Abstract

This paper studies the effect of pupil displacements on the best achievable performance of retinal imaging adaptive optics (AO) systems, using 52 trajectories of horizontal and vertical displacements sampled at 80 Hz by a pupil tracker (PT) device on 13 different subjects. This effect is quantified in the form of minimal root mean square (rms) of the residual phase affecting image formation, as a function of the delay between PT measurement and wavefront correction. It is shown that simple dynamic models identified from data can be used to predict horizontal and vertical pupil displacements with greater accuracy (in terms of average rms) over short-term time horizons. The potential impact of these improvements on residual wavefront rms is investigated. These results allow to quantify the part of disturbances corrected by retinal imaging systems that are caused by relative displacements of an otherwise fixed or slowy-varying subject-dependent aberration. They also suggest that prediction has a limited impact on wavefront rms and that taking into account PT measurements in real time improves the performance of AO retinal imaging systems.

© 2016 Optical Society of America

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References

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2015 (1)

2012 (2)

B. Sahin, B. Lamory, X. Levecq, F. Harms, and C. Dainty, “Adaptive optics with pupil tracking for high resolution retinal imaging,” Biomed. Opt. Express 3(2), 225–239 (2012).
[Crossref] [PubMed]

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[Crossref]

2011 (2)

A. Garcia-Rissmann, C. Kulcsár, H.-F. Raynaud, Y. El Mrabet, B. Sahin, and B. Lamory, “Adaptive prediction of human eye pupil position and effects on wavefront errors,” Proc. SPIE Ophthalmic Technologies XXI, 78851W (2011).

S. Niu, J. Shen, C. Liang, Y. Zhang, and B. Li, “High-resolution retinal imaging with micro adaptive optics systems,” Appl. Opt. 50(22), 4365–4375 (2011).
[Crossref] [PubMed]

2010 (3)

C. Li, N. Sredar, K.M. Ivers, H. Queener, and J. Porter, “A correction algorithm to simultaneously control dual deformable mirrors in a woofer-tweeter adaptive optics system,” Opt. Express 18(16), 16671–16684 (2010).
[Crossref] [PubMed]

P. Godara, A.M. Dubis, A. Roorda, J.L. Duncan, and J. Carroll, “Adaptive optics retinal imaging: emerging clinical applications,” Optometry Vision Sci. 87(12), 930–941 (2010).
[Crossref]

J. Arines, P. Prado, and S. Bará, “Pupil tracking with a Hartmann-Shack wavefront sensor,” J. Biomed. Opt. 15(3), 036022 (2010).
[Crossref] [PubMed]

2009 (1)

O. V. Komogortsev and J. I. Khan, “Eye movement prediction by oculomotor plant Kalman filter with brainstem control,” J. Cont. Th. Appl. 7(1), 14–22 (2009).
[Crossref]

2008 (1)

2006 (1)

2005 (2)

S. Gruppetta, F. Lacombe, and P. Puget, “Study of the dynamic aberrations of the human tear film,” Opt. Express 13(19), 7631–7636 (2005).
[Crossref] [PubMed]

J.-R. Liang, S. Moshel, A. Z. Zivotofsky, A. Caspi, R. Engbert, R. Kliegl, and S. Havlin, “Scaling of horizontal and vertical fixational eye movements,” Phys. Rev. E 71, 031909 (2005).
[Crossref]

2004 (3)

Z. Ramdane-Cherif, A. Nait-Ali, and O. Krebs, “An autoregressive (AR) model applied to eye tremor movement, clinical application in schizophrenia,” J. Med. Syst. 28(5), 489–495 (2004).
[Crossref] [PubMed]

S. Martinez-Conde, S. L. Macknik, and D. H. Hubel, “The role of fixational eye movements in visual perception,” Nat. Rev. Neurosci. 5, 229–240 (2004).
[Crossref] [PubMed]

