Abstract

The linear 2-dim irreducible representations of the dihedral groups (Dn) are interpreted as classical linear operators of geometrical optics. It is shown that the 2-dim irreducible representation of D4 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

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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).

2003

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).

M. Viana, “Invariance conditions for random curvature models,” Methodol. Comput. Appl. Probab. 5, 439–453 (2003).
[CrossRef]

1999

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]

J. Wagemans, “Parallel visual processes in symmetry perception: Normality and pathology,” Doc. Ophthalmol. 95, 359–370 (1999).
[CrossRef] [PubMed]

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]

1998

1997

V. Lakshminarayanan, S. Varadharajan, “Expressions for aberration coefficients using nonlinear transforms,” Optom. Vision Sci. 74, 676–686 (1997).
[CrossRef]

C. Campbell, “The refractive group,” Optom. Vision Sci. 74, 381–387 (1997).
[CrossRef]

1995

J. P. Szlyk, W. Seiple, W. Xie, “Symmetry discrimination in patients with retinitis pigmentosa,” Vision Res. 35, 1633–1640 (1995).
[CrossRef] [PubMed]

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]

M. E. Marhic, “Roots of the identity operator and optics,” J. Opt. Soc. Am. A 12, 1448–1459 (1995).
[CrossRef]

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]

1994

1993

1992

D. Pauleikhoff, R. Wormald, L. Wright, A. Wessing, A. Bird, “Macular disease in an elderly population,” Ger. J. Ophthalmol. 1, 12–15 (1992).
[PubMed]

1990

1985

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

H. Saunders, “The Algebra of Sphero-Cylinders,” Ophthalmic Physiol. Opt. 5, 157–163 (1985).
[CrossRef] [PubMed]

1981

H. Bacry, M. Cadilhac, “Metaplectic groups and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

1976

W. E. Humphrey, “A remote subjective refractor employing continuously variable sphere-cylinder corrections,” Opt. Eng. 15, 286–291 (1976).
[CrossRef]

1965

E. J. Hannan, “Group representations and applied probability,” J. Appl. Probab. 2, 1–68 (1965).
[CrossRef]

1957

A. T. James, “The relationship algebra of an experimental design,” Ann. Math. Stat. 28, 993–1002 (1957).
[CrossRef]

Andersson, S.

S. Andersson, Normal Statistical Models Given by Group Symmetry (Deutsche Mathematiker–Vereinigung Seminar Lecture Notes, Günzburg, Germany, 1992).

Bacry, H.

H. Bacry, M. Cadilhac, “Metaplectic groups and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Bence, S. J.

K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical Methods for Physics and Engineering, 2nd ed. (Cambridge U. Press, New York, 2002).
[CrossRef]

Bennett, A. G.

A. G. Bennett, R. B. Rabbetts, Clinical Visual Optics (Butterworth-Heinemann, 1984).

Bird, A.

D. Pauleikhoff, R. Wormald, L. Wright, A. Wessing, A. Bird, “Macular disease in an elderly population,” Ger. J. Ophthalmol. 1, 12–15 (1992).
[PubMed]

Bradley, A.

Cadilhac, M.

H. Bacry, M. Cadilhac, “Metaplectic groups and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Campbell, C.

C. Campbell, “The refractive group,” Optom. Vision Sci. 74, 381–387 (1997).
[CrossRef]

C. Campbell, “Ray vector fields,” J. Opt. Soc. Am. A 11, 618–622 (1994).
[CrossRef]

Diaconis, P.

P. Diaconis, Group Representation in Probability and Statistics (Institute of Mathematical Statistics, Hayward, California, 1988).

Dutta, A. B.

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]

Eaton, M. L.

M. L. Eaton, Group Invariance Applications in Statistics (Institute of Mathematical Statistics–American Statistical Association, Hayward, California, 1989).

Fisher, C.

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]

Ghatak, A. K.

V. Lakshminarayanan, A. K. Ghatak, K. Thyagarajan, Lagrangian Optics (Kluwer, 2001).

Guillemin, V.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984).

