Abstract

We develop a sparse matrix approximation method to decompose a wave front into a basis set of actuator influence functions for an active optical system consisting of a deformable mirror and a segmented primary mirror. The wave front used is constructed by Zernike polynomials to simulate the output of a phase-retrieval algorithm. Results of a Monte Carlo simulation of the optical control loop are compared with the standard, nonsparse approach in terms of accuracy and precision, as well as computational speed and memory. The sparse matrix approximation method can yield more than a 50-fold increase in the speed and a 20-fold reduction in matrix size and a commensurate decrease in required memory, with less than 10% degradation in solution accuracy. Our method is also shown to be better than when elements are selected for the sparse matrix on a magnitude basis alone. We show that the method developed is a viable alternative to use of the full control matrix in a phase-retrieval-based active optical control system.

© 2001 Optical Society of America

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References

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  1. B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, R. G. Paxman, “Comparison of Shack–Hartmann wavefront sensing and phase-diverse phase retrieval,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, ed., Proc. SPIE3126, 307–320 (1997).
    [CrossRef]
  2. P. Hariharan, Optical Interferometry (Academic, Sydney, Australia, 1985).
  3. L. Salas, “Variable separation in curvature sensing: fast method for solving the irradiance transport equation in the context of optical telescopes,” Appl. Opt. 35, 1593–1596 (1996).
    [CrossRef] [PubMed]
  4. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  5. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
    [CrossRef]
  6. R. G. Lyon, J. E. Dorband, J. M. Hollis, “Hubble space telescope faint object camera calculated point-spread functions,” Appl. Opt. 36, 1752–1765 (1997).
    [CrossRef] [PubMed]
  7. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1728–1736 (1993).
    [CrossRef]
  8. 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]
  9. R. Schneider, P. L. Levin, M. Spasojevic, “Multiscale compression of BEM equations for electrostatic systems,” IEEE Trans. Dielectr. Electr. Insul. 3, 482–493 (1996).
    [CrossRef]
  10. M. A. Player, J. Van Weereld, A. R. Allen, D. A. L. Collie, “Truncated-Newton algorithm for three-dimensional electrical impedance tomography,” Electron. Lett. 35, 2189–2191 (1999).
    [CrossRef]
  11. E. W. Justh, M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, P. S. Krishnaprasad, “Adaptive optics with advanced phase-contrast techniques. II. High-resolution wave-front control,” J. Opt. Soc. Am. A 18, 1300–1311 (2001).
    [CrossRef]
  12. G. W. Carhart, M. A. Vorontsov, M. Cohen, G. Cauwenberghs, R. T. Edwards, “Adaptive wavefront correction using a VLSI implementation of the parallel gradient descent algorithm,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3760, 61–67 (1999).
    [CrossRef]
  13. S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (McGraw-Hill, New York, 1959).
  14. Xinetics Inc., 37 MacArthur Ave., Devens, Mass. 01432; http://www.tiac.net/users/xinetics .
  15. M. A. Ealey, J. A. Wellman, “Xinetics low cost deformable mirrors with actuator replacement cartridges,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 680–687 (1994).
    [CrossRef]
  16. G. Strang, Linear Algebra and Its Applications (Academic, New York, 1976).
  17. R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  18. G. H. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996).
  19. W. H. Press, B. P. Flannery, S. A. Teulkolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).
  20. M. Snir, S. W. Otto, S. Huss-Lederman, D. W. Walker, J. Dongarra, MPI: The Complete Reference (MIT, Cambridge, Mass., 1997).
  21. For a description of the Highly Parallel Integrated Virtual Environment (HIVE), see http://newton.gsfc.nasa.gov/thehive/ .

2001 (1)

1999 (1)

M. A. Player, J. Van Weereld, A. R. Allen, D. A. L. Collie, “Truncated-Newton algorithm for three-dimensional electrical impedance tomography,” Electron. Lett. 35, 2189–2191 (1999).
[CrossRef]

1997 (1)

1996 (2)

L. Salas, “Variable separation in curvature sensing: fast method for solving the irradiance transport equation in the context of optical telescopes,” Appl. Opt. 35, 1593–1596 (1996).
[CrossRef] [PubMed]

R. Schneider, P. L. Levin, M. Spasojevic, “Multiscale compression of BEM equations for electrostatic systems,” IEEE Trans. Dielectr. Electr. Insul. 3, 482–493 (1996).
[CrossRef]

1995 (1)

1993 (1)

1982 (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[CrossRef]

1976 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Allen, A. R.

M. A. Player, J. Van Weereld, A. R. Allen, D. A. L. Collie, “Truncated-Newton algorithm for three-dimensional electrical impedance tomography,” Electron. Lett. 35, 2189–2191 (1999).
[CrossRef]

Beresnev, L. A.

