# American Institute of Mathematical Sciences

February  2011, 5(1): 75-93. doi: 10.3934/ipi.2011.5.75

## An algorithm for recovering unknown projection orientations and shifts in 3-D tomography

 1 Department of Mathematics and Physics, Lappeenranta University of Technology, Lappeenranta, Finland 2 Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014 Helsinki,

Received  March 2010 Revised  July 2010 Published  February 2011

It is common for example in Cryo-electron microscopy of viruses, that the orientations at which the projections are acquired, are totally unknown. We introduce here a moment based algorithm for recovering them in the three-dimensional parallel beam tomography. In this context, there is likely to be also unknown shifts in the projections. They will be estimated simultaneously. Also stability properties of the algorithm are examined. Our considerations rely on recent results that guarantee a solution to be almost always unique. A similar analysis can also be done in the two-dimensional problem.
Citation: 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
##### References:
 [1] S. Basu and Y. Bresler, Uniqueness of tomography with unknown view angles, IEEE Transactions on Image Processing, 9 (2000), 1094-1106. doi: 10.1109/83.846251. [2] S. Basu and Y. Bresler, Feasibility of tomography with unknown view angles, IEEE Transactions on Image Processing, 9 (2000), 1107-1122. doi: 10.1109/83.846252. [3] S. Basu and Y. Bresler, The stability of nonlinear least squares problems and the Cramér-Rao Bound, IEEE Transactions on Signal Processing, 48 (2000), 3426-3436. doi: 10.1109/78.887032. [4] D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra," Undergraduate Texts in Mathematics, 2nd edition, Springer-Verlag, New York, 1997. [5] D. Cox, J. Little and D. O'Shea, "Using Algebraic Geometry," Graduate Texts in Mathematics, 1st edition, Springer-Verlag, New York, 1998. [6] D. N. P. Doan, K. C. Lee , P. Laurinmäki, S. Butcher, S-M. Wong and T. Dokland, Three-dimensional reconstruction of hibiscus chlorotic ringspot virus, Journal of Structural Biology, 144 (2003), 253-261. doi: 10.1016/j.jsb.2003.10.001. [7] P. C. Doerschuk and J. E. Johnson, Ab initio reconstruction and experimental design for cryo electron microscopy, IEEE Transactions on Information Theory, 46 (2000), 1714-1729. doi: 10.1109/18.857786. [8] A. V. Fiacco, "Introduction to Sensitivity and Stability Analysis in Nonlinear Programming," Academic, New York, 1983. [9] J. Frank, "Three-dimensional Electron Microscopy of Macromolecular Assemblies," CA: Academic, San Diego, 1996. [10] M. S. Gelfand and A. B. Goncharov, Spational rotational alignment of identical particles given their projections: Theory and practice, in "Mathematical Problems of Tomography," volume 81 of Translations of Mathematical Monographs (eds. I. M. Gelfand and S. G. Gindikin), American Mathematical Society, Providence, Rhode Island, (1990), 97-122. (Translated from the Russian by S. Gelfand) [11] M. Giaquinta