Abstract

By analyzing the relationship between eigenvalues of the product of the Jacobian matrix of performance functions and a damping factor, we believe that we have developed a new analytical method for setting an adequate initial value for the damping factor in the damped least-squares problem. The value of a damping factor should be almost that of a median of the eigenvalues. The effectiveness of the method is shown by some numerical experiments.

© 1994 Optical Society of America

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References

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  1. T. H. Jamieson, Optimization Techniques in Lens Design (Hilger, London, 1971).
  2. A. K. Rigler, R. J. Pegis, “Optimization methods in optics,” in The Computer in Optical Research—Methods and Applications (Springer-Verlag, Berlin, 1980), pp. 211–268.
  3. K. A. Levenberg, “A method for the solution of certain nonlinear problems in least squares,” Q. J. Appl. Math. 2, 164–168 (1944).
  4. D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
    [Crossref]
  5. A. Girard, “Calcul automatique en optique geometrique,” Rev. Opt. 37, 225–241 (1958).
  6. C. G. Wynne, “Lens designing by electronic digital computer I,” Proc. Phys. Soc. London 73, 777–787 (1959).
    [Crossref]
  7. C. G. Wynne, P. M. J. H. Wormell, “Lens design by computer,” Appl. Opt. 2, 1233–1238 (1963).
    [Crossref]
  8. Ref. 1, pp. 61–62.
  9. M. J. Kidger, C. G. Wynne, “Experiments with lens optimization procedures,” Opt. Acta 14, 279–288 (1967).
    [Crossref]
  10. D. F. Shanno, “Parameter solution for modified Newton methods for function minimization,” SLAM J. Numer. Anal. 7, 366–372 (1970).
    [Crossref]
  11. A. Jones, “Spiral—a new algorithm for nonlinear parameter estimation using least squares,” Comput. J. 13, 301–308 (1970).
    [Crossref]
  12. J. Meiron, “Damped least-squares method for automatic lens design,” J. Opt. Soc. Am. 55, 1105–1109 (1965).
    [Crossref]
  13. D. R. Buchele, “Damping factor for the least-squares method of optical design,” Appl. Opt. 7, 2433–2435 (1968).
    [Crossref] [PubMed]
  14. H. Matsui, K. Tanaka, “Solution of the damped least-squares problem by using a combination of eigenvalue and singular value decompositions,” Appl. Opt. 31, 2241–2243 (1992).
    [Crossref] [PubMed]
  15. G. H. Golub, C. Reinsch, “Singular value decomposition and least-squares solutions,” Numer. Math. 14, 403–420 (1970).
    [Crossref]
  16. For example, see C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N. J., 1974), p. 237.
  17. For example, see W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, 1974), p. 192.
  18. Y. Matsui, Theory of Lens Design (Kyoritsu, Tokyo, 1972), pp. 114–119 (in Japanese).

1992 (1)

1970 (3)

G. H. Golub, C. Reinsch, “Singular value decomposition and least-squares solutions,” Numer. Math. 14, 403–420 (1970).
[Crossref]

D. F. Shanno, “Parameter solution for modified Newton methods for function minimization,” SLAM J. Numer. Anal. 7, 366–372 (1970).
[Crossref]

A. Jones, “Spiral—a new algorithm for nonlinear parameter estimation using least squares,” Comput. J. 13, 301–308 (1970).
[Crossref]

1968 (1)

1967 (1)

M. J. Kidger, C. G. Wynne, “Experiments with lens optimization procedures,” Opt. Acta 14, 279–288 (1967).
[Crossref]

1965 (1)

1963 (2)

C. G. Wynne, P. M. J. H. Wormell, “Lens design by computer,” Appl. Opt. 2, 1233–1238 (1963).
[Crossref]

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
[Crossref]

1959 (1)

C. G. Wynne, “Lens designing by electronic digital computer I,” Proc. Phys. Soc. London 73, 777–787 (1959).
[Crossref]

1958 (1)

A. Girard, “Calcul automatique en optique geometrique,” Rev. Opt. 37, 225–241 (1958).

1944 (1)

K. A. Levenberg, “A method for the solution of certain nonlinear problems in least squares,” Q. J. Appl. Math. 2, 164–168 (1944).

Buchele, D. R.

