In this paper, we investigate the problems of anomaly detection and localization from noisy tomographic data. These are characteristic of a class of problems that cannot be optimally solved because they involve hypothesis testing over hypothesis spaces with extremely large cardinality. Our multiscale hypothesis testing approach addresses the key issues associated with this class of problems. A multiscale hypothesis test is a hierarchical sequence of composite hypothesis tests that discards large portions of the hypothesis space with minimal computational burden and zooms in on the likely true hypothesis. For the anomaly detection and localization problems, hypothesis zooming corresponds to spatial zooming - anomalies are successively localized to finer and finer spatial scales. The key challenges we address include how to hierarchically divide a large hypothesis space and how to process the data at each stage of the hierarchy to decide which parts of the hypothesis space deserve more attention. For the latter, we pose and solve a nonlinear optimization problem for a decision statistic that maximally disambiguates composite hypotheses. With no more computational complexity, our optimized statistic shows substantial improvement over conventional approaches. We provide examples that demonstrate this and quantify how much performance is sacrificed by the use of a suboptimal method as compared to that achievable if the optimal approach were computationally feasible.
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