## Wave front sensor-less adaptive optics: a model-based approach using sphere packings

Optics Express, Vol. 14, Issue 4, pp. 1339-1352 (2006)

http://dx.doi.org/10.1364/OE.14.001339

Acrobat PDF (236 KB)

### Abstract

Certain adaptive optics systems do not employ a wave front sensor but rather maximise a photodetector signal by appropriate control of an adaptive element. The maximisation procedure must be optimised if the system is to work efficiently. Such optimisation is often implemented empirically, but further insight can be obtained by using an appropriate mathematical model. In many practical systems aberrations can be accurately represented by a small number of modes of an orthogonal basis, such as the Zernike polynomials. By heuristic reasoning we develop a model for the operation of such systems and demonstrate a link with the geometrical problems of sphere packings and coverings. This approach aids the optimisation of control algorithms and is illustrated by application to direct search and hill climbing algorithms. We develop an efficient scheme using a direct maximisation calculation that permits the measurement of *N* Zernike modes with only *N* +1 intensity measurements.

© 2006 Optical Society of America

## 1. Introduction

*a priori*knowledge of the optimised function. Rather than provide prescriptions for specific optical applications, we concentrate on a general, mathematically constructive approach that may elucidate algorithm design. The paper is structured as follows. Firstly, we describe the mathematical model of the optical system based upon an aberration expansion in terms of Zernike polynomials and we derive a function describing the resulting photodetector signal. This function is shown to have a well defined maximum and spherical symmetry, which can be used to our advantage in algorithm design; in particular, this aids the choice of the aberrations that are sequentially applied by the adaptive element. In Section 4, we use the spherical topology to show how lattice sphere packings assist the choice of candidate solutions to ensure the most efficient coverage of the search space. These optimum lattices are used in Section 5 as the basis for global, exhaustive search algorithms. This is extended in the subsequent section to a more efficient, multi-level exhaustive search. In Section 7, we apply a similar approach to a steepest ascent hill-climbing algorithm, a local search method where a set of candidates is chosen to optimally search the immediate neighbourhood at each stage. By using local properties of the function and following the direction of steepest gradient, this method converges more rapidly than the exhaustive search. In Section 8, we describe a direct maximisation algorithm that has much better convergence properties than the search algorithms. Rather than performing a search through different candidates either globally or locally, this algorithm takes into account the global topology of the function. This enables direct calculation of the position of the maximum using the smallest possible number of function evaluations or, equivalently, the fewest intensity measurements.

## 2. Mathematical model

*r*,

*θ*), where

*r*and

*θ*are polar coordinates in the pupil plane of the lens. The coordinates are normalised such that the pupil has a radius of 1. In the pupil plane of the lens is a phase mask, which could in practice be an adaptive element, that subtracts a phase function Ψ(

*r*,

*θ*) from the input wave front. Fourier diffraction theory[12] shows that the signal measured by the photodetector is given by

*N*Zernike polynomials[13], each denoted by

*Z*

_{n}(

*r*,

*θ*):

*N*element vector

**a**, whose elements are the Zernike mode coefficients

*a*

_{n}. Similarly, the correction Ψ introduced by the adaptive element is represented by the vector

**b**, whose elements are the coefficients

*b*

_{n}. We then define

**c**=

**a**-

**b**and

*c*

_{n}is the coefficient of the

*n*th mode. The function

*f*(

**c**) is independent of the overall intensity and is equivalent to the Strehl ratio [13]. Due to the orthogonality of the Zernike modes, for small ∣

**c**∣ we find that

**c**∣

^{2}also represents the variance of the corrected wave front [13]. The system is considered to be well corrected if ∣

**c**∣ <

*ε*, where

*ε*

^{2}is a small quantity equal to the maximum acceptable wave front variance. Equation (6) provides the equivalent requirement that

*f*(

**c**) > 1-

*ε*

^{2}. From Eq. (6) we see that

*f*(

**c**) is isotropic for small arguments - a property that is illustrated further in Fig. 2. It follows that the contours of constant

*f*(

**c**) are

*N*-dimensional spheres centred on the origin. This is a different expression of the well known result that the Strehl ratio depends only on the variance of the aberration and not its form[13]. It is also important to note that for large arguments

*f*(

**c**) becomes small. This is a consequence of the rapid variation of the exponential term in the integral of Eq. (5). The global maximum at

**c**=

**0**is therefore much larger than any of the surrounding local maxima. These properties of

*f*(

**c**) can be used to our advantage when designing maximisation algorithms.

## 3. Strategy for algorithm design

*a priori*mathematical knowledge of the problem. The first step in the design of an optimisation algorithm is the specification of the problem. This can be broken down into three components: 1) The objective - the purpose to be fulfilled by the algorithm; 2) The evaluation (or merit) function - a measure indicating the quality of a candidate solution; 3) The representation - a model of the problem that specifies the alternative candidate solutions [14]. We define these components in a general manner that is not specific to a particular algorithm.

