## Direct and inverse discrete Zernike transform

Optics Express, Vol. 17, Issue 26, pp. 24269-24281 (2009)

http://dx.doi.org/10.1364/OE.17.024269

Acrobat PDF (859 KB)

### Abstract

An invertible discrete Zernike transform, DZT is proposed and implemented. Three types of non-redundant samplings, random, hybrid (perturbed deterministic) and deterministic (spiral) are shown to provide completeness of the resulting sampled Zernike polynomial expansion. When completeness is guaranteed, then we can obtain an orthonormal basis, and hence the inversion only requires transposition of the matrix formed by the basis vectors (modes). The discrete Zernike modes are given for different sampling patterns and number of samples. The DZT has been implemented showing better performance, numerical stability and robustness than the standard Zernike expansion in numerical simulations. Non-redundant (critical) sampling along with an invertible transformation can be useful in a wide variety of applications.

© 2009 OSA

## 1. Introduction

2. R. Navarro and E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. **24**(14), 951–953 (1999). [CrossRef] [PubMed]

3. R. J. Noll, “Phase estimates from slope–type wave–front sensors,” J. Opt. Soc. Am. **68**(1), 139–140 (1978). [CrossRef]

5. J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. **19**(9), 1510–1518 (1980). [CrossRef] [PubMed]

7. J. Alda and G. D. Boreman, “Zernike-based matrix model of deformable mirrors: optimization of aperture size,” Appl. Opt. **32**, 2431–2438 (1993). [CrossRef] [PubMed]

8. G. D. Love, “Wave-front correction and production of Zernike modes with a liquid-crystal spatial light modulator,” Appl. Opt. **36**(7), 1517–1520 (1997). [CrossRef] [PubMed]

9. C.-J. Kim, “Polynomial fit of interferograms,” Appl. Opt. **21**(24), 4521–4525 (1982). [CrossRef] [PubMed]

11. B. Qi, H. Chen, and N. Dong, “Wavefront fitting of interferograms with Zernike polynomials,” Opt. Eng. **41**(7), 1565–1569 (2002). [CrossRef]

12. J. Nam and J. Rubinstein, “Numerical reconstruction of optical surfaces,” J. Opt. Soc. Am. A **25**(7), 1697–1709 (2008). [CrossRef]

13. J. Schwiegerling, J. Greivenkamp, and J. Miller, “Representation of videokeratoscopic height data with Zernike polynomials,” J. Opt. Soc. Am. A **12**(10), 2105–2113 (1995). [CrossRef]

14. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. **66**(3), 207–211 (1976). [CrossRef]

*δ*is the Kronecker delta function. The radial order

_{m0}*n*is integer positive, and the angular frequency

*m*can only take on values

*-n, -n*+ 2,

*-n + 4, ... n*.

**W**=

**ZC,**where

**Z**is a matrix whose columns are sampled Zernike polynomials and

**W**and

**C**are column vectors corresponding to samples and coefficients of the wavefront respectively. The lack of completeness implies that

**Z**does not exist and hence we cannot solve Eq. (1) to obtain the coefficients. The standard way to overcome this problem is to oversample the wavefront and estimate a number of coefficients

^{−1}*J*lower than the number of samples

*I (J < I)*. The coefficients are then estimated by least squares fit to the data. In matrix notation, this is equivalent to compute the Moore-Penrose pseudoinverse of

**Z**. (

**Z**is not square as it has more rows –samples- than columns –coefficients-):

15. J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. **71**(8), 989–992 (1981). [CrossRef]

17. O. Soloviev and G. Vdovin, “Hartmann-Shack test with random masks for modal wavefront reconstruction,” Opt. Express **13**(23), 9570–9584 (2005). [CrossRef] [PubMed]

18. M. Ares and S. Royo, “Comparison of cubic B-spline and Zernike-fitting techniques in complex wavefront reconstruction,” Appl. Opt. **45**(27), 6954–6964 (2006). [CrossRef] [PubMed]

17. O. Soloviev and G. Vdovin, “Hartmann-Shack test with random masks for modal wavefront reconstruction,” Opt. Express **13**(23), 9570–9584 (2005). [CrossRef] [PubMed]

19. L. Diaz-Santana, G. Walker, and S. X. Bará, “Sampling geometries for ocular aberrometry: A model for evaluation of performance,” Opt. Express **13**(22), 8801–8818 (2005). [CrossRef] [PubMed]

