Abstract

It is shown that the aberration estimated at any point of the pupil using wavefront slope aberrometers such as Hartmann–Shack wavefront sensors or laser ray tracers is a spatial average of the actual aberration weighted by a characteristic function that depends on the aberrometer design and on the estimation procedure. This characteristic function, whose explicit form is given here for wavefront slope aberrometers using either modal or zonal estimators, may be useful in analyzing some basic aspects of the aberrometer performance. It is also instrumental in establishing the links between the statistical properties of the actual and the estimated aberrations. Explicit formulas are given to show in terms of this function how the bias arises in the first- and second-order statistics of the retrieved aberrations. This approach is mathematically equivalent to the analysis of the effects of modal coupling (cross-coupling and aliasing). It may provide, however, some complementary insight.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).
  2. F. Merkle, "Adaptive optics," in International Trends in Optics, J.W.Goodman, ed. (Academic, 1991), Chap. 26, pp. 375-390.
  3. J. Primot, G. Rousset, and J. C. Fontanella, "Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images," J. Opt. Soc. Am. A 7, 1589-1608 (1990).
    [CrossRef]
  4. D. Dayton, B. Pierson, B. Spielbusch, and J. Gonglewski, "Atmospheric structure function measurements with a Shack-Hartmann wave-front sensor," Opt. Lett. 17, 1737-1739 (1992).
    [CrossRef] [PubMed]
  5. T. W. Nicholls, G. D. Boreman, and J. C. Dainty, "Use of a Shack-Hartmann wave-front sensor to measure deviations from a Kolmogorov phase spectrum," Opt. Lett. 20, 2460-2462 (1995).
    [CrossRef] [PubMed]
  6. E. E. Silbaugh, B. M. Welsh, and M. C. Roggemann, "Characterization of atmospheric turbulence phase statistics using wave-front slope measurements," J. Opt. Soc. Am. A 13, 2453-2460 (1996).
    [CrossRef]
  7. C. Rao, W. Jiang, and N. Ling, "Measuring the power-law exponent of an atmospheric turbulence phase power spectrum with a Shack-Hartmann wave-front sensor," Opt. Lett. 24, 1008-1010 (1999).
    [CrossRef]
  8. T. Kohno and S. Tanaka, "Figure measurement of concave mirror by fiber-grating Hartmann test," Opt. Rev. 1, 118-120 (1994).
    [CrossRef]
  9. N. S. Prasad, S. M. Doyle, and M. K. Giles, "Collimation and beam alignment: testing and estimation using liquid-crystal televisions," Opt. Eng. (Bellingham) 35, 1815-1819 (1996).
    [CrossRef]
  10. H. J. Tiziani and J. H. Chen, "Shack-Hartmann sensor for fast infrared wave-front testing," J. Mod. Opt. 44, 535-541 (1997).
    [CrossRef]
  11. G. Artzner, "Aspherical wavefront measurements: Shack-Hartmann numerical and practical experiments," Pure Appl. Opt. 7, 435-448 (1998).
    [CrossRef]
  12. J. Pfund, N. Lindlein, J. Schwider, R. Burow, Th. Blumel, and K.-E. Elssner, "Absolute sphericity measurement: a comparative study of the use of interferometry and a Shack-Hartmann sensor," Opt. Lett. 23, 742-744 (1998).
    [CrossRef]
  13. J. Ares, T. Mancebo, and S. Bará, "Position and displacement sensing with Shack-Hartmann wavefront sensors," Appl. Opt. 39, 1511-1520 (2000).
    [CrossRef]