**b**that gives the maximum photodetector signal or

*f*(

**c**) = 1. Usually this would be relaxed so that we instead search for a solution

**b**that is arbitrarily near to the maximum, such that

*f*(

**c**) > 1-

*ε*

^{2}. We note that the objective will be fulfilled when ∣

**a**-

**b**∣ <

*ε*. This is equivalent to specifying that

**a**lies within an

*N*-dimensional sphere of radius

*ε*centred on the candidate solution

**b**. The obvious choice for the evaluation function is the photodetector signal, since this has a well behaved maximum at the correct solution

**b**=

**a**.

**b**=

**a**lies within a finite region ∑ of

**R**

^{N}, where

**R**denotes the field of real numbers. In a practical system using a global search method, the solution space ∑ might be determined by the adaptive element’s capabilities – which aberrations it can correct – or the range of input aberrations. When using a local search, ∑ might encompass the candidate solutions ‘near’ to the present estimate. We can define the representation to consist of a set

*B*of candidate solutions, each represented by a vector

**b**, that is a finite subset of the points in ∑.

*B*of candidate solutions to provide the best efficiency. To guarantee finding the correct solution, we need to ensure that every point in ∑ lies within a distance e of at least one of the candidate solutions in

*B*. This would ensure that ∑ is ‘covered’ by the overlapping

*N*-dimensional spheres centred on the candidate solutions. To obtain the best efficiency, we must choose

*B*to have the fewest candidate solutions whilst still covering ∑. It follows that finding the optimum representation is equivalent to finding the optimum covering of ∑ with overlapping

*N*-dimensional spheres. The search for such sphere packings is a mathematical problem that has been extensively investigated [15]. It is important to note that only in the best case would a random set

*B*provide the same efficiency as the optimum sphere packing.

## 4. Sphere coverings and the representation

**R**

^{N}with equal, overlapping

*N*-dimensional spheres (hereafter referred to simply as ‘spheres’). The efficiency of a covering is quantified in terms of its ‘thickness’, which is equivalent to the average number of spheres that cover a point of the space. The covering problem therefore asks for the arrangement of spheres that has the minimal thickness, otherwise termed the thinnest covering. This is non-trivial and has been the subject of lengthy mathematical investigation. Optimal coverings of

**R**

^{N}, which include both lattice or non-lattice arrangements of spheres, have only been proven for

*N*≤ 3 and optimal lattice coverings have only been found for

*N*≤ 5 [15, 16

16. T. C. Hales, “An overview of the Kepler conjecture,” http://xxx.lanl.gov/ math.MG/9811071 (1999).

*N*≤ 24, although they are not necessarily proven to be optimal [15].

*N*= 1, when only a single Zernike mode is present. The region bounded by a 1-dimensional sphere of radius e is simply a line segment of length 2

*ε*. The thinnest covering therefore consists of spheres arranged in a line with their centres spaced by 2

*ε*; thus the candidate solutions in

*B*should consist of integer multiples of 2

*ε*. Since the spheres do not overlap, this is a perfect covering with a thickness of 1. One can think of this process as dividing up the

*b*

_{1}-axis into sections of width 2

*ε*in order to find the section containing the maximum intensity. It is obvious that the maximum error would be

*ε*.

