Jeffrey M. Druce

Mojtaba Kadkhodaie Elyaderani, Swayambhoo Jain, Jeffrey Druce et al.

This paper considers the problem of estimating an unknown high dimensional signal from noisy linear measurements, {when} the signal is assumed to possess a \emph{group-sparse} structure in a {known,} fixed dictionary. We consider signals generated according to a natural probabilistic model, and establish new conditions under which the set of indices of the non-zero groups of the signal (called the group-level support) may be accurately estimated via the group Lasso. Our results strengthen existing coherence-based analyses that exhibit the well-known "square root" bottleneck, allowing for the number of recoverable nonzero groups to be nearly as large as the total number of groups. We also establish a sufficient recovery condition relating the number of nonzero groups and the signal to noise ratio (quantified in terms of the ratio of the squared Euclidean norms of nonzero groups and the variance of the random additive {measurement} noise), and validate this trend empirically. Finally, we examine the implications of our results in the context of a structural health monitoring application, where the group Lasso approach facilitates demixing of a propagating acoustic wavefield, acquired on the material surface by a scanning laser Doppler vibrometer, into antithetical components, one of which indicates the locations of internal material defects.

Jeffrey M. Druce, Jarvis D. Haupt, Stefano Gonella

This paper proposes a strategy for the detection and triangulation of structural anomalies in solid media. The method revolves around the construction of sparse representations of the medium's dynamic response, obtained by learning instructive dictionaries which form a suitable basis for the response data. The resulting sparse coding problem is recast as a modified dictionary learning task with additional spatial sparsity constraints enforced on the atoms of the learned dictionaries, which provides them with a prescribed spatial topology that is designed to unveil anomalous regions in the physical domain. The proposed methodology is model agnostic, i.e., it forsakes the need for a physical model and requires virtually no a priori knowledge of the structure's material properties, as all the inferences are exclusively informed by the data through the layers of information that are available in the intrinsic salient structure of the material's dynamic response. This characteristic makes the approach powerful for anomaly identification in systems with unknown or heterogeneous property distribution, for which a model is unsuitable or unreliable. The method is validated using both synthetically