Carhart, G. W.

E. W. Justh, M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, P. S. Krishnaprasad, “Adaptive optics with advanced phase-contrast techniques. II. High-resolution wave-front control,” J. Opt. Soc. Am. A 18, 1300–1311 (2001).
[CrossRef]

G. W. Carhart, M. A. Vorontsov, M. Cohen, G. Cauwenberghs, R. T. Edwards, “Adaptive wavefront correction using a VLSI implementation of the parallel gradient descent algorithm,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3760, 61–67 (1999).
[CrossRef]

Carrara, D. A.

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, R. G. Paxman, “Comparison of Shack–Hartmann wavefront sensing and phase-diverse phase retrieval,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, ed., Proc. SPIE3126, 307–320 (1997).
[CrossRef]

Cauwenberghs, G.

G. W. Carhart, M. A. Vorontsov, M. Cohen, G. Cauwenberghs, R. T. Edwards, “Adaptive wavefront correction using a VLSI implementation of the parallel gradient descent algorithm,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3760, 61–67 (1999).
[CrossRef]

Cohen, M.

G. W. Carhart, M. A. Vorontsov, M. Cohen, G. Cauwenberghs, R. T. Edwards, “Adaptive wavefront correction using a VLSI implementation of the parallel gradient descent algorithm,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3760, 61–67 (1999).
[CrossRef]

Collie, D. A. L.

M. A. Player, J. Van Weereld, A. R. Allen, D. A. L. Collie, “Truncated-Newton algorithm for three-dimensional electrical impedance tomography,” Electron. Lett. 35, 2189–2191 (1999).
[CrossRef]

Dongarra, J.

M. Snir, S. W. Otto, S. Huss-Lederman, D. W. Walker, J. Dongarra, MPI: The Complete Reference (MIT, Cambridge, Mass., 1997).

Dorband, J. E.

Ealey, M. A.

M. A. Ealey, J. A. Wellman, “Xinetics low cost deformable mirrors with actuator replacement cartridges,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 680–687 (1994).
[CrossRef]

Edwards, R. T.

G. W. Carhart, M. A. Vorontsov, M. Cohen, G. Cauwenberghs, R. T. Edwards, “Adaptive wavefront correction using a VLSI implementation of the parallel gradient descent algorithm,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3760, 61–67 (1999).
[CrossRef]

Ellerbroek, B. L.

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, R. G. Paxman, “Comparison of Shack–Hartmann wavefront sensing and phase-diverse phase retrieval,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, ed., Proc. SPIE3126, 307–320 (1997).
[CrossRef]

Fienup, J. R.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teulkolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996).

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[CrossRef]

Hariharan, P.

P. Hariharan, Optical Interferometry (Academic, Sydney, Australia, 1985).

Hollis, J. M.

Huss-Lederman, S.

M. Snir, S. W. Otto, S. Huss-Lederman, D. W. Walker, J. Dongarra, MPI: The Complete Reference (MIT, Cambridge, Mass., 1997).

Justh, E. W.

Kibblewhite, E. J.

Krishnaprasad, P. S.

Lee, D. J.

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, R. G. Paxman, “Comparison of Shack–Hartmann wavefront sensing and phase-diverse phase retrieval,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, ed., Proc. SPIE3126, 307–320 (1997).
[CrossRef]

Levin, P. L.

R. Schneider, P. L. Levin, M. Spasojevic, “Multiscale compression of BEM equations for electrostatic systems,” IEEE Trans. Dielectr. Electr. Insul. 3, 482–493 (1996).
[CrossRef]

Lyon, R. G.

Noll, R.

Otto, S. W.

M. Snir, S. W. Otto, S. Huss-Lederman, D. W. Walker, J. Dongarra, MPI: The Complete Reference (MIT, Cambridge, Mass., 1997).

Paxman, R. G.

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, R. G. Paxman, “Comparison of Shack–Hartmann wavefront sensing and phase-diverse phase retrieval,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, ed., Proc. SPIE3126, 307–320 (1997).
[CrossRef]

Player, M. A.

M. A. Player, J. Van Weereld, A. R. Allen, D. A. L. Collie, “Truncated-Newton algorithm for three-dimensional electrical impedance tomography,” Electron. Lett. 35, 2189–2191 (1999).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teulkolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Salas, L.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Schneider, R.

R. Schneider, P. L. Levin, M. Spasojevic, “Multiscale compression of BEM equations for electrostatic systems,” IEEE Trans. Dielectr. Electr. Insul. 3, 482–493 (1996).
[CrossRef]

Snir, M.

M. Snir, S. W. Otto, S. Huss-Lederman, D. W. Walker, J. Dongarra, MPI: The Complete Reference (MIT, Cambridge, Mass., 1997).

