Near-optimal detection of geometric objects by fast multiscale methods

We construct detectors for "geometric" objects in noisy data. Examples include a detector for presence of a line segment of unknown length, position, and orientation in two-dimensional image data with additive white Gaussian noise. We focus on the following two issues. i) The optimal detection threshold-i.e., the signal strength below which no method of detection can be successful for large dataset size n. ii) The optimal computational complexity of a near-optimal detector, i.e., the complexity required to detect signals slightly exceeding the detection threshold. We describe a general approach to such problems which covers several classes of geometrically defined signals; for example, with one-dimensional data, signals having elevated mean on an interval, and, in d-dimensional data, signals with elevated mean on a rectangle, a ball, or an ellipsoid. In all these problems, we show that a naive or straightforward approach leads to detector thresholds and algorithms which are asymptotically far away from optimal. At the same time, a multiscale geometric analysis of these classes of objects allows us to derive asymptotically optimal detection thresholds and fast algorithms for near-optimal detectors.