## Near-optimal detection of geometric objects by fast multiscale methods (2005)

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Venue: | IEEE Trans. Inform. Theory |

Citations: | 24 - 8 self |

### BibTeX

@ARTICLE{Arias-castro05near-optimaldetection,

author = {Ery Arias-castro and David L. Donoho and Xiaoming Huo and Senior Member},

title = {Near-optimal detection of geometric objects by fast multiscale methods},

journal = {IEEE Trans. Inform. Theory},

year = {2005},

volume = {51}

}

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### Abstract

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 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. 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-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. Index Terms—Beamlets, detecting hot spots, detecting line segments, Hough transform, image processing, maxima of Gaussian processes, multiscale geometric analysis, Radon transform. I.