Spasojevic, M.

R. Schneider, P. L. Levin, M. Spasojevic, “Multiscale compression of BEM equations for electrostatic systems,” IEEE Trans. Dielectr. Electr. Insul. 3, 482–493 (1996).
[CrossRef]

Strang, G.

G. Strang, Linear Algebra and Its Applications (Academic, New York, 1976).

Teulkolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teulkolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Thelen, B. J.

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, R. G. Paxman, “Comparison of Shack–Hartmann wavefront sensing and phase-diverse phase retrieval,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, ed., Proc. SPIE3126, 307–320 (1997).
[CrossRef]

Timoshenko, S.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (McGraw-Hill, New York, 1959).

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996).

Van Weereld, J.

M. A. Player, J. Van Weereld, A. R. Allen, D. A. L. Collie, “Truncated-Newton algorithm for three-dimensional electrical impedance tomography,” Electron. Lett. 35, 2189–2191 (1999).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teulkolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Vorontsov, M. A.

E. W. Justh, M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, P. S. Krishnaprasad, “Adaptive optics with advanced phase-contrast techniques. II. High-resolution wave-front control,” J. Opt. Soc. Am. A 18, 1300–1311 (2001).
[CrossRef]

G. W. Carhart, M. A. Vorontsov, M. Cohen, G. Cauwenberghs, R. T. Edwards, “Adaptive wavefront correction using a VLSI implementation of the parallel gradient descent algorithm,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3760, 61–67 (1999).
[CrossRef]

Vuilleumier, R.

Walker, D. W.

M. Snir, S. W. Otto, S. Huss-Lederman, D. W. Walker, J. Dongarra, MPI: The Complete Reference (MIT, Cambridge, Mass., 1997).

Wellman, J. A.

M. A. Ealey, J. A. Wellman, “Xinetics low cost deformable mirrors with actuator replacement cartridges,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 680–687 (1994).
[CrossRef]

Wild, W. J.

Woinowsky-Krieger, S.

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (McGraw-Hill, New York, 1959).

Appl. Opt. (3)

Electron. Lett. (1)

M. A. Player, J. Van Weereld, A. R. Allen, D. A. L. Collie, “Truncated-Newton algorithm for three-dimensional electrical impedance tomography,” Electron. Lett. 35, 2189–2191 (1999).
[CrossRef]

IEEE Trans. Dielectr. Electr. Insul. (1)

R. Schneider, P. L. Levin, M. Spasojevic, “Multiscale compression of BEM equations for electrostatic systems,” IEEE Trans. Dielectr. Electr. Insul. 3, 482–493 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Other (11)

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, R. G. Paxman, “Comparison of Shack–Hartmann wavefront sensing and phase-diverse phase retrieval,” in Adaptive Optics and Applications, R. K. Tyson, R. Q. Fugate, ed., Proc. SPIE3126, 307–320 (1997).
[CrossRef]

P. Hariharan, Optical Interferometry (Academic, Sydney, Australia, 1985).

G. W. Carhart, M. A. Vorontsov, M. Cohen, G. Cauwenberghs, R. T. Edwards, “Adaptive wavefront correction using a VLSI implementation of the parallel gradient descent algorithm,” in High-Resolution Wavefront Control: Methods, Devices, and Applications, J. D. Gonglewski, M. A. Vorontsov, eds., Proc. SPIE3760, 61–67 (1999).
[CrossRef]

S. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (McGraw-Hill, New York, 1959).

Xinetics Inc., 37 MacArthur Ave., Devens, Mass. 01432; http://www.tiac.net/users/xinetics .

M. A. Ealey, J. A. Wellman, “Xinetics low cost deformable mirrors with actuator replacement cartridges,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 680–687 (1994).
[CrossRef]

G. Strang, Linear Algebra and Its Applications (Academic, New York, 1976).

G. H. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996).

W. H. Press, B. P. Flannery, S. A. Teulkolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

M. Snir, S. W. Otto, S. Huss-Lederman, D. W. Walker, J. Dongarra, MPI: The Complete Reference (MIT, Cambridge, Mass., 1997).

For a description of the Highly Parallel Integrated Virtual Environment (HIVE), see http://newton.gsfc.nasa.gov/thehive/ .

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Figures (14)

Fig. 1
Fig. 1

Active optical system. Images detected at the focal-plane camera are processed by the phase-retrieval algorithm or wave-front sensing to estimate the wave front. The wave front is processed by the control software to provide commands for the actuators.

Fig. 2
Fig. 2

(a) Example of a Zernike polynomial wave front before correction. (b) Example of a residual wave front after correction. The contrast is enhanced to show the influence of the actuators.

