OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 26 — Dec. 21, 2009
  • pp: 24269–24281
« Show journal navigation

Direct and inverse discrete Zernike transform

Rafael Navarro, Justo Arines, and Ricardo Rivera  »View Author Affiliations


Optics Express, Vol. 17, Issue 26, pp. 24269-24281 (2009)
http://dx.doi.org/10.1364/OE.17.024269


View Full Text Article

Acrobat PDF (859 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

Despite the importance of the basis of Zernike polynomials, they present several drawbacks in practical applications. This is a reason why different authors prefer to use other alternative basis functions (Fourier, splines, etc). A major problem is that in real applications one works with discrete (sampled) arrays of data rather than with continuous functions, and then Zernike polynomials do not form a complete basis on the sampled circle [5

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

]. As an example, the RMS error is probably the most common metric used to determine the difference between two wavefronts or between a wavefront and some ideal reference. By orthonormality of ZPs, one can compute that RMS error either point by point (wi) or between coefficients (cnm) of the Zernike expansion. However, with a discrete number of samples, one finds that these two metrics provide different results in general:

iwi2/In,mcnm2.
(5)

This is because sampled ZPs are neither complete nor orthogonal, and as a result the discrete expansion is not invertible.

In the discrete case, Eq. (1) can be written as 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−1 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 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-):

C=(ZTZ)1ZTW.
(6)

The tilde means estimated, since the wavefront expansion is approximated. This estimation is optimal under a least squares criterion (minimum RMS error), but may not be exact (due to mode coupling and aliasing [15

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

17

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]

]) and always requires a redundant sampling. Therefore, the estimation of the coefficients of the Zernike expansion is still an open problem, which has attracted the interest of many researchers.

Different studies have focused on alternative basis functions (Fourier, splines, etc [18

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]

].), estimation methods, or sampling patterns (random, optimized designs, etc [17

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

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]

].). In addition to these general problems, wavefront sensing (estimation from slope measurements [20

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

]) is especially interesting in many applications: atmospheric turbulence measurements [21

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]

], optical testing [2

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

], adaptive optics [6

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

], ophthalmology [22

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]

], etc. In wavefront sensing, the coefficients are estimated from wavefront partial derivatives (ignoring constant factors):
C=(DZTDZ)1DZM
(7)
where vector 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 DZ 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 M (and columns of DZ) are 2I (double number of samples).

In summary, the estimation of the coefficients of the Zernike expansion relies on redundancy (oversampling). Leaving wavefront sensing (Eq. (7) apart, and focusing on the more basic problem of inverting Eq. (1), to compute the coefficients of the Zernike expansion, our goal is to find an invertible discrete transform. When invertible, the discrete transform can be expressed as a square matrix (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

For continuous signals, since ZPs form a complete orthonormal basis, we can solve Eq. (1) for the coefficients. Orthonormality implies that we can compute the coefficients as the projections (inner product) of the function W on each basis function:

cnm=0102πW(ρ,θ)Znm(ρ,θ)ρdθdρ.
(8)

In what follows, we study how to construct an orthonormal basis functions for the discrete case, so that we can apply a similar recipe to compute the coefficients, i.e. by simple transposition of the elements of the basis.

In real applications, we will have a discrete (sampled) function Wi=jcjZij (W=ZC in matrix notation), where we merged indexes n, m into a single index j=(n(n+2)+m)/2 (starting with j = 0). For a given maximum order N of the polynomials, the total number of modes will be J=1+N(N+3)/2 . Let us investigate if there exist conditions where we can invert matrix Z and solve for C=Z1W. A necessary condition is 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.

We have computed both the rank and condition number of matrix Z for different sampling patterns and number of samples. Some of these sampling patterns are depicted in Fig. 1
Fig. 1 Examples of sampling patterns with 91 points providing singular (hexagonal and hexapolar, upper panels) and invertible (random and spiral, lower panels) (Z).
. 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 (Rank(Z)<I and Det(Z)=0) as shown in Table 1

Table 1. Rank of matrix Z for different sampling schemes (rows) and number of samples (columns). Square (Sq), Hexagonal (H)

table-icon
View This Table
| View All Tables
. Perhaps, these types of regular patterns are not appropriate to sample Zernike polynomials. In polar coordinates these schemes are not so regular and highly redundant (both in ρ and θ).

Different studies [15

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

17

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

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]

] have shown a strong influence of the sampling pattern on wavefront reconstruction from wavefront slopes. In particular, random patterns show better performance [17

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

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]

]. Perhaps, the subjacent effect has to do with the fact that random patterns are non redundant in 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.