20. W. H. Southwell, “Wave–front estimation from wave–front slope measurements,” J. Opt. Soc. Am. **70**(8), 998–1006 (1980). [CrossRef]

21. E. E. Silbaugh, B. M. Welsh, and M. C. Roggemann, “Characterization of Atmospheric Turbulence Phase Statics Using Wave-Front Slope Measurements,” J. Opt. Soc. Am. A **13**(12), 2453–2460 (1996). [CrossRef]

2. R. Navarro and E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. **24**(14), 951–953 (1999). [CrossRef] [PubMed]

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

**M**contains the measurements (displacements along X and along Y of spots at the image of the sensor, which are proportional to wavefront slopes) and

**D**is the matrix formed by the partial derivatives of the ZPs with respect to X and Y. Then, we have a matrix form similar to Eq. (6), but with the additional problem that the derivatives of ZPs do not form a basis even for continuous signals. This is somehow compensated by the fact that the dimensions of

_{Z}**M**(and columns of

**D**) are 2

_{Z}*I*(double number of samples).

*I*=

*J*), and that matrix must have an inverse. Furthermore, if the basis vectors (matrix columns) are orthonormal, then the inverse transform (matrix) is equal to its transpose, which ensures numerical stability. For this purpose, we will depart from the Zernike polynomial basis on a continuous circle, and study the problem of completeness in the discrete case; in particular, the role of the sampling pattern. Then we find that determined types of non redundant sampling patterns do ensure completeness. Here we introduce the spiral pattern and compare it with random sampling grids. Once the problem of completeness is overcome, then it is straightforward to apply the Gram-Schmidt method to obtain an orthonormal basis over the sampled circular pupil. Then, we compute the discrete Zernike modes for different sampling patterns and number of samples. Finally, we implement the direct and inverse DZT (discrete Zernike transform) and evaluate its performance with some examples.

## 2. Discrete Zernike transform

*W*on each basis function:

*n, m*into a single index

*j*= 0). For a given maximum order N of the polynomials, the total number of modes will be

**Z**and solve for

**Z**to be square, so that the number of coefficients equals the number of samples

*J = I*. In what follows we will refer to this case as critical sampling.

**Z**for different sampling patterns and number of samples. Some of these sampling patterns are depicted in Fig. 1 . Square and hexagonal patterns are typical in Hartmann-Shack wavefront sensors, whereas hexapolar patterns are often used in optical design. All these sampling patterns yield singular

**Z**matrices (

*ρ*and

*θ*).

15. J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. **71**(8), 989–992 (1981). [CrossRef]

17. O. Soloviev and G. Vdovin, “Hartmann-Shack test with random masks for modal wavefront reconstruction,” Opt. Express **13**(23), 9570–9584 (2005). [CrossRef] [PubMed]

19. L. Diaz-Santana, G. Walker, and S. X. Bará, “Sampling geometries for ocular aberrometry: A model for evaluation of performance,” Opt. Express **13**(22), 8801–8818 (2005). [CrossRef] [PubMed]

**13**(23), 9570–9584 (2005). [CrossRef] [PubMed]

19. L. Diaz-Santana, G. Walker, and S. X. Bará, “Sampling geometries for ocular aberrometry: A model for evaluation of performance,” Opt. Express **13**(22), 8801–8818 (2005). [CrossRef] [PubMed]

*ρ*and

*θ*(i.e. the values of coordinates are never repeated) which may help to avoid

**Z**to be singular. Since all Zernike polynomials are different from 0, singularity of

**Z**must come from redundancy between sampled basis functions. Then, let us suppose that such redundancy in the

**Z**matrix has to do with redundancy (repetition) of values of coordinates at the sampling points, and verify this idea through numerical simulation.

### 2.1. Random and deterministic sampling patterns

**Z**matrix obtained for each of them, in order to choose the best realization.

^{−8}to 10

^{−2}in pupil radius units.