Fig. 3
Fig. 3

Distribution of binned negative values. Distribution of rows 38 and 174 of S as a function of the bin midpoint. The distribution is defined as the number of elements falling in a bin divided by the bin width. After sorting, we binned each sequential 512 elements. The distribution value d k = 512/m(S i,k+512 - S i,k ), where m is 67,600; the number of points in a row i is either 38 or 174; and k = 0, 512, 1024, … , 67,584. The last distribution value, d 67,584, is equal to 15/m(S i,k67,599 - S i,67,584). The bin k midpoint is 1/2(S i,k+512 + S i,k ) and is always negative in this figure. To show the difference in distributions, this figure is on a log–log scale. Rows 38 and 174 correspond to actuators in the center of a segment and in the central obscuration, respectively.

Fig. 4
Fig. 4

Algorithm flow chart used to find the positive and negative thresholds for row i.

Fig. 5
Fig. 5

Simulated optical control loop. The wave-front simulator (1) generates wave-front data (2). The wave-front error (3) is calculated from the wave-front data. The actuator fitting program (4) generates an updated actuator vector a (k+1). The updated actuator vector is used to generate a corrected wave front (5). Wave-front data, wave-front error data, and actuator data from each iteration k of the control loop are saved for analysis.

Fig. 6
Fig. 6

Eigenvalues of R T R ordered along the ordinate from least to greatest. The abscissa contains the dimensionless index of the eigenvalue. The ratio of largest to smallest eigenvalue, 1.818 × 106, is the two-norm condition number.17

Fig. 7
Fig. 7

Trial algorithm: relative wave-front error for control-loop iteration 1. Error is a function of density parameter ρ. Wave-front relative error at iteration k is defined in Eq. (3). Maximum, minimum, and mean values for 100 wave-front cases are shown. The error bars represent the standard deviation of the data point ordinate value. A nonmonotonic region of the curve is evident at ρ = 0.2.

Fig. 8
Fig. 8

Corrected algorithm: relative wave-front error for control-loop iteration 1. The error is a function of density parameter ρ. See Fig. 7’s caption. The error shows a monotonic decrease with increasing ρ and is of smaller magnitude than the first algorithm relative wave-front error.

Fig. 9
Fig. 9

Corrected algorithm: mean relative wave-front error for control-loop iteration k, 1–4. The error is a function of the density parameter ρ. Wave-front relative error at iteration k is defined in Eq. (3). The mean error decreases monotonically with increasing control-loop iterations.

Fig. 10
Fig. 10

Corrected algorithm: row error as a function of density parameter ρ. Row error is defined in Eq. (19).

Fig. 11
Fig. 11

Corrected algorithm: actuator error for control-loop iteration 1 as a function of density parameter ρ. Actuator error is defined in Eq. (20).

Fig. 12
Fig. 12

Corrected algorithm: mean actuator error for control-loop iteration k, 1–4 as a function of density parameter ρ. Actuator error is defined in Eq. (20).

Fig. 13
Fig. 13

Corrected algorithm: computation time as a function of density parameter ρ. See Section 5 and Appendix A for timing results and computer hardware discussion. The mean values for 400 wave-front control iterations are plotted. The ordinate is time in seconds to compute the matrix–vector product Ŝw (k). For ρ = 1, Ŝ is replaced by the full matrix S. The error bars represent the standard deviation of the ordinate data value.

Fig. 14
Fig. 14

Corrected algorithm: matrix size and overhead as a function of density parameter ρ. The overhead is the size of the storage devoted to sparse data location and length.

Equations (21)

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

rj=exp-αx-xjsinαx-xj+π/4,
wk=wk-1+Rak,
εw=wkw0.
ak=ak-1-RTR-1RTwk-1,
SRTR-1RT;
ak=ak-1-Swk-1,
Ui,j=Si,k,  k0,, m-1,
Ui,l+1Ui,l,  l0,, m-2.
Ui,nti<0,  Ui,nti-1=0,
Ui,pti>0,  Ui,pti+1=0.
Bi=k=pti+1nti-1 Ui,k.
if Bi<0,  ptipti-1,
if Bi>0,  ntinti+1,
if Bi=0,  ntinti+1,  ptipti-1.
Sˆi,j=Si,jif Si,jUi,nti or Ui,ptiSi,j0otherwise.
εc=|flx-x||x|,
|flri · rj-ri · rj|mu|ri| · |rj|,
εdot=|flri · rj-ri · rj||ri · rj|mu |ri| · |rj||ri · rj|.
εChuκ2RTR,
εr=1ni=0n-1ωij=0m-1 Sˆij-Sij,
εa=i=0n-1asi-ai2ωi2i=0n-1aiωi21/2,

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