## Deformable mirror model for open-loop adaptive optics using multivariate adaptive regression splines

Optics Express, Vol. 18, Issue 7, pp. 6492-6505 (2010)

http://dx.doi.org/10.1364/OE.18.006492

Acrobat PDF (460 KB)

### Abstract

Open-loop adaptive optics is a technique in which the turbulent wavefront is measured before it hits the deformable mirror for correction. We present a technique to model a deformable mirror working in open-loop based on multivariate adaptive regression splines (MARS), a non-parametric regression technique. The model’s input is the wavefront correction to apply to the mirror and its output is the set of voltages to shape the mirror. We performed experiments with an electrostrictive deformable mirror, achieving positioning errors of the order of 1.2% RMS of the peak-to-peak wavefront excursion. The technique does not depend on the physical parameters of the device; therefore it may be included in the control scheme of any type of deformable mirror.

© 2010 OSA

## 1. Introduction

1. F. Hammer, F. Sayede, E. Gendron, T. Fusco, D. Burgarella, V. Cayatte, J. M. Conan, F. Courbin, H. Flores, I. Guinouard, L. Jocou, A. Lancon, G. Monnet, M. Mouhcine, F. Rigaud, D. Rouan, G. Rousset, V. Buat, and F. Zamkotsian, “The FALCON Concept: Multi-Object Spectroscopy Combined with MCAO in Near-IR,” Proc. ESO Workshop (2002).

2. F. Assémat, E. Gendron, and F. Hammer, “The FALCON concept: multi-object adaptive optics and atmospheric tomography for integral field spectroscopy - principles and performance on an 8-m telescope,” Mon. Not. R. Astron. Soc. **376**(1), 287–312 (2007). [CrossRef]

4. J. Friedman, “Multivariate adaptive regression splines,” Ann. Stat. **19**(1), 1–67 (1991). [CrossRef]

5. C. Hom, P. Dean, and S. Winzer, “Simulating electrostrictive DM: I nonlinear static analysis,” Smart Mater. Struct. **8**(5), 691–699 (1999). [CrossRef]

8. T. Bifano, P. Bierden, H. Zhu, S. Cornelissen, and J. Kim, “Megapixel wavefront correctors,” Proc. SPIE **5490**, 1472–1481 (2004). [CrossRef]

8. T. Bifano, P. Bierden, H. Zhu, S. Cornelissen, and J. Kim, “Megapixel wavefront correctors,” Proc. SPIE **5490**, 1472–1481 (2004). [CrossRef]

11. J. Stewart, A. Diouf, Y. Zhou, and T. Bifano, “Open-loop control of a MEMS deformable mirror for large-amplitude wavefront control,” J. Opt. Soc. Am. A **24**(12), 3827–3833 (2007). [CrossRef]

## 2. Multivariate adaptive regression splines

4. J. Friedman, “Multivariate adaptive regression splines,” Ann. Stat. **19**(1), 1–67 (1991). [CrossRef]

13. S. Sekulic and B. R. Kowalski, “MARS: a tutorial,” J. Chemometr. **6**(4), 199–216 (1992). [CrossRef]

*e*is an error vector of dimension (

*n x 1*).

15. Q.-S. Xu, M. Daszykowski, B. Walczak, F. Daeyaert, M. R. de Jonge, J. Heeres, L. M. H. Koymans, P. J. Lewi, H. M. Vinkers, P. A. Janssen, and D. L. Massart, “Multivariate adaptive regression splines - studies of HIV reverse transcriptase inhibitors,” Chemom. Intell. Lab. Syst. **72**(1), 27–34 (2004). [CrossRef]

*a priori*assumptions about the underlying functional relationship between dependent and independent variables. Instead, this relation is uncovered from a set of coefficients and piecewise polynomials of degree

*q*(basis functions) that are entirely driven from the regression data

*(y, X)*. The MARS regression model is constructed by fitting basis functions to distinct intervals of the independent variables. Generally, piecewise polynomials, also called splines, have pieces smoothly connected together. In MARS terminology, the joining points of the polynomials are called knots, nodes or breakdown points. These will be denoted by the small letter

*t*. For a spline of degree

*q*each segment is a polynomial function. MARS uses two-sided truncated power functions as spline basis functions, described by Eqs. (2) and 3: Where

*t*. The dashed line represents the left-sided spline,

*t*. The two-sided truncated functions of the dependent variable are basis functions, linear or nonlinear, that describe the underlying phenomena. The MARS model of a dependent variable

