Abstract

Karhunen–Loève functions represent the best choice for modal wavefront reconstruction. They are usually built up as a linear combination of Zernike polynomials by using principal component analysis methods; thus they are ordered by covariance. Using Shannon information theory, we provide a best reordering procedure based on the concept of mutual information. This enhances reconstruction efficiency, allowing us to reduce the basis dimension while maintaining the same fitting error in wavefront reconstruction.

© 2010 Optical Society of America

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References

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  1. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207-211 (1976).
    [CrossRef]
  2. F. Roddier, Adaptive Optics in Astronomy (Cambridge Univ. Press, 1999).
    [CrossRef]
  3. G. M. Dai, “Modal wave-front reconstruction with Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A 13, 1218-1255 (1996).
    [CrossRef]
  4. P. K. Baheti and M. A. Neifeld, “Recognition using information-optimal adaptive feature-specific imaging,” J. Opt. Soc. Am. A 1055-1070 (2009).
    [CrossRef]
  5. A. Ashok, P. K. Baheti, and M. A. Neifeld, “Compressive imaging system design using task-specific information,” Appl. Opt. 47, 4457-4471 (2008).
    [CrossRef] [PubMed]
  6. C. E. Shannon, “A Mathematical Theory of Communication,” Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).
  7. A. I. Khinchin, Mathematical Foundations of Information Theory (Dover, 1957).
  8. S. Haykin, Communication Systems (Wiley, 2000).
  9. O. von der Lühe, D. Soltau, T. Berkefeld, and T. Schelenz, “KAOS: adaptive optics system for the Vacuum Tower Telescope at Teide Observatory,” Proc. SPIE 4853, 187-193(2003).

2009 (1)

P. K. Baheti and M. A. Neifeld, “Recognition using information-optimal adaptive feature-specific imaging,” J. Opt. Soc. Am. A 1055-1070 (2009).
[CrossRef]

2008 (1)

2003 (1)

O. von der Lühe, D. Soltau, T. Berkefeld, and T. Schelenz, “KAOS: adaptive optics system for the Vacuum Tower Telescope at Teide Observatory,” Proc. SPIE 4853, 187-193(2003).

1996 (1)

1976 (1)

1948 (1)

C. E. Shannon, “A Mathematical Theory of Communication,” Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).

Ashok, A.

Baheti, P. K.

P. K. Baheti and M. A. Neifeld, “Recognition using information-optimal adaptive feature-specific imaging,” J. Opt. Soc. Am. A 1055-1070 (2009).
[CrossRef]

A. Ashok, P. K. Baheti, and M. A. Neifeld, “Compressive imaging system design using task-specific information,” Appl. Opt. 47, 4457-4471 (2008).
[CrossRef] [PubMed]

Berkefeld, T.

O. von der Lühe, D. Soltau, T. Berkefeld, and T. Schelenz, “KAOS: adaptive optics system for the Vacuum Tower Telescope at Teide Observatory,” Proc. SPIE 4853, 187-193(2003).

Dai, G. M.

Haykin, S.

S. Haykin, Communication Systems (Wiley, 2000).

Khinchin, A. I.

A. I. Khinchin, Mathematical Foundations of Information Theory (Dover, 1957).

Neifeld, M. A.

P. K. Baheti and M. A. Neifeld, “Recognition using information-optimal adaptive feature-specific imaging,” J. Opt. Soc. Am. A 1055-1070 (2009).
[CrossRef]

A. Ashok, P. K. Baheti, and M. A. Neifeld, “Compressive imaging system design using task-specific information,” Appl. Opt. 47, 4457-4471 (2008).
[CrossRef] [PubMed]

Noll, R. J.

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge Univ. Press, 1999).
[CrossRef]

Schelenz, T.

O. von der Lühe, D. Soltau, T. Berkefeld, and T. Schelenz, “KAOS: adaptive optics system for the Vacuum Tower Telescope at Teide Observatory,” Proc. SPIE 4853, 187-193(2003).

Shannon, C. E.

C. E. Shannon, “A Mathematical Theory of Communication,” Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).

Soltau, D.

