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. 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 the beamlet dictionary; the resulting information is stored in a beamlet pyramid. 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 exploiting the first four components of the beamlet framework, we can formulate beamlet-based algorithms which are able to identify and extract beamlets and chains of beamlets with special properties. In this paper we describe a four-level hierarchy of beamlet algorithms. The first level consists of simple procedures which ignore the structure of the beamlet pyramid and beamlet graph; the second level exploits only the parent-child dependence between scales; the third level incorporates collinearity and co-curvity relationships; and the fourth level allows global optimization over the full space of polygons in an image. These algorithms can be shown in practice to have suprisingly powerful and apparently unprecedented capabilities, for example in detection of very faint curves in very noisy data. We compare this framework with important antecedents in image processing (Brandt and Dym; Horn and collaborators; Götze and Druckenmiller) and in geometric measure theory (Jones; David and Semmes; and Lerman).