*y*with

*M*basis functions (terms) can be written as Eq. (4):where

*X*containing n objects and

*p*explanatory variables, there are

4. J. Friedman, “Multivariate adaptive regression splines,” Ann. Stat. **19**(1), 1–67 (1991). [CrossRef]

*M*is the number of basis functions in Eq. (4), and the parameter

*d*is a penalty for each basis function included into the model. It can be also regarded as a smoothing parameter. Large values of

*d*lead to fewer basis functions and therefore smoother function estimates. For more details about the selection of the d parameter, see [4

**19**(1), 1–67 (1991). [CrossRef]

*d*equals 2, and the maximum interaction level of the spline basis functions is restricted to 3.The main steps of the MARS algorithm as applied here can be summarised as follows:Forward stepwise selection:

- 1. Start with the simplest model, i.e. with the constant coefficient only.
- 2. Explore the space of the basis functions for each explanatory variable.
- 3. Determine the pair of basis functions that minimises the prediction error, and include them in the model.
- 4. Go to step 2 until a model with predetermined complexity is derived.

- 5. Search the entire set of basis functions (excluding the constant) and delete from the model the one that contributes least to the overall goodness of fit using the GCV criterion.
- 6. Repeat 5 until GCV reaches its minimum.

**19**(1), 1–67 (1991). [CrossRef]

### 2.1. ANOVA decomposition of the MARS model

### 2.2. Prediction ability of the MARS model

*n-1*objects. Then the model is used to predict the value for the object left out. When all objects have been left out once, RMSECV is given by Eq. (9):where

*i*-th object,

*i*-th object with the model built without the

*i*-th object.

*n*objects.

## 3. Nonlinear behavior of the DM

18. J. W. Evans, B. Macintosh, L. Poyneer, K. Morzinski, S. Severson, D. Dillon, D. Gavel, and L. Reza, “Demonstrating sub-nm closed loop MEMS flattening,” Opt. Express **14**(12), 5558–5570 (2006). [CrossRef] [PubMed]

## 4. Actuator’s area of influence

*R*(or ‘cross-correlation’), which is calculated as follows.

*C*is the covariance matrix of the set of data, the cross correlation

*R*is defined as in Eq. (11):

*R*represents one actuator, and the values at positions

*(i,j)*in

*R*correspond to the correlation between actuators

*i*and

*j*Using this definition, it follows naturally that the diagonal elements of

*R*will be unitary.

*a priori*knowledge of the complexity of a DM model.

## 5. Data taking methodology

_{offset}= 50 volts), applying positive and negative voltages around this plateau voltage, in order to model the DM in accordance with a typical AO system. Our interferometer was calibrated for each run with this plateau position, which was subtracted automatically for all phase maps. This is why our data has a zero mean value, around which there are positive and negative excursions.

*X*and

*Y*coordinates are pixel numbers). Edge actuators have largely asymmetrical influence functions, unlike the rest of the actuators; therefore the fitting procedure with the Gaussian did not produce satisfactory results for the edge actuators. We decided to identify the actuator positions manually for the edge actuators, assuming they are located under the ‘tallest’ area of the influence function (where the gradient changes sign). As can be seen in Fig. 4, the position of these actuators do not comply with a square grid as the rest of actuators do: this is a result of the boundary conditions of the membrane; but it is not relevant that it is not a square grid, as long as we sample the height of the actuators at coordinates that respond to the different positions that the actuators can adopt.

*Z*coordinate of the phase maps at actuator positions and we use these values to represent the DM surface. We are also sampling six points of the surface in areas of the DM we left static, which we use to fit a plane that represents the position of the DM membrane not deformed by the actuators. The position of this plane accounts for any drifts throughout the run and is subtracted from the

*Z*coordinate measured at actuator positions. We found this method to be simple and give consistent results. This is the reason one sees a tilted surrounding in Fig. 2, given the fact that the 9 x 5 actuators were poked one by one and it took time to complete the run. During this period, there were drifts in the measured value of the DM surface, which were compensated in the mirror data (this is why the aperture circle is not tilted), showing the effect of the compensation in the outer areas of the phase maps, which otherwise have a null value.

*height*(

*Z*coordinate) inputs to produce 55 voltage outputs. We exercise the DM to its limits, in order to gather realistic data for an AO system: the DM is first ‘raised’ to half its range and from that point the 55 actuators are given random voltages, to shape the DM surface to a completely random phase. The range of the random values is the maximum permitted without damaging the surface of the mirror. For our Xinetics DM this is achieved by applying +/− 12 volts to the actuators on top of the raised half range height. This is similar to an AO system that needs to compensate positive and negative phases. As an example, Fig. 6 presents the first 9 random positions of one of our runs. It can be seen that there is an average value of 0 Angstrom (because the interferometer subtracts the purely raised mirror as its reference) and positive (redder) and negative (bluer) ‘bumps’.