Girard, A.

A. Girard, “Calcul automatique en optique geometrique,” Rev. Opt. 37, 225–241 (1958).

Golub, G. H.

G. H. Golub, C. Reinsch, “Singular value decomposition and least-squares solutions,” Numer. Math. 14, 403–420 (1970).
[Crossref]

Hanson, R. J.

For example, see C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N. J., 1974), p. 237.

Jamieson, T. H.

T. H. Jamieson, Optimization Techniques in Lens Design (Hilger, London, 1971).

Jones, A.

A. Jones, “Spiral—a new algorithm for nonlinear parameter estimation using least squares,” Comput. J. 13, 301–308 (1970).
[Crossref]

Kidger, M. J.

M. J. Kidger, C. G. Wynne, “Experiments with lens optimization procedures,” Opt. Acta 14, 279–288 (1967).
[Crossref]

Lawson, C. L.

For example, see C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N. J., 1974), p. 237.

Levenberg, K. A.

K. A. Levenberg, “A method for the solution of certain nonlinear problems in least squares,” Q. J. Appl. Math. 2, 164–168 (1944).

Marquardt, D. W.

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
[Crossref]

Matsui, H.

Matsui, Y.

Y. Matsui, Theory of Lens Design (Kyoritsu, Tokyo, 1972), pp. 114–119 (in Japanese).

Meiron, J.

Pegis, R. J.

A. K. Rigler, R. J. Pegis, “Optimization methods in optics,” in The Computer in Optical Research—Methods and Applications (Springer-Verlag, Berlin, 1980), pp. 211–268.

Reinsch, C.

G. H. Golub, C. Reinsch, “Singular value decomposition and least-squares solutions,” Numer. Math. 14, 403–420 (1970).
[Crossref]

Rigler, A. K.

A. K. Rigler, R. J. Pegis, “Optimization methods in optics,” in The Computer in Optical Research—Methods and Applications (Springer-Verlag, Berlin, 1980), pp. 211–268.

Shanno, D. F.

D. F. Shanno, “Parameter solution for modified Newton methods for function minimization,” SLAM J. Numer. Anal. 7, 366–372 (1970).
[Crossref]

Tanaka, K.

Welford, W. T.

For example, see W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, 1974), p. 192.

Wormell, P. M. J. H.

Wynne, C. G.

M. J. Kidger, C. G. Wynne, “Experiments with lens optimization procedures,” Opt. Acta 14, 279–288 (1967).
[Crossref]

C. G. Wynne, P. M. J. H. Wormell, “Lens design by computer,” Appl. Opt. 2, 1233–1238 (1963).
[Crossref]

C. G. Wynne, “Lens designing by electronic digital computer I,” Proc. Phys. Soc. London 73, 777–787 (1959).
[Crossref]

Appl. Opt. (3)

Comput. J. (1)

A. Jones, “Spiral—a new algorithm for nonlinear parameter estimation using least squares,” Comput. J. 13, 301–308 (1970).
[Crossref]

J. Opt. Soc. Am. (1)

J. Soc. Ind. Appl. Math. (1)

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
[Crossref]

Numer. Math. (1)

G. H. Golub, C. Reinsch, “Singular value decomposition and least-squares solutions,” Numer. Math. 14, 403–420 (1970).
[Crossref]

Opt. Acta (1)

M. J. Kidger, C. G. Wynne, “Experiments with lens optimization procedures,” Opt. Acta 14, 279–288 (1967).
[Crossref]

Proc. Phys. Soc. London (1)

C. G. Wynne, “Lens designing by electronic digital computer I,” Proc. Phys. Soc. London 73, 777–787 (1959).
[Crossref]

Q. J. Appl. Math. (1)

K. A. Levenberg, “A method for the solution of certain nonlinear problems in least squares,” Q. J. Appl. Math. 2, 164–168 (1944).

Rev. Opt. (1)

A. Girard, “Calcul automatique en optique geometrique,” Rev. Opt. 37, 225–241 (1958).

SLAM J. Numer. Anal. (1)

D. F. Shanno, “Parameter solution for modified Newton methods for function minimization,” SLAM J. Numer. Anal. 7, 366–372 (1970).
[Crossref]

Other (6)

For example, see C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N. J., 1974), p. 237.

