Abstract

Adaptive optics systems for future large optical telescopes may require thousands of sensors and actuators. Optimal reconstruction of phase errors using relative measurements requires feedback from every sensor to each actuator, resulting in computational scaling for n actuators of n2. The optimum local reconstructor is investigated, wherein each actuator command depends only on sensor information in a neighboring region. The resulting performance degradation on “global” modes is quantified analytically, and two approaches are considered for recovering global performance. Combining local and global estimators in a two-layer hierarchic architecture yields computations scaling with n4/3; extending this approach to multiple layers yields linear scaling. An alternative approach that maintains a local structure is to allow actuator commands to depend on both local sensors and prior local estimates. This iterative approach is equivalent to a temporal low-pass filter on global information and gives a scaling of n3/2. The algorithms are simulated by using data from the Palomar Observatory adaptive optics system. The analysis is general enough to also be applicable to active optics or other systems with many sensors and actuators.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Hardy, Adaptive Optics for Astronomical Telescopes, Oxford Series on Optical and Imaging Sciences 16 (Oxford U. Press, New York, 1998).
  2. R. Dekany, J. E. Nelson, B. Bauman, “Design considerations for CELT adaptive optics,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 212–225 (2000).
    [CrossRef]
  3. J. Nelson, T. Mast, eds., “Conceptual design for a 30-meter telescope,” (University of California and California Institute of Technology, Berkeley, Calif., 2002).
  4. R. H. Hudgin, “Optimal wave-front estimation,” J. Opt. Soc. Am. 67, 378–382 (1977).
    [CrossRef]
  5. K. Freischlad, C. Zeiss, “Wavefront integration from difference data,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 212–218 (1992).
    [CrossRef]
  6. L. A. Poyneer, D. T. Gavel, J. M. Brase, “Fast wavefront reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. A 19, 2100–2111 (2002).
    [CrossRef]
  7. B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. A 19, 1803–1816 (2002).
    [CrossRef]
  8. L. Gilles, C. R. Vogel, B. L. Ellerbroek, “Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction,” J. Opt. Soc. Am. A 19, 1817–1822 (2002).
    [CrossRef]
  9. W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, “Sparse matrix wave-front estimators for adaptive-optics systems for large ground-based telescopes,” Opt. Lett. 20, 955–957 (1995).
    [CrossRef] [PubMed]
  10. T. P. Murphy, R. G. Lyon, J. E. Dorband, J. M. Hollis, “Sparse matrix approximation method for an active optical control system,” Appl. Opt. 40, 6505–6514 (2001).
    [CrossRef]
  11. K. Li, E. B. Kosmatopoulos, P. A. Ioannou, H. Ryaciotaki-Boussalis, “Large segmented telescopes: centralized, decentralized and overlapping control designs,” IEEE Control Syst. Mag., October2000, 59–72.
  12. R. D’Andrea, C. Langbort, R. Chandra, “A state space approach to control of interconnected systems,” in Mathematical Systems Theory in Biology, Communication, Computation and Finance, IMA Vol. 134 in Mathematics and Its Application, J. Rosenthal, D. S. Gillian, eds. (Springer-Verlag, New York, 2003), pp. 157–182.
  13. D. M. Young, Iterative Solution of Large Linear Systems (Academic, New York, 1971).
  14. F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1035–1044 (2002).
    [CrossRef]
  15. M. Troy, R. Dekany, G. Brack, B. Oppenheimer, E. Bloemhof, T. Trinh, F. Dekens, F. Shi, T. Hayward, B. Brandl, “Palomar adaptive optics project: status and performance,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 31–40 (2000).
    [CrossRef]
  16. J.-P. Gaffard, G. Ledanois, “Adaptive optics transfer function modeling,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 34–45 (1991).
    [CrossRef]
  17. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  18. T. Truong, G. Brack, T. Trinh, M. Troy, F. Shi, R. G. Dekany, “Real-time wavefront processors for the next generation of adaptive optics systems: a design and analysis,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 911–922 (2002).
    [CrossRef]

2002

2001

2000

K. Li, E. B. Kosmatopoulos, P. A. Ioannou, H. Ryaciotaki-Boussalis, “Large segmented telescopes: centralized, decentralized and overlapping control designs,” IEEE Control Syst. Mag., October2000, 59–72.

1995

1977

1976

Bauman, B.

R. Dekany, J. E. Nelson, B. Bauman, “Design considerations for CELT adaptive optics,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 212–225 (2000).
[CrossRef]

Bloemhof, E.