and S. Hildebrandt, "Calculus of Variations I," A Series of Comprehensive Studies in Mathematics, Springer-Verlag, Berlin, 1996. [12] A. B. Goncharov, Integral geometry and three-dimensional reconstruction of randomly oriented identical particles from their electron microphotos, Acta Applicandae Mathematicae, 11 (1988), 199-211. doi: 10.1007/BF00140118. [13] A. B. Goncharov, Three-dimensional reconstruction of arbitrarily arranged identical particles given their projections, in "Mathematical Problems of Tomography," volume 81 of Translations of Mathematical Monographs (eds. I. M. Gelfand and S. G. Gindikin), American Mathematical Society, Providence, Rhode Island, (1990), 67-96. (Translated from the Russian by S. Gelfand) [14] M. Hedley and H. Yan, Motion artifact suppression: A review of post-processing techniques, Magnetic Resonance Imaging, 10 (1992), 627-635. doi: 10.1016/0730-725X(92)90014-Q. [15] S. G. Krantz and H. R. Parks, "A Primer of Real Analytic Functions," Birkhäuser, Basel, 1992. [16] L. Lamberg, Unique recovery of unknown projection orientations in three-dimensional tomography, Inverse Problems and Imaging, 2 (2008), 547-575. doi: 10.3934/ipi.2008.2.547. [17] L. Lamberg and L. Ylinen, Two-dimensional tomography with unknown view angles, Inverse Problems and Imaging, 1 (2007), 623-642. [18] P. D. Lauren and N. Nandhakumar, Estimating the viewing parameters of random, noisy projections of asymmetric objects for tomographic reconstruction, IEEE Transactions on Pattern Analysis and Machine Intelligence, 19 (1997), 417-430. doi: 10.1109/34.589202. [19] E. Lehmann, "Theory of Point Estimation," Springer, New York, 1998. [20] S. P. Mallick, S. Agarwal, D. J. Kriegman, S. J. Belongie, B. Carragher and C. S. Potter, Structure and View Estimation for Tomographic Reconstruction: A Bayesian Approach, Computer Vision and Pattern Recognition, 2 (2006), 2253-2260. [21] F. Natterer, "The Mathematics of Computerized Tomography," John Wiley & Sons Inc, Stuttgart, 1986. [22] V. M. Panaretos, On random tomography with unobservable projection angles,, Submitted to the Annals of Statistic., (). [23] D. B. Salzman, A method of general moments for orienting 2D projections of unknown 3D objects, Computer Vision, Graphics, and Image Processing, 50 (1990), 129-156. doi: 10.1016/0734-189X(90)90038-W. [24] I. R. Shafarevich, "Basic Algebraic Geometry," Springer-Verlag, Berlin, 1974. (Translated from the Russian by K. A. Hirsch) [25] D. C. Solmon, The X-ray transform, Journal of Mathematical Analysis and Applications, 56 (1976), 61-83. doi: 10.1016/0022-247X(76)90008-1. [26] M. Van Heel, Angular reconstitution: A posteriori assignment of projection directions for 3D reconstruction, Ultramicroscopy, 21 (1987), 111-123. doi: 10.1016/0304-3991(87)90078-7. [27] H. L. Van Trees, "Detection, Estimation, and Modulation Theory, Part 1," John Wiley, New York, 1968. [28] G. Wahba, "Spline Models for Observational Data," Society for Industrial and Applied Mathematics, Philadelphia, 1990.

