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SECTION B

Geodesic regression on spheres from a numerical optimization viewpoint

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Pages 1935-1945 | Received 27 Aug 2013, Accepted 10 Jan 2014, Published online: 28 Mar 2014

References

  • P.A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, NJ, 2008.
  • E. Batzies, L. Machado, and F.S. Leite, The geometric mean and the geodesic fitting problem on the Grassmann manifold, Preprints of Department of Mathematics, University of Coimbra, 2013, pp. 1–20.
  • W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, New York, 1975.
  • S.R. Buss and J.P. Fillmore, Spherical averages and applications to spherical splines and interpolation, ACM Trans. Graph. 2 (2001), pp. 95–126. doi: 10.1145/502122.502124
  • M.P. do Carmo, Riemannian Geometry, Mathematics: Theory and Applications, äuser, Boston, MA, 1992.
  • P. Crouch, G. Kun, and F.S. Leite, The De Casteljau algorithm on lie groups and spheres, J. Dyn. Control Syst. 5 (1999), pp. 397–429. doi: 10.1023/A:1021770717822
  • G. Farin, Curves and Surfaces for Computer Aided Geometric Design, Academic Press, San Diego, CA, 1990.
  • A.L. Gould, A regression technique for angular variates, Biometrics 25 (1969), pp. 683–700. doi: 10.2307/2528567
  • K. Hüper and F.S. Leite, On the geometry of rolling and interpolation curves on Sn, SOn and Grassmann manifolds, J. Dyn. Control Syst. 13 (2007), pp. 467–502. doi: 10.1007/s10883-007-9027-3
  • O. Jackson and P. Crouch, Dynamic interpolation and application to flight control, J. Guidance Control Dyn. 14 (1995), pp. 814–822. doi: 10.2514/3.20717
  • P.E. Jupp and J.T. Kent, Fitting smooth paths to spherical data, Appl. Stat. 36 (1987), pp. 34–46. doi: 10.2307/2347843
  • H. Karcher, Riemannian center of mass and mollifier smoothing, Commun. Pure Appl. Math. 30 (1977), pp. 509–541. doi: 10.1002/cpa.3160300502
  • A. Kume, I.L. Dryden, and H. Le, Shape space smoothing splines for planar landmark data, Biometrika 94 (2007), pp. 513–528. doi: 10.1093/biomet/asm047
  • P. Lancaster and K. Salkauskas, Curve and Surface Fitting, Academic Press, Boston, MA, 1990.
  • L. Machado, Least squares problems on Riemannian manifolds, Ph.D. thesis, University of Coimbra, 2006.
  • L. Machado and F.S. Leite, Fitting smooth paths on Riemannian manifolds, Int. J. Appl. Math. Stat. 4 (2006), pp. 25–53.
  • L. Machado and F.S. Leite, Approximating sets of symmetric and positive-definite matrices by geodesics, Conference Papers in Mathematics 2013, 2013, pp. Article ID 425608, 7 pages.
  • L. Machado and F.S. Leite, Interpolation and polynomial fitting on the SPD manifold, CDC 2013: 52nd IEEE Conference on Decision and Control, Florence, Italy, December 10–13, 2013, pp. 1150–1155.
  • L. Machado, F.S. Leite, and K. Krakowski, Higher-order smoothing splines versus least squares problems on Riemannian manifolds, J. Dyn. Control Syst. 16 (2010), pp. 121–148. doi: 10.1007/s10883-010-9080-1
  • J.W. Milnor, Morse Theory, Princeton University Press, Princeton, NJ, 1963.
  • C. Moler, J. Little, and S. Bangert, Matlab User's Guide – The Language of Technical Computing. The MathWorks Inc., Sherborn, MA, 2001.
  • L. Noakes, G. Heinzinger, and B. Paden, Cubic splines on curved spaces, IMA J. Math. Control Inf. 6 (1989), pp. 465–473. doi: 10.1093/imamci/6.4.465
  • J. Nocedal and S.J. Wright, Numerical Optimization, 2nd ed., Springer Series in Operations Research, Springer-Verlag, New York, 2006.
  • M.J.D. Powell, Approximation Theory and Methods, Cambridge University Press, London, 1981.
  • Q. Rentmeesters, A gradient method for geodesic data fitting on some symmetric Riemannian manifolds, 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, FL, USA, December 12–15, 2011, pp. 7141–7146.
  • R. Thompson and R.M. Clark, Fitting polar wander paths, Phys. Earth Planet. Inter. 27 (1981), pp. 1–7. doi: 10.1016/0031-9201(81)90082-0

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