*N*= 2, it is tempting to choose each element of a candidate solution

**b**to be an integer multiple of 2

*ε*. The vectors in

*B*would then point to the vertices of a regular square grid, a scaled version of the integer lattice (usually referred to as

**Z**

_{N}) [15]. This is illustrated in Fig. 3(a). As discussed in Section 3, the correct solution will only be found if it lies within a sphere of radius

*ε*centred on a candidate solution

**b**. It can be seen that significant portions of the plane, and hence potential solutions, do not lie within one of the spheres. Indeed, only 79% of the plane is covered and therefore the representation is incomplete. We could overcome this by reducing the spacing of the lattice to

**a**lies within at least one sphere and the representation is complete. However, this scaled integer lattice (with thickness 1.57) does not provide the thinnest possible covering, which is instead given by the hexagonal lattice shown in Fig. 3(c) [17

17. R. Kershner, “The number of circles covering a set,” Am. J. Math. **61**665–671 (1939). [CrossRef]

_{1}of the incomplete integer lattice in

*N*dimensions is equivalent the ratio between the volume of a sphere and the cube that it inscribes. The volume of a unit radius sphere is given by [15]

^{N}, the thickness follows as

_{2}is now given by the ratio of the volume of a sphere and its circumscribed cube, which has volume (2/

^{N}. Hence,

*N*≤ 23, the best known coverings are provided by the lattice known as

*N*. The difference between the thickness of the best integer lattice covering and the optimal known lattice covering increases markedly with

*N*. Indeed, for

*N*= 10 their ratio is approximately 50. With the incomplete lattice covering, it is clear that as

*N*increases, the proportion of the possible solutions that are covered decreases dramatically. This trend continues for larger

*N*, as can be seen from the asymptotic behaviours of these thicknesses, given by log(Θ

_{1}) ~ -

*N*log(

*N*), log(Θ

_{2}) ~

*N*and log(Θ

_{3}) ~

*N*.

*ε*. We note that other lattices, known as quantisers, have been found that would alternatively minimise the mean square error of the measurement [15].

## 5. Exhaustive search using sphere coverings

**Z**

_{N}(with appropriate spacing to ensure coverage) and the lattice

*f*(

**c**) > 0.9. The set

*B*of candidate solutions consisted of the centres of the spheres required to cover ∑; this included the lattice points within ∑ and an extra ‘layer’ of spheres necessary to complete the covering near the periphery. We therefore used lattice points within a radius ρ +

*ε*of the origin; these were found using an appropriate search algorithm [18

18. E. Vitterbo and J. Boutros, “A universal lattice code decoder for fading channels,” IEEE Trans. Inf. Theory **45**1639–1642 (1999). [CrossRef]

*K*, for

*N*≤ 6 are shown in Fig. 5 (for the calculations throughout this paper we take

*N*= 1 to be Zernike mode

*i*= 4,

*N*= 2 to include modes

*i*= 4 and

*i*= 5,

*N*= 3 to include modes

*i*= 4 to

*i*= 6, etc.). As expected, the lattice

**Z**

_{N}and in both cases

*K*increased exponentially with

*N*.

## 6. Exhaustive search with branch and bound

*ε*, we performed a coarse search using spheres of larger radius before performing progressively more local searches. The total number of evaluations was therefore reduced whilst still obtaining the same final accuracy. It can be seen from Fig. 2 that the variation in

*f*(

**c**) is small for ∣

**c**∣ ≈ 1 so a search algorithm based upon spheres of this radius or smaller would still be appropriate. Again we chose ∑ to be a spherical region of radius ρ = 1.07 centred on the origin and we took

^{3}

*ε*and was chosen to permit a convenient three level search). For the first, coarsest search we chose to cover ∑ with spheres of radius 1.5

^{2}

*ε*= 0.71 and we proceeded to find the candidate with the maximum intensity. For the second search, the sphere centred on this candidate was covered with smaller spheres of radius 1.5

*ε*= 0.47. The final search used spheres of radius

*ε*= 0.32, yielding a result with the same accuracy as the simple exhaustive search algorithm. Since the ratio of the search space radius and the covering sphere radius was the same at each step, the same number of evaluations was performed at each of the three levels. Such a three level search, based upon

*N*≤ 6. For each

*N*, one thousand random solutions were tested by evaluating

*f*(

**c**) using Eq. 5. In each case, the solution was found within the required tolerance of

*ε*. The total numbers of evaluations,

*K*, for each search are shown in Fig. 5. For

*N*> 2, the branch and bound method was significantly more efficient than a simple exhaustive search.