*I*equispaced samples, the interval will be

*ρ*. If the comb is 2D (2-dimensional) we obtain a pure polar sampling, which is redundant in both coordinates. A way to avoid redundancy is to apply 1D Dirac combs to both coordinates; or in other words to make

*ρ*proportional to

*θ*and set

*N*cycles covering a circular area with radius

_{C}24. N. U. Mayall and S. Vasilevskis, “Quantitative tests of the Lick Observatory 120-Inch mirror,” Astron. J. **65**, 304–317 (1960). [CrossRef]

*θ*(

*N*is chosen to be integer, but in some cases this could result in a redundant sampling. If that happens (see below) we add 1/2 cycle to break periodicity: Thus, we have two cases

_{c}*ρ*is never repeated, and with the additional condition that the sampling is not periodic in

*N*= 7 and

*N*= 12 and represent the two possible cases of

*N*. In the first case we have

_{c}*J*=

*I*= 36; then

*N*= 3,

_{c}*NSPC*= 11.667. Since this is not an integer number, the sampling is non redundant. In the second example,

*N*= 12 and

*I*=

*J*= 91. If we choose an integer value

*N*= 5,

_{c}*NSPC*= 18 and the sampling would be periodic in

*θ*; i.e. redundant. We can avoid that redundancy by adding 0.5 cycles so that

*N*= 5.5, then

_{c}*NSPC*= 16.36.

25. R. Navarro, “Objective refraction from aberrometry: theory,” J. Biomed. Opt. **14**(2), 024021 (2009). [CrossRef] [PubMed]

*k*>>1 (in particular for

*k*= ∞ the initial point will be at the origin). Finally, the last sample of the spiral has to strictly meet the condition

*ρ*< 1 to avoid partial occlusion of the marginal samples by the pupil. One possible criterion is to keep the area covered by this last sample equal to the average. As an approximation, here we impose the radial distance of the last sample to the pupil edge to be equal to half the width of the last cycle:

_{Ι}*N*

_{c}:*I*= 36 and

*i = 1, 2,...I*and

*N*of Zernike polynomials, we want as many samples as Zernike modes,

*N*integer when

_{c}*NSPC*is non integer; or add 0.5 to avoid periodicity if

*NSPC*integer). Finally choose a value for

*k*to have the spiral sampling completely determined.

*N*= 12,

*I*= 91, hexagonal, hexapolar, random and spiral (square and perturbed regular patterns are not included.)

### 2.2. Completeness of sampled Zernike polynomials

**Z**obtained for the different sampling patterns analyzed and for different number of samples. First of all, only random and spiral patterns permit to set an arbitrary number of samples which provides total flexibility to match the number of samples to any (maximum) order

*N*of Zernike polynomials. This is the reason why some rows in Table 1 are incomplete. The 2D regular patterns considered here are centered at the origin (i.e. they include the central sample) and they can only match determined orders, except for the case N = 7 (I = 36), where we had to remove the central sample, otherwise we had 37 samples. The Table shows that random spiral and perturbed patterns (except for perturbed hexapolar) provide maximum rank (completeness), whereas regular 2D patterns yield lower ranks. Among them, square and hexagonal seem equivalent, but the hexapolar shows the lowest value for 36 samples.

*I*= 36, regular 2D patterns give values between 0 (hexapolar) and 10

^{−18}(square and hexagonal) and much higher (3x10

^{−4}) for random and spiral patterns. The effect of random perturbations on matrix conditioning is shown in Fig. 2 . For hexagonal and square patterns, conditioning improves with the magnitude of perturbation, reaching values similar to random and spiral patterns for perturbations between 10

^{−3}and 10

^{−2}. The hexapolar sampling improves but never reaches acceptable values, which suggests that it is not suitable for modal analysis. The spiral is basically unaffected, except for large perturbations, for which the pattern becomes nearly random. We want to remark that perturbations with standard deviations above 10

^{−5}seem enough to obtain conditioning levels (10

^{−7}) enough in theory to invert

**Z**. These levels of perturbation are 4 orders of magnitude lower than sample coordinates, which are negligible in practice.

^{−4}obtained for random and spiral patterns for

*I*= 36 permit us to compute

**Z**with a reasonable accuracy. However, as the number of samples increases, the conditioning gets worse. For example, in the case of

^{−1}*I*= 91 samples, spiral and random patterns yield condition numbers of the order of 10

^{−8}. This means that numerical instability will increase and accuracy in the inversion of

**Z**will decrease with

*I*.

**Z**matrix, which is confirmed by the lower values both in rank and condition number of

**Z**. Non-polar sampling (square, hexagonal) has an intermediate level of redundancy, which can be improved by introducing small perturbations to the regular sampling grid. On the other hand either fully random or spiral patterns seem to guarantee completeness. The later has the advantage of being deterministic and regular. Nevertheless, completeness does not ensure an accurate inversion in practice. In fact for real applications where

*I*is of the order or 10

^{2}or higher, the matrix inversion will be instable numerically.