Hannan, E. J.

E. J. Hannan, “Group representations and applied probability,” J. Appl. Probab. 2, 1–68 (1965).
[CrossRef]

Hellige, J. B.

J. B. Hellige, Hemispheric Asymmetry (Harvard U. Press, 1993).

Hobson, M. P.

K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical Methods for Physics and Engineering, 2nd ed. (Cambridge U. Press, New York, 2002).
[CrossRef]

Horner, D.

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.

Humphrey, W. E.

W. E. Humphrey, “A remote subjective refractor employing continuously variable sphere-cylinder corrections,” Opt. Eng. 15, 286–291 (1976).
[CrossRef]

Jagannathan, R.

James, A. T.

A. T. James, “The relationship algebra of an experimental design,” Ann. Math. Stat. 28, 993–1002 (1957).
[CrossRef]

James, G.

G. James, M. Liebeck, Representations and Characters of Groups (Cambridge U. Press, 1993).

Kauderer, M.

Kharhoff, M.

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).

Kim, N.

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).

Lakshminarayanan, V.

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).

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]

V. Lakshminarayanan, S. Varadharajan, “Expressions for aberration coefficients using nonlinear transforms,” Optom. Vision Sci. 74, 676–686 (1997).
[CrossRef]

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]

V. Lakshminarayanan, A. K. Ghatak, K. Thyagarajan, Lagrangian Optics (Kluwer, 2001).

Lee, H.

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]

H. Lee, “The covariance structure of concomitants of ordered symmetrically dependent observations,” Ph.D. thesis (The University of Illinois at Chicago, 1998).

Liebeck, M.

G. James, M. Liebeck, Representations and Characters of Groups (Cambridge U. Press, 1993).

Marhic, M. E.

McMahon, T.

Mukunda, N.

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]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Nachbin, L.

L. Nachbin, The Haar Integral (Van Nostrand, 1965).

Olkin, I.

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]

M. Viana, I. Olkin, “Symmetrically dependent models arising in visual assessment data,” Technical Report 1998-11 (Stanford University, 1998).

M. Viana, I. Olkin, “Symmetrically dependent models arising in visual assessment data,” Biometrics 56, 1188–1191 (2000).

Pauleikhoff, D.

D. Pauleikhoff, R. Wormald, L. Wright, A. Wessing, A. Bird, “Macular disease in an elderly population,” Ger. J. Ophthalmol. 1, 12–15 (1992).
[PubMed]

Rabbetts, R. B.

A. G. Bennett, R. B. Rabbetts, Clinical Visual Optics (Butterworth-Heinemann, 1984).

Raghuram, A.

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).

Rao, C. R.

C. R. Rao, Linear Statistical Inference and Its Applications (Wiley, 1973).
[CrossRef]

Riley, K. F.

K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical Methods for Physics and Engineering, 2nd ed. (Cambridge U. Press, New York, 2002).
[CrossRef]

Rock, I.

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]

Salmon, T. O.

Saunders, H.

H. Saunders, “The Algebra of Sphero-Cylinders,” Ophthalmic Physiol. Opt. 5, 157–163 (1985).
[CrossRef] [PubMed]

Seiple, W.

J. P. Szlyk, W. Seiple, W. Xie, “Symmetry discrimination in patients with retinitis pigmentosa,” Vision Res. 35, 1633–1640 (1995).
[CrossRef] [PubMed]

Serre, J.-P.

J.-P. Serre, Linear Representations of Finite Groups (Springer-Verlag, 1977).
[CrossRef]

Simon, R.

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]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Sridhar, R.

Sternberg, S.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984).

Sudarshan, E. C. G.

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Swaddle, J. P.

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]

Szlyk, J.

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]

Szlyk, J. P.

J. P. Szlyk, W. Seiple, W. Xie, “Symmetry discrimination in patients with retinitis pigmentosa,” Vision Res. 35, 1633–1640 (1995).
[CrossRef] [PubMed]

Thibos, L. N.

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]

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.

Thyagarajan, K.

