Sparse multidimensional modal analysis using a multigrid dictionary refinement

Abstract

We address the problem of multidimensional modal estimation using sparse estimation techniques coupled with an efficient multigrid approach. Modal dictionaries are obtained by discretizing modal functions (damped complex exponentials). To get a good resolution, it is necessary to choose a fine discretization grid resulting in intractable computational problems due to the huge size of the dictionaries. The idea behind the multigrid approach amounts to refine the dictionary over several levels of resolution. The algorithm starts from a coarse grid and adaptively improves the resolution in dependence of the active set provided by sparse approximation methods. The proposed method is quite general in the sense that it allows one to process in the same way mono-and multidimensional signals. We show through simulations that, as compared to high-resolution modal estimation methods, the proposed sparse modal method can greatly enhance the estimation accuracy for noisy signals and shows good robustness with respect to the choice of the number of components.

Sparse modal estimation of 2-D NMR signals

“…Rather, we propose to start with a coarse one (N low) and to adaptively refine it through a multi-grid scheme. This principle proposed in [12] is sketched on figure 1. The main idea is the adaptation of the dictionary as a function of the previous dictionary and the estimated coefficients.…”

“…The method we propose here for pairing of 2-D modes consists in exploiting the sparse approximation principle for R-D signals [12]. We denote by F a and F b the number of estimated modesâ i andb k , respectively.…”

“…This complexity is further increased in the case of 2-D signals in which we are confronted with 2-D grids. We have recently proposed a multi-grid sparse approximation method [12] where the estimation is started with a coarse grid which is then iteratively refined by inserting new grid points leading to a multiresolution-like scheme. For 1-D signals, this method allows achieving a good resolution while maintaining reasonable the dictionary size.…”

High resolution sparse estimation of exponentially decaying signals

“…Such techniques have successfully been applied to line spectral data, and the topic has attracted notable attention in the recent literature (see, e.g., [5][6][7][8][9][10][11]). Although these algorithms appear quite different from each other, they share the property that the considered dictionary grid should be selected sufficiently fine to allow for a sparse signal representation (see also [12,13]), which, if extended to also consider damped modes, necessitates a large dictionary matrix containing elements with a sufficiently fine grid over the range of both the potential frequencies and damping candidates (see, e.g., [2,14,15]); this will be particularly noticeable if treating large data sets, or data sets with multiple measurement dimensions, such as in NMR measurements. In order to mitigate this problem, we here propose a novel dictionary learning approach wherein we iteratively decompose the signal with a fixed small dictionary, adaptively learning the dictionary elements best suited to enhance sparsity.…”

Editorial

“…Sahnoun et al [8] address multi-dimensional modal estimation using sparse estimation techniques in combination with an efficient multigrid approach. To overcome huge size in the necessary dictionaries, they refine their dictionaries over several levels of resolution.…”