### 2.3. Orthogonalization

*I*=

*J*). However, the condition number

**Z**is low so that the estimation of

**Z**could be instable numerically and noisy, getting worse as the number of samples increases. Only a complete and orthogonal basis would warrant an adequate modal representation of the signal. The orthogonalization of sampled ZPs was proposed and successfully implemented before as an alternative wavefront estimation method [26

^{−1}26. D. Malacara-Hernandez, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. **29**(6), 672–675 (1990). [CrossRef]

27. D. J. Fischer, J. T. O'Bryan, R. Lopez, and H. P. Stahl, “Vector formulation for interferogram surface fitting,” Appl. Opt. **32**(25), 4738–4743 (1993). [CrossRef] [PubMed]

**Z**was lower than the number of samples, then the resulting number of orthogonal basis functions was lower than the number of samples; hence the resulting DZT transform was not complete and the initial samples cannot be exactly recovered from the coefficients. Here, we apply the same Gram-Schmidt orthogonalization procedure, but the difference is that now we first ensure that

*J*is the number of orthogonal basis functions; i.e. discrete Zernike models.

**Z**is decomposed into a product [23

23. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK User's Guide, 3rd Ed., (SIAM, Philadelphia, 1999), http://www.netlib.org/lapack/lug/lapack_lug.html.

**Q**is the matrix formed with the new orthonormal basis vectors and

**Q**to the

**Z**basis. The importance of an orthonormal basis is that it guarantees that

**Q**direct and inverse transform (the Discrete Zernike Transform). Nevertheless, we want to remark that the DZT will depend not only on the number of samples

*I*, but also on the sampling scheme. For each sampling scheme

*s*we will have a different

**Z**matrix and hence a different basis change operator

**R**and sampling-distinctive direct

**Q**and inverse

**Q**discrete Zernike transforms

^{T}*DZTs*:

## 3. Modes of the DZT

**Q**matrix. Zernike modes are highly significant in optics since each mode corresponds to a type of aberration: piston (

*n*= 0,

*m*= 0), tilt (

*n*= 1,

*m*= ± 1), defocus (

*n*= 2,

*m*= 0), and so on. Each mode corresponds to a Zernike polynomial defined on a continuous circle of unit radius. Sampled polynomials do not form an orthogonal basis anymore, but if we apply a complete critical sampling scheme

*s*, we can find a new orthonormal basis. The modes or basis functions for that particular sampling are the columns of matrix

**Q**. These are linear combinations of the sampled Zernike polynomials, but these linear combinations may differ substantially from the originals, mainly for the higher orders. Figures 3and 4 compare the modes of the orthonormal DZT for random, R, perturbed hexagonal (with perturbation σ = 10

^{−3}) H and spiral, S, sampling patterns. The three upper rows correspond to 36 (n ≤ 7) samples, and the lower rows to 91 (n ≤ 12) samples. The bottom row (∞ number of samples) shows the continuous Zernike polynomials. (For the case H36 the central sample was removed, otherwise we would have 37 sampling points). Only modes with non-negative angular frequency (m ≥ 0) are shown up to radial order n = 7.

*n*= 7 in the upper rows.

## 4. Computer simulation and results

28. J. Arines, E. Pailos, P. Prado, and S. Bará, “The contribution of the fixational eye movements to the variability of the measured ocular aberration,” Ophthalmic Physiol. Opt. **29**(3), 281–287 (2009). [CrossRef] [PubMed]

*I*= 91 samples and tested for the different patterns. Different conditions were implemented with 91 and 182 wavefront modes (non cero coefficients), both with different levels of noise (0. 1%, 3% and 5%) added to the samples. The metric used was always RMS values (Eq. (5) or RMS differences (errors).

^{T}) (Eq. (10) to estimate the continuous

*I*>>

*J*. Using the DZT (and the complete schemes R and S), the results are greatly improved (the errors are of the order of 10

^{−14}μm). Note that now we applied matrix R to the continuous original coefficients to compute the RMS error in the discrete Q basis. Using the DZT the error also increases with the presence of higher order modes and noise, but improves by one (182 modes, central columns) or three (noise, right columns) orders of magnitude compared to the standard method. If we reconstruct the wavefront from the estimated coefficients, we observe that the standard method is affected by both under sampling and noise, but the DZT, Q transform is unaffected and the initial measurements are recovered with high fidelity.