## 7. Steepest ascent hill climbing

*ε*centred on the present solution. This surface needs to be covered by the spheres centred on the adjacent candidate solutions. Just as the intersection of two spheres forms a circle, the intersection of two

*N*-spheres forms an (

*N*-1)-sphere. The problem of covering this neighbourhood is therefore equivalent to the problem of covering the surface of an

*N*-sphere with an arrangement of (

*N*-1)-spheres. The minimum number of neighbouring candidates that would be required whilst still spanning

*N*dimensions is

*N*+1 and, if regularly spaced, they would be positioned at the vertices of an

*N*-dimensional regular simplex (i.e. an equilateral triangle for

*N*= 2, a regular tetrahedron for

*N*= 3, etc.). The vectors representing the simplex vertices are derived in Appendix B. This arrangement covers the neighbourhood only if the distance,

*s*, from the present solution to the candidates satisfies 0 <

*s*= 2

*ε*/

*N*(see Appendix C). The efficiency of such a SAHC algorithm is illustrated in Fig. 6 for

*s*= 2

*ε*/

*N*and

*N*≤ 6. In each trial a randomly oriented vector of magnitude 1.5 was used as the initial candidate. For each data point,

*K*was taken as the mean number of evaluated candidates from one hundred trials. Since the step size

*s*∝ 1/

*N*and the number of evaluations per step varies as

*N*, we expect

*K*to vary as

*N*

^{2}; this quadratic dependence is confirmed by Fig. 6. When

*s*was increased beyond 2

*ε*/

*N*, it was observed that the algorithm failed to find the solution using some initial candidates.

*f*(

**c**), we were able to use fixed step sizes to achieve convergence to the desired precision.

## 8. Direct maximisation

*F*(

**c**) is known entirely. We can take advantage of this

*a priori*knowledge in constructing a scheme to find the maximum directly, rather than using a search strategy. This leads to significantly faster optimisation than the search algorithms.

*F*(

**c**) has a well defined maximum and is isotropic in the surrounding region. Since its value also becomes small away from the maximum, a good estimate of the location of the maximum can be found from a first moment (centre of mass) calculation. This estimate, denoted by the vector

**W**, could be formulated as

*N*-dimensional integral over a suitably large region ∑ and d

*V*is the volume element at

**b**. In practice, Eq. (11

11. M. J. Booth, T. Wilson, H.-B. Sun, T. Ota, and S. Kawata, “Methods for the characterisation of deformable membrane mirrors,” Appl. Opt. **44**5131–5139 (2005). [CrossRef] [PubMed]

**W**as

_{m}are integration weights that depend upon the particular numerical integration scheme [19]. The vectors

**b**

_{m}are the

*M*integration abcissæ, the locations where the integrand is evaluated, each of which represents the aberration introduced by the adaptive element for a particular intensity measurement. If a large number of appropriately distributed abcissæ are used then

**W**≈

**a**. A simple integration scheme could, for example, involve the regular distribution of the

**b**

_{m}throughout ∑, based upon a lattice such as

**Z**

_{N}, using weights γ

_{m}= 1. This would be a multidimensional analogue of block integration with one variable. More advanced schemes, such as Gaussian quadrature, might alternatively be employed [19].

**b**

_{m}for measurement of

*N*modes would be at the

*N*+ 1 vertices of a regular simplex. In this case, since a small number of abcissæ are used, the approximation errors in Eq. (12) become significant and the approximate equality

**W**≈

**a**no longer holds. However, with suitable choice of the vector length ∣

**b**

_{m}∣ there remains a linear relationship between

**W**and

**a**so we can calculate a as

**S**, whose element

*S*

_{ik}is defined as

*E*as

*E*with ∣

**a**∣ for ∣

**b**

_{m}∣ =0.5 and

*N*= 6. Each data point shows the mean and standard deviation from a sample of 1000 randomly oriented input aberrations of a given magnitude. The mean value of

*E*was within the measurement tolerance of

**a**∣ ≤ 1.09. The same analysis was performed for other combinations of modes for

*N*≤ 6. In all cases

*E*showed ∣

**a**∣ similar dependence on a and the ranges over which the mean value of

*E*was within the measurement tolerance all lay between ∣

**a**∣ ≤ 1.05 and ∣

**a**∣ ≤ 1.26.

*N*Zernike modes using only

*N*+ 1 intensity measurements. This is the minimum number of measurements possible to retrieve the

*N*+ 1 unknown parameters in the system (i.e., the

*N*modal coefficients and the total intensity,

*I*

_{0}).

