Fast X-Ray and Beamlet Transforms for Three-Dimensional Data

Three-dimensional volumetric data are becoming increasingly available in a wide range of scientific and technical disciplines. With the right tools, we can expect such data to yield valuable insights about many important systems in our three-dimensional world. In this paper, we develop tools for the analysis of 3-D data which may contain structures built from lines, line segments, and filaments. These tools come in two main forms: (a) Monoscale: the X-ray transform, offering the collection of line integrals along a wide range of lines running through the image – at all different orientations and positions; and (b) Multiscale: the (3-D) beamlet transform, offering the collection of line integrals along line segments which, in addition to ranging through a wide collection of locations and positions, also occupy a wide range of scales. We describe three principles for computing these transforms: exact (slow) evaluation, approximate, recursive evaluation based on a multiscale divide-and-conquer approach, and fast exact evaluation based on the use of the two-dimensional Fast Slant Stack algorithm (Averbuch et al. 2001) applied to slices of sheared arrays. We compare these different computational strategies from the viewpoint of analysing the small 3-D datasets available currently, and the larger 3-D datasets surely to become available in the near future, as storage and processing power continue their exponential growth. We also describe several basic applications of these tools, for example in finding faint structures buried in noisy data.