## 5. Discussion and conclusions

*I*≥ 2

*J*). In applications such as wavefront control by deformable mirrors this has an important cost, because mirrors use to have a limited number of actuators. Roughly speaking we could consider an actuator as a sample

*I*. On the other hand one should be able to control the highest number of modes

*J*for optimal performance. Completeness and orthogonality guarantee the possibility to reach that maximum,

*J*=

*I*. The Q transform implies to work in a discrete domain, and matrix

**R**changes from the continuous to the discrete domain. However, the inverse transformation

**R**from the discrete to the continuum may have a bad condition number in general. If we apply a complete sampling, then we can compute

^{−1}**Z**, which implies that

^{−1}**R**also exists. Nevertheless, condition numbers get worse with increasing number of samples (from 10

^{−1}**to 10**

^{−4}**when passing from 36 to 91 samples), and this might affect numerical accuracy when trying to extrapolate from the discrete to the continuous wavefront. However there might be applications were it would be possible to work in the discrete plane, as for example wavefront modulation with discrete spatial light modulators, deformable mirrors, or modal iterative algorithms among others. In these cases the bad conditioning of**

^{−8}**R**would not spoil the discrete coefficients.

^{−1}## Acknowledgements

## References and links

1. | V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in |

2. | R. Navarro and E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. |

3. | R. J. Noll, “Phase estimates from slope–type wave–front sensors,” J. Opt. Soc. Am. |

4. | R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. |

5. | J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. |

6. | R. K. Tyson, |

7. | J. Alda and G. D. Boreman, “Zernike-based matrix model of deformable mirrors: optimization of aperture size,” Appl. Opt. |

8. | G. D. Love, “Wave-front correction and production of Zernike modes with a liquid-crystal spatial light modulator,” Appl. Opt. |

9. | C.-J. Kim, “Polynomial fit of interferograms,” Appl. Opt. |

10. | H. van Brug, “Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry,” Appl. Opt. |

11. | B. Qi, H. Chen, and N. Dong, “Wavefront fitting of interferograms with Zernike polynomials,” Opt. Eng. |

12. | J. Nam and J. Rubinstein, “Numerical reconstruction of optical surfaces,” J. Opt. Soc. Am. A |

13. | J. Schwiegerling, J. Greivenkamp, and J. Miller, “Representation of videokeratoscopic height data with Zernike polynomials,” J. Opt. Soc. Am. A |

14. | R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. |

15. | J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. |

16. | J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. |

17. | O. Soloviev and G. Vdovin, “Hartmann-Shack test with random masks for modal wavefront reconstruction,” Opt. Express |

18. | M. Ares and S. Royo, “Comparison of cubic B-spline and Zernike-fitting techniques in complex wavefront reconstruction,” Appl. Opt. |

19. | L. Diaz-Santana, G. Walker, and S. X. Bará, “Sampling geometries for ocular aberrometry: A model for evaluation of performance,” Opt. Express |

20. | W. H. Southwell, “Wave–front estimation from wave–front slope measurements,” J. Opt. Soc. Am. |

21. | E. E. Silbaugh, B. M. Welsh, and M. C. Roggemann, “Characterization of Atmospheric Turbulence Phase Statics Using Wave-Front Slope Measurements,” J. Opt. Soc. Am. A |

22. | J. Liang, B. Grimm, S. Goelz, and J. F. Bille, “Objective measurement of wave aberrations of the human eye with use of a Hartmann-Shack wave-front sensor,” J. Opt. Soc. Am. A |

23. | E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK User's Guide, 3rd Ed., (SIAM, Philadelphia, 1999), http://www.netlib.org/lapack/lug/lapack_lug.html. |

24. | N. U. Mayall and S. Vasilevskis, “Quantitative tests of the Lick Observatory 120-Inch mirror,” Astron. J. |

25. | R. Navarro, “Objective refraction from aberrometry: theory,” J. Biomed. Opt. |

26. | D. Malacara-Hernandez, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. |

27. | D. J. Fischer, J. T. O'Bryan, R. Lopez, and H. P. Stahl, “Vector formulation for interferogram surface fitting,” Appl. Opt. |

28. | J. Arines, E. Pailos, P. Prado, and S. Bará, “The contribution of the fixational eye movements to the variability of the measured ocular aberration,” Ophthalmic Physiol. Opt. |

