Motion detection in diffraction tomography by common circle methods
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- by Michael Quellmalz, Peter Elbau, Otmar Scherzer and Gabriele Steidl
- Math. Comp. 93 (2024), 747-784
- DOI: https://doi.org/10.1090/mcom/3869
- Published electronically: July 5, 2023
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Abstract:
The method of common lines is a well-established reconstruction technique in cryogenic electron microscopy (cryo-EM), which can be used to extract the relative orientations of an object given tomographic projection images from different directions.
In this paper, we deal with an analogous problem in optical diffraction tomography. Based on the Fourier diffraction theorem, we show that rigid motions of the object, i.e., rotations and translations, can be determined by detecting common circles in the Fourier-transformed data. We introduce two methods to identify common circles. The first one is motivated by the common line approach for projection images and detects the relative orientation by parameterizing the common circles in the two images. The second one assumes a smooth motion over time and calculates the angular velocity of the rotational motion via an infinitesimal version of the common circle method. Interestingly, using the stereographic projection, both methods can be reformulated as common line methods, but these lines are, in contrast to those used in cryo-EM, not confined to pass through the origin and allow for a full reconstruction of the relative orientations. Numerical proof-of-the-concept examples demonstrate the performance of our reconstruction methods.
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Bibliographic Information
- Michael Quellmalz
- Affiliation: TU Berlin, Straße des 17. Juni 136, D-10587 Berlin, Germany
- MR Author ID: 1120022
- ORCID: 0000-0001-6206-5705
- Email: quellmalz@math.tu-berlin.de
- Peter Elbau
- Affiliation: University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
- MR Author ID: 701033
- ORCID: 0000-0001-5894-5793
- Email: peter.elbau@univie.ac.at
- Otmar Scherzer
- Affiliation: University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria; Johann Radon Institute for Computational and Applied Mathematics (RICAM), Altenbergerstraße 69. A-4040 Linz, Austria; and Christian Doppler Laboratory for Mathematical Modeling and Simulation of Next Generations of Ultrasound Devices (MaMSi), Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
- MR Author ID: 291193
- Email: otmar.scherzer@univie.ac.at
- Gabriele Steidl
- Affiliation: TU Berlin, Straße des 17. Juni 136, D-10587 Berlin, Germany
- MR Author ID: 166780
- Email: steidl@math.tu-berlin.de
- Received by editor(s): September 20, 2022
- Received by editor(s) in revised form: April 11, 2023, and May 3, 2023
- Published electronically: July 5, 2023
- Additional Notes: The second and third authors were supported by the Austrian Science Fund (FWF), with SFB F68 “Tomography Across the Scales”, project F6804-N36 and F6807-N36.
Funding by the DFG under the SFB “Tomography Across the Scales” (STE 571/19-1, project number: 495365311) is gratefully acknowledged. The financial support by the Austrian Federal Ministry for Digital and Economic Affairs, the National Foundation for Research, Technology and Development and the Christian Doppler Research Association is gratefully acknowledged. This research was funded in whole, or in part, by the Austrian Science Fund (FWF) P 34981.
For the purpose of open access, the authors have applied a CC BY public copyright license to any Authors Accepted Manuscript version arising from this submission. - © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 747-784
- MSC (2000): Primary 92C55, 78A46, 94A08, 42B05
- DOI: https://doi.org/10.1090/mcom/3869
- MathSciNet review: 4678583