  14. J. Liang, B. Grimm, S. Goelz, and J. Bille, "Objective measurement of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor," J. Opt. Soc. Am. A 11, 1949-1957 (1994).
    [CrossRef]
  15. J. Liang and D. R. Williams, "Aberrations and retinal image quality of the normal human eye," J. Opt. Soc. Am. A 14, 2873-2883 (1997).
    [CrossRef]
  16. P. M. Prieto, F. Vargas-Martin, S. Goelz, and P. Artal, "Analysis of the performance of the Hartmann-Shack sensor in the human eye," J. Opt. Soc. Am. A 17, 1388-1398 (2000).
    [CrossRef]
  17. R. Navarro and M. A. Losada, "Aberrations and relative efficiency of light pencils in the living human eye," Optom. Vision Sci. 74, 540-547 (1997).
    [CrossRef]
  18. R. Navarro, E. Moreno, and C. Dorronsoro, "Monochromatic aberrations and point-spread functions of the human eye across the visual field," J. Opt. Soc. Am. A 15, 2522-2529 (1998).
    [CrossRef]
  19. R. Navarro and E. Moreno-Barriuso, "Laser ray-tracing method for optical testing," Opt. Lett. 24, 951-953 (1999).
    [CrossRef]
  20. R. H. Webb, C. M. Penney, and K. P. Thompson, "Measurement of ocular wavefront distortion with a spatially resolved refractometer," Appl. Opt. 31, 3678-3686 (1992).
    [CrossRef] [PubMed]
  21. J. C. He, S. Marcos, R. H. Webb, and S. A. Burns, "Measurement of the wavefront aberration of the eye by a fast psychophysical procedure," J. Opt. Soc. Am. A 15, 2449-2456 (1998).
    [CrossRef]
  22. R. H. Webb, C. M. Penney, J. Sobiech, P. R. Staver, and S. A. Burns, "SRR (spatially resolved refractometer): a null-seeking aberrometer," Appl. Opt. 42, 736-744 (2003).
    [CrossRef] [PubMed]
  23. E. Moreno-Barriuso, S. Marcos, R. Navarro, and S. A. Burns, "Comparing laser ray tracing, spatially resolved refractometer and Hartmann-Shack sensor to measure the ocular wavefront aberration," Optom. Vision Sci. 78, 152-156 (2001).
    [CrossRef]
  24. E. Moreno-Barriuso and R. Navarro, "Laser ray tracing versus Hartmann-Shack sensor for measuring optical aberrations in the human eye," J. Opt. Soc. Am. A 17, 974-985 (2000).
    [CrossRef]
  25. S. Marcos, L. Díaz-Santana, L. Llorente, and C. Dainty, "Ocular aberrations with ray tracing and Shack-Hartmann wave-front sensors: Does polarization play a role?" J. Opt. Soc. Am. A 19, 1063-1072 (2002).
    [CrossRef]
  26. L. Llorente, L. Díaz-Santana, D. Lara-Saucedo, and S. Marcos, "Aberrations of the human eye in visible and near infrared illumination," Optom. Vision Sci. 80, 26-35 (2003).
    [CrossRef]
  27. P. Rodríguez, R. Navarro, J. Arines, and S. Bará, "A new calibration set of phase plates for ocular aberrometers," J. Refract. Surg. 22, 275-284 (2006).
    [PubMed]
  28. R. Cubalchini, "Modal wavefront estimation from phase derivative measurements," J. Opt. Soc. Am. 69, 972-977 (1979).
    [CrossRef]
  29. W. H. Southwell, "Wave-front estimation from wave-front slope measurements," J. Opt. Soc. Am. 70, 998-1006 (1980).
    [CrossRef]
  30. R. H. Hudgin, "Wave-front reconstruction for compensated imaging," J. Opt. Soc. Am. 67, 375-378 (1977).
    [CrossRef]
  31. S. N. Bezdid'ko, "The use of Zernike polynomials in optics," Sov. J. Opt. Technol. 41, 425-429 (1974).
  32. M. Born and E. Wolf, Principles of Optics, pp. 464-466, 767-772, (Cambridge U. Press, 1998).