## 9. Conclusions

*N*, was increased. The number of measurements required in the exhaustive search methods increased exponentially with

*N*, whereas the SAHC method required a number that increases as

*N*

^{2}. The direct maximisation method was significantly more efficient, requiring only

*N*+1 measurements for N modes, over the range of input aberrations used here. This improvement over the other methods was possible since the calculation effectively took into account

*a priori*knowledge about the form of the function being maximised. This is applicable to any system where the aberration can be accurately represented by the

*N*orthonormal modes.

## Appendix A: Zernike polynomials

21. R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. **66**207–277 (1976). [CrossRef]

^{2}. The mode indexing schemes, using the single index

*i*or the dual indices (

*n*,

*m*), are explained by Neil

*et al*. [20

20. M. A. A. Neil, M. J. Booth, and T. Wilson, “New modal wavefront sensor: a theoretical analysis,” J. Opt. Soc. Am. A **17**1098–1107 (2000). [CrossRef]

## Appendix B: Simplex construction

**b**

_{n}that represent the vertices of a regular

*N*-dimensional simplex proceeds as follows. Firstly, note that when the centroid of the simplex is at the origin

**b**

_{1}= (1,0,0...). Using equation 16 we find that

**b**

_{n})

_{m}represents the

*m*th element of

**b**

_{n}. The symmetry of the simplex means that the coefficients (

**b**

_{n})1 for

*n*> 1 are equal, so we can determine the first coordinate of the remaining vectors as

**b**

_{2}to have only two non-zero coordinates in the first two dimensions and hence lies in the plane defined by the first two coordinates. The second coordinate is simply obtained by

## Appendix C: Calculation of step size for SAHC method

*N*-dimensional sphere of radius

*ε*is centred at the origin. Its surface, denoted by

*T*, must be covered by the

*N*+ 1 overlapping spheres, also of radius

*ε*, centred on the vertices of a regular simplex whose centroid is positioned at the origin. The distance from the origin to each simplex vertex is

*s*and the vertices are described by the vectors

*s*

**b**

_{n}, where we use the vectors derived in Appendix B. There are

*N*+1 similar points on

*T*, given by the vectors -

*ε*

**b**

_{n}, that are equidistant from the adjacent vertices (the directions of these vectors correspond, in

*N*= 2, to the midpoints of the triangle’s edges, and in

*N*= 3, to the centroids of the tetrahedron’s faces). If we ensure that one of these points is covered, we can conclude that the whole of

*T*will be covered. To satisfy this, the distance between the point -

*ε*

**b**

_{1}and the vertex

*s*

**b**

_{2}must be less than

*ε*. Hence,

## Acknowledgments

## References and links

1. | J. W. Hardy, |

2. | M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope”, Proc. Nat. Acad. Sci. |

3. | A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Technol. |

4. | L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multipho-ton scanning microscopy using a deformable mirror,” J. Microsc. |

5. | P. N. Marsh, D. Burns, and J. M. Girkin, “Practical implementation of adaptive optics in multiphoton microscopy,” Opt. Express |

6. | O. Albert, L. Sherman, G. Mourou, T. B. Norris, and G. Vdovin, “Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy,” Opt. Lett. |

7. | W. Lubeigt, G. Valentine, J. Girkin, E. Bente, and D. Burns, “Active transverse mode control and optimization of an all-solid-state laser using an intracavity adaptive-optic mirror,” Opt. Express |

8. | E. Theofanidou, L. Wilson, W. J. Hossack, and J. Arlt, “Spherical aberration correction for optical tweezers,” Opt. Commun. |

9. | A. C. F. Gonte and R. Dandliker, “Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror,” Opt. Eng. |

10. | M. Vorontsov, “Decoupled stochastic parallel gradient descent optimization for adaptive optics: integrated approach for wave-front sensor information fusion,” J. Opt. Soc. Am. A |

11. | M. J. Booth, T. Wilson, H.-B. Sun, T. Ota, and S. Kawata, “Methods for the characterisation of deformable membrane mirrors,” Appl. Opt. |

12. | T. Wilson and C. J. R. Sheppard, |

13. | M. Born and E. Wolf, |

14. | Michalewicz and D. B. Fogel, |

15. | J. H. Conway and N. J. A. Sloan, |

16. | T. C. Hales, “An overview of the Kepler conjecture,” http://xxx.lanl.gov/ math.MG/9811071 (1999). |

