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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Stephen A. Burns
  • Vol. 22, Iss. 11 — Nov. 1, 2005
  • pp: 2483–2489

Dihedral representations and statistical geometric optics. I. Spherocylindrical lenses

Vasudevan Lakshminarayanan and Marlos Viana  »View Author Affiliations

JOSA A, Vol. 22, Issue 11, pp. 2483-2489 (2005)

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The linear 2-dim irreducible representations of the dihedral groups ( D n ) are interpreted as classical linear operators of geometrical optics. It is shown that the 2-dim irreducible representation of D 4 is simply the refractive group described by Campbell [Optom. Vision Sci. 74, 381 (1997) ]. The dihedral Fourier-inverse mechanism is introduced and shown to provide a systematic connection between the standard refractive data and their vector space representation, as proposed by Thibos et al. [Vision Sci. Appl. 2, 14 (1994) ].

© 2005 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(000.3870) General : Mathematics
(030.6600) Coherence and statistical optics : Statistical optics
(080.0080) Geometric optics : Geometric optics
(080.2720) Geometric optics : Mathematical methods (general)
(170.4460) Medical optics and biotechnology : Ophthalmic optics and devices

ToC Category:
Geometrical optics

Original Manuscript: September 22, 2004
Revised Manuscript: January 28, 2005
Manuscript Accepted: March 16, 2005
Published: November 1, 2005

Vasudevan Lakshminarayanan and Marlos Viana, "Dihedral representations and statistical geometric optics. I. Spherocylindrical lenses," J. Opt. Soc. Am. A 22, 2483-2489 (2005)