  33. L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, "Standards for reporting the optical aberrations of eyes," in Vision Science and Its Applications 2000, V.Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.
  34. J. Herrmann, "Cross coupling and aliasing in modal wave-front estimation," J. Opt. Soc. Am. 71, 989-992 (1981).
    [CrossRef]
  35. L. Díaz-Santana, G. Walker, and S. X. Bará, "Sampling geometries for ocular aberrometry: a model for evaluation of performance," Opt. Express 13, 8801-8818 (2005).
    [CrossRef]
  36. O. Soloviev and G. Vdovin, "Hartmann-Shack test with random masks for modal wavefront reconstruction," Opt. Express 13, 9570-9584 (2005).
    [CrossRef]
  37. S. Bará, P. Prado, J. Arines, and J. Ares, "Estimation-induced correlations of the Zernike coefficients of the eye aberration," Opt. Lett. 31, 2646-2648 (2006).
    [CrossRef]
  38. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  39. V. I. Tatarskii, The Propagation of Waves in the Turbulent Atmosphere (Nauka, Moscow, 1967), pp. 385-390 (in Russian).
  40. M. R. Teague, "Irradiance moments: their propagation and use for unique retrieval of phase," J. Opt. Soc. Am. 72, 1199-1209 (1982).
    [CrossRef]
  41. S. Bará, "Measuring eye aberrations with Hartmann-Shack wave-front sensors: Should the irradiance distribution across the eye pupil be taken into account?" J. Opt. Soc. Am. A 20, 2237-2245 (2003).
    [CrossRef]
  42. P. Ehrenfest, "Notes on the approximate validity of quantum mechanics," Z. Phys. 45, 455-457 (1927) (in German).
    [CrossRef]
  43. R. J. Cook, "Beam wander in a turbulent medium: An application of Ehrenfest's theorem," J. Opt. Soc. Am. 65, 942-948 (1975).
    [CrossRef]
  44. S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, "Comparison of centroid computation algorithms in a Shack-Hartmann sensor," Mon. Not. R. Astron. Soc. 371, 323-336 (2006).
    [CrossRef]
  45. R. Irwan and R. G. Lane, "Analysis of optimal centroid estimation applied to Shack-Hartmann sensing," Appl. Opt. 38, 6737-6743 (1999).
    [CrossRef]
  46. D. A. Montera, B. M. Welsh, M. C. Roggemann, and D. W. Ruck, "Use of artificial neural networks for Hartmann-sensor lenslet centroid estimation," Appl. Opt. 35, 5747-5757 (1996).
    [CrossRef] [PubMed]
  47. J.-M. Ruggiu, C. J. Solomon, and G. Loos, "Gram-Charlier matched filter for Shack-Hartmann sensing at low light levels," Opt. Lett. 23, 235-237 (1998).
    [CrossRef]
  48. J. Arines and J. Ares, "Minimum variance centroid thresholding," Opt. Lett. 27, 497-499 (2002).
    [CrossRef]
  49. V. Laude, S. Olivier, C. Dirson, and J.-P. Huignard, "Hartmann wave-front scanner," Opt. Lett. 24, 1796-1798 (1999).
    [CrossRef]
  50. S. Olivier, V. Laude, and J.-P. Huignard, "Liquid-crystal Hartmann wave-front scanner," Appl. Opt. 39, 3838-3846 (2000).
    [CrossRef]
  51. J. Primot, "Theoretical description of Shack-Hartmann wave-front sensor," Opt. Commun. 222, 81-92 (2003).
    [CrossRef]
  52. E. P. Wallner, "Optimal wave-front correction using slope measurements," J. Opt. Soc. Am. 73, 1771-1776 (1983).
    [CrossRef]
  53. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211 (1976).
    [CrossRef]