## 6. Results and analysis

- • Qualitative test: we generate some arbitrary combination of Zernike polynomials and feed the model with the values of the combination of polynomials evaluated at actuator coordinates. The model outputs the voltages for the 55 actuators, which we run with the DM, measuring the surface shape with our interferometer
- • Quantitative test: we produced an additional 1000 new random positions with the same parameters as the ones used for training and feed the model with them. The model produces the predicted voltages to achieve such random shapes, which we run with the DM, acquiring phase maps.

*GoTo*’ error, or the difference in position between the desired shape and the achieved phase by the mirror. We decided on using purely random voltages to test our model to its limits, since the spatial frequency response of uniformly distributed random data is more stringent at high spatial frequencies compared to Kolmogorov turbulence for example.

21. E. Deconinck, M. H. Zhang, F. Petitet, E. Dubus, I. Ijjaali, D. Coomans, and Y. Vander Heyden, “Boosted regression trees, multivariate adaptive regression splines and their two-step combinations with multiple linear regression or partial least squares to predict blood–brain barrier passage: A case study,” Anal. Chim. Acta **609**(1), 13–23 (2008). [CrossRef] [PubMed]

*Y*-axis (where the majority of the actuators modeled are) and incorporate spatial frequencies that would be too high for this mirror to shape. The results of this experiment are clearly seen along the right panels of Fig. 7. For the first polynomial (at the top of the figure), the difference between theory and experiment is fairly small, but when increasing the order of the polynomial and thus the spatial bandwidth, the fitting error becomes significant, although the areas with limited spatial frequencies are still well modeled (see for instance the central part of the plot at the bottom-right panel). It is interesting to note that these results were obtained with our MARS model, which was never trained with Zernike polynomials. This allows us to establish that our training method is general and would be appropriate for a mirror in an AO system.

*GoTo*’ error was obtained with a new 1000 trials random run. The MARS model was fed with the facesheet positions from this run and the voltages produced were used for a new run. The phase maps of both runs were subtracted (at actuator coordinates), to produce the results in Figs. 8 and 9 . Figure 8 shows the maxima and minima of each trial, to confirm the large span in the data. Figure 9 is the main result of this paper, presented as a

*GoTo*error in Angstrom (RMS value for the 55 actuators) and as a percentage of the full-range of actuators excursion (from Fig. 8)

22. B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE **3126**, 569–579 (1997). [CrossRef]

## 7. Conclusions

*GoTo*error of 1.2% of the full-range of actuator positions. This model has the benefit of not developing a physical model of the DM; therefore the DM modeling strategy can be the same, regardless of the type of mirror in operation. MARS produced a series of simple equations, involving only sums and multiplications – they are fast to compute, unlike an iterative solution, so it should not add any significant latency to a real-time AO computer.

## Acknowledgments

## References and links

1. | F. Hammer, F. Sayede, E. Gendron, T. Fusco, D. Burgarella, V. Cayatte, J. M. Conan, F. Courbin, H. Flores, I. Guinouard, L. Jocou, A. Lancon, G. Monnet, M. Mouhcine, F. Rigaud, D. Rouan, G. Rousset, V. Buat, and F. Zamkotsian, “The FALCON Concept: Multi-Object Spectroscopy Combined with MCAO in Near-IR,” Proc. ESO Workshop (2002). |

2. | F. Assémat, E. Gendron, and F. Hammer, “The FALCON concept: multi-object adaptive optics and atmospheric tomography for integral field spectroscopy - principles and performance on an 8-m telescope,” Mon. Not. R. Astron. Soc. |

3. | D. Guzmán, A. Guesalaga, R. Myers, R. Sharples, T. Morris, A. Basden, C. Saunter, N. Dipper, L. Young, L. Rodríguez, M. Reyes, and Y. Martin, “Deformable mirror controller for open-loop adaptive optics” Proc. SPIE |

4. | J. Friedman, “Multivariate adaptive regression splines,” Ann. Stat. |

5. | C. Hom, P. Dean, and S. Winzer, “Simulating electrostrictive DM: I nonlinear static analysis,” Smart Mater. Struct. |

6. | D. Andersen, M. Fischer, R. Conan, M. Fletcher, and J. P. Veran, “VOLT: the Victoria Open Loop Testbed” Proc. SPIE |