17. | R. Kershner, “The number of circles covering a set,” Am. J. Math. |

18. | E. Vitterbo and J. Boutros, “A universal lattice code decoder for fading channels,” IEEE Trans. Inf. Theory |

19. | W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, |

20. | M. A. A. Neil, M. J. Booth, and T. Wilson, “New modal wavefront sensor: a theoretical analysis,” J. Opt. Soc. Am. A |

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

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

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

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: November 15, 2005

Revised Manuscript: January 20, 2006

Manuscript Accepted: February 13, 2006

Published: February 20, 2006

**Virtual Issues**

Vol. 1, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Martin Booth, "Wave front sensor-less adaptive optics: a model-based approach using sphere packings," Opt. Express **14**, 1339-1352 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-4-1339

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### References

- J. W. Hardy, Adaptive Optics for Astronomical Telescopes, (Oxford University Press, 1998).
- M. J. Booth, M. A. A. Neil, R. Ju¡skaitis and T.Wilson, "Adaptive aberration correction in a confocal microscope," Proc. Nat. Acad. Sci. 99, 5788-5792 (2002). [CrossRef] [PubMed]
- A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine and J. M. Girkin, "Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy," Microsc. Res. Technol. 67, 36-44 (2005). [CrossRef]
- L. Sherman, J. Y. Ye, O. Albert and T. B. Norris,"Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror," J. Microsc. 206, 65-71 (2002). [CrossRef] [PubMed]
- P. N. Marsh, D. Burns and J. M. Girkin, "Practical implementation of adaptive optics in multiphoton microscopy," Opt. Express 11, 1123-1130 (2003). [CrossRef] [PubMed]
- O. Albert, L. Sherman, G. Mourou, T. B. Norris and G. Vdovin, "Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy," Opt. Lett. 25, 52-54 (2000). [CrossRef]
- W. Lubeigt, G. Valentine, J. Girkin, E. Bente, and D. Burns, "Active transverse mode control and optimization of an all-solid-state laser using an intracavity adaptive-optic mirror," Opt. Express 10, 550-555 (2002). [PubMed]
- E. Theofanidou, L. Wilson,W. J. Hossack and J. Arlt, "Spherical aberration correction for optical tweezers," Opt. Commun. 236, 145-150 (2004). [CrossRef]
- A. C. F. Gonte and R. Dandliker, "Optimization of single-mode fiber coupling efficiency with an adaptive membrane mirror," Opt. Eng. 41, 1073-1076 (2002). [CrossRef]
- M. Vorontsov, "Decoupled stochastic parallel gradient descent optimization for adaptive optics: integrated approach for wave-front sensor information fusion," J. Opt. Soc. Am. A 19, 356-368 (2002). [CrossRef]
- M. J. Booth, T. Wilson, H.-B. Sun, T. Ota and S. Kawata, "Methods for the characterisation of deformable membrane mirrors," Appl. Opt. 44, 5131-5139 (2005). [CrossRef] [PubMed]
- T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy, (Academic Press, London, 1984).
- M. Born and E. Wolf, Principles of Optics, 6th Edition, (Pergamon Press, 1983).
- Z. Michalewicz and D. B. Fogel, How to Solve It: Modern Heuristics, (Springer, Berlin, 2000).
- J. H. Conway and N. J. A. Sloan, Sphere Packings, Lattices and Groups, 3rd Edition, (Springer-Verlag, 1998).
- T. C. Hales, "An overview of the Kepler conjecture," http://xxx.lanl.gov/ math.MG/9811071 (1999).
- R. Kershner, "The number of circles covering a set," Am. J. Math. 61, 665-671 (1939). [CrossRef]
- E. Vitterbo and J. Boutros, "A universal lattice code decoder for fading channels," IEEE Trans. Inf. Theory 45, 1639-1642 (1999). [CrossRef]
- W. Press, S. Teukolsky, W. Vetterling and B. Flannery, Numerical Recipes in C, 2nd Edition, (Cambridge University Press, 1992).
- M. A. A. Neil, M. J. Booth and T. Wilson, "New modal wavefront sensor: a theoretical analysis," J. Opt. Soc. Am. A 17, 1098-1107 (2000). [CrossRef]
- R. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-277 (1976). [CrossRef]

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