2006 (3)

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, "Comparison of centroid computation algorithms in a Shack-Hartmann sensor," Mon. Not. R. Astron. Soc. 371, 323-336 (2006).
[CrossRef]

P. Rodríguez, R. Navarro, J. Arines, and S. Bará, "A new calibration set of phase plates for ocular aberrometers," J. Refract. Surg. 22, 275-284 (2006).
[PubMed]

S. Bará, P. Prado, J. Arines, and J. Ares, "Estimation-induced correlations of the Zernike coefficients of the eye aberration," Opt. Lett. 31, 2646-2648 (2006).
[CrossRef]

2005 (2)

2003 (4)

R. H. Webb, C. M. Penney, J. Sobiech, P. R. Staver, and S. A. Burns, "SRR (spatially resolved refractometer): a null-seeking aberrometer," Appl. Opt. 42, 736-744 (2003).
[CrossRef] [PubMed]

S. Bará, "Measuring eye aberrations with Hartmann-Shack wave-front sensors: Should the irradiance distribution across the eye pupil be taken into account?" J. Opt. Soc. Am. A 20, 2237-2245 (2003).
[CrossRef]

J. Primot, "Theoretical description of Shack-Hartmann wave-front sensor," Opt. Commun. 222, 81-92 (2003).
[CrossRef]

L. Llorente, L. Díaz-Santana, D. Lara-Saucedo, and S. Marcos, "Aberrations of the human eye in visible and near infrared illumination," Optom. Vision Sci. 80, 26-35 (2003).
[CrossRef]

2002 (2)

2001 (1)

E. Moreno-Barriuso, S. Marcos, R. Navarro, and S. A. Burns, "Comparing laser ray tracing, spatially resolved refractometer and Hartmann-Shack sensor to measure the ocular wavefront aberration," Optom. Vision Sci. 78, 152-156 (2001).
[CrossRef]

2000 (4)

1999 (4)

1998 (5)

1997 (3)

R. Navarro and M. A. Losada, "Aberrations and relative efficiency of light pencils in the living human eye," Optom. Vision Sci. 74, 540-547 (1997).
[CrossRef]

H. J. Tiziani and J. H. Chen, "Shack-Hartmann sensor for fast infrared wave-front testing," J. Mod. Opt. 44, 535-541 (1997).
[CrossRef]

J. Liang and D. R. Williams, "Aberrations and retinal image quality of the normal human eye," J. Opt. Soc. Am. A 14, 2873-2883 (1997).
[CrossRef]

1996 (3)

1995 (1)

1994 (2)

1992 (2)

1990 (1)

J. Primot, G. Rousset, and J. C. Fontanella, "Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images," J. Opt. Soc. Am. A 7, 1589-1608 (1990).
[CrossRef]

1983 (1)

1982 (1)

1981 (1)

1980 (1)

1979 (1)

1977 (1)

1976 (1)

1975 (1)

1974 (1)

S. N. Bezdid'ko, "The use of Zernike polynomials in optics," Sov. J. Opt. Technol. 41, 425-429 (1974).

1927 (1)

P. Ehrenfest, "Notes on the approximate validity of quantum mechanics," Z. Phys. 45, 455-457 (1927) (in German).
[CrossRef]

Appl. Opt. (6)

J. Mod. Opt. (1)

H. J. Tiziani and J. H. Chen, "Shack-Hartmann sensor for fast infrared wave-front testing," J. Mod. Opt. 44, 535-541 (1997).
[CrossRef]

J. Opt. Soc. Am. (8)

J. Opt. Soc. Am. A (10)

S. Bará, "Measuring eye aberrations with Hartmann-Shack wave-front sensors: Should the irradiance distribution across the eye pupil be taken into account?" J. Opt. Soc. Am. A 20, 2237-2245 (2003).
[CrossRef]

S. Marcos, L. Díaz-Santana, L. Llorente, and C. Dainty, "Ocular aberrations with ray tracing and Shack-Hartmann wave-front sensors: Does polarization play a role?" J. Opt. Soc. Am. A 19, 1063-1072 (2002).
[CrossRef]

E. E. Silbaugh, B. M. Welsh, and M. C. Roggemann, "Characterization of atmospheric turbulence phase statistics using wave-front slope measurements," J. Opt. Soc. Am. A 13, 2453-2460 (1996).
[CrossRef]