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  1. V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984).
  2. V. Lakshminarayanan, A. K. Ghatak, K. Thyagarajan, Lagrangian Optics (Kluwer, 2001).
  3. V. Lakshminarayanan, S. Varadharajan, “Calculation of aberration coefficients: a matrix approach method,” in Basic and Clinical Applications of Vision Science, V. Lakshminarayanan, ed. (Kluwer, 1997), pp. 111–114. [CrossRef]
  4. V. Lakshminarayanan, S. Varadharajan, “Expressions for aberration coefficients using nonlinear transforms,” Optom. Vision Sci. 74, 676–686 (1997). [CrossRef]
  5. E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985). [CrossRef]
  6. A. B. Dutta, N. Mukunda, R. Simon, “The real and symplectic groups in quantum mechanics and optics,” Pramana J. Phys. 45, 471–497 (1995). [CrossRef]
  7. M. Kauderer, “Fourier-optics approach to the symplectic group,” J. Opt. Soc. Am. A 7, 231–239 (1990). [CrossRef]
  8. H. Bacry, M. Cadilhac, “Metaplectic groups and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981). [CrossRef]
  9. A. T. James, “The relationship algebra of an experimental design,” Ann. Math. Stat. 28, 993–1002 (1957). [CrossRef]
  10. L. Nachbin, The Haar Integral (Van Nostrand, 1965).
  11. E. J. Hannan, “Group representations and applied probability,” J. Appl. Probab. 2, 1–68 (1965). [CrossRef]
  12. P. Diaconis, Group Representation in Probability and Statistics (Institute of Mathematical Statistics, Hayward, California, 1988).
  13. M. L. Eaton, Group Invariance Applications in Statistics (Institute of Mathematical Statistics–American Statistical Association, Hayward, California, 1989).
  14. R. A. Wijsman, Invariant Measures on Groups and Their Use in Statistics, Vol. 14 (Institute of Mathematical Statistics, Hayward, California, 1990).
  15. S. Andersson, Normal Statistical Models Given by Group Symmetry (Deutsche Mathematiker–Vereinigung Seminar Lecture Notes, Günzburg, Germany, 1992).
  16. M. Viana, Symmetry Studies—An Introduction (IMPA Institute for Pure and Applied Mathematics, Rio de Janeiro, Brazil, 2003).
  17. M. Viana, Lecture Notes on Symmetry Studies (EURANDOM, Technische Universiteit Eindhoven, Eindhoven, The Netherlands, 2005).
  18. C. Campbell, “The refractive group,” Optom. Vision Sci. 74, 381–387 (1997). [CrossRef]
  19. J.-P. Serre, Linear Representations of Finite Groups (Springer-Verlag, 1977). [CrossRef]
  20. K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical Methods for Physics and Engineering, 2nd ed. (Cambridge U. Press, New York, 2002). [CrossRef]
  21. C. R. Rao, Linear Statistical Inference and Its Applications (Wiley, 1973). [CrossRef]
  22. J. P. Szlyk, W. Seiple, W. Xie, “Symmetry discrimination in patients with retinitis pigmentosa,” Vision Res. 35, 1633–1640 (1995). [CrossRef] [PubMed]
  23. J. Szlyk, I. Rock, C. Fisher, “Level of processing in the perception of symmetrical forms viewed from different angles,” Spatial Vis. 9, 139–150 (1995). [CrossRef]
  24. T. O. Salmon, L. N. Thibos, A. Bradley, “Comparison of the eye’s wave-front aberration measured psychophysically and with the Shack–Hartmann wave-front sensor,” J. Opt. Soc. Am. A 15, 2457–2464 (1998). [CrossRef]
  25. V. Lakshminarayanan, R. Sridhar, R. Jagannathan, “Lie algebraic treatment of dioptric power and optical aberrations,” J. Opt. Soc. Am. A 15, 2497–2503 (1998). [CrossRef]
  26. G. James, M. Liebeck, Representations and Characters of Groups (Cambridge U. Press, 1993).
  27. M. E. Marhic, “Roots of the identity operator and optics,” J. Opt. Soc. Am. A 12, 1448–1459 (1995). [CrossRef]
  28. A. G. Bennett, R. B. Rabbetts, Clinical Visual Optics (Butterworth-Heinemann, 1984).
  29. C. Campbell, “Ray vector fields,” J. Opt. Soc. Am. A 11, 618–622 (1994). [CrossRef]
  30. W. E. Humphrey, “A remote subjective refractor employing continuously variable sphere-cylinder corrections,” Opt. Eng. 15, 286–291 (1976). [CrossRef]
  31. H. Saunders, “The Algebra of Sphero-Cylinders,” Ophthalmic Physiol. Opt. 5, 157–163 (1985). [CrossRef] [PubMed]
  32. L. N. Thibos, W. Wheeler, D. Horner, “A vector method for the analysis of astigmatic refractive error,” in Vision Science and Its Applications, Vol. 2 of 1994 OSA Technical Digest Series (Optical Society of America, 1994), pp. 14–17.
  33. A. Raghuram, N. Kim, M. Kharhoff, V. Lakshminarayanan, “The role of symmetry in perception of human faces: preliminary results,” Optom. Vision Sci. 80, 194 (2003).
  34. D. Pauleikhoff, R. Wormald, L. Wright, A. Wessing, A. Bird, “Macular disease in an elderly population,” Ger. J. Ophthalmol. 1, 12–15 (1992). [PubMed]
  35. J. Wagemans, “Parallel visual processes in symmetry perception: Normality and pathology,” Doc. Ophthalmol. 95, 359–370 (1999). [CrossRef] [PubMed]
  36. J. P. Swaddle, “Visual signalling by asymmetry: a review of perceptual processes,” Philos. Trans. R. Soc. London, Ser. B 354, 1383–1393 (1999). [CrossRef] [PubMed]
  37. C. W. Tyler, Human Symmetry Perception and Its Computational Analysis [Lawrence Erlbaum (Reprint), 2002].
  38. J. B. Hellige, Hemispheric Asymmetry (Harvard U. Press, 1993).
  39. M. Viana, “Invariance conditions for random curvature models,” Methodol. Comput. Appl. Probab. 5, 439–453 (2003). [CrossRef]
  40. M. A. G. Viana, I. Olkin, T. McMahon, “Multivariate assessment of computer analyzed corneal topographers,” J. Opt. Soc. Am. A 10, 1826–1834 (1993). [CrossRef]
  41. H. Lee, M. Viana, “The joint covariance structure of ordered symmetrically dependent observations and their concomitants of order statistics,” Stat. Probab. Lett. 43, 411–414 (1999). [CrossRef]
  42. M. Viana, I. Olkin, “Symmetrically dependent models arising in visual assessment data,” Technical Report 1998-11 (Stanford University, 1998).
  43. M. Viana, I. Olkin, “Symmetrically dependent models arising in visual assessment data,” Biometrics 56, 1188–1191 (2000).
  44. H. Lee, “The covariance structure of concomitants of ordered symmetrically dependent observations,” Ph.D. thesis (The University of Illinois at Chicago, 1998).

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