7. | E. Laag, D. Gavel, and M. Ammons, “Open-loop woofer-tweeter control on the LAO multi-conjugate adaptive optics testbed” in |

8. | T. Bifano, P. Bierden, H. Zhu, S. Cornelissen, and J. Kim, “Megapixel wavefront correctors,” Proc. SPIE |

9. | C. Blain, O. Guyon, R. Conan, and C. Bradley, “Simple iterative method for open-loop control of MEMS deformable mirrors”, Proc. SPIE 7015, 701534–701534–8 (2008). |

10. | K. Morzinski, K. Harpsoe, D. Gavel, and M. Ammons, “The open-loop control of MEMS: modeling and experimental results”, Proc. SPIE |

11. | J. Stewart, A. Diouf, Y. Zhou, and T. Bifano, “Open-loop control of a MEMS deformable mirror for large-amplitude wavefront control,” J. Opt. Soc. Am. A |

12. | J. Hardy, “Wavefront Correctors” in |

13. | S. Sekulic and B. R. Kowalski, “MARS: a tutorial,” J. Chemometr. |

14. | L. Breiman, J. H. Friedman, R. A. Olshen, and C. G. Stone, Classification and Regression Trees., Wadsworth International Group, Belmont, CA (1984) |

15. | Q.-S. Xu, M. Daszykowski, B. Walczak, F. Daeyaert, M. R. de Jonge, J. Heeres, L. M. H. Koymans, P. J. Lewi, H. M. Vinkers, P. A. Janssen, and D. L. Massart, “Multivariate adaptive regression splines - studies of HIV reverse transcriptase inhibitors,” Chemom. Intell. Lab. Syst. |

16. | P. Craven and G. Wahba, “Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation,” Numer. Math. |

17. | D. L. Massart, B. Vandeginste, L. Buydens, S. De Jong, P. Lewi, and J. Smeyers-Verbeke, In: “Handbook of Chemometrics and Qualimetrics” vol. 20 A., Elsevier, Amsterdam (1997) |

18. | J. W. Evans, B. Macintosh, L. Poyneer, K. Morzinski, S. Severson, D. Dillon, D. Gavel, and L. Reza, “Demonstrating sub-nm closed loop MEMS flattening,” Opt. Express |

19. | Y.F. Li, S.H. Ng, M. Xie, T.N. Goh. “A systematic comparison of metamodeling techniques for simulation optimization in Decision Support Systems”. Applied Soft Computing, In Press, Corrected Proof, Available online 24 December 2009. doi:10.1016/j.asoc.2009.11.034 |

20. | M. Carlin, T. Kavli, and B. Lillekjendlie, “A comparison of four methods for non-linear data modelling,” Chemom. Intell. Lab. Syst. |

21. | E. Deconinck, M. H. Zhang, F. Petitet, E. Dubus, I. Ijjaali, D. Coomans, and Y. Vander Heyden, “Boosted regression trees, multivariate adaptive regression splines and their two-step combinations with multiple linear regression or partial least squares to predict blood–brain barrier passage: A case study,” Anal. Chim. Acta |

22. | B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE |

**OCIS Codes**

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

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(010.1285) Atmospheric and oceanic optics : Atmospheric correction

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: January 11, 2010

Revised Manuscript: February 19, 2010

Manuscript Accepted: February 22, 2010

Published: March 15, 2010

**Citation**

Dani Guzmán, Francisco Javier de Cos Juez, Fernando Sánchez Lasheras, Richard Myers, and Laura Young, "Deformable mirror model for open-loop adaptive optics using multivariate adaptive regression splines," Opt. Express **18**, 6492-6505 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-6492

