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Abstract This paper addresses the overdetermined problem of inverting the n -dimensional cone (or Compton) transform that integrates a function over conical surfaces in R n . The study of the cone transform originates from Compton camera imaging, a nuclear imaging method for the passive detection of gamma-ray sources. We present a new identity relating the n -dimensional cone and Radon transforms through spherical convolutions with arbitrary weight functions. This relationship, which generalizes a previously obtained identity, leads to various inversion formulas in n -dimensions under a mild assumption on the geometry of detectors. We present two such formulas along with the results of their numerical implementation using synthetic phantoms. Compared to our previously discovered inversion techniques, the new formulas are more stable and simpler to implement numerically.
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Microlocal properties of seven-dimensional lemon and apple Radon transforms with applications in Compton scattering tomography
Abstract We present a microlocal analysis of two novel Radon transforms of interest in Compton scattering tomography, which map compactly supported L 2 functions to their integrals over seven-dimensional sets of apple and lemon surfaces. Specifically, we show that the apple and lemon transforms are elliptic Fourier integral operators, which satisfy the Bolker condition. After an analysis of the full seven-dimensional case, we focus our attention on n D subsets of apple and lemon surfaces with fixed central axis, where n < 7. Such subsets of surface integrals have applications in airport baggage and security screening. When the data dimensionality is restricted, the apple transform is shown to violate the Bolker condition, and there are artifacts which occur on apple–cylinder intersections. The lemon transform is shown to satisfy the Bolker condition, when the support of the function is restricted to the strip 0 < z < 1 .
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- Award ID(s):
- 1712207
- PAR ID:
- 10349713
- Date Published:
- Journal Name:
- Inverse Problems
- Volume:
- 38
- Issue:
- 6
- ISSN:
- 0266-5611
- Page Range / eLocation ID:
- 064001
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1088/1361-6420/ac65ad
