Abstract

We describe generalized projection procedures for the design of arbitrary filter functions for correlators. More specifically, serial and parallel implementations of projection-based algorithms are employed. The novelty of this procedure lies in its generality and its ability to handle wide varieties of constraints by the same procedure. The procedure is demonstrated by the design of filters for the 4-f linear correlator, the phase-extraction correlator, and variants thereof. The filters are subject to a variety of constraints, including rotation-invariant pattern recognition and class discrimination. Examples are given to show the versatility, flexibility, and applicability of the design process to a variety of pattern-recognition tasks. Satisfactory results are also obtained because of the combination with the special nonlinear correlators proposed for pattern recognition.

© 1995 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. D. Casasent, D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77–84 (1977).
    [CrossRef]
  4. M. Fleisher, U. Mahlab, J. Shamir, “Target location measurement by optical correlators: a performance criterion,” Appl. Opt. 31, 230–235 (1992).
    [CrossRef] [PubMed]
  5. J. Rosen, J. Shamir, “Application of the projection-onto-constraint-sets algorithm for optical pattern recognition,” Opt. Lett. 16, 752–754 (1991).
    [CrossRef] [PubMed]
  6. J. Rosen, “Learning in correlators based on projections onto constraint sets,” Opt. Lett. 18, 1183–1185 (1993).
    [CrossRef] [PubMed]
  7. T. Kotzer, N. Cohen, J. Shamir, “Image reconstruction by a novel parallel projection onto constraint sets method,” EE Pub. 919 (Dept. of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, June1994).
  8. T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Multi-distance, Multi-projection, parallel projection method,” presented at the International Conference on Optical Computing-OC94, Edinburgh, Scotland, August, 1994.
  9. T. Kotzer, N. Cohen, J. Shamir, “Signal synthesis and reconstruction by projection methods in a hyper space,” presented at the Annual Meeting of the Optical Society of America, Dallas, Tex., October 1994.
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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  15. T. Kotzer, N. Cohen, J. Shamir, “A projection algorithm for consistent and inconsistent constraints,” EE Pub. 920 (Dept. of Electrical Engineering, Technion Israel Institute of Technology, Haifa, August1994).
  16. T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Summed distance error reduction of simultaneous multiprojections and applications,” EE Pub. 909 (Dept. of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, Israel, August1994).
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    [CrossRef]
  20. T. Kotzer, J. Rosen, J. Shamir, “Multiple-object input in nonlinear correlation,” Appl. Opt. 32, 1919–1932 (1993).
    [CrossRef] [PubMed]
  21. E. Silvera, T. Kotzer, J. Shamir, “Adaptive pattern recognition with rotation, scale and shift invariance,” Appl. Opt. 34, 1891–1900 (1995).
    [CrossRef] [PubMed]
  22. J. Rosen, T. Kotzer, J. Shamir, “Optical implementation of phase extraction pattern recognition,” Opt. Commun. 83, 10–14 (1991).
    [CrossRef]
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    [CrossRef]
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1995 (1)

1994 (1)

Y. Censor, T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numerical Algorithms 8, 221–239 (1994).
[CrossRef]

1993 (2)

1992 (3)

1991 (3)

1990 (1)

1989 (1)

1988 (1)

1984 (3)

1982 (2)

Y. Hsu, H. H. Arsenault, G. April, “Rotation invariant digital pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4012–4015 (1982).
[CrossRef] [PubMed]

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—theory,” IEEE Trans. Med. Imag. TMI-1, 81–94 (1982).
[CrossRef]

1977 (1)

D. Casasent, D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77–84 (1977).
[CrossRef]

Akhiezer, N. I.

N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space (Ungar, New York, 1961), Vol. I.

April, G.

Arsenault, H. H.

Casasent, D.

D. Casasent, D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77–84 (1977).
[CrossRef]

Censor, Y.

