Applications of Beamlets to Detection and Extraction of Lines, Curves and Objects in Very Noisy Images

Beamlets are a special dyadically organized collection of line segments, exhibiting a range of lengths, positions and orientations. This collection is relatively compact: there are O(n log2(n)) beamlets, compared to O(n) line segments. This collection is relatively expressive: up to a certain tolerance, it does not take more than O(log2(n)) beamlets to approximate a single line segment. Because of these two properties, chains of a relatively few beamlets can build quite general curves. The beamlet pyramid is a multiscale data structure which stores all the line integrals of the image over all beamlets. By summing a relatively few coefficients from the beamlet pyramid, one can obtain integrals of the image along quite general polygonal curves. We consider the beamlet pyramid a natural platform on which to build new methods for detecting linear and curvilinear features in very noisy data. The general approach is to solve such problems by adaptively constructing chains of beamlets which extremize certain integrals over the image. We give examples in the problems of detecting the presence of line segments in very noisy data; detecting curves in noisy data; and in the problem of extracting objects in very noisy data. We are able, in examples, to detect objects which seem practically invisible to the unaided eye.