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##### References:
 [1] S. Basu and Y. Bresler, Uniqueness of tomography with unknown view angles, IEEE Transactions on Image Processing, 9 (2000), 1094-1106. doi: 10.1109/83.846251. [2] S. Basu and Y. Bresler, Feasibility of tomography with unknown view angles, IEEE Transactions on Image Processing, 9 (2000), 1107-1122. doi: 10.1109/83.846252. [3] S. Basu and Y. Bresler, The stability of nonlinear least squares problems and the Cramér-Rao Bound, IEEE Transactions on Signal Processing, 48 (2000), 3426-3436. doi: 10.1109/78.887032. [4] D. Cox, J. Little and D. O'Shea, "Ideals, Varieties and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra," Undergraduate Texts in Mathematics, 2nd edition, Springer-Verlag, New York, 1997. [5] D. Cox, J. Little and D. O'Shea, "Using Algebraic Geometry," Graduate Texts in Mathematics, 1st edition, Springer-Verlag, New York, 1998. [6] D. N. P. Doan, K. C. Lee , P. Laurinmäki, S. Butcher, S-M. Wong and T. Dokland, Three-dimensional reconstruction of hibiscus chlorotic ringspot virus, Journal of Structural Biology, 144 (2003), 253-261. doi: 10.1016/j.jsb.2003.10.001. [7] P. C. Doerschuk and J. E. Johnson, Ab initio reconstruction and experimental design for cryo electron microscopy, IEEE Transactions on Information Theory, 46 (2000), 1714-1729. doi: 10.1109/18.857786. [8] A. V. Fiacco, "Introduction to Sensitivity and Stability Analysis in Nonlinear Programming," Academic, New York, 1983. [9] J. Frank, "Three-dimensional Electron Microscopy of Macromolecular Assemblies," CA: Academic, San Diego, 1996. [10] M. S. Gelfand and A. B. Goncharov, Spational rotational alignment of identical particles given their projections: Theory and practice, in "Mathematical Problems of Tomography," volume 81 of Translations of Mathematical Monographs (eds. I. M. Gelfand and S. G. Gindikin), American Mathematical Society, Providence, Rhode Island, (1990), 97-122. (Translated from the Russian by S. Gelfand) [11] M. Giaquinta and S. Hildebrandt, "Calculus of Variations I," A Series of Comprehensive Studies in Mathematics, Springer-Verlag, Berlin, 1996. [12] A. B. Goncharov, Integral geometry and three-dimensional reconstruction of randomly oriented identical particles from their electron microphotos, Acta Applicandae Mathematicae, 11 (1988), 199-211. doi: 10.1007/BF00140118. [13] A. B. Goncharov, Three-dimensional reconstruction of arbitrarily arranged identical particles given their projections, in "Mathematical Problems of Tomography," volume 81 of Translations of Mathematical Monographs (eds. I. M. Gelfand and S. G. Gindikin), American Mathematical Society, Providence, Rhode Island, (1990), 67-96. (Translated from the Russian by S. Gelfand) [14] M. Hedley and H. Yan, Motion artifact suppression: A review of post-processing techniques, Magnetic Resonance Imaging, 10 (1992), 627-635. doi: 10.1016/0730-725X(92)90014-Q. [15] S. G. Krantz and H. R. Parks, "A Primer of Real Analytic Functions," Birkhäuser, Basel, 1992. [16] L. Lamberg, Unique recovery of unknown projection orientations in three-dimensional tomography, Inverse Problems and Imaging, 2 (2008), 547-575. doi: 10.3934/ipi.2008.2.547. [17] L. Lamberg and L. Ylinen, Two-dimensional tomography with unknown view angles, Inverse Problems and Imaging, 1 (2007), 623-642. [18] P. D. Lauren and N. Nandhakumar, Estimating the viewing parameters of random, noisy projections of asymmetric objects for tomographic reconstruction, IEEE Transactions on Pattern Analysis and Machine Intelligence, 19 (1997), 417-430. doi: 10.1109/34.589202. [19] E. Lehmann, "Theory of Point Estimation," Springer, New York, 1998. [20] S. P. Mallick, S. Agarwal, D. J. Kriegman, S. J. Belongie, B. Carragher and C. S. Potter, Structure and View Estimation for Tomographic Reconstruction: A Bayesian Approach, Computer Vision and Pattern Recognition, 2 (2006), 2253-2260. [21] F. Natterer, "The Mathematics of Computerized Tomography," John Wiley & Sons Inc, Stuttgart, 1986. [22] V. M. Panaretos, On random tomography with unobservable projection angles,, Submitted to the Annals of Statistic., (). [23] D. B. Salzman, A method of general moments for orienting 2D projections of unknown 3D objects, Computer Vision, Graphics, and Image Processing, 50 (1990), 129-156. doi: 10.1016/0734-189X(90)90038-W. [24] I. R. Shafarevich, "Basic Algebraic Geometry," Springer-Verlag, Berlin, 1974. (Translated from the Russian by K. A. Hirsch) [25] D. C. Solmon, The X-ray transform, Journal of Mathematical Analysis and Applications, 56 (1976), 61-83. doi: 10.1016/0022-247X(76)90008-1. [26] M. Van Heel, Angular reconstitution: A posteriori assignment of projection directions for 3D reconstruction, Ultramicroscopy, 21 (1987), 111-123. doi: 10.1016/0304-3991(87)90078-7. [27] H. L. Van Trees, "Detection, Estimation, and Modulation Theory, Part 1," John Wiley, New York, 1968. [28] G. Wahba, "Spline Models for Observational Data," Society for Industrial and Applied Mathematics, Philadelphia, 1990.
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