D.R. Iskander, M.J. Collins, M.R. Morelande, and M. Zhu, “Analyzing the dynamic wavefront aberrations in the human eye,” IEEE Trans. Biomed. Eng. 51(11), 1969–1980 (2004).
[Crossref] [PubMed]

2002 (2)

2001 (3)

E.J. Fernández, I. Iglesias, and P. Artal, “Closed-loop adaptive optics in the human eye,” Opt. Lett. 26(10), 746–748 (2001).
[Crossref]

J.-F. Le Gargasson, M. Glanc, and P. Léna, “Retinal imaging with adaptive optics,” CR. Acad. Sci. Paris Applied Physics 2, 1131–1138 (2001).

H. Hofer, P. Artal, B. Singer, J.L Aragón, and D.R. Williams, “Dynamics of the eye’s wave aberration,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 18(3), 497–506 (2001).
[Crossref] [PubMed]

2000 (1)

A. Roorda, “Adaptive optics ophtalmoscopy,” J. Refract. Surg. 16, S602–S607 (2000).
[PubMed]

1997 (1)

1994 (1)

1954 (1)

Anderson, B. D. O.

B. D. O. Anderson and J. B. Moore, Optimal filtering (London Prentice Hall, 1979).

Aragón, J.L

H. Hofer, P. Artal, B. Singer, J.L Aragón, and D.R. Williams, “Dynamics of the eye’s wave aberration,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 18(3), 497–506 (2001).
[Crossref] [PubMed]

Arines, J.

J. Arines, P. Prado, and S. Bará, “Pupil tracking with a Hartmann-Shack wavefront sensor,” J. Biomed. Opt. 15(3), 036022 (2010).
[Crossref] [PubMed]

Armington, J.C.

Artal, P.

E.J. Fernández, I. Iglesias, and P. Artal, “Closed-loop adaptive optics in the human eye,” Opt. Lett. 26(10), 746–748 (2001).
[Crossref]

H. Hofer, P. Artal, B. Singer, J.L Aragón, and D.R. Williams, “Dynamics of the eye’s wave aberration,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 18(3), 497–506 (2001).
[Crossref] [PubMed]

Bará, S.

J. Arines, P. Prado, and S. Bará, “Pupil tracking with a Hartmann-Shack wavefront sensor,” J. Biomed. Opt. 15(3), 036022 (2010).
[Crossref] [PubMed]

Baumann, B.

Ben-Amara, F.

M. Ficocelli and F. Ben-Amara, “Online Tuning of Retinal Imaging Adaptive Optics Systems,” IEEE Trans. Control Syst. Technol. 20(3), 747–754 (2012).
[Crossref]

Bigelow, C.EE.

Bille, J.F.

Bradley, A.

Burns, S.

Burns, S.A.

Carroll, J.

P. Godara, A.M. Dubis, A. Roorda, J.L. Duncan, and J. Carroll, “Adaptive optics retinal imaging: emerging clinical applications,” Optometry Vision Sci. 87(12), 930–941 (2010).
[Crossref]

Caspi, A.

J.-R. Liang, S. Moshel, A. Z. Zivotofsky, A. Caspi, R. Engbert, R. Kliegl, and S. Havlin, “Scaling of horizontal and vertical fixational eye movements,” Phys. Rev. E 71, 031909 (2005).
[Crossref]

Cheng, X.

Chteau, N.

C. Viard, K. Nakashima, B. Lamory, M. Pques, X. Levecq, and N. Chteau, “Imaging microscopic structures in pathological retinas using a flood-illumination adaptive optics retinal camera,” Proc. SPIE 7885, Ophthalmic TechnologiesXXI, 788509 (2011).
[Crossref]

Collins, M.J.

D.R. Iskander, M.J. Collins, M.R. Morelande, and M. Zhu, “Analyzing the dynamic wavefront aberrations in the human eye,” IEEE Trans. Biomed. Eng. 51(11), 1969–1980 (2004).
[Crossref] [PubMed]

Conan, J.-M.