P. M. Prieto, F. Vargas-Martin, S. Goelz, and P. Artal, "Analysis of the performance of the Hartmann-Shack sensor in the human eye," J. Opt. Soc. Am. A 17, 1388-1398 (2000).
[CrossRef]

E. Moreno-Barriuso and R. Navarro, "Laser ray tracing versus Hartmann-Shack sensor for measuring optical aberrations in the human eye," J. Opt. Soc. Am. A 17, 974-985 (2000).
[CrossRef]

J. Liang, B. Grimm, S. Goelz, and J. Bille, "Objective measurement of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor," J. Opt. Soc. Am. A 11, 1949-1957 (1994).
[CrossRef]

J. C. He, S. Marcos, R. H. Webb, and S. A. Burns, "Measurement of the wavefront aberration of the eye by a fast psychophysical procedure," J. Opt. Soc. Am. A 15, 2449-2456 (1998).
[CrossRef]

R. Navarro, E. Moreno, and C. Dorronsoro, "Monochromatic aberrations and point-spread functions of the human eye across the visual field," J. Opt. Soc. Am. A 15, 2522-2529 (1998).
[CrossRef]

J. Liang and D. R. Williams, "Aberrations and retinal image quality of the normal human eye," J. Opt. Soc. Am. A 14, 2873-2883 (1997).
[CrossRef]

J. Primot, G. Rousset, and J. C. Fontanella, "Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images," J. Opt. Soc. Am. A 7, 1589-1608 (1990).
[CrossRef]

J. Refract. Surg. (1)

P. Rodríguez, R. Navarro, J. Arines, and S. Bará, "A new calibration set of phase plates for ocular aberrometers," J. Refract. Surg. 22, 275-284 (2006).
[PubMed]

Mon. Not. R. Astron. Soc. (1)

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, "Comparison of centroid computation algorithms in a Shack-Hartmann sensor," Mon. Not. R. Astron. Soc. 371, 323-336 (2006).
[CrossRef]

Opt. Commun. (1)

J. Primot, "Theoretical description of Shack-Hartmann wave-front sensor," Opt. Commun. 222, 81-92 (2003).
[CrossRef]

Opt. Eng. (Bellingham) (1)

N. S. Prasad, S. M. Doyle, and M. K. Giles, "Collimation and beam alignment: testing and estimation using liquid-crystal televisions," Opt. Eng. (Bellingham) 35, 1815-1819 (1996).
[CrossRef]

Opt. Express (2)

Opt. Lett. (9)

S. Bará, P. Prado, J. Arines, and J. Ares, "Estimation-induced correlations of the Zernike coefficients of the eye aberration," Opt. Lett. 31, 2646-2648 (2006).
[CrossRef]

R. Navarro and E. Moreno-Barriuso, "Laser ray-tracing method for optical testing," Opt. Lett. 24, 951-953 (1999).
[CrossRef]

J. Arines and J. Ares, "Minimum variance centroid thresholding," Opt. Lett. 27, 497-499 (2002).
[CrossRef]

D. Dayton, B. Pierson, B. Spielbusch, and J. Gonglewski, "Atmospheric structure function measurements with a Shack-Hartmann wave-front sensor," Opt. Lett. 17, 1737-1739 (1992).
[CrossRef] [PubMed]

T. W. Nicholls, G. D. Boreman, and J. C. Dainty, "Use of a Shack-Hartmann wave-front sensor to measure deviations from a Kolmogorov phase spectrum," Opt. Lett. 20, 2460-2462 (1995).
[CrossRef] [PubMed]

J.-M. Ruggiu, C. J. Solomon, and G. Loos, "Gram-Charlier matched filter for Shack-Hartmann sensing at low light levels," Opt. Lett. 23, 235-237 (1998).
[CrossRef]