For this purpose, we have implemented three cases with (i) pure random sampling; (ii) small random perturbations of regular sampling grids; and (iii) non-redundant deterministic sampling.

2.1. Random and deterministic sampling patterns

The perturbed regular sampling patterns (ii) were implemented by adding small random Cartesian displacements (εx,εy) to the sampling points of regular grids. These perturbations have a Gaussian distribution with zero mean, and their magnitude is determined by the standard deviation σ. We have performed simulations with perturbations ranging from 10−8 to 10−2 in pupil radius units.

Finally, we designed regular (deterministic) non redundant sampling patterns (iii). Regular sampling patterns are commonly obtained by convolution of the function to be sampled with a Dirac comb. Let us start with the angular coordinate. To sample the interval [0, θmax] with I equispaced samples, the interval will be δθ=θmax/(I1). We could apply a similar sampling to 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 θmax=2πNC. In this way we obtain a rolled 1D pattern, which is a spiral with NC cycles covering a circular area with radiusρmaxθmax. To completely avoid redundancy, we have to be careful with the periodicity of the angular variable, i.e. we need to guarantee that the number of samples per cycle NSPC=2π/δθ is non integer. The difference between polar and spiral patterns is that the former is a purely 2-dimensional whereas the spiral is obtained by rolling a 1D pattern. Despite their different nature both can adequately cover a circular domain. The linear spiral, however, has the problem that the density of samples per unit of area is high at the center and decreases towards the edge. One way to avoid that problem is to use an array of spirals to form an helical pattern [24

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

]. Here, however, the goal was to avoid redundancy, and we implemented a sampling scheme based on a single Fermat or parabolic spiral, where the radial coordinate is proportional to the square root of θ (ρ(θ)=θ/θmax) to ensure that ρ1. With the parabolic spiral we can keep the density of samples nearly constant when the angle is sampled uniformly. In a first approximation, this occurs when the number of cycles is proportional to the square root of the number of samples NcI/π. Usually Nc 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 Nc=int(I/π) or Nc=int(I/π)+0.5; where “int” means nearest integer. In terms of the number of cycles δθ=2πNc/(I1). By definition, the radial coordinate ρ is never repeated, and with the additional condition that the sampling is not periodic in 2π (i.e. the number of samples per cycle is not integer, NSPC=2π/δθ=(I1)/Nci), then we avoid any redundancy in both radial and angular coordinates. The two examples implemented here correspond to orders N = 7 and N = 12 and represent the two possible cases of Nc. In the first case we have J = I = 36; then Nc = 3, δθ=0.5386 radians and 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 Nc = 5, δθ=0.349 but then we will have 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 Nc = 5.5, then δθ=0.384 radians and NSPC = 16.36.

In some applications [25

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

] it may be convenient to avoid the origin, starting the spiral with an initial sample at θ1=δθ/k with 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 ρΙ < 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: 1ρI=1/2(ρIρINSPC); solving for ρI=2/3+1/3ρINSPC; and in terms of Nc: ρI=2/3+1/3(Nc1)/Nc. (In the examples ρI=36 = 0.9388 for I = 36 and

ρI=91 = 0.9682 respectively.) Now, the sampling grid is fully determined by θi=δθ/k+(i1)δθ with i = 1, 2,...I and ρi=ρIθi/θI. Therefore, given a maximum order N of Zernike polynomials, we want as many samples as Zernike modes, I=J=1+N(N+3)/2 ; then assign a number of cycles (first option Nc integer when 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.

Figure 1 shows some of the sampling patterns analyzed for N = 12, I = 91, hexagonal, hexapolar, random and spiral (square and perturbed regular patterns are not included.)

2.2. Completeness of sampled Zernike polynomials

Table 1 shows the rank of 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.

Condition numbers of the order of 10−4 obtained for random and spiral patterns for I = 36 permit us to compute Z−1 with a reasonable accuracy. However, as the number of samples increases, the conditioning gets worse. For example, in the case of 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.

In summary, the completeness of sampled Zernike polynomial basis is strongly dependent on sampling pattern. The above results support the relationship between redundancy, low efficiency of sampling and lack of completeness. Taking into account the symmetry of ZPs where radial and angular parts are separable, polar (or hexapolar) sampling schemes are expected to have the highest redundancy in the 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 102 or higher, the matrix inversion will be instable numerically.

