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Updated: May 28, 2026

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A Method for 3D Reconstruction and Virtual Reality Analysis of Glial and Neuronal Cells
Published on: September 28, 2019
Gel'fand-Graev's reconstruction formula in the 3D real space
Yangbo Ye1, Hengyong Yu, Ge Wang
1Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, USA. yey@math.uiowa.edu
Medical Physics
|October 8, 2011
Summary
The Gel
Area of Science:
- Mathematics
- Medical Imaging
- Tomography
Background:
- Classical work by Gel'fand and Graev on integral transforms.
- The Gel'fand-Graev inversion formula connects projection data to image reconstruction via the Hilbert transform.
Purpose of the Study:
- To explore key results of Gel'fand and Graev's work for future research.
- To present the Gel'fand-Graev inversion formula in a 3D context using the 1D x-ray transform.
Main Methods:
- Utilizing the Gel'fand-Graev inversion formula.
- Applying the formula in a 3D setting with the 1D x-ray transform.
- Leveraging pseudodifferential operators as a theoretical tool.
Main Results:
- Established a fundamental mathematical link between the Gel'fand-Graev formula and differential backprojection/backprojection filtration.
- Demonstrated this link for the 1D x-ray transform in 3D real space.
Conclusions:
- Highlights the significance of theoretical mathematics in tomographic imaging.
- Underscores the potential value of pure mathematical findings for practical applications.
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