Y. Censor, T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numerical Algorithms 8, 221–239 (1994).
[CrossRef]

T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Multi-distance, Multi-projection, parallel projection method,” presented at the International Conference on Optical Computing-OC94, Edinburgh, Scotland, August, 1994.

T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Summed distance error reduction of simultaneous multiprojections and applications,” EE Pub. 909 (Dept. of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, Israel, August1994).

Cohen, N.

T. Kotzer, N. Cohen, J. Shamir, “A projection algorithm for consistent and inconsistent constraints,” EE Pub. 920 (Dept. of Electrical Engineering, Technion Israel Institute of Technology, Haifa, August1994).

T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Summed distance error reduction of simultaneous multiprojections and applications,” EE Pub. 909 (Dept. of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, Israel, August1994).

T. Kotzer, N. Cohen, J. Shamir, “Signal synthesis and reconstruction by projection methods in a hyper space,” presented at the Annual Meeting of the Optical Society of America, Dallas, Tex., October 1994.

T. Kotzer, N. Cohen, J. Shamir, “Extended and alternative projections onto convex constraint sets: theory and applications,” EE Pub. 900, (Dept. of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, November1993).

T. Kotzer, N. Cohen, J. Shamir, “Image reconstruction by a novel parallel projection onto constraint sets method,” EE Pub. 919 (Dept. of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, June1994).

T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Multi-distance, Multi-projection, parallel projection method,” presented at the International Conference on Optical Computing-OC94, Edinburgh, Scotland, August, 1994.

Elfving, T.

Y. Censor, T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numerical Algorithms 8, 221–239 (1994).
[CrossRef]

Flannery, D. L.

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

Fleisher, M.

Gianino, P. D.

Glazman, I. M.

N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space (Ungar, New York, 1961), Vol. I.

Hahn, W. B.

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

Hendrix, C.

Horner, J. L.

Hsu, Y.

Javidi, B.

Kotzer, T.

E. Silvera, T. Kotzer, J. Shamir, “Adaptive pattern recognition with rotation, scale and shift invariance,” Appl. Opt. 34, 1891–1900 (1995).
[CrossRef] [PubMed]

T. Kotzer, J. Rosen, J. Shamir, “Multiple-object input in nonlinear correlation,” Appl. Opt. 32, 1919–1932 (1993).
[CrossRef] [PubMed]

T. Kotzer, J. Rosen, J. Shamir, “Phase extraction pattern recognition,” Appl. Opt. 31, 1126–1137 (1992).
[CrossRef] [PubMed]

J. Rosen, T. Kotzer, J. Shamir, “Optical implementation of phase extraction pattern recognition,” Opt. Commun. 83, 10–14 (1991).
[CrossRef]

T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Summed distance error reduction of simultaneous multiprojections and applications,” EE Pub. 909 (Dept. of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, Israel, August1994).

T. Kotzer, N. Cohen, J. Shamir, “Signal synthesis and reconstruction by projection methods in a hyper space,” presented at the Annual Meeting of the Optical Society of America, Dallas, Tex., October 1994.

T. Kotzer, N. Cohen, J. Shamir, “Extended and alternative projections onto convex constraint sets: theory and applications,” EE Pub. 900, (Dept. of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, November1993).

T. Kotzer, N. Cohen, J. Shamir, “A projection algorithm for consistent and inconsistent constraints,” EE Pub. 920 (Dept. of Electrical Engineering, Technion Israel Institute of Technology, Haifa, August1994).

T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Multi-distance, Multi-projection, parallel projection method,” presented at the International Conference on Optical Computing-OC94, Edinburgh, Scotland, August, 1994.

T. Kotzer, N. Cohen, J. Shamir, “Image reconstruction by a novel parallel projection onto constraint sets method,” EE Pub. 919 (Dept. of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, June1994).

Kumar, B. V. K. V.

Levi, A.

Mahlab, U.

Pierra, G.

G. Pierra, “Decomposition through formalization in a product space,” Math. Prog. 28, 96–115 (1984).
[CrossRef]

Psaltis, D.