J. Jarosz, S. Meimon, J.-M. Conan, and M. Paques, “High temporal resolution ocular aberrometry with pupil tracking,” Proc. SPIE 8930, Ophthalmic TechnologiesXXIV, 89300D (2014).

Dainty, C.

B. Sahin, B. Lamory, X. Levecq, F. Harms, and C. Dainty, “Adaptive optics with pupil tracking for high resolution retinal imaging,” Biomed. Opt. Express 3(2), 225–239 (2012).
[Crossref] [PubMed]

B. Sahin, B. Lamory, X. Levecq, L. Vabre, and C. Dainty, “Retinal imaging system with adaptive optics enhanced with pupil tracking,” Proc. SPIE 7885, Ophthalmic TechnologiesXXI, 788517 (2011).
[Crossref]

Dubis, A.M.

P. Godara, A.M. Dubis, A. Roorda, J.L. Duncan, and J. Carroll, “Adaptive optics retinal imaging: emerging clinical applications,” Optometry Vision Sci. 87(12), 930–941 (2010).
[Crossref]

Duncan, J.L.

P. Godara, A.M. Dubis, A. Roorda, J.L. Duncan, and J. Carroll, “Adaptive optics retinal imaging: emerging clinical applications,” Optometry Vision Sci. 87(12), 930–941 (2010).
[Crossref]

El Mrabet, Y.

A. Garcia-Rissmann, C. Kulcsár, H.-F. Raynaud, Y. El Mrabet, B. Sahin, and B. Lamory, “Adaptive prediction of human eye pupil position and effects on wavefront errors,” Proc. SPIE Ophthalmic Technologies XXI, 78851W (2011).

Engbert, R.

J.-R. Liang, S. Moshel, A. Z. Zivotofsky, A. Caspi, R. Engbert, R. Kliegl, and S. Havlin, “Scaling of horizontal and vertical fixational eye movements,” Phys. Rev. E 71, 031909 (2005).
[Crossref]

Felberer, F.

Ferguson, R.D.

Fernández, E.J.

Ficocelli, M.

M. Ficocelli and F. Ben-Amara, “Online Tuning of Retinal Imaging Adaptive Optics Systems,” IEEE Trans. Control Syst. Technol. 20(3), 747–754 (2012).
[Crossref]

Garcia-Rissmann, A.

A. Garcia-Rissmann, C. Kulcsár, H.-F. Raynaud, Y. El Mrabet, B. Sahin, and B. Lamory, “Adaptive prediction of human eye pupil position and effects on wavefront errors,” Proc. SPIE Ophthalmic Technologies XXI, 78851W (2011).

Glanc, M.

J.-F. Le Gargasson, M. Glanc, and P. Léna, “Retinal imaging with adaptive optics,” CR. Acad. Sci. Paris Applied Physics 2, 1131–1138 (2001).

Godara, P.

P. Godara, A.M. Dubis, A. Roorda, J.L. Duncan, and J. Carroll, “Adaptive optics retinal imaging: emerging clinical applications,” Optometry Vision Sci. 87(12), 930–941 (2010).
[Crossref]

Goelz, S.

Grimm, B.

Gruppetta, S.

Haindl, R.

Hammer, D.X.

Harms, F.

Havlin, S.

J.-R. Liang, S. Moshel, A. Z. Zivotofsky, A. Caspi, R. Engbert, R. Kliegl, and S. Havlin, “Scaling of horizontal and vertical fixational eye movements,” Phys. Rev. E 71, 031909 (2005).
[Crossref]

Hitzenberger, C. K.

Hofer, H.

H. Hofer, P. Artal, B. Singer, J.L Aragón, and D.R. Williams, “Dynamics of the eye’s wave aberration,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 18(3), 497–506 (2001).
[Crossref] [PubMed]

Hong, X.

Hubel, D. H.

S. Martinez-Conde, S. L. Macknik, and D. H. Hubel, “The role of fixational eye movements in visual perception,” Nat. Rev. Neurosci. 5, 229–240 (2004).
[Crossref] [PubMed]

Iftimia, N.V.

Iglesias, I.

Iskander, D.R.

D.R. Iskander, M.J. Collins, M.R. Morelande, and M. Zhu, “Analyzing the dynamic wavefront aberrations in the human eye,” IEEE Trans. Biomed. Eng. 51(11), 1969–1980 (2004).
[Crossref] [PubMed]

Ivers, K.M.

Jarosz, J.

J. Jarosz, S. Meimon, J.-M. Conan, and M. Paques, “High temporal resolution ocular aberrometry with pupil tracking,” Proc. SPIE 8930, Ophthalmic TechnologiesXXIV, 89300D (2014).

Khan, J. I.

O. V. Komogortsev and J. I. Khan, “Eye movement prediction by oculomotor plant Kalman filter with brainstem control,” J. Cont. Th. Appl. 7(1), 14–22 (2009).
[Crossref]

Kliegl, R.

J.-R. Liang, S. Moshel, A. Z. Zivotofsky, A. Caspi, R. Engbert, R. Kliegl, and S. Havlin, “Scaling of horizontal and vertical fixational eye movements,” Phys. Rev. E 71, 031909 (2005).
[Crossref]

Komogortsev, O. V.

O. V. Komogortsev and J. I. Khan, “Eye movement prediction by oculomotor plant Kalman filter with brainstem control,” J. Cont. Th. Appl. 7(1), 14–22 (2009).
[Crossref]

Krebs, O.

Z. Ramdane-Cherif, A. Nait-Ali, and O. Krebs, “An autoregressive (AR) model applied to eye tremor movement, clinical application in schizophrenia,” J. Med. Syst. 28(5), 489–495 (2004).
[Crossref] [PubMed]

Kulcsár, C.

A. Garcia-Rissmann, C. Kulcsár, H.-F. Raynaud, Y. El Mrabet, B. Sahin, and B. Lamory, “Adaptive prediction of human eye pupil position and effects on wavefront errors,” Proc. SPIE Ophthalmic Technologies XXI, 78851W (2011).

Lacombe, F.

Lamory, B.

B. Sahin, B. Lamory, X. Levecq, F. Harms, and C. Dainty, “Adaptive optics with pupil tracking for high resolution retinal imaging,” Biomed. Opt. Express 3(2), 225–239 (2012).
[Crossref] [PubMed]

A. Garcia-Rissmann, C. Kulcsár, H.-F. Raynaud, Y. El Mrabet, B. Sahin, and B. Lamory, “Adaptive prediction of human eye pupil position and effects on wavefront errors,” Proc. SPIE Ophthalmic Technologies XXI, 78851W (2011).

B. Sahin, B. Lamory, X. Levecq, L. Vabre, and C. Dainty, “Retinal imaging system with adaptive optics enhanced with pupil tracking,” Proc. SPIE 7885, Ophthalmic TechnologiesXXI, 788517 (2011).
[Crossref]

C. Viard, K. Nakashima, B. Lamory, M. Pques, X. Levecq, and N. Chteau, “Imaging microscopic structures in pathological retinas using a flood-illumination adaptive optics retinal camera,” Proc. SPIE 7885, Ophthalmic TechnologiesXXI, 788509 (2011).
[Crossref]

Le Gargasson, J.-F.

J.-F. Le Gargasson, M. Glanc, and P. Léna, “Retinal imaging with adaptive optics,” CR. Acad. Sci. Paris Applied Physics 2, 1131–1138 (2001).

Léna, P.

J.-F. Le Gargasson, M. Glanc, and P. Léna, “Retinal imaging with adaptive optics,” CR. Acad. Sci. Paris Applied Physics 2, 1131–1138 (2001).

Levecq, X.

B. Sahin, B. Lamory, X. Levecq, F. Harms, and C. Dainty, “Adaptive optics with pupil tracking for high resolution retinal imaging,” Biomed. Opt. Express 3(2), 225–239 (2012).
[Crossref] [PubMed]

B. Sahin, B. Lamory, X. Levecq, L. Vabre, and C. Dainty, “Retinal imaging system with adaptive optics enhanced with pupil tracking,” Proc. SPIE 7885, Ophthalmic TechnologiesXXI, 788517 (2011).
[Crossref]

C. Viard, K. Nakashima, B. Lamory, M. Pques, X. Levecq, and N. Chteau, “Imaging microscopic structures in pathological retinas using a flood-illumination adaptive optics retinal camera,” Proc. SPIE 7885, Ophthalmic TechnologiesXXI, 788509 (2011).
[Crossref]

Li, B.

Li, C.

Liand, J.

Liang, C.

Liang, J.

Liang, J.-R.

J.-R. Liang, S. Moshel, A. Z. Zivotofsky, A. Caspi, R. Engbert, R. Kliegl, and S. Havlin, “Scaling of horizontal and vertical fixational eye movements,” Phys. Rev. E 71, 031909 (2005).
[Crossref]

Macknik, S. L.

S. Martinez-Conde, S. L. Macknik, and D. H. Hubel, “The role of fixational eye movements in visual perception,” Nat. Rev. Neurosci. 5, 229–240 (2004).
[Crossref] [PubMed]

Martinez-Conde, S.

S. Martinez-Conde, S. L. Macknik, and D. H. Hubel, “The role of fixational eye movements in visual perception,” Nat. Rev. Neurosci. 5, 229–240 (2004).
[Crossref] [PubMed]

Meimon, S.

J. Jarosz, S. Meimon, J.-M. Conan, and M. Paques, “High temporal resolution ocular aberrometry with pupil tracking,” Proc. SPIE 8930, Ophthalmic TechnologiesXXIV, 89300D (2014).

Miller, D.T.

Moore, J. B.

B. D. O. Anderson and J. B. Moore, Optimal filtering (London Prentice Hall, 1979).

Morelande, M.R.

D.R. Iskander, M.J. Collins, M.R. Morelande, and M. Zhu, “Analyzing the dynamic wavefront aberrations in the human eye,” IEEE Trans. Biomed. Eng. 51(11), 1969–1980 (2004).
[Crossref] [PubMed]

Moshel, S.

J.-R. Liang, S. Moshel, A. Z. Zivotofsky, A. Caspi, R. Engbert, R. Kliegl, and S. Havlin, “Scaling of horizontal and vertical fixational eye movements,” Phys. Rev. E 71, 031909 (2005).
[Crossref]

Nait-Ali, A.

Z. Ramdane-Cherif, A. Nait-Ali, and O. Krebs, “An autoregressive (AR) model applied to eye tremor movement, clinical application in schizophrenia,” J. Med. Syst. 28(5), 489–495 (2004).
[Crossref] [PubMed]

Nakashima, K.

C. Viard, K. Nakashima, B. Lamory, M. Pques, X. Levecq, and N. Chteau, “Imaging microscopic structures in pathological retinas using a flood-illumination adaptive optics retinal camera,” Proc. SPIE 7885, Ophthalmic TechnologiesXXI, 788509 (2011).
[Crossref]

Niu, S.

Paques, M.

J. Jarosz, S. Meimon, J.-M. Conan, and M. Paques, “High temporal resolution ocular aberrometry with pupil tracking,” Proc. SPIE 8930, Ophthalmic TechnologiesXXIV, 89300D (2014).

Pircher, M.

Porter, J.

Pques, M.

C. Viard, K. Nakashima, B. Lamory, M. Pques, X. Levecq, and N. Chteau, “Imaging microscopic structures in pathological retinas using a flood-illumination adaptive optics retinal camera,” Proc. SPIE 7885, Ophthalmic TechnologiesXXI, 788509 (2011).
[Crossref]

Prado, P.

J. Arines, P. Prado, and S. Bará, “Pupil tracking with a Hartmann-Shack wavefront sensor,” J. Biomed. Opt. 15(3), 036022 (2010).
[Crossref] [PubMed]

Pronzato, L.

E. Walter and L. Pronzato, Identification of Parametric Models from Experimental Data (SpringerUK, 1997).

Puget, P.

Qi, X.

Queener, H.

Ramdane-Cherif, Z.

Z. Ramdane-Cherif, A. Nait-Ali, and O. Krebs, “An autoregressive (AR) model applied to eye tremor movement, clinical application in schizophrenia,” J. Med. Syst. 28(5), 489–495 (2004).
[Crossref] [PubMed]

Ratliff, F.

Raynaud, H.-F.

A. Garcia-Rissmann, C. Kulcsár, H.-F. Raynaud, Y. El Mrabet, B. Sahin, and B. Lamory, “Adaptive prediction of human eye pupil position and effects on wavefront errors,” Proc. SPIE Ophthalmic Technologies XXI, 78851W (2011).

Rechenmacher, M.

Riggs, L. A.

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Optometry Vision Sci. (1)

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Phys. Rev. E (1)

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Proc. SPIE Ophthalmic Technologies (1)

A. Garcia-Rissmann, C. Kulcsár, H.-F. Raynaud, Y. El Mrabet, B. Sahin, and B. Lamory, “Adaptive prediction of human eye pupil position and effects on wavefront errors,” Proc. SPIE Ophthalmic Technologies XXI, 78851W (2011).

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Figures (10)

Fig. 1
Fig. 1 Pupil-Tracker (PT): acquisition (left) and system’s description
Fig. 2
Fig. 2 Example of pupil position measurements for one of the subjects in our sample (dashed lines delimit saccades zones).
Fig. 3
Fig. 3 Example 1: PT measurements during fixation for the same subject at different instants (top and bottom). The time unit is Δt = 12.5 ms. Signal is a mixture of saccades, drift, tremor, blink and small head movements. Noise measurement (3σ) is 20 μm.
Fig. 4
Fig. 4 Statistics of displacements as a function of the delay between PT measurements. The unit delay is Δt = 12.5 ms. The curves represent the displacement values δ pm under which we have 99%, 95% and 90% (dashed, dot-dashed and dotted lines, respectively) of the data, when the delay between two successive positions varies from Δt to 6Δt. The average value calculated considering all data is also shown (solid line).
Fig. 5
Fig. 5 Comparison of predictors for Δt and 2Δt, in a good improvement case. Solid line: measurements; dashed line: predictions.
Fig. 6
Fig. 6 Comparison of predictors for Δt and 2Δt, in a case without improvement. Solid line: measurements; dashed line: predictions.
Fig. 7
Fig. 7 Statistics of phase screens generated from real and synthetic Zernike coefficients. The latter shows a more concentrated distribution, while the histogram generated from real data exhibits a large scattering. Dashed lines denote median values.
Fig. 8
Fig. 8 Impact of random displacements of the pupil on the residual rms. The difference between reference and shifted phase screens are piston and tip-tilt subtracted before the computation of the rms values. The curve represents the average value and the error bars are the standard deviations calculated from 500 realizations (a number of 50 randomly selected phase screens were used here, and 10 random directions used for each displacement amplitude).
Fig. 9
Fig. 9 Residual wavefront error (in rms, piston/tip/tilt removed) as a function of the PT frame rate. The curves correspond to upper limits considering all 99%, 95% and 90% smallest absolute value of the one-step displacements (dashed, dot-dashed and dotted lines). The mean and median computed over all one-step displacements are shown as circles and stars, respectively. The red and green lines represent the linear fits as explained in the text. The arrow shows the current WFS rate.
Fig. 10
Fig. 10 Residual wavefront errors obtained from prediction models, expressed in rms (nm), for a prediction delay of Δt. The dashed line denotes the median value of the histogram.

