticity of these random vectors to estimate the number of clusters k. We compare our method to classical spectral clustering on synthetic data, and show that it reaches equal performance while being faster by a factor at least two for large datasets

Sparse time-frequency distributions of chirps from a compressed sensing perspective

International audienceConsidering that multicomponent chirp signals are sparse in the time-frequency domain, it is possible to attach to them highly localized distributions thanks to a compressed sensing approach based on very few measurements in the ambiguity plane. The principle of the technique is described, with emphasis on the choice of the measurement subset for which an optimality criterion is proposed

Stochastic discrete scale invariance: Renormalization group operators and Iterated Function Systems

International audienceWe revisit here the notion of discrete scale invariance. Initially defined for signal indexed by the positive reals, we present a generalized version of discrete scale invariant signals relying on a renormalization group approach. In this view, the signals are seen as fixed point of a renormalization operator acting on a space of signal. We recall how to show that these fixed point present discrete scale invariance. As an illustration we use the random iterated function system as generators of random processes of the interval that are dicretely scale invariant

Time-frequency localization from sparsity constraints

4 pages, 3 figures, 1 table, submitted to IEEE Int. Conf. on Acoust., Speech and Signal Proc. ICASSP 2008.In the case of multicomponent AM-FM signals, the idealized representation which consists of weighted trajectories on the time-frequency (TF) plane, is intrinsically sparse. Recent advances in optimal recovery from sparsity constraints thus suggest to revisit the issue of TF localization by exploiting sparsity, as adapted to the specific context of (quadratic) TF distributions. Based on classical results in TF analysis, it is argued that the relevant information is mostly concentrated in a restricted subset of Fourier coefficients of the Wigner-Ville distribution neighbouring the origin of the ambiguity plane. Using this incomplete information as the primary constraint, the desired distribution follows as the minimum l1-norm solution in the transformed TF domain. Possibilities and limitations of the approach are demonstrated via controlled numerical experiments, its performance is assessed in various configurations and the results are compared with standard techniques. It is shown that improved representations can be obtained, though at a computational cost which is significantly increased

Time-Frequency Surrogates for Nonstationary Signal Analysis

International audienceThe purpose of the present communication is, after a brief outline of the use of simple surrogates as introduced so far for stationarity tests, to deal with constructions of surrogates in ''time-frequency domains''. Indeed, for transient detection or cross-correlations analysis, one need to construct directly ''surrogate time-frequency distributions'', as opposed to distributions of surrogate time series, and keeping the `geometrical' structure in the plane of the quadratic distribution

Revisiting and testing stationarity

6 pages, 4 figures, 10 references. To be presented at the 2008 Euro American Workshop on Information Optics will be held during June 1 - 5, 2008 at Les Tresoms in Annecy, France.The concept of stationarity is revisited from an operational perspective that explicitly takes into account the observation scale. A general framework is described for testing such a relative stationarity via the introduction of stationarized surrogate data