Optimal Two-Stage Search for Sparse Targets Using Convex Criteria

We consider the problem of estimating and detecting sparse signals over a large area of an image or other medium. We introduce a novel cost function that captures the tradeoff between allocating energy to signal regions, called regions of interest (ROI), versus exploration of other regions. We show that minimizing our cost guarantees reduction of both the error probability over the unknown ROI and the mean square error (MSE) in estimating the ROI content. Two solutions to the resource allocation problem, subject to a total resource constraint, are derived. Asymptotic analysis shows that the estimated ROI converges to the true ROI. We show that our adaptive sampling method outperforms exhaustive search and are nearly optimal in terms of MSE performance. An illustrative example of our method in radar imaging is given.