Beamlets and Multiscale Image Analysis
We describe a framework for multiscale image analysis in which line segments play a role analogous to the role played by points in wavelet analysis.
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Statistics Department, Stanford University; School of Industrial and Systems Engineering, Georgi a Institute of Technology
Abstract. We describe a framework for multiscale image analysis in which lin e segme nts playa role analogous to the role played by points in wavelet analysis. The framework has five key components. The beamlet dictionary is a dyadicallyorganized collection of line segments, occupying a range of dyadic locations and scales, and occurring at a range of orientations. The beamlet transform of an image f (x , y) is the collection of integrals of f over each segment in th e beamlet dictionary; the resulting information is stored in a beamlet pyram id. The beamlet graph is the graph structure with pixel corners as vertices and beamlets as edges ; a path through this graph corresponds to a polygon in the original image. By exploit ing the first four component s of the beamlet fram ework, we can formulate beamlet-based algorithms which ar e able to identify and ext ract beamlets and chains of beamlets with special properties. In this paper we describe a four-lev el hierarchy of beamlet algorithms. The first level consists of simple procedures which ignore the structure of the beamlet pyramid and beamlet graph; th e second level exploits only t he parent-child dependence between scales ; the third level incorporates colline arity and co-curvity relationships; and the fourth level allows global optimization over the full sp ace of polygons in an image. These algorit hms can be shown in practice to have suprisingly powerful and apparently unprecedented capabilities, for example in det ection of very faint cur ves in very noisy data. We compare this framework with important antecedents in image processing (Brandt and Dym; Horn and collaborat ors; G5tze and Druckenmiller) and in geometric measure theory (Jones; David and Semmes; and Lerman) .
Key Words and Phrases. Multis cale Line Segments. Multiscale Radon Transform. Line Segment Extraction. Curve Extraction. Obj ect Extraction. Acknowledgements. This work has been partially supported by AFOSR MURI 95-P49620-96-1-0028, by National Science Found ation grants DMS 98-72890 (KDI) , DMS 95-05151 , and Ecs-97-07111, and by DARPA ACMP BAA 98-04. The authors would like to thank Achi Brandt , Emmanuel Cand es, Scott S. Chen , Raphy Coifman , Davi Geiger, David Horn, Hiroshi Ishikawa , Ian Jermyn, Peter Jones, Gilad Lerman, Ofer Levi, Frank Natterer, Michael Saunders, Stephen Semmes, and Jean-Luc Starck for helpful comment s, preprints, and references. T. J. Barth, et al. (eds.), Multiscale and Multiresolution © Springer-Verlag Berlin Heidelberg 2002
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David Donoh o and Xiaoming Huo
Introduction
In t he last t en years, multiscale thinking in general and wavelet analysis in part icular have become qu it e popular. The journal App lied and Computa tional Harmo nic Analysis, founded only in 1993, has quickly beco me one of the most- cit ed journals in t he mathematical sciences, and leading figures on wavelet resear ch became among t he most cite d author
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