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Analysis of Point Clouds and Pixel Arrays Using Multiscale Coverings

We study the approximation properties of three dimensional beamlets. This is relevant for the analysis of three dimensional point clouds and pixel arrays containing filaments, which is of interest in Astrophysics for example. We also introduce similar, though not as groomed dictionaries designed to approximate other objects such as sheets in higher dimension. These dictionaries are organized into graphs so that connected components and paths are in correspondence with tubular regions used for covering purposes. Once such a graphical struture is available, looking for a certain type of object (say filaments) in a point cloud or pixel array may be translated into finding certain features within that graph, such as connected components or paths. We include numerical experiments that illustrate the use of the beamlet graph in characterizing the filamentarity content of 3D datasets.