J. Pfund, N. Lindlein, J. Schwider, R. Burow, Th. Blumel, and K.-E. Elssner, "Absolute sphericity measurement: a comparative study of the use of interferometry and a Shack-Hartmann sensor," Opt. Lett. 23, 742-744 (1998).
[CrossRef]

C. Rao, W. Jiang, and N. Ling, "Measuring the power-law exponent of an atmospheric turbulence phase power spectrum with a Shack-Hartmann wave-front sensor," Opt. Lett. 24, 1008-1010 (1999).
[CrossRef]

V. Laude, S. Olivier, C. Dirson, and J.-P. Huignard, "Hartmann wave-front scanner," Opt. Lett. 24, 1796-1798 (1999).
[CrossRef]

Opt. Rev. (1)

T. Kohno and S. Tanaka, "Figure measurement of concave mirror by fiber-grating Hartmann test," Opt. Rev. 1, 118-120 (1994).
[CrossRef]

Optom. Vision Sci. (3)

R. Navarro and M. A. Losada, "Aberrations and relative efficiency of light pencils in the living human eye," Optom. Vision Sci. 74, 540-547 (1997).
[CrossRef]

E. Moreno-Barriuso, S. Marcos, R. Navarro, and S. A. Burns, "Comparing laser ray tracing, spatially resolved refractometer and Hartmann-Shack sensor to measure the ocular wavefront aberration," Optom. Vision Sci. 78, 152-156 (2001).
[CrossRef]

L. Llorente, L. Díaz-Santana, D. Lara-Saucedo, and S. Marcos, "Aberrations of the human eye in visible and near infrared illumination," Optom. Vision Sci. 80, 26-35 (2003).
[CrossRef]

Pure Appl. Opt. (1)

G. Artzner, "Aspherical wavefront measurements: Shack-Hartmann numerical and practical experiments," Pure Appl. Opt. 7, 435-448 (1998).
[CrossRef]

Sov. J. Opt. Technol. (1)

S. N. Bezdid'ko, "The use of Zernike polynomials in optics," Sov. J. Opt. Technol. 41, 425-429 (1974).

Z. Phys. (1)

P. Ehrenfest, "Notes on the approximate validity of quantum mechanics," Z. Phys. 45, 455-457 (1927) (in German).
[CrossRef]

Other (6)

M. Born and E. Wolf, Principles of Optics, pp. 464-466, 767-772, (Cambridge U. Press, 1998).

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, R. Webb, and VSIA Standards Taskforce Members, "Standards for reporting the optical aberrations of eyes," in Vision Science and Its Applications 2000, V.Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232-244.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

V. I. Tatarskii, The Propagation of Waves in the Turbulent Atmosphere (Nauka, Moscow, 1967), pp. 385-390 (in Russian).

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).

F. Merkle, "Adaptive optics," in International Trends in Optics, J.W.Goodman, ed. (Academic, 1991), Chap. 26, pp. 375-390.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Upper row: shape of the generalized aperture functions Ω ( r ) for (left to right): square subpupils of size d, circular subpupils of radius R = d 2 , and Gaussian sampling beams of rms width σ = d 12 1 2 . Lower row, left to right: shape of the directional derivatives Ω ( r ) x of the upper row functions. The brightness of each image has been normalized to 1. The scale bar on the right indicates the normalized value associated with each gray level.

Fig. 2
Fig. 2

Gray level plot of H ( r , r ) for a HS with 69 unvignetted, evenly illuminated square subpupils of width d = 0.1818 (in length units normalized to the pupil radius) with a 100% fill factor distributed in a square array. The value of H ( r , r ) as a function of r for two different estimation points r is represented, one at the geometrical center of the pupil (left) and the other in the upper-right pupil quadrant (right). Their positions are indicated in each case by the brightest spot within the pupil. See other parameters in the text.

Fig. 3
Fig. 3

Gray level plot of H ( r , r ) for the same HS wavefront sensor as in Fig. 2, but with subpupils of 70% size.