O. von der Lühe, D. Soltau, T. Berkefeld, and T. Schelenz, “KAOS: adaptive optics system for the Vacuum Tower Telescope at Teide Observatory,” Proc. SPIE 4853, 187-193(2003).

von der Lühe, O.

O. von der Lühe, D. Soltau, T. Berkefeld, and T. Schelenz, “KAOS: adaptive optics system for the Vacuum Tower Telescope at Teide Observatory,” Proc. SPIE 4853, 187-193(2003).

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

C. E. Shannon, “A Mathematical Theory of Communication,” Bell Syst. Tech. J. 27, 379-423, 623-656 (1948).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

G. M. Dai, “Modal wave-front reconstruction with Zernike polynomials and Karhunen-Loève functions,” J. Opt. Soc. Am. A 13, 1218-1255 (1996).
[CrossRef]

P. K. Baheti and M. A. Neifeld, “Recognition using information-optimal adaptive feature-specific imaging,” J. Opt. Soc. Am. A 1055-1070 (2009).
[CrossRef]

Proc. SPIE (1)

O. von der Lühe, D. Soltau, T. Berkefeld, and T. Schelenz, “KAOS: adaptive optics system for the Vacuum Tower Telescope at Teide Observatory,” Proc. SPIE 4853, 187-193(2003).

Other (3)

F. Roddier, Adaptive Optics in Astronomy (Cambridge Univ. Press, 1999).
[CrossRef]

A. I. Khinchin, Mathematical Foundations of Information Theory (Dover, 1957).

S. Haykin, Communication Systems (Wiley, 2000).

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Figures (6)

Fig. 1
Fig. 1

AO loop scheme. The control loop can be though of as composed by three blocks: the wavefront sensor (WFS), the control system (CC) and the deformable mirror (DM). The incoming wavefront phase is sensed by the wavefront sensor and corrected by the deformable mirror. If the time lag is high, the residual uncorrected wavefront aberrations are not negligible.

Fig. 2
Fig. 2

Example of MI estimated by using reconstructed phase histogram.

Fig. 3
Fig. 3

Zernike coefficient distribution.

Fig. 4
Fig. 4

Example of fitting error measured on a single frame. The Zernike base, ordered by MI, yields a better reconstruction with respect to the standard Zernike and KL ordered by MI, with the steepest trend toward zero.

Fig. 5
Fig. 5

Performance factor for the KL basis ordered by MI and the Zernike base ordered by MI. The KL base has been built by using 100 Zernike coefficients. Performance factors show a better fitting error for the Zernike base with respect to the KL base.

Fig. 6
Fig. 6

Histogram of the PF for Zernike and KL bases.

Tables (1)

Tables Icon

Table 1 Performance Factors

Equations (11)

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Z n m ( r , α ) = n + 1 R n m { 2 cos ( m α ) 2 sin ( m α ) 1 ( m = 0 ) } ,
R n m = s = 0 ( n m ) / 2 ( 1 ) s ( n s ) ! s ! [ ( n + m ) / 2 s ] ! [ ( n m ) / 2 s ] ! r n 2 s ,
φ ( r ) = j a j Z j ( r ) ,
A = [ a 1 , a 2 , a 3 , ] ,
H ( X ) = E [ I ( x k ) ] = k = 0 K 1 p k I ( x k ) = k = 0 K 1 p k log ( 1 / p k ) ,
I ( X ; Y ) = H ( X ) H ( X | Y ) ,
H ( X | Y ) = k = 0 K 1 j = 0 J 1 p ( x j , y k ) log [ 1 p ( x j | y k ) ] ,
I ( X ; Y ) = H ( X ) + H ( Y ) H ( X , Y ) ,
H ( X , Y ) = j = 0 J 1 k = 0 K 1 p ( x j , y k ) log [ 1 p ( x j , y k ) ] .
Δ φ n = i , j | φ ( n ) i j φ ( reference ) i j | 2 ,
PF = Δ φ Ze Δ φ new basis Δ φ Ze .

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