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Published online by Cambridge University Press: 25 June 2025
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 phenomena 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 different strategies for computing these transforms and several basic applications, for example in finding faint structures buried in noisy data.
1. Introduction
In field after field, we are currently seeing new initiatives aimed at gathering large high-resolution three-dimensional datasets. While three-dimensional data have always been crucial to understanding the physical world we live in, this transition to ubiquitous 3-D data gathering seems novel. The driving force is undoubtedly the pervasive influence of increasing storage capacity and computer processing power, which affects our ability to create new 3-D measurement instruments, but which also makes it possible to analyze the massive volumes of data that inevitably result when 3-D data are being gathered.
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