For example, see W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, 1974), p. 192.

Y. Matsui, Theory of Lens Design (Kyoritsu, Tokyo, 1972), pp. 114–119 (in Japanese).

T. H. Jamieson, Optimization Techniques in Lens Design (Hilger, London, 1971).

A. K. Rigler, R. J. Pegis, “Optimization methods in optics,” in The Computer in Optical Research—Methods and Applications (Springer-Verlag, Berlin, 1980), pp. 211–268.

Ref. 1, pp. 61–62.

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of the solution vectors obtained by the least-squares, DLS, and SD methods.

Fig. 2
Fig. 2

Cross-sectional view of the first numerical example.

Fig. 3
Fig. 3

Comparison of the rates of convergence of the optimizations by three different damping factors in case 1 of the first example.

Fig. 4
Fig. 4

Comparison of the rates of convergence of the optimizations by three different damping factors in case 2 of the first example.

Fig. 5
Fig. 5

Comparison of the rates of convergence of the optimizations by three different damping factors in case 3 of the first example.

Fig. 6
Fig. 6

Comparison of the rates of convergence of the optimizations by three different damping factors in case 4 of the first example.

Fig. 7
Fig. 7

Topography of the merit function of the first example together with the loci of optimization steps in those cases in which the damping factors all equal 1.0.

Fig. 8
Fig. 8

Topography of the merit function of the first example together with the loci of optimization steps in those cases in which all the damping factors equal 1 × 10−4.

Fig. 9
Fig. 9

Topography of the merit function of the first example together with the loci of the optimization steps when the damping factors are 1 × 10−8, 1 × 10−8, 1 × 10−10, 1 × 10−9 for cases 1, 2, 3, and 4, respectively.

Fig. 10
Fig. 10

Cross-sectional view of the second numerical example. The dashed-dotted line indicates an optical axis.

Fig. 11
Fig. 11

Distribution of the eigenvalues of (ATA) of the second example.

Fig. 12
Fig. 12

Comparison of the rates of convergence of the optimizations by two different damping factors for the second example.

Equations (56)