V. Lakshminarayanan, A. K. Ghatak, K. Thyagarajan, Lagrangian Optics (Kluwer, 2001).

Tyler, C. W.

C. W. Tyler, Human Symmetry Perception and Its Computational Analysis [Lawrence Erlbaum (Reprint), 2002].

Varadharajan, S.

V. Lakshminarayanan, S. Varadharajan, “Expressions for aberration coefficients using nonlinear transforms,” Optom. Vision Sci. 74, 676–686 (1997).
[CrossRef]

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]

Viana, M.

M. Viana, “Invariance conditions for random curvature models,” Methodol. Comput. Appl. Probab. 5, 439–453 (2003).
[CrossRef]

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]

M. Viana, I. Olkin, “Symmetrically dependent models arising in visual assessment data,” Biometrics 56, 1188–1191 (2000).

M. Viana, Symmetry Studies—An Introduction (IMPA Institute for Pure and Applied Mathematics, Rio de Janeiro, Brazil, 2003).

M. Viana, Lecture Notes on Symmetry Studies (EURANDOM, Technische Universiteit Eindhoven, Eindhoven, The Netherlands, 2005).

M. Viana, I. Olkin, “Symmetrically dependent models arising in visual assessment data,” Technical Report 1998-11 (Stanford University, 1998).

Viana, M. A. G.

Wagemans, J.

J. Wagemans, “Parallel visual processes in symmetry perception: Normality and pathology,” Doc. Ophthalmol. 95, 359–370 (1999).
[CrossRef] [PubMed]

Wessing, A.

D. Pauleikhoff, R. Wormald, L. Wright, A. Wessing, A. Bird, “Macular disease in an elderly population,” Ger. J. Ophthalmol. 1, 12–15 (1992).
[PubMed]

Wheeler, W.

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.

Wijsman, R. A.

R. A. Wijsman, Invariant Measures on Groups and Their Use in Statistics, Vol. 14 (Institute of Mathematical Statistics, Hayward, California, 1990).

Wormald, R.

D. Pauleikhoff, R. Wormald, L. Wright, A. Wessing, A. Bird, “Macular disease in an elderly population,” Ger. J. Ophthalmol. 1, 12–15 (1992).
[PubMed]

Wright, L.

D. Pauleikhoff, R. Wormald, L. Wright, A. Wessing, A. Bird, “Macular disease in an elderly population,” Ger. J. Ophthalmol. 1, 12–15 (1992).
[PubMed]

Xie, W.

J. P. Szlyk, W. Seiple, W. Xie, “Symmetry discrimination in patients with retinitis pigmentosa,” Vision Res. 35, 1633–1640 (1995).
[CrossRef] [PubMed]

Ann. Math. Stat.

A. T. James, “The relationship algebra of an experimental design,” Ann. Math. Stat. 28, 993–1002 (1957).
[CrossRef]

Doc. Ophthalmol.

J. Wagemans, “Parallel visual processes in symmetry perception: Normality and pathology,” Doc. Ophthalmol. 95, 359–370 (1999).
[CrossRef] [PubMed]

Ger. J. Ophthalmol.

D. Pauleikhoff, R. Wormald, L. Wright, A. Wessing, A. Bird, “Macular disease in an elderly population,” Ger. J. Ophthalmol. 1, 12–15 (1992).
[PubMed]

J. Appl. Probab.

E. J. Hannan, “Group representations and applied probability,” J. Appl. Probab. 2, 1–68 (1965).
[CrossRef]

J. Opt. Soc. Am. A

Methodol. Comput. Appl. Probab.

M. Viana, “Invariance conditions for random curvature models,” Methodol. Comput. Appl. Probab. 5, 439–453 (2003).
[CrossRef]

Ophthalmic Physiol. Opt.

H. Saunders, “The Algebra of Sphero-Cylinders,” Ophthalmic Physiol. Opt. 5, 157–163 (1985).
[CrossRef] [PubMed]

Opt. Acta

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[CrossRef]

Opt. Eng.

