American Institute of Mathematical Sciences

Advanced
November  2008, 2(4): 547-575. doi: 10.3934/ipi.2008.2.547

Unique recovery of unknown projection orientations in three-dimensional tomography

Lars Lamberg 1,

1. 

Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014 Helsinki,

Received  September 2008 Revised  October 2008 Published  November 2008

We consider uniqueness of three-dimensional parallel beam tomography in which both the object being imaged and the projection orientations are unknown. This problem occurs in certain practical applications, for example in cryo electron microscopy of viral particles, where the projection orientations may be completely unknown due to the random orientations of the particles being imaged. We show that only three projections are needed to guarantee unique recovery of the unknown projection orientations (up to a common orthogonal transformation), if the object belongs to a certain generic set of objects. In particular, the uniqueness holds for almost all objects. We also show that, if the object belongs to that generic set, $k+1$ projections at unknown orientations suffice to determine uniquely also the geometric moments of the object of order less or equal to $k$. As a consequence, the object belonging to that generic set, is uniquely determined (up to an orthogonal transformation) by almost any infinitely many projections at unknown orientations. We show that the uniqueness problem is related to some properties of certain homogeneous polynomials that depend on the projection orientations and the geometric moments of objects. Here certain theorems of algebraic geometry turn out to be useful. We also provide a system of equations, that is uniquely solvable and gives the desired projection orientations and object's geometric moments.
Keywords: Homogeneous system, Projective space, Geometric moment, Projection orientation, Resultant, Bezout's theorem, Tomography with unknown projection orientations.
Mathematics Subject Classification: Primary: 65R32, 44A12.
Citation: Lars Lamberg. Unique recovery of unknown projection orientations in three-dimensional tomography. Inverse Problems and Imaging, 2008, 2 (4) : 547-575. doi: 10.3934/ipi.2008.2.547
[1]

Jaakko Ketola, Lars Lamberg. An algorithm for recovering unknown projection orientations and shifts in 3-D tomography. Inverse Problems and Imaging, 2011, 5 (1) : 75-93. doi: 10.3934/ipi.2011.5.75
[2]

Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: Extension and analysis. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022014
[3]

Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1D case. Kinetic and Related Models, 2020, 13 (6) : 1243-1280. doi: 10.3934/krm.2020045
[4]

Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43
[5]

Raffaele Chiappinelli. Eigenvalues of homogeneous gradient mappings in Hilbert space and the Birkoff-Kellogg theorem. Conference Publications, 2007, 2007 (Special) : 260-268. doi: 10.3934/proc.2007.2007.260
[6]

Walter Briec, Bernardin Solonandrasana. Some remarks on a successive projection sequence. Journal of Industrial and Management Optimization, 2006, 2 (4) : 451-466. doi: 10.3934/jimo.2006.2.451
[7]

Nimish Shah, Lei Yang. Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5247-5287. doi: 10.3934/dcds.2020227
[8]

Tim Kreutzmann, Andreas Rieder. Geometric reconstruction in bioluminescence tomography. Inverse Problems and Imaging, 2014, 8 (1) : 173-197. doi: 10.3934/ipi.2014.8.173
[9]

Lars Lamberg, Lauri Ylinen. Two-Dimensional tomography with unknown view angles. Inverse Problems and Imaging, 2007, 1 (4) : 623-642. doi: 10.3934/ipi.2007.1.623
[10]

Aki Pulkkinen, Ville Kolehmainen, Jari P. Kaipio, Benjamin T. Cox, Simon R. Arridge, Tanja Tarvainen. Approximate marginalization of unknown scattering in quantitative photoacoustic tomography. Inverse Problems and Imaging, 2014, 8 (3) : 811-829. doi: 10.3934/ipi.2014.8.811
[11]

Julian Koellermeier, Giovanni Samaey. Projective integration schemes for hyperbolic moment equations. Kinetic and Related Models, 2021, 14 (2) : 353-387. doi: 10.3934/krm.2021008
[12]

Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155
[13]

Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067
[14]

Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555
[15]

Qingzhi Yang. The revisit of a projection algorithm with variable steps for variational inequalities. Journal of Industrial and Management Optimization, 2005, 1 (2) : 211-217. doi: 10.3934/jimo.2005.1.211
[16]

Ya-zheng Dang, Jie Sun, Su Zhang. Double projection algorithms for solving the split feasibility problems. Journal of Industrial and Management Optimization, 2019, 15 (4) : 2023-2034. doi: 10.3934/jimo.2018135
[17]

Thomas Schuster, Joachim Weickert. On the application of projection methods for computing optical flow fields. Inverse Problems and Imaging, 2007, 1 (4) : 673-690. doi: 10.3934/ipi.2007.1.673
[18]

Dang Van Hieu. Projection methods for solving split equilibrium problems. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2331-2349. doi: 10.3934/jimo.2019056
[19]

Henk Broer, Konstantinos Efstathiou, Olga Lukina. A geometric fractional monodromy theorem. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 517-532. doi: 10.3934/dcdss.2010.3.517
[20]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure and Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511

2021 Impact Factor: 1.483

Tools

Metrics

  • PDF downloads (122)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy