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 & 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. doi: 10.1109/83.846251. Google Scholar

[2]

S. Basu and Y. Bresler, Feasibility of tomography with unknown view angles,, IEEE Transactions on Image Processing, 9 (2000), 1107. doi: 10.1109/83.846252. Google Scholar

[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. doi: 10.1109/78.887032. Google Scholar

[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, (1997). Google Scholar

[5]

D. Cox, J. Little and D. O'Shea, "Using Algebraic Geometry,", Graduate Texts in Mathematics, (1998). Google Scholar

[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. doi: 10.1016/j.jsb.2003.10.001. Google Scholar

[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. doi: 10.1109/18.857786. Google Scholar

[8]

A. V. Fiacco, "Introduction to Sensitivity and Stability Analysis in Nonlinear Programming,", Academic, (1983). Google Scholar

[9]

J. Frank, "Three-dimensional Electron Microscopy of Macromolecular Assemblies,", CA: Academic, (1996). Google Scholar

[10]

M. S. Gelfand and A. B. Goncharov, Spational rotational alignment of identical particles given their projections: Theory and practice,, in, 81 (1990), 97. Google Scholar

[11]

M. Giaquinta and S. Hildebrandt, "Calculus of Variations I,", A Series of Comprehensive Studies in Mathematics, (1996). Google Scholar

[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. doi: 10.1007/BF00140118. Google Scholar

[13]

A. B. Goncharov, Three-dimensional reconstruction of arbitrarily arranged identical particles given their projections,, in, 81 (1990), 67. Google Scholar

[14]

M. Hedley and H. Yan, Motion artifact suppression: A review of post-processing techniques,, Magnetic Resonance Imaging, 10 (1992), 627. doi: 10.1016/0730-725X(92)90014-Q. Google Scholar

[15]

S. G. Krantz and H. R. Parks, "A Primer of Real Analytic Functions,", Birkhäuser, (1992). Google Scholar

[16]

L. Lamberg, Unique recovery of unknown projection orientations in three-dimensional tomography,, Inverse Problems and Imaging, 2 (2008), 547. doi: 10.3934/ipi.2008.2.547. Google Scholar

[17]

L. Lamberg and L. Ylinen, Two-dimensional tomography with unknown view angles,, Inverse Problems and Imaging, 1 (2007), 623. Google Scholar

[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. doi: 10.1109/34.589202. Google Scholar

[19]

E. Lehmann, "Theory of Point Estimation,", Springer, (1998). Google Scholar

[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. Google Scholar

[21]

F. Natterer, "The Mathematics of Computerized Tomography,", John Wiley & Sons Inc, (1986). Google Scholar

[22]

V. M. Panaretos, On random tomography with unobservable projection angles,, Submitted to the Annals of Statistic., (). Google Scholar

[23]

D. B. Salzman, A method of general moments for orienting 2D projections of unknown 3D objects,, Computer Vision, 50 (1990), 129. doi: 10.1016/0734-189X(90)90038-W. Google Scholar

[24]

I. R. Shafarevich, "Basic Algebraic Geometry,", Springer-Verlag, (1974). Google Scholar

[25]

D. C. Solmon, The X-ray transform,, Journal of Mathematical Analysis and Applications, 56 (1976), 61. doi: 10.1016/0022-247X(76)90008-1. Google Scholar

[26]

M. Van Heel, Angular reconstitution: A posteriori assignment of projection directions for 3D reconstruction,, Ultramicroscopy, 21 (1987), 111. doi: 10.1016/0304-3991(87)90078-7. Google Scholar

[27]

H. L. Van Trees, "Detection, Estimation, and Modulation Theory, Part 1,", John Wiley, (1968). Google Scholar

[28]

G. Wahba, "Spline Models for Observational Data,", Society for Industrial and Applied Mathematics, (1990). Google Scholar

show all references

References:
[1]