**OCIS Codes**

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

(220.1010) Optical design and fabrication : Aberrations (global)

(080.1005) Geometric optics : Aberration expansions

(110.7348) Imaging systems : Wavefront encoding

**History**

Original Manuscript: July 9, 2009

Revised Manuscript: November 20, 2009

Manuscript Accepted: December 4, 2009

Published: December 18, 2009

**Citation**

Rafael Navarro, Justo Arines, and Ricardo Rivera, "Direct and inverse discrete Zernike transform," Opt. Express **17**, 24269-24281 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-24269

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### References

- V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, New York, 2007).
- R. Navarro and E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. 24(14), 951–953 (1999). [CrossRef] [PubMed]
- R. J. Noll, “Phase estimates from slope–type wave–front sensors,” J. Opt. Soc. Am. 68(1), 139–140 (1978). [CrossRef]
- R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69(7), 972–977 (1979). [CrossRef]
- J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19(9), 1510–1518 (1980). [CrossRef] [PubMed]
- R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, 1991).
- J. Alda and G. D. Boreman, “Zernike-based matrix model of deformable mirrors: optimization of aperture size,” Appl. Opt. 32, 2431–2438 (1993). [CrossRef] [PubMed]
- G. D. Love, “Wave-front correction and production of Zernike modes with a liquid-crystal spatial light modulator,” Appl. Opt. 36(7), 1517–1520 (1997). [CrossRef] [PubMed]
- C.-J. Kim, “Polynomial fit of interferograms,” Appl. Opt. 21(24), 4521–4525 (1982). [CrossRef] [PubMed]
- H. van Brug, “Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry,” Appl. Opt. 36(13), 2788–2790 (1997). [CrossRef] [PubMed]
- B. Qi, H. Chen, and N. Dong, “Wavefront fitting of interferograms with Zernike polynomials,” Opt. Eng. 41(7), 1565–1569 (2002). [CrossRef]
- J. Nam and J. Rubinstein, “Numerical reconstruction of optical surfaces,” J. Opt. Soc. Am. A 25(7), 1697–1709 (2008). [CrossRef]
- J. Schwiegerling, J. Greivenkamp, and J. Miller, “Representation of videokeratoscopic height data with Zernike polynomials,” J. Opt. Soc. Am. A 12(10), 2105–2113 (1995). [CrossRef]
- R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). [CrossRef]
- J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. 71(8), 989–992 (1981). [CrossRef]
- J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70(1), 28–35 (1980). [CrossRef]
- O. Soloviev and G. Vdovin, “Hartmann-Shack test with random masks for modal wavefront reconstruction,” Opt. Express 13(23), 9570–9584 (2005). [CrossRef] [PubMed]
- M. Ares and S. Royo, “Comparison of cubic B-spline and Zernike-fitting techniques in complex wavefront reconstruction,” Appl. Opt. 45(27), 6954–6964 (2006). [CrossRef] [PubMed]
- L. Diaz-Santana, G. Walker, and S. X. Bará, “Sampling geometries for ocular aberrometry: A model for evaluation of performance,” Opt. Express 13(22), 8801–8818 (2005). [CrossRef] [PubMed]
- W. H. Southwell, “Wave–front estimation from wave–front slope measurements,” J. Opt. Soc. Am. 70(8), 998–1006 (1980). [CrossRef]
- E. E. Silbaugh, B. M. Welsh, and M. C. Roggemann, “Characterization of Atmospheric Turbulence Phase Statics Using Wave-Front Slope Measurements,” J. Opt. Soc. Am. A 13(12), 2453–2460 (1996). [CrossRef]
- J. Liang, B. Grimm, S. Goelz, and J. F. Bille, “Objective measurement of wave aberrations of the human eye with use of a Hartmann-Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994). [CrossRef]
- E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK User's Guide, 3rd Ed., (SIAM, Philadelphia, 1999), http://www.netlib.org/lapack/lug/lapack_lug.html .
- N. U. Mayall and S. Vasilevskis, “Quantitative tests of the Lick Observatory 120-Inch mirror,” Astron. J. 65, 304–317 (1960). [CrossRef]
- R. Navarro, “Objective refraction from aberrometry: theory,” J. Biomed. Opt. 14(2), 024021 (2009). [CrossRef] [PubMed]
- D. Malacara-Hernandez, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990). [CrossRef]
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