2.3. Orthogonalization

In the Gram-Schmidt orthogonalization process, the initial matrix 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.

]:
Z=QR,
(9)
where Q is the matrix formed with the new orthonormal basis vectors and R=QTZ is an upper triangular matrix passing from the Q to the Z basis. The importance of an orthonormal basis is that it guarantees that Q1=QTso that the inversion is easy and exact (optimum condition number = 1). Now we can express both the 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 QT discrete Zernike transforms DZTs:

DZT(cj)=Wi=jcjQij and DZT1(Wi)=cj=iWiQij.
(10)

3. Modes of the DZT

Let us analyze the resulting orthonormal basis functions; i.e. columns of the 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
Fig. 4 Modes of the DZT (continued).
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.

If we compare the discrete and continuous (bottom row) modes we can see clear differences. Many times we observe change of polarity (sign reversals) of different modes, depending on the sampling pattern and number of samples. Tilt Q11, for instance, shows a sign reversal for random and spiral patterns for the low sampling rate (36), but for 91 samples we obtain the opposite case (no reversals except for the hexagonal one). In general, similarities between discrete and continuous modes increase with the number of samples as expected. The differences tend to increase with the order of polynomials. This is patent for the highest order modes n = 7 in the upper rows.

In conclusion, the continuous Zernike polynomial expansion is not well suited in practical applications, since one has to deal with sampled signals or measurements. For discrete signals it is possible to find the modes (basis vectors), but they are specific for each sampling pattern and sampling rate. If the sampling pattern is complete, then we can obtain an invertible transform with the same number of modes than samples. For incomplete sampling, it is possible to find a subspace and the modes Q of that subspace, but then J < I and the inverse transform is ill-conditioned.

4. Computer simulation and results

We implemented a computer simulation to test these theoretical results. We used an ocular wavefront aberration pattern taken from the data set previously used in another study [28

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]

]. We focused on the case of 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).

First of all, we confirmed the inequality of Eq. (5). Even for the noise-free case and with equal number of modes and samples (91), the difference between the left and right sides of Eq. (5) is about 7.5%-10.7% depending on the sampling pattern. As expected, the direct computation on the sampled wavefront yields lower RMS values (underestimation). The lowest and highest biases were obtained with perturbed hexagonal (7.49%) and spiral (10.66%) samplings respectively.

5. Discussion and conclusions

The discrete Zernike modes do change with the sampling pattern (see Fig. 3
Fig. 3 Modes of the DZT, Qnm(only m ≥ 0 are shown), for different sampling schemes: random (R), perturbed hexagonal (H) and spiral (S). The three upper rows correspond to I = 36 samples and the three lower rows to I = 91. Bottom row represents the continuous (I = ∞) Zernike modes.
), which has physical consequences. For example, the spherical aberration of a standard lens (Z40 bottom row in Fig. 3) is different from that of a segmented mirror (or other optical systems). If one has a mirror with 36 hexagonal facets the spherical aberration looks different: Q40 for H36. The same applies for defocus, astigmatism and the rest of aberration modes. In fact, the aberration modes change both with the sampling type and the sampling rate, especially the highest orders.

Acknowledgements

This work was supported by the Comisión Interministerial de Ciencia y Tecnología, Spain, under Grant FIS2008-00697. R. Rivera acknowledges support by Alban, the European Union Program of High Level Scholarships for Latin America, scholarship Nº E07D402088CL.

References and links

1.

V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, New York, 2007).

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]

4.

R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69(7), 972–977 (1979). [CrossRef]

5.

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

6.

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

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]

10.

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]

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]

15.

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

16.

J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70(1), 28–35 (1980). [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]

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]

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]

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. 65, 304–317 (1960). [CrossRef]

25.

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

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]

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]

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, 3rd ed., D. Malacara, ed. (Wiley, New York, 2007).
  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]
  4. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69(7), 972–977 (1979). [CrossRef]
  5. J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19(9), 1510–1518 (1980). [CrossRef] [PubMed]
  6. R. K. Tyson, Principles of Adaptive Optics (Academic, Boston, 1991).
  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]
  10. 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]
  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]
  15. J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. 71(8), 989–992 (1981). [CrossRef]
  16. J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70(1), 28–35 (1980). [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]
  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]
  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]
  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. 65, 304–317 (1960). [CrossRef]
  25. R. Navarro, “Objective refraction from aberrometry: theory,” J. Biomed. Opt. 14(2), 024021 (2009). [CrossRef] [PubMed]
  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]
  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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 4
 
Fig. 3
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited