Abstract

The active control systems of segmented mirror telescopes are vulnerable to a malfunction of a few (or even one) of their segment edge sensors, the effects of which can propagate through the entire system and seriously compromise the overall telescope image quality. Since there are thousands of such sensors in the extremely large telescopes now under development, it is essential to develop fast and efficient algorithms that can identify bad sensors so that they can be removed from the control loop. Such algorithms are nontrivial; for example, a simple residual-to-the-fit test will often fail to identify a bad sensor. We propose an algorithm that can reliably identify a single bad sensor and we extend it to the more difficult case of multiple bad sensors. Somewhat surprisingly, the identification of a fixed number of bad sensors does not necessarily become more difficult as the telescope becomes larger and the number of sensors in the control system increases.

© 2009 Optical Society of America

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References

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  1. J. E. Nelson, T. S. Mast, and S. M. Faber, “The design of the Keck Observatory and Telescope,” Keck Observatory Report 90 (W. M. Keck Observatory, Kamuela, Hawaii 1985).
  2. J. A. Booth, M. T. Adams, E. S. Barker, F. N. Bash, J. R. Fowler, J. M. Good, G. J. Hill, P. W. Kelton, D. L. Lambert, P. J. MacQueen, P. Palunas, L. W. Ramsey, and G. L. Wesley, “The Hobby-Eberly Telescope: performance upgrades, status and plans,” Proc. SPIE 5489, 288-299 (2004).
    [CrossRef]
  3. S. Buous, J. Menzies, and H. Gajjar, “SALT segmented primary mirror: commissioning capacitive edge sensing system and performance comparison with inductive sensor,” Proc. SPIE 7012, 70123G (2008).
    [CrossRef]
  4. J. M. Rodriguez-Espinoza, P. Alvarez, and F. Sanchez, “The GTC, an advanced 10 m telescope for the ORM,” Astrophys. Space Sci. 263, 355-360 (1998).
    [CrossRef]
  5. J. Nelson and G. H. Sanders, “TMT status report,” Proc. SPIE 6267, 626728 (2006).
    [CrossRef]
  6. T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., Vol. 57 of European Southern Observatory Conference and Workshop Proceedings (SPIE, 2000), pp. 72-82.
  7. R. Gilmozzi and J. Spyromilio, “The 42 m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
    [CrossRef]
  8. See, e.g., http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse.
  9. R. W. Cohen, T. S. Mast, and J. E. Nelson, “Performance of the W. M. Keck Telescope active mirror control system,” Proc. SPIE 2199, 105-116 (1994).
    [CrossRef]
  10. G. Chanan, D. G. MacMartin, J. Nelson, and T. Mast, “Control and alignment of segmented-mirror telescopes: matrices, modes, and error propagation,” Appl. Opt. 43, 1223-1232(2004).
    [CrossRef] [PubMed]

2008

S. Buous, J. Menzies, and H. Gajjar, “SALT segmented primary mirror: commissioning capacitive edge sensing system and performance comparison with inductive sensor,” Proc. SPIE 7012, 70123G (2008).
[CrossRef]

R. Gilmozzi and J. Spyromilio, “The 42 m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
[CrossRef]

2006

J. Nelson and G. H. Sanders, “TMT status report,” Proc. SPIE 6267, 626728 (2006).
[CrossRef]

2004

J. A. Booth, M. T. Adams, E. S. Barker, F. N. Bash, J. R. Fowler, J. M. Good, G. J. Hill, P. W. Kelton, D. L. Lambert, P. J. MacQueen, P. Palunas, L. W. Ramsey, and G. L. Wesley, “The Hobby-Eberly Telescope: performance upgrades, status and plans,” Proc. SPIE 5489, 288-299 (2004).
[CrossRef]

G. Chanan, D. G. MacMartin, J. Nelson, and T. Mast, “Control and alignment of segmented-mirror telescopes: matrices, modes, and error propagation,” Appl. Opt. 43, 1223-1232(2004).
[CrossRef] [PubMed]

1998

J. M. Rodriguez-Espinoza, P. Alvarez, and F. Sanchez, “The GTC, an advanced 10 m telescope for the ORM,” Astrophys. Space Sci. 263, 355-360 (1998).
[CrossRef]

1994

R. W. Cohen, T. S. Mast, and J. E. Nelson, “Performance of the W. M. Keck Telescope active mirror control system,” Proc. SPIE 2199, 105-116 (1994).
[CrossRef]

Adams, M. T.

J. A. Booth, M. T. Adams, E. S. Barker, F. N. Bash, J. R. Fowler, J. M. Good, G. J. Hill, P. W. Kelton, D. L. Lambert, P. J. MacQueen, P. Palunas, L. W. Ramsey, and G. L. Wesley, “The Hobby-Eberly Telescope: performance upgrades, status and plans,” Proc. SPIE 5489, 288-299 (2004).
[CrossRef]

Alvarez, P.

J. M. Rodriguez-Espinoza, P. Alvarez, and F. Sanchez, “The GTC, an advanced 10 m telescope for the ORM,” Astrophys. Space Sci. 263, 355-360 (1998).
[CrossRef]

Andersen, T.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., Vol. 57 of European Southern Observatory Conference and Workshop Proceedings (SPIE, 2000), pp. 72-82.

Ardeberg, A.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., Vol. 57 of European Southern Observatory Conference and Workshop Proceedings (SPIE, 2000), pp. 72-82.

Barker, E. S.

J. A. Booth, M. T. Adams, E. S. Barker, F. N. Bash, J. R. Fowler, J. M. Good, G. J. Hill, P. W. Kelton, D. L. Lambert, P. J. MacQueen, P. Palunas, L. W. Ramsey, and G. L. Wesley, “The Hobby-Eberly Telescope: performance upgrades, status and plans,” Proc. SPIE 5489, 288-299 (2004).
[CrossRef]

Bash, F. N.

J. A. Booth, M. T. Adams, E. S. Barker, F. N. Bash, J. R. Fowler, J. M. Good, G. J. Hill, P. W. Kelton, D. L. Lambert, P. J. MacQueen, P. Palunas, L. W. Ramsey, and G. L. Wesley, “The Hobby-Eberly Telescope: performance upgrades, status and plans,” Proc. SPIE 5489, 288-299 (2004).
[CrossRef]

Beckers, J.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., Vol. 57 of European Southern Observatory Conference and Workshop Proceedings (SPIE, 2000), pp. 72-82.

Booth, J. A.

J. A. Booth, M. T. Adams, E. S. Barker, F. N. Bash, J. R. Fowler, J. M. Good, G. J. Hill, P. W. Kelton, D. L. Lambert, P. J. MacQueen, P. Palunas, L. W. Ramsey, and G. L. Wesley, “The Hobby-Eberly Telescope: performance upgrades, status and plans,” Proc. SPIE 5489, 288-299 (2004).
[CrossRef]

Buous, S.

S. Buous, J. Menzies, and H. Gajjar, “SALT segmented primary mirror: commissioning capacitive edge sensing system and performance comparison with inductive sensor,” Proc. SPIE 7012, 70123G (2008).
[CrossRef]

Chanan, G.

Cohen, R. W.

R. W. Cohen, T. S. Mast, and J. E. Nelson, “Performance of the W. M. Keck Telescope active mirror control system,” Proc. SPIE 2199, 105-116 (1994).
[CrossRef]

Faber, S. M.

J. E. Nelson, T. S. Mast, and S. M. Faber, “The design of the Keck Observatory and Telescope,” Keck Observatory Report 90 (W. M. Keck Observatory, Kamuela, Hawaii 1985).

Flicker, R.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., Vol. 57 of European Southern Observatory Conference and Workshop Proceedings (SPIE, 2000), pp. 72-82.

Fowler, J. R.

J. A. Booth, M. T. Adams, E. S. Barker, F. N. Bash, J. R. Fowler, J. M. Good, G. J. Hill, P. W. Kelton, D. L. Lambert, P. J. MacQueen, P. Palunas, L. W. Ramsey, and G. L. Wesley, “The Hobby-Eberly Telescope: performance upgrades, status and plans,” Proc. SPIE 5489, 288-299 (2004).
[CrossRef]

Gajjar, H.

S. Buous, J. Menzies, and H. Gajjar, “SALT segmented primary mirror: commissioning capacitive edge sensing system and performance comparison with inductive sensor,” Proc. SPIE 7012, 70123G (2008).
[CrossRef]

Gilmozzi, R.

R. Gilmozzi and J. Spyromilio, “The 42 m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
[CrossRef]

Gontcharov, A.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., Vol. 57 of European Southern Observatory Conference and Workshop Proceedings (SPIE, 2000), pp. 72-82.

Good, J. M.

J. A. Booth, M. T. Adams, E. S. Barker, F. N. Bash, J. R. Fowler, J. M. Good, G. J. Hill, P. W. Kelton, D. L. Lambert, P. J. MacQueen, P. Palunas, L. W. Ramsey, and G. L. Wesley, “The Hobby-Eberly Telescope: performance upgrades, status and plans,” Proc. SPIE 5489, 288-299 (2004).
[CrossRef]

Hill, G. J.

J. A. Booth, M. T. Adams, E. S. Barker, F. N. Bash, J. R. Fowler, J. M. Good, G. J. Hill, P. W. Kelton, D. L. Lambert, P. J. MacQueen, P. Palunas, L. W. Ramsey, and G. L. Wesley, “The Hobby-Eberly Telescope: performance upgrades, status and plans,” Proc. SPIE 5489, 288-299 (2004).
[CrossRef]

Jessen, N. C.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., Vol. 57 of European Southern Observatory Conference and Workshop Proceedings (SPIE, 2000), pp. 72-82.

Kelton, P. W.

J. A. Booth, M. T. Adams, E. S. Barker, F. N. Bash, J. R. Fowler, J. M. Good, G. J. Hill, P. W. Kelton, D. L. Lambert, P. J. MacQueen, P. Palunas, L. W. Ramsey, and G. L. Wesley, “The Hobby-Eberly Telescope: performance upgrades, status and plans,” Proc. SPIE 5489, 288-299 (2004).
[CrossRef]

Lambert, D. L.

J. A. Booth, M. T. Adams, E. S. Barker, F. N. Bash, J. R. Fowler, J. M. Good, G. J. Hill, P. W. Kelton, D. L. Lambert, P. J. MacQueen, P. Palunas, L. W. Ramsey, and G. L. Wesley, “The Hobby-Eberly Telescope: performance upgrades, status and plans,” Proc. SPIE 5489, 288-299 (2004).
[CrossRef]

MacMartin, D. G.

MacQueen, P. J.

J. A. Booth, M. T. Adams, E. S. Barker, F. N. Bash, J. R. Fowler, J. M. Good, G. J. Hill, P. W. Kelton, D. L. Lambert, P. J. MacQueen, P. Palunas, L. W. Ramsey, and G. L. Wesley, “The Hobby-Eberly Telescope: performance upgrades, status and plans,” Proc. SPIE 5489, 288-299 (2004).
[CrossRef]

Mannery, E.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., Vol. 57 of European Southern Observatory Conference and Workshop Proceedings (SPIE, 2000), pp. 72-82.

Mast, T.

Mast, T. S.

R. W. Cohen, T. S. Mast, and J. E. Nelson, “Performance of the W. M. Keck Telescope active mirror control system,” Proc. SPIE 2199, 105-116 (1994).
[CrossRef]

J. E. Nelson, T. S. Mast, and S. M. Faber, “The design of the Keck Observatory and Telescope,” Keck Observatory Report 90 (W. M. Keck Observatory, Kamuela, Hawaii 1985).

Menzies, J.

S. Buous, J. Menzies, and H. Gajjar, “SALT segmented primary mirror: commissioning capacitive edge sensing system and performance comparison with inductive sensor,” Proc. SPIE 7012, 70123G (2008).
[CrossRef]

Nelson, J.

Nelson, J. E.

R. W. Cohen, T. S. Mast, and J. E. Nelson, “Performance of the W. M. Keck Telescope active mirror control system,” Proc. SPIE 2199, 105-116 (1994).
[CrossRef]

J. E. Nelson, T. S. Mast, and S. M. Faber, “The design of the Keck Observatory and Telescope,” Keck Observatory Report 90 (W. M. Keck Observatory, Kamuela, Hawaii 1985).

Owner-Petersen, M.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., Vol. 57 of European Southern Observatory Conference and Workshop Proceedings (SPIE, 2000), pp. 72-82.

Palunas, P.

J. A. Booth, M. T. Adams, E. S. Barker, F. N. Bash, J. R. Fowler, J. M. Good, G. J. Hill, P. W. Kelton, D. L. Lambert, P. J. MacQueen, P. Palunas, L. W. Ramsey, and G. L. Wesley, “The Hobby-Eberly Telescope: performance upgrades, status and plans,” Proc. SPIE 5489, 288-299 (2004).
[CrossRef]

Ramsey, L. W.

J. A. Booth, M. T. Adams, E. S. Barker, F. N. Bash, J. R. Fowler, J. M. Good, G. J. Hill, P. W. Kelton, D. L. Lambert, P. J. MacQueen, P. Palunas, L. W. Ramsey, and G. L. Wesley, “The Hobby-Eberly Telescope: performance upgrades, status and plans,” Proc. SPIE 5489, 288-299 (2004).
[CrossRef]

Riewaldt, H.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., Vol. 57 of European Southern Observatory Conference and Workshop Proceedings (SPIE, 2000), pp. 72-82.

Rodriguez-Espinoza, J. M.

J. M. Rodriguez-Espinoza, P. Alvarez, and F. Sanchez, “The GTC, an advanced 10 m telescope for the ORM,” Astrophys. Space Sci. 263, 355-360 (1998).
[CrossRef]

Sanchez, F.

J. M. Rodriguez-Espinoza, P. Alvarez, and F. Sanchez, “The GTC, an advanced 10 m telescope for the ORM,” Astrophys. Space Sci. 263, 355-360 (1998).
[CrossRef]

Sanders, G. H.

J. Nelson and G. H. Sanders, “TMT status report,” Proc. SPIE 6267, 626728 (2006).
[CrossRef]

Spyromilio, J.

R. Gilmozzi and J. Spyromilio, “The 42 m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
[CrossRef]

Wesley, G. L.

J. A. Booth, M. T. Adams, E. S. Barker, F. N. Bash, J. R. Fowler, J. M. Good, G. J. Hill, P. W. Kelton, D. L. Lambert, P. J. MacQueen, P. Palunas, L. W. Ramsey, and G. L. Wesley, “The Hobby-Eberly Telescope: performance upgrades, status and plans,” Proc. SPIE 5489, 288-299 (2004).
[CrossRef]

Appl. Opt.

Astrophys. Space Sci.

J. M. Rodriguez-Espinoza, P. Alvarez, and F. Sanchez, “The GTC, an advanced 10 m telescope for the ORM,” Astrophys. Space Sci. 263, 355-360 (1998).
[CrossRef]

Proc. SPIE

J. Nelson and G. H. Sanders, “TMT status report,” Proc. SPIE 6267, 626728 (2006).
[CrossRef]

J. A. Booth, M. T. Adams, E. S. Barker, F. N. Bash, J. R. Fowler, J. M. Good, G. J. Hill, P. W. Kelton, D. L. Lambert, P. J. MacQueen, P. Palunas, L. W. Ramsey, and G. L. Wesley, “The Hobby-Eberly Telescope: performance upgrades, status and plans,” Proc. SPIE 5489, 288-299 (2004).
[CrossRef]

S. Buous, J. Menzies, and H. Gajjar, “SALT segmented primary mirror: commissioning capacitive edge sensing system and performance comparison with inductive sensor,” Proc. SPIE 7012, 70123G (2008).
[CrossRef]

R. Gilmozzi and J. Spyromilio, “The 42 m European ELT: status,” Proc. SPIE 7012, 701219 (2008).
[CrossRef]

R. W. Cohen, T. S. Mast, and J. E. Nelson, “Performance of the W. M. Keck Telescope active mirror control system,” Proc. SPIE 2199, 105-116 (1994).
[CrossRef]

Other

J. E. Nelson, T. S. Mast, and S. M. Faber, “The design of the Keck Observatory and Telescope,” Keck Observatory Report 90 (W. M. Keck Observatory, Kamuela, Hawaii 1985).

See, e.g., http://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse.

T. Andersen, A. Ardeberg, J. Beckers, R. Flicker, A. Gontcharov, N. C. Jessen, E. Mannery, M. Owner-Petersen, and H. Riewaldt, “The proposed 50 m Swedish Extremely Large Telescope,” in Proceedings of the Backaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, and R. Gilmozzi, eds., Vol. 57 of European Southern Observatory Conference and Workshop Proceedings (SPIE, 2000), pp. 72-82.

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Figures (10)

Fig. 1
Fig. 1

Keck primary mirror surface error as a function of the error in the malfunctioning sensor(s) for one, two, or three malfunctioning sensors. Both the surface error and the sensor error are in units of the standard deviation (σ) of the noise in the readings of the good sensors. The surface error does not go to zero as the error in the malfunctioning sensors goes to zero because of noise in the properly functioning sensors.

Fig. 2
Fig. 2

Hypothetical distribution of sensor readings for the Keck active control system. The sensor readings are randomly distributed with zero mean and unit standard deviation, except for sensor number 149, which has an anomalous reading of 15 units.

Fig. 3
Fig. 3

Residuals to the fit for the sensor distribution in Fig. 2. The malfunctioning sensor does not have a correspondingly large residual; in fact, it is not even the maximum. This figure illustrates the difficulty of the malfunctioning sensor detection problem. In practice, the zero points corresponding to the sensor readings are not known, so one is forced to look for the signature of a malfunctioning sensor in the residuals to the fit, where it is much less obvious.

Fig. 4
Fig. 4

Maximum over i of | r i j / r j j | for i j , as a function of j. Here r i j is the ( i , j ) element of the residual matrix R for the Keck telescope active control system. If this parameter is greater than unity for a given sensor, then the “maximum residual” method will fail to identify a malfunction in that sensor, even in the limit of infinite signal to noise. This is the case for 18 of the 168 sensors. The high sensor numbers of the problematic sensors reflect their location at the periphery of the primary mirror (see Fig. 5).

Fig. 5
Fig. 5

Solid circles show the positions of the 18 sensors on the Keck telescope that would give false negatives according to the maximum residual test. The nearest-neighbor open circles give the positions of the sensors that would give the corresponding false positives. Note that, in some cases, two different nearest neighbors both give false positives. (The map is not quite symmetrical because the distributions of the sensors and actuators (not shown) have different symmetries.)

Fig. 6
Fig. 6

Same as Fig. 4, but for the TMT active control system. In this case, the “maximum residual” method fails for 78 of 2772 sensors for which the plotted statistic exceeds unity.

Fig. 7
Fig. 7

Same as Fig. 4, but now the plotted function is the maximum over i of the ratio | r i j / r i i r j j | for i j , as a function of j, again for the Keck telescope. Note the different denominator compared to Fig. 4. In the current figure, this ratio never exceeds unity for any sensor. As a result, this MWR algorithm will, in the limit of infinite signal to noise, always identify a single malfunctioning sensor.

Fig. 8
Fig. 8

Same as Fig. 7, but for the TMT active control system. Again, the fact that the plotted statistic is always strictly less than unity shows that the MWR algorithm will, in the limit of infinite signal to noise, always identify a malfunctioning single sensor.

Fig. 9
Fig. 9

Same as Fig. 3, but for the weighted sensor residual, with the weighting as described in the text. This time, the malfunctioning sensor has the MWR. In the limit where the error of the (single) bad sensor is large compared to the uncertainty in the readings of the good sensors, the algorithm is guaranteed to work.

Fig. 10
Fig. 10

TMT primary mirror surface error as a function of the error in the malfunctioning sensor(s) for one, two, or three malfunctioning sensors. Both the surface error and the sensor error are in units of the standard deviation (σ) of the noise in the readings of the good sensors. Compared to Keck (see Fig. 1), the consequences of a given number of bad sensors for TMT are relatively smaller for small sensor errors and absolutely smaller for large sensor errors. See text for explanation.

Tables (3)

Tables Icon

Table 1 Maximum Weighted Residual Algorithm Performance for Single Bad Sensors (1000 Trials)

Tables Icon

Table 2 Keck Maximum Weighted Residual Algorithm Performance for 0, 1, 2, or 3 Bad Sensors (1000 Trials)

Tables Icon

Table 3 Thirty Meter Telescope Maximum Weighted Residual Algorithm Performance for 0, 1, 2, or 3 Bad Sensors (1000 Trials)

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

Aa = b .
Bb = a ,
b = b A a .
R = I AB ,
i r i j r i k = i r j i r i k = r j k
i r i j 2 = r j j ,
Rb = b .
p i = b · r i / r i i ,
p i = j r j i b j / r i i ,
= j k r j i r j k b k / r i i ,
= k r i k b k / r i i ,
= b i / r i i ,
s i , j k = | r i j r i i ± r i k r i i | ,
b ( 1 ) = b ( 0 ) ( b ( 0 ) · r ^ j 1 ) r ^ j 1 ,

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