W. E. Humphrey, “A remote subjective refractor employing continuously variable sphere-cylinder corrections,” Opt. Eng. 15, 286–291 (1976).
[CrossRef]

Optom. Vision Sci.

V. Lakshminarayanan, S. Varadharajan, “Expressions for aberration coefficients using nonlinear transforms,” Optom. Vision Sci. 74, 676–686 (1997).
[CrossRef]

C. Campbell, “The refractive group,” Optom. Vision Sci. 74, 381–387 (1997).
[CrossRef]

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).

Philos. Trans. R. Soc. London, Ser. B

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]

Phys. Rev. A

H. Bacry, M. Cadilhac, “Metaplectic groups and Fourier optics,” Phys. Rev. A 23, 2533–2536 (1981).
[CrossRef]

Pramana J. Phys.

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]

Spatial Vis.

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]

Stat. Probab. Lett.

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]

Vision Res.

J. P. Szlyk, W. Seiple, W. Xie, “Symmetry discrimination in patients with retinitis pigmentosa,” Vision Res. 35, 1633–1640 (1995).
[CrossRef] [PubMed]

Other

L. Nachbin, The Haar Integral (Van Nostrand, 1965).

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, UK, 1984).

V. Lakshminarayanan, A. K. Ghatak, K. Thyagarajan, Lagrangian Optics (Kluwer, 2001).

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]

J.-P. Serre, Linear Representations of Finite Groups (Springer-Verlag, 1977).
[CrossRef]

K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical Methods for Physics and Engineering, 2nd ed. (Cambridge U. Press, New York, 2002).
[CrossRef]

C. R. Rao, Linear Statistical Inference and Its Applications (Wiley, 1973).
[CrossRef]

P. Diaconis, Group Representation in Probability and Statistics (Institute of Mathematical Statistics, Hayward, California, 1988).

M. L. Eaton, Group Invariance Applications in Statistics (Institute of Mathematical Statistics–American Statistical Association, Hayward, California, 1989).

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Figures (1)

Fig. 1
Fig. 1

Refractive profile for s = 4.25 , c = 1.5 , α = 20 deg (outer contour) and s = 2.75 , c = 1.00 , α = 10 deg (inner contour).

Equations (39)