M. Troy, R. Dekany, G. Brack, B. Oppenheimer, E. Bloemhof, T. Trinh, F. Dekens, F. Shi, T. Hayward, B. Brandl, “Palomar adaptive optics project: status and performance,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 31–40 (2000).
[CrossRef]

Brack, G.

M. Troy, R. Dekany, G. Brack, B. Oppenheimer, E. Bloemhof, T. Trinh, F. Dekens, F. Shi, T. Hayward, B. Brandl, “Palomar adaptive optics project: status and performance,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 31–40 (2000).
[CrossRef]

T. Truong, G. Brack, T. Trinh, M. Troy, F. Shi, R. G. Dekany, “Real-time wavefront processors for the next generation of adaptive optics systems: a design and analysis,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 911–922 (2002).
[CrossRef]

Brack, G. L.

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1035–1044 (2002).
[CrossRef]

Brandl, B.

M. Troy, R. Dekany, G. Brack, B. Oppenheimer, E. Bloemhof, T. Trinh, F. Dekens, F. Shi, T. Hayward, B. Brandl, “Palomar adaptive optics project: status and performance,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 31–40 (2000).
[CrossRef]

Brase, J. M.

Burruss, R. S.

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1035–1044 (2002).
[CrossRef]

Chandra, R.

R. D’Andrea, C. Langbort, R. Chandra, “A state space approach to control of interconnected systems,” in Mathematical Systems Theory in Biology, Communication, Computation and Finance, IMA Vol. 134 in Mathematics and Its Application, J. Rosenthal, D. S. Gillian, eds. (Springer-Verlag, New York, 2003), pp. 157–182.

D’Andrea, R.

R. D’Andrea, C. Langbort, R. Chandra, “A state space approach to control of interconnected systems,” in Mathematical Systems Theory in Biology, Communication, Computation and Finance, IMA Vol. 134 in Mathematics and Its Application, J. Rosenthal, D. S. Gillian, eds. (Springer-Verlag, New York, 2003), pp. 157–182.

Dekany, R.

M. Troy, R. Dekany, G. Brack, B. Oppenheimer, E. Bloemhof, T. Trinh, F. Dekens, F. Shi, T. Hayward, B. Brandl, “Palomar adaptive optics project: status and performance,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 31–40 (2000).
[CrossRef]

R. Dekany, J. E. Nelson, B. Bauman, “Design considerations for CELT adaptive optics,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 212–225 (2000).
[CrossRef]

Dekany, R. G.

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1035–1044 (2002).
[CrossRef]

T. Truong, G. Brack, T. Trinh, M. Troy, F. Shi, R. G. Dekany, “Real-time wavefront processors for the next generation of adaptive optics systems: a design and analysis,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 911–922 (2002).
[CrossRef]

Dekens, F.

M. Troy, R. Dekany, G. Brack, B. Oppenheimer, E. Bloemhof, T. Trinh, F. Dekens, F. Shi, T. Hayward, B. Brandl, “Palomar adaptive optics project: status and performance,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 31–40 (2000).
[CrossRef]

Dorband, J. E.

Ellerbroek, B. L.

Freischlad, K.

K. Freischlad, C. Zeiss, “Wavefront integration from difference data,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 212–218 (1992).
[CrossRef]

Gaffard, J.-P.

J.-P. Gaffard, G. Ledanois, “Adaptive optics transfer function modeling,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 34–45 (1991).
[CrossRef]

Gavel, D. T.

Gilles, L.

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes, Oxford Series on Optical and Imaging Sciences 16 (Oxford U. Press, New York, 1998).

Hayward, T.

M. Troy, R. Dekany, G. Brack, B. Oppenheimer, E. Bloemhof, T. Trinh, F. Dekens, F. Shi, T. Hayward, B. Brandl, “Palomar adaptive optics project: status and performance,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 31–40 (2000).
[CrossRef]

Hollis, J. M.

Hudgin, R. H.

Ioannou, P. A.

K. Li, E. B. Kosmatopoulos, P. A. Ioannou, H. Ryaciotaki-Boussalis, “Large segmented telescopes: centralized, decentralized and overlapping control designs,” IEEE Control Syst. Mag., October2000, 59–72.

Kibblewhite, E. J.

Kosmatopoulos, E. B.

K. Li, E. B. Kosmatopoulos, P. A. Ioannou, H. Ryaciotaki-Boussalis, “Large segmented telescopes: centralized, decentralized and overlapping control designs,” IEEE Control Syst. Mag., October2000, 59–72.

Langbort, C.