S. Basu and Y. Bresler, Uniqueness of tomography with unknown view angles,, IEEE Transactions on Image Processing, 9 (2000), 1094. doi: 10.1109/83.846251. Google Scholar

[2]

S. Basu and Y. Bresler, Feasibility of tomography with unknown view angles,, IEEE Transactions on Image Processing, 9 (2000), 1107. doi: 10.1109/83.846252. Google Scholar

[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. doi: 10.1109/78.887032. Google Scholar

[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, (1997). Google Scholar

[5]

D. Cox, J. Little and D. O'Shea, "Using Algebraic Geometry,", Graduate Texts in Mathematics, (1998). Google Scholar

[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. doi: 10.1016/j.jsb.2003.10.001. Google Scholar

[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. doi: 10.1109/18.857786. Google Scholar

[8]

A. V. Fiacco, "Introduction to Sensitivity and Stability Analysis in Nonlinear Programming,", Academic, (1983). Google Scholar

[9]

J. Frank, "Three-dimensional Electron Microscopy of Macromolecular Assemblies,", CA: Academic, (1996). Google Scholar

[10]

M. S. Gelfand and A. B. Goncharov, Spational rotational alignment of identical particles given their projections: Theory and practice,, in, 81 (1990), 97. Google Scholar

[11]

M. Giaquinta and S. Hildebrandt, "Calculus of Variations I,", A Series of Comprehensive Studies in Mathematics, (1996). Google Scholar

[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. doi: 10.1007/BF00140118. Google Scholar

[13]

A. B. Goncharov, Three-dimensional reconstruction of arbitrarily arranged identical particles given their projections,, in, 81 (1990), 67. Google Scholar

[14]

M. Hedley and H. Yan, Motion artifact suppression: A review of post-processing techniques,, Magnetic Resonance Imaging, 10 (1992), 627. doi: 10.1016/0730-725X(92)90014-Q. Google Scholar

[15]

S. G. Krantz and H. R. Parks, "A Primer of Real Analytic Functions,", Birkhäuser, (1992). Google Scholar

[16]

L. Lamberg, Unique recovery of unknown projection orientations in three-dimensional tomography,, Inverse Problems and Imaging, 2 (2008), 547. doi: 10.3934/ipi.2008.2.547. Google Scholar

[17]

L. Lamberg and L. Ylinen, Two-dimensional tomography with unknown view angles,, Inverse Problems and Imaging, 1 (2007), 623. Google Scholar

[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. doi: 10.1109/34.589202. Google Scholar

[19]

E. Lehmann, "Theory of Point Estimation,", Springer, (1998). Google Scholar

[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. Google Scholar

[21]

F. Natterer, "The Mathematics of Computerized Tomography,", John Wiley & Sons Inc, (1986). Google Scholar

[22]

V. M. Panaretos, On random tomography with unobservable projection angles,, Submitted to the Annals of Statistic., (). Google Scholar

[23]

D. B. Salzman, A method of general moments for orienting 2D projections of unknown 3D objects,, Computer Vision, 50 (1990), 129. doi: 10.1016/0734-189X(90)90038-W. Google Scholar

[24]

I. R. Shafarevich, "Basic Algebraic Geometry,", Springer-Verlag, (1974). Google Scholar

[25]

D. C. Solmon, The X-ray transform,, Journal of Mathematical Analysis and Applications, 56 (1976), 61. doi: 10.1016/0022-247X(76)90008-1. Google Scholar

[26]

M. Van Heel, Angular reconstitution: A posteriori assignment of projection directions for 3D reconstruction,, Ultramicroscopy, 21 (1987), 111. doi: 10.1016/0304-3991(87)90078-7. Google Scholar

[27]

H. L. Van Trees, "Detection, Estimation, and Modulation Theory, Part 1,", John Wiley, (1968). Google Scholar

[28]

G. Wahba, "Spline Models for Observational Data,", Society for Industrial and Applied Mathematics, (1990). Google Scholar

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