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Minimize ϕ ( X ) = F ( X ) T F ( X ) ,
( A T A + i = 1 m F i H i ) Δ X DLS = A T F 0 ,
( A T A + ρ I ) Δ X DLS = A T F 0 ,
( A T A + D ) Δ X DLS = A T F 0 ,
A = UTV T ,
A T A = VSV T ,
( S + D ) Δ X DLS = Y ,
Δ X DLS = V T Δ X DLS ,
Y = T T U T F 0 .
Δ x DLS , k = y k / ( s k + d k ) ,
Δ X DLS = V Δ X DLS .
Δ x DLS , k = y k / ( s k + ρ ) .
( A T A ) Δ X LS = A T F 0 ,
Δ X LS = V Δ X LS ,
Δ x LS , k = y k / s k .
Δ X SD = grad ϕ = A T F 0 ,
Δ X SD = V Δ X SD ,
Δ x SD , k = y k .
Δ X DLS = V T Δ X DLS ,
Δ X LS = V T Δ X LS ,
Δ X SD = V T Δ X SD ,
cos θ DLS = Δ X DLS Δ X LS Δ X DLS Δ X LS ,
= Δ X DLS Δ X LS Δ X DLS Δ X LS ,
= i = 1 n [ y i 2 / s i ( s i + ρ ) ] { i = 1 n [ y i / ( s i + ρ ) ] 2 } 1 / 2 [ i = 1 n ( y i / s i ) 2 ] 1 / 2 .
cos θ SD = Δ X SD Δ X LS Δ X SD Δ X LS ,
= Δ X SD Δ X LS Δ X SD Δ X LS ,
= i = 1 n ( y i 2 / s i ) [ i = 1 n ( y i 2 ) ] 1 / 2 [ i = 1 n ( y i / s i ) 2 ] 1 / 2 .
d d ρ cos θ DLS 0 .
d d ρ Δ X DLS 2 = 2 Δ X DLS T ( S + ρ I ) 1 Δ X DLS < 0 ,
= 2 i = 1 n [ y i 2 / ( s i + ρ ) 3 ] < 0 ,
θ DLS = θ SD / 2 .
Δ X H = Δ X SD Δ X LS + Δ X LS Δ X SD Δ X LS Δ X SD .
Δ x H , k = ( i = 1 n y i 2 ) 1 / / 2 ( y k / s k ) 1 / 2 + [ i = 1 n ( y i / s i ) 2 ] 1 / 2 y k [ i = 1 n ( y i / s i ) 2 ] 1 / 2 ( i = 1 n y i 2 ) 1 / 2 .
Δ x DLS , j Δ x H , j = Δ x DLS , k Δ x H , k = C , j , k ( 1 , 2 , , n ) ,
d k = s k C { [ i = 1 n ( y i / s i ) 2 ] 1 / 2 s k + ( i = 1 n y i 2 ) 1 / 2 } s k .
s j / ( s j + ρ ) ( i = 1 n y i 2 ) 1 / 2 ( y j / s j ) + [ i = 1 n ( y i / s i ) 2 ] 1 / 2 y j = s k / ( s k + ρ ) ( i = 1 n y i 2 ) 1 / 2 ( y k / s k ) + [ i = 1 n ( y i / s i ) 2 ] 1 / 2 y k .
ρ = s j s k [ i = 1 n ( y i / s i ) 2 ] 1 / 2 [ i = 1 n ( y i 2 ) ] 1 / 2 , 1 j , k n , j k .
s j = s min , s k = s max ,
s max 2 ρ 2 = s max 2 i = 1 n ( y i 2 ) i = 1 n { [ 1 ( s min / s i ) 2 ] y i 2 } > 0 ,
s min 2 ρ 2 = s min 2 i = 1 n ( y i 2 ) i = 1 n { [ 1 ( s max / s i ) 2 ] y i 2 } < 0 .
s min < ρ < s max ,
s j 2 ρ 2 = s j 2 i = 1 n ( y i 2 ) i = 1 n { [ 1 ( s k / s i ) 2 ] y i 2 } 0 .
Δ X DLS = Δ X LS / 2 .
s j = d j , j ( 1 , 2 , , n ) .
Δ X DLS = { i = 1 n [ y i / ( s i + ρ ) ] 2 } 1 / 2 > { i = 1 n [ y i / ( 2 s i ) ] 2 } 1 / 2 = Δ X LS / 2
Δ X DLS < Δ X LS / 2 ,
s min < ρ < s max .
Δ X DLS Δ X LS / 2 .
ρ = s ( n + 1 ) / 2 , n = odd , = [ s ( n / 2 ) + s ( n / 2 ) + 1 ] / 2 , n = even .
ϕ = [ w a ( I I t ) ] 2 + [ w b ( I I I I t ) ] 2 ,
I t = 4.6 , I I t = 2.2 , w a = 10 , w b = 10 .
ϕ = 0 , when 1 / r 1 = 1.3912 , 1 / r 2 = 0.4152 ,
ϕ = 0 , when 1 / r 1 = 0.6745 , 1 / r 2 = 0.2198 ,
Case 1 : 1 / r 1 = 1.8 , 1 / r 2 = 0.5 ; Case 2 : 1 / r 1 = 1.9 , 1 / r 2 = 0.1 ; Case 3 : 1 / r 1 = 1.4 , 1 / r 2 = 0.7 ; Case 4 : 1 / r 1 = 0.4 , 1 / r 2 = 0.5 .
Case 1 : ϕ = 2126.1 , s 1 = 8.6 × 10 13 , s 2 = 2.2 × 10 8 ; Case 2 : ϕ = 3404.6 , s 1 = 1.4 × 10 12 , s 2 = 2.4 × 10 8 ; Case 3 : ϕ = 153.5 , s 1 = 3.0 × 10 14 , s 2 = 1.1 × 10 9 ; Case 4 : ϕ = 500.7 , s 1 = 2.3 × 10 11 , s 2 = 2.0 × 10 9 .
Case 1 : ρ = 1.0 , 1 × 10 4 , 1 × 10 8 ; Case 2 : ρ = 1.0 , 1 × 10 4 , 1 × 10 8 ; Case 3 : ρ = 1.0 , 1 × 10 4 , 1 × 10 10 ; Case 4 : ρ = 1.0 , 1 × 10 4 , 1 × 10 9 .

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