R. D’Andrea, C. Langbort, R. Chandra, “A state space approach to control of interconnected systems,” in Mathematical Systems Theory in Biology, Communication, Computation and Finance, IMA Vol. 134 in Mathematics and Its Application, J. Rosenthal, D. S. Gillian, eds. (Springer-Verlag, New York, 2003), pp. 157–182.

Ledanois, G.

J.-P. Gaffard, G. Ledanois, “Adaptive optics transfer function modeling,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 34–45 (1991).
[CrossRef]

Li, K.

K. Li, E. B. Kosmatopoulos, P. A. Ioannou, H. Ryaciotaki-Boussalis, “Large segmented telescopes: centralized, decentralized and overlapping control designs,” IEEE Control Syst. Mag., October2000, 59–72.

Lyon, R. G.

MacMartin, D. G.

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1035–1044 (2002).
[CrossRef]

Murphy, T. P.

Nelson, J. E.

R. Dekany, J. E. Nelson, B. Bauman, “Design considerations for CELT adaptive optics,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 212–225 (2000).
[CrossRef]

Noll, R. J.

Oppenheimer, B.

M. Troy, R. Dekany, G. Brack, B. Oppenheimer, E. Bloemhof, T. Trinh, F. Dekens, F. Shi, T. Hayward, B. Brandl, “Palomar adaptive optics project: status and performance,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 31–40 (2000).
[CrossRef]

Poyneer, L. A.

Ryaciotaki-Boussalis, H.

K. Li, E. B. Kosmatopoulos, P. A. Ioannou, H. Ryaciotaki-Boussalis, “Large segmented telescopes: centralized, decentralized and overlapping control designs,” IEEE Control Syst. Mag., October2000, 59–72.

Shi, F.

T. Truong, G. Brack, T. Trinh, M. Troy, F. Shi, R. G. Dekany, “Real-time wavefront processors for the next generation of adaptive optics systems: a design and analysis,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 911–922 (2002).
[CrossRef]

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1035–1044 (2002).
[CrossRef]

M. Troy, R. Dekany, G. Brack, B. Oppenheimer, E. Bloemhof, T. Trinh, F. Dekens, F. Shi, T. Hayward, B. Brandl, “Palomar adaptive optics project: status and performance,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 31–40 (2000).
[CrossRef]

Trinh, T.

M. Troy, R. Dekany, G. Brack, B. Oppenheimer, E. Bloemhof, T. Trinh, F. Dekens, F. Shi, T. Hayward, B. Brandl, “Palomar adaptive optics project: status and performance,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 31–40 (2000).
[CrossRef]

T. Truong, G. Brack, T. Trinh, M. Troy, F. Shi, R. G. Dekany, “Real-time wavefront processors for the next generation of adaptive optics systems: a design and analysis,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 911–922 (2002).
[CrossRef]

Troy, M.

T. Truong, G. Brack, T. Trinh, M. Troy, F. Shi, R. G. Dekany, “Real-time wavefront processors for the next generation of adaptive optics systems: a design and analysis,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 911–922 (2002).
[CrossRef]

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1035–1044 (2002).
[CrossRef]

M. Troy, R. Dekany, G. Brack, B. Oppenheimer, E. Bloemhof, T. Trinh, F. Dekens, F. Shi, T. Hayward, B. Brandl, “Palomar adaptive optics project: status and performance,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 31–40 (2000).
[CrossRef]

Truong, T.

T. Truong, G. Brack, T. Trinh, M. Troy, F. Shi, R. G. Dekany, “Real-time wavefront processors for the next generation of adaptive optics systems: a design and analysis,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 911–922 (2002).
[CrossRef]

Vogel, C. R.

Vuilleumier, R.

Wild, W. J.

Young, D. M.

D. M. Young, Iterative Solution of Large Linear Systems (Academic, New York, 1971).

Zeiss, C.

K. Freischlad, C. Zeiss, “Wavefront integration from difference data,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 212–218 (1992).
[CrossRef]

Appl. Opt.

IEEE Control Syst. Mag.

K. Li, E. B. Kosmatopoulos, P. A. Ioannou, H. Ryaciotaki-Boussalis, “Large segmented telescopes: centralized, decentralized and overlapping control designs,” IEEE Control Syst. Mag., October2000, 59–72.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Other

R. D’Andrea, C. Langbort, R. Chandra, “A state space approach to control of interconnected systems,” in Mathematical Systems Theory in Biology, Communication, Computation and Finance, IMA Vol. 134 in Mathematics and Its Application, J. Rosenthal, D. S. Gillian, eds. (Springer-Verlag, New York, 2003), pp. 157–182.