Fig. 4
Fig. 4

Gray level plot of H ( r , r ) for a LRT aberrometer with the same square sampling pattern as the previous ones and with a Gaussian rms spot size σ = d 12 1 2 .

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

ρ c ( z ) = ρ c ( 0 ) + z i ( r ) W ( r ) d 2 r ,
m S = Ω S ( r ) W ( r ) d 2 r + ν S ,
m S = [ Ω S ( r ) ] W ( r ) d 2 r + ν S .
Ω s q ( r ) = 1 d 2 rect ( r d ) = { d 2 , if x d 2 and y d 2 0 , otherwise ,
Ω c i r ( r ) = 1 π R 2 circ ( r R ) = { 1 π R 2 , if r R 0 , otherwise ,
Ω G a ( r ) = 1 π σ 2 exp [ r 2 σ 2 ] ,
Ω s q ( r ) = 1 d 2 rect ( y d ) [ δ ( x + d 2 ) δ ( x d 2 ) ] x ̂ + 1 d 2 rect ( x d ) [ δ ( y + d 2 ) δ ( y d 2 ) ] y ̂ ,
Ω c i r ( r ) = 1 π R 2 ( x R ) δ ( r R ) x ̂ + 1 π R 2 ( y R ) δ ( r R ) y ̂ = 1 π R 3 δ ( r R ) r ,
Ω G a ( r ) = 2 π σ 4 exp [ r 2 σ 2 ] r ,
m S = 1 d [ 1 d y S d 2 y S + d 2 W ( x S + d 2 , y ) d y 1 d y S d 2 y S + d 2 W ( x S d 2 , y ) d y ] x ̂ + 1 d [ 1 d x S d 2 x S + d 2 W ( x , y S + d 2 ) d x 1 d x S d 2 x S + d 2 W ( x , y S d 2 ) d x ] y ̂ + ν S .
m S = [ Ω S ( r ) α ] W ( r ) d 2 r + ν S ,
[ Ω ( r ) ] S = Ω S ( r ) α ,
m = Ω ( r ) W ( r ) d 2 r + ν .
W ( r ) = i = 1 a i Z i ( r ) = Z T ( r ) a ,
W ̂ ( r ) = i = 1 M a ̂ i Z i ( r ) = Z M T ( r ) a ̂ ,
W ̂ ( r ) = H ( r , r ) W ( r ) d 2 r + Z M T ( r ) R ν ,
H ( r , r ) = Z M T ( r ) R Ω ( r ) .
W ̂ ( r ) W ( r ) = [ H ( r , r ) δ ( r r ) ] W ( r ) d 2 r ,
W ̂ ( r ) = H ( r , r ) W ( r ) d 2 r .
B W ̂ ( r 1 , r 2 ) = r r H ( r 1 , r ) H ( r 2 , r ) B W ( r , r ) d 2 r d 2 r + Z M T ( r 1 ) RC ν R T Z M ( r 2 ) ,
D W ̂ ( r 1 , r 2 ) = 1 2 r r [ H ( r 1 , r ) H ( r 2 , r ) ] [ H ( r 1 , r ) H ( r 2 , r ) ] D W ( r , r ) d 2 r d 2 r + [ Z M T ( r 1 ) Z M T ( r 2 ) ] RC ν R T [ Z M ( r 1 ) Z M ( r 2 ) ] ,
ϕ ̂ ( κ ) = A ( r ) W ̂ ( r ) exp { i 2 π κ r } d 2 r ,
ϕ ̂ ( κ ) = G ( κ , κ ) ϕ ( κ ) d 2 κ + Q M T ( κ ) R ν ,
G ( κ , κ ) = r r A ( r ) A ( r ) H ( r , r ) exp { i 2 π [ κ r + κ r ] } d 2 r d 2 r .
Φ ̂ ( κ ) = κ κ G ( κ , κ ) G ( κ , κ ) Φ ( κ , κ ) d 2 κ d 2 κ + Q M T ( κ ) RC ν R T Q M * ( κ ) .

Metrics