Tables (5)

Tables Icon

Table 1 Comparison of the performance of the predictors for all trajectories. Each cell gives the mean, standard deviation and maximum value of the series of 52 horizontal and vertical prediction errors rms (in μm) computed for all trajectories; the success rate (r) of both Kalman predictors over the dummy predictor for the 52 trajectories is displayed in the two last lines. Saccades and abrupt changes in the trajectories have been masked out in the estimation process, as described in the text. H: horizontal; V: vertical. Δt = 12.5 ms.

Tables Icon

Table 2 p-values for Student tests of assumption ‘the mean rms of SKF+RLS or EK prediction error is smaller than the mean rms of the dummy predictor’ for horizontal and vertical displacements ranging from Δt to 5Δt. Values below 0.0001 are rounded to zero. Any value above 0.05 indicates that the difference is not significant.

Tables Icon

Table 3 Prediction error along trajectories: mean, standard deviation, maximum and minimum values of the percentage of sample times in which the Kalman filters give a smaller prediction error than the dummy predictor. Any value above 50% of mean percentage indicates a better prediction than with dummy.

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Table 4 Relative performance improvements and degradations: mean, standard deviation and maximum value of rms over the 52 trajectories in percentage points and μm (in brackets). For dummy with respect to (w.r.t.) SKF+RLS for example, p% of improvement means that dummy decreases the prediction error rms of SKF+RLS by p%, and p% of degradation means that dummy increases the prediction error rms obtained by SKF+RLS by p%. Horizontal and vertical displacement are considered, with one- and two-step ahead predictions.

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Table 5 Mean wavefront residuals (nm rms) obtained from the prediction models, for three different time delays. These numbers are to be compared with the non PT case, corresponding to an approximate delay of 5Δt and which leads to an mean and median rms of 12.1 nm and 7.7 nm, respectively.

Equations (30)

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δ p k = p k p k 1
δ p k m = p k m p k 1 m
p k + = p k + j = 1 δ p k + j .
p ^ k + | k = p k m + j = 1 δ p ^ k + j | k
p ^ k + | k = p k m .
x k + 1 = A x k + v k
y k = C x k + w k
η k + 1 = a η k + v k
A = ( a h 0 0 a v ) , C = ( 1 0 0 1 ) .
x ^ k | k = x ^ k | k 1 + H k ( y k C x ^ k | k 1 )
x ^ k + 1 | k = A x ^ k | k
θ = ( a h a v )
y k = δ p k = r k θ + w k .
Σ k mc = Σ k 1 mc Σ k 1 mc r k r k Σ k 1 mc 1 + r k Σ k 1 mc r k
θ ^ k = θ ^ k 1 + L k e k
Σ 0 mc = ( r ¯ k 0 r ¯ k 0 ) 1 θ ^ 0 = Σ 0 mc r ¯ k 0 y ¯ k 0
r ¯ k 0 = ( r 0 r k 0 ) , y ¯ k 0 = ( y 0 y k 0 ) .
x k e = ( x 1 , k e x 2 , k e ) = ( x k θ )
x 1 , k + 1 e = A ( x 2 , k e ) x 1 , k e + v k
x 2 , k + 1 e = x 2 , k e
x k + 1 e = f e ( x k e ) + v k e
f ( x k e ) = ( A ( x 2 , k e ) x 1 , k e x 2 , k e ) and v k e = ( v k 0 )
x ^ k | k e = x ^ k | k 1 e + L k ( y k f e ( x ^ k | k 1 e ) )
x ^ k + 1 | k e = f e ( x ^ k | k e )
f e ( x k e ) f e ( x ^ k | k e ) + A k ( x k e x ^ k | k e )
A k = f e ( x k e ) x k e | x ^ k | k e
A k = ( A ( x ^ 2 , k | k e ) diag ( x ^ 1 , k | k e ) 0 I )
y k = C x k e + w k
wavefront rms α × ( PT frame rate ) β [ nm ]

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