D. M. Young, Iterative Solution of Large Linear Systems (Academic, New York, 1971).

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1035–1044 (2002).
[CrossRef]

M. Troy, R. Dekany, G. Brack, B. Oppenheimer, E. Bloemhof, T. Trinh, F. Dekens, F. Shi, T. Hayward, B. Brandl, “Palomar adaptive optics project: status and performance,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 31–40 (2000).
[CrossRef]

J.-P. Gaffard, G. Ledanois, “Adaptive optics transfer function modeling,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 34–45 (1991).
[CrossRef]

T. Truong, G. Brack, T. Trinh, M. Troy, F. Shi, R. G. Dekany, “Real-time wavefront processors for the next generation of adaptive optics systems: a design and analysis,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 911–922 (2002).
[CrossRef]

J. W. Hardy, Adaptive Optics for Astronomical Telescopes, Oxford Series on Optical and Imaging Sciences 16 (Oxford U. Press, New York, 1998).

R. Dekany, J. E. Nelson, B. Bauman, “Design considerations for CELT adaptive optics,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 212–225 (2000).
[CrossRef]

J. Nelson, T. Mast, eds., “Conceptual design for a 30-meter telescope,” (University of California and California Institute of Technology, Berkeley, Calif., 2002).

K. Freischlad, C. Zeiss, “Wavefront integration from difference data,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 212–218 (1992).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Actuator (○) and sensor (+) layout for Palomar adaptive optics (AO). The circles indicate the outer edge of the mirror and the region obscured by the telescope secondary mirror. Only those subapertures used in reconstruction are shown.

Fig. 2
Fig. 2

Control block diagram. Displacement x is influenced by control u and disturbance w. Control consists of estimation xˆ=Ky and control u=C(s)xˆ.

Fig. 3
Fig. 3

Hierarchy schematic for AO geometry with d=4. Global superelements are shaded, and global displacements are indicated with solid circles.

Fig. 4
Fig. 4

Schematic of multiple-layer hierarchy in one spatial dimension. Each layer ℓ aggregates information y(l-1)=Ψ(l)y(l) to pass up to the next-coarser layer and uses interpolation and averaging of the estimate from the next-coarser layer, x^g(l)=Φ(l)x^(l-1), to correct the information missing from the local estimate x^l(l)=K(l)y(l) at layer ℓ.

Fig. 5
Fig. 5

Modal loop gain of different spatial extent of local control. Basis functions are Zernike modes ordered with increasing spatial wave number.

Fig. 6
Fig. 6

Modal loop gain comparing local control with d=4 and two-layer hierarchic control with 21 global variables.

Fig. 7
Fig. 7

Modal performance on Palomar AO simulation comparing the least-squares optimum with local control (d=4), two-layer hierarchic control, and iterative control (N=1).

Fig. 8
Fig. 8

Simulated performance versus computational improvement for Palomar AO with sparse, hierarchic, and iterative controllers. Performance is determined by the rms residual over the first 91 Zernike modes for a fixed sample rate.

Fig. 9
Fig. 9

Simulated performance versus noise multiplier for Palomar AO with sparse, hierarchic, and iterative controllers.

Equations (27)

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

x=u+w
y=Ax+η,
A#=limρ0(ATA+ρI)-1AT.
xˆ-x=(KA-I)x+Kη
=-m=1Mμm(μmTx)+Kη,
K=QAT(AQAT+R)-1
=(ATR-1A+Q-1)-1ATR-1.
yjΩisx(yj)-x(xi)<d/2,
x^i|yΩi=x^i|y-m=1Mi(μ˜mi)i[(μ˜mi)Tx]+Kη.
g(kx, ky)=1-sinc(kxd)sinc(kyd).
g11=1-sinc2(πδ)
π2δ2/3.
xˆ=x^l+x^g=Ky+Φ[Aξ#(Ψy)].
x^(1)=K(1)y(1),K(1)=(A(1))#.
x^(l)=K(l)y(l)+Φ(l)x^(l-1),
y(l)=Ψ(l+1)y(l+1),
k=014k=43
x^k=Kyk+Gx^k-1,
ζ1=Kyk+Gx^k-1,
ξm+1=Kyk+Gζm,m=1,, N-2,
x^k=Kyk+GζN-1.
G=I-KA.
(Gxˆ)i=m=1M(μ˜mi)i[(μ˜mi)Txˆ].
xˆ(z)=(I-Gz-1)-1KAx(z)=H(z)x(z).
H(z)=Vσizz-(1-σi)VT.
fcfs=π6 δ2N.
x^k=i=0N-1GiKy+GNx^K-1,

Metrics