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### References

- F. Hammer, F. Sayede, E. Gendron, T. Fusco, D. Burgarella, V. Cayatte, J. M. Conan, F. Courbin, H. Flores, I. Guinouard, L. Jocou, A. Lancon, G. Monnet, M. Mouhcine, F. Rigaud, D. Rouan, G. Rousset, V. Buat, and F. Zamkotsian, “The FALCON Concept: Multi-Object Spectroscopy Combined with MCAO in Near-IR,” Proc. ESO Workshop (2002).
- F. Assémat, E. Gendron, and F. Hammer, “The FALCON concept: multi-object adaptive optics and atmospheric tomography for integral field spectroscopy - principles and performance on an 8-m telescope,” Mon. Not. R. Astron. Soc. 376(1), 287–312 (2007). [CrossRef]
- D. Guzmán, A. Guesalaga, R. Myers, R. Sharples, T. Morris, A. Basden, C. Saunter, N. Dipper, L. Young, L. Rodríguez, M. Reyes, and Y. Martin, “Deformable mirror controller for open-loop adaptive optics” Proc. SPIE 7015, 70153X–70153X–12 (2008).
- J. Friedman, “Multivariate adaptive regression splines,” Ann. Stat. 19(1), 1–67 (1991). [CrossRef]
- C. Hom, P. Dean, and S. Winzer, “Simulating electrostrictive DM: I nonlinear static analysis,” Smart Mater. Struct. 8(5), 691–699 (1999). [CrossRef]
- D. Andersen, M. Fischer, R. Conan, M. Fletcher, and J. P. Veran, “VOLT: the Victoria Open Loop Testbed” Proc. SPIE 7015, 7015OH-7015OH-11 (2008).
- E. Laag, D. Gavel, and M. Ammons, “Open-loop woofer-tweeter control on the LAO multi-conjugate adaptive optics testbed” in Adaptive optics for industry and medicine, C. Dainty. (Imperial College Press, 2008), pp. 143–148.
- T. Bifano, P. Bierden, H. Zhu, S. Cornelissen, and J. Kim, “Megapixel wavefront correctors,” Proc. SPIE 5490, 1472–1481 (2004). [CrossRef]
- C. Blain, O. Guyon, R. Conan, and C. Bradley, “Simple iterative method for open-loop control of MEMS deformable mirrors”, Proc. SPIE 7015, 701534–701534–8 (2008).
- K. Morzinski, K. Harpsoe, D. Gavel, and M. Ammons, “The open-loop control of MEMS: modeling and experimental results”, Proc. SPIE 6467, 6467OG-6467OG-10 (2007).
- J. Stewart, A. Diouf, Y. Zhou, and T. Bifano, “Open-loop control of a MEMS deformable mirror for large-amplitude wavefront control,” J. Opt. Soc. Am. A 24(12), 3827–3833 (2007). [CrossRef]
- J. Hardy, “Wavefront Correctors” in Adaptive Optics for Astronomical Telescopes (Oxford 1998), pp. 176–212.
- S. Sekulic and B. R. Kowalski, “MARS: a tutorial,” J. Chemometr. 6(4), 199–216 (1992). [CrossRef]
- L. Breiman, J. H. Friedman, R. A. Olshen, and C. G. Stone, Classification and Regression Trees., Wadsworth International Group, Belmont, CA (1984)
- Q.-S. Xu, M. Daszykowski, B. Walczak, F. Daeyaert, M. R. de Jonge, J. Heeres, L. M. H. Koymans, P. J. Lewi, H. M. Vinkers, P. A. Janssen, and D. L. Massart, “Multivariate adaptive regression splines - studies of HIV reverse transcriptase inhibitors,” Chemom. Intell. Lab. Syst. 72(1), 27–34 (2004). [CrossRef]
- P. Craven and G. Wahba, “Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation,” Numer. Math. 31, 317–403 (1979).
- D. L. Massart, B. Vandeginste, L. Buydens, S. De Jong, P. Lewi, and J. Smeyers-Verbeke, In: “Handbook of Chemometrics and Qualimetrics” vol. 20 A., Elsevier, Amsterdam (1997)
- J. W. Evans, B. Macintosh, L. Poyneer, K. Morzinski, S. Severson, D. Dillon, D. Gavel, and L. Reza, “Demonstrating sub-nm closed loop MEMS flattening,” Opt. Express 14(12), 5558–5570 (2006). [CrossRef] [PubMed]
- Y.F. Li, S.H. Ng, M. Xie, T.N. Goh. “A systematic comparison of metamodeling techniques for simulation optimization in Decision Support Systems”. Applied Soft Computing, In Press, Corrected Proof, Available online 24 December 2009. doi:10.1016/j.asoc.2009.11.034
- M. Carlin, T. Kavli, and B. Lillekjendlie, “A comparison of four methods for non-linear data modelling,” Chemom. Intell. Lab. Syst. 23(1), 163–177 (1994). [CrossRef]
- E. Deconinck, M. H. Zhang, F. Petitet, E. Dubus, I. Ijjaali, D. Coomans, and Y. Vander Heyden, “Boosted regression trees, multivariate adaptive regression splines and their two-step combinations with multiple linear regression or partial least squares to predict blood–brain barrier passage: A case study,” Anal. Chim. Acta 609(1), 13–23 (2008). [CrossRef] [PubMed]
- B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997). [CrossRef]

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