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I = P 1 + P 2 + + P h
x 2 = ( x x ) = ( x P 1 x ) + ( x P 2 x ) + + ( x P h x )
x ̂ ( β ) = τ G x ( τ ) β ( τ ) ,
β j ( η k ) = [ cos ( w j k ) sin ( w j k ) sin ( w j k ) cos ( w j k ) ] ,
β j ( η k τ ) = [ cos ( w j k ) sin ( w j k ) sin ( w j k ) cos ( w j k ) ] ,
β ( 1 ) = [ 1 0 0 1 ] , β ( η ) = [ 0 1 1 0 ] ,
β ( η 2 ) = [ 1 0 0 1 ] , β ( η 3 ) = [ 0 1 1 0 ] ,
β ( τ ) = [ 1 0 0 1 ] , β ( η τ ) = [ 0 1 1 0 ] ,
β ( η 2 τ ) = [ 1 0 0 1 ] , β ( η 3 τ ) = [ 0 1 1 0 ] ,
[ χ 1 η η 2 η 3 τ η τ η 2 τ η 3 τ χ 1 1 1 1 1 1 1 1 1 χ 2 1 1 1 1 1 1 1 1 χ 3 1 1 1 1 1 1 1 1 χ 4 1 1 1 1 1 1 1 1 χ 5 2 0 2 0 0 0 0 0 ] .
β ̃ j ( η k ) = [ ω k j 0 0 ω ¯ k j ] , β ̃ j ( η k τ ) = [ 0 ω k j ω ¯ k 0 ] ,
( σ , τ ) × α ( σ , τ ) = { ( σ σ , τ ) when τ = 1 ( σ σ 1 , τ τ ) when τ = ( 12 ) } ,
P i = n i G τ D 4 χ ¯ i ( τ ) ϕ ( τ ) , j = 1 , , 5 ,
P 2 = 1 8 [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] ,
P 3 = 1 8 [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] ,
P 4 = 1 8 [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] ,
P 5 = 1 2 [ 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 ] .
x 2 = x P 1 x + + x P 5 x ,
[ source x P x DF P 1 ( u + r + R + ρ + h + d + v + D ) 2 8 1 P 2 ( u + r + R + ρ h d v D ) 2 8 1 P 3 ( u + r R + ρ h + d v + D ) 2 8 1 P 4 ( u + r R + ρ + h d + v D ) 2 8 1 P 5 [ ( u R ) 2 + ( r ρ ) 2 + ( h v ) 2 + ( d D ) 2 ] 2 4 total u 2 + r 2 + R 2 + ρ 2 + h 2 + d 2 + v 2 + D 2 8 ] .
x ( η k τ l ) = k r + l a , k = 0 , 1 , 2 , 3 , l = 0 , 1 ,
I = I 8 I N = i = 1 5 P i ( A + Q ) ,
I 8 = P 1 + + P 5 , I N = A + Q ,
π ( θ ) = ( n n ) [ κ s cos 2 ( θ α ) + κ f sin 2 ( θ α ) ] , 0 θ 2 π , 0 α π ,
F = [ s + c sin 2 ( α ) c sin ( α ) cos ( α ) c sin ( α ) cos ( α ) s + c cos 2 ( α ) ] = [ S C + C x C x S + S + ] .
x ̂ ( η ) = { F if η = β , tr F if η β , }
x ( τ ) = η n η G tr [ η ( τ 1 ) x ̂ ( η ) ] , τ D 4 ,
[ Rotational Axial Coefficients ̱ Coefficients ̱ j x ( η j ) x ( η j τ ) 0 3 ( 2 s + c ) 4 c cos ( 2 α ) 4 1 0 c sin ( 2 α ) 4 2 ( 2 s + c ) 4 c cos ( 2 α ) 4 3 0 c sin ( 2 α ) 4 ] .
C 0 = c cos ( 2 α ) , C 45 = c sin ( 2 α ) , M = [ s + ( s + c ) ] 2 = s + c 2 ,
x ( 1 ) = 3 M 2 , x ( η 2 ) = M 2 , x ( τ ) = C 0 ,
x ( η τ ) = C 45 , x ( η 2 τ ) = C 0 , x ( η 3 τ ) = C 45 ,
( s , c , α ) = ( 4.25 , 1.5 , 20 deg ) F 1 = [ 4.0745 0.48207 0.48207 2.9255 ] ,
( s , c , α ) = ( 2.75 , 1.0 , 10 deg ) F 2 = [ 2.7198 0.17101 0.17101 1.7802 ]
F 1 : 1 4 [ j x ( η j ) x ( η j τ ) 0 21.0 1.1491 1 0 0.96414 2 7.0 1.1491 3 0 0.96414 ] ,
F 2 : 1 4 [ j x ( η j ) x ( η j τ ) 0 13.500 0.93970 1 0 0.34202 2 4.5000 0.93970 3 0 0.34202 ] .
F 1 = 1 4 [ 21 β ( 1 ) 7 β ( η 2 ) + 1.1491 β ( τ ) + 0.69414 β ( η τ ) 1.1491 β ( η 2 τ ) 0.69414 β ( η 3 τ ) ] ,
F 2 = 1 4 [ 13.5 β ( 1 ) 4.5 β ( η 2 ) 0.9397 β ( τ ) 0.342 β ( η τ ) + 0.9397 β ( η 2 τ ) + 0.342 β ( η 3 τ ) ] ,
β ( 1 ) = [ 1 0 0 1 ] , β ( η 2 ) = [ 1 0 0 1 ] ,
β ( τ ) = [ 1 0 0 1 ] , β ( η τ ) = [ 0 1 1 0 ] ,
β ( η 2 τ ) = [ 1 0 0 1 ] , β ( η 3 τ ) = [ 0 1 1 0 ] .

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