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An algorithm for recovering unknown projection orientations and shifts in 3D 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, 
References:
[1] 
S. Basu and Y. Bresler, Uniqueness of tomography with unknown view angles, IEEE Transactions on Image Processing, 9 (2000), 10941106. 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), 11071122. doi: 10.1109/83.846252. 
[3] 
S. Basu and Y. Bresler, The stability of nonlinear least squares problems and the CramérRao Bound, IEEE Transactions on Signal Processing, 48 (2000), 34263436. 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, SpringerVerlag, New York, 1997. 
[5] 
D. Cox, J. Little and D. O'Shea, "Using Algebraic Geometry," Graduate Texts in Mathematics, 1st edition, SpringerVerlag, New York, 1998. 
[6] 
D. N. P. Doan, K. C. Lee , P. Laurinmäki, S. Butcher, SM. Wong and T. Dokland, Threedimensional reconstruction of hibiscus chlorotic ringspot virus, Journal of Structural Biology, 144 (2003), 253261. 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), 17141729. 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, "Threedimensional 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), 97122. (Translated from the Russian by S. Gelfand) 
[11] 
M. Giaquinta and S. Hildebrandt, "Calculus of Variations I," A Series of Comprehensive Studies in Mathematics, SpringerVerlag, Berlin, 1996. 
[12] 
A. B. Goncharov, Integral geometry and threedimensional reconstruction of randomly oriented identical particles from their electron microphotos, Acta Applicandae Mathematicae, 11 (1988), 199211. doi: 10.1007/BF00140118. 
[13] 
A. B. Goncharov, Threedimensional 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), 6796. (Translated from the Russian by S. Gelfand) 
[14] 
M. Hedley and H. Yan, Motion artifact suppression: A review of postprocessing techniques, Magnetic Resonance Imaging, 10 (1992), 627635. doi: 10.1016/0730725X(92)90014Q. 
[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 threedimensional tomography, Inverse Problems and Imaging, 2 (2008), 547575. doi: 10.3934/ipi.2008.2.547. 
[17] 
L. Lamberg and L. Ylinen, Twodimensional tomography with unknown view angles, Inverse Problems and Imaging, 1 (2007), 623642. 
[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), 417430. 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), 22532260. 
[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), 129156. doi: 10.1016/0734189X(90)90038W. 
[24] 
I. R. Shafarevich, "Basic Algebraic Geometry," SpringerVerlag, Berlin, 1974. (Translated from the Russian by K. A. Hirsch) 
[25] 
D. C. Solmon, The Xray transform, Journal of Mathematical Analysis and Applications, 56 (1976), 6183. doi: 10.1016/0022247X(76)900081. 
[26] 
M. Van Heel, Angular reconstitution: A posteriori assignment of projection directions for 3D reconstruction, Ultramicroscopy, 21 (1987), 111123. doi: 10.1016/03043991(87)900787. 
[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. 
show all references
References:
[1] 
S. Basu and Y. Bresler, Uniqueness of tomography with unknown view angles, IEEE Transactions on Image Processing, 9 (2000), 10941106. 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), 11071122. doi: 10.1109/83.846252. 
[3] 
S. Basu and Y. Bresler, The stability of nonlinear least squares problems and the CramérRao Bound, IEEE Transactions on Signal Processing, 48 (2000), 34263436. 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, SpringerVerlag, New York, 1997. 
[5] 
D. Cox, J. Little and D. O'Shea, "Using Algebraic Geometry," Graduate Texts in Mathematics, 1st edition, SpringerVerlag, New York, 1998. 
[6] 
D. N. P. Doan, K. C. Lee , P. Laurinmäki, S. Butcher, SM. Wong and T. Dokland, Threedimensional reconstruction of hibiscus chlorotic ringspot virus, Journal of Structural Biology, 144 (2003), 253261. 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), 17141729. 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, "Threedimensional 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), 97122. (Translated from the Russian by S. Gelfand) 
[11] 
M. Giaquinta and S. Hildebrandt, "Calculus of Variations I," A Series of Comprehensive Studies in Mathematics, SpringerVerlag, Berlin, 1996. 
[12] 
A. B. Goncharov, Integral geometry and threedimensional reconstruction of randomly oriented identical particles from their electron microphotos, Acta Applicandae Mathematicae, 11 (1988), 199211. doi: 10.1007/BF00140118. 
[13] 
A. B. Goncharov, Threedimensional 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), 6796. (Translated from the Russian by S. Gelfand) 
[14] 
M. Hedley and H. Yan, Motion artifact suppression: A review of postprocessing techniques, Magnetic Resonance Imaging, 10 (1992), 627635. doi: 10.1016/0730725X(92)90014Q. 
[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 threedimensional tomography, Inverse Problems and Imaging, 2 (2008), 547575. doi: 10.3934/ipi.2008.2.547. 
[17] 
L. Lamberg and L. Ylinen, Twodimensional tomography with unknown view angles, Inverse Problems and Imaging, 1 (2007), 623642. 
[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), 417430. 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), 22532260. 
[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), 129156. doi: 10.1016/0734189X(90)90038W. 
[24] 
I. R. Shafarevich, "Basic Algebraic Geometry," SpringerVerlag, Berlin, 1974. (Translated from the Russian by K. A. Hirsch) 
[25] 
D. C. Solmon, The Xray transform, Journal of Mathematical Analysis and Applications, 56 (1976), 6183. doi: 10.1016/0022247X(76)900081. 
[26] 
M. Van Heel, Angular reconstitution: A posteriori assignment of projection directions for 3D reconstruction, Ultramicroscopy, 21 (1987), 111123. doi: 10.1016/03043991(87)900787. 
[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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