Near-Optimal Detection of Geometric Objects by Fast Multiscale Methods Ery Arias-Castro a , David Donoho a , Xiaoming Huo b August 18, 2003 Abstract 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 Gaussian white noise. We focus on 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 1-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. Key Words and Phrases. Multiscale Geometric Analysis. Image Processing. Detect- ing Line Segments. Detecting Hot Spots. Maxima of Gaussian Processes. Beamlets. Hough Transform. Radon Transform. Acknowledgments. This work has been partially supported by National Science Foundation grants DMS 00-77261, DMS 01-40587 and DMS 95–05151, by AFOSR MURI 95–P49620–96–1– 0028, and by DARPA ACMP. The authors would like to thank Achi Brandt, Emmanuel Cand` es, Raphy Coifman, Michael Elad, Gary Hewer, Peter Jones, David Siegmund, and Tse Leung Lai for valuable references and discussions. a : Department of Statistics, Stanford University. {acery,donoho}@stat.stanford.edu b : School of Industrial and Systems Engineering, Georgia Institute of Technology. xiaom- ing@isye.gatech.edu 1