D. Casasent, D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77–84 (1977).
[CrossRef]

Rosen, J.

Shamir, J.

E. Silvera, T. Kotzer, J. Shamir, “Adaptive pattern recognition with rotation, scale and shift invariance,” Appl. Opt. 34, 1891–1900 (1995).
[CrossRef] [PubMed]

T. Kotzer, J. Rosen, J. Shamir, “Multiple-object input in nonlinear correlation,” Appl. Opt. 32, 1919–1932 (1993).
[CrossRef] [PubMed]

M. Fleisher, U. Mahlab, J. Shamir, “Target location measurement by optical correlators: a performance criterion,” Appl. Opt. 31, 230–235 (1992).
[CrossRef] [PubMed]

T. Kotzer, J. Rosen, J. Shamir, “Phase extraction pattern recognition,” Appl. Opt. 31, 1126–1137 (1992).
[CrossRef] [PubMed]

J. Rosen, J. Shamir, “Application of the projection-onto-constraint-sets algorithm for optical pattern recognition,” Opt. Lett. 16, 752–754 (1991).
[CrossRef] [PubMed]

J. Rosen, T. Kotzer, J. Shamir, “Optical implementation of phase extraction pattern recognition,” Opt. Commun. 83, 10–14 (1991).
[CrossRef]

J. Rosen, J. Shamir, “Circular harmonic phase filter for efficient rotation invariant pattern recognition,” Appl. Opt. 27, 2895–2899 (1988).
[CrossRef] [PubMed]

T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Multi-distance, Multi-projection, parallel projection method,” presented at the International Conference on Optical Computing-OC94, Edinburgh, Scotland, August, 1994.

T. Kotzer, N. Cohen, J. Shamir, “Image reconstruction by a novel parallel projection onto constraint sets method,” EE Pub. 919 (Dept. of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, June1994).

T. Kotzer, N. Cohen, J. Shamir, “Signal synthesis and reconstruction by projection methods in a hyper space,” presented at the Annual Meeting of the Optical Society of America, Dallas, Tex., October 1994.

T. Kotzer, N. Cohen, J. Shamir, “Extended and alternative projections onto convex constraint sets: theory and applications,” EE Pub. 900, (Dept. of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, November1993).

T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Summed distance error reduction of simultaneous multiprojections and applications,” EE Pub. 909 (Dept. of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, Israel, August1994).

T. Kotzer, N. Cohen, J. Shamir, “A projection algorithm for consistent and inconsistent constraints,” EE Pub. 920 (Dept. of Electrical Engineering, Technion Israel Institute of Technology, Haifa, August1994).

Shei, W.

Silvera, E.

Stark, H.

Wang, J.

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—theory,” IEEE Trans. Med. Imag. TMI-1, 81–94 (1982).
[CrossRef]

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—theory,” IEEE Trans. Med. Imag. TMI-1, 81–94 (1982).
[CrossRef]

Appl. Opt. (9)

IEEE Trans. Med. Imag. (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—theory,” IEEE Trans. Med. Imag. TMI-1, 81–94 (1982).
[CrossRef]

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

Math. Prog. (1)

G. Pierra, “Decomposition through formalization in a product space,” Math. Prog. 28, 96–115 (1984).
[CrossRef]

Numerical Algorithms (1)

Y. Censor, T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numerical Algorithms 8, 221–239 (1994).
[CrossRef]

Opt. Commun. (1)

J. Rosen, T. Kotzer, J. Shamir, “Optical implementation of phase extraction pattern recognition,” Opt. Commun. 83, 10–14 (1991).
[CrossRef]

Opt. Eng. (1)

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

Opt. Lett. (3)

Proc. IEEE (1)

D. Casasent, D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77–84 (1977).
[CrossRef]

Other (7)

T. Kotzer, N. Cohen, J. Shamir, “Image reconstruction by a novel parallel projection onto constraint sets method,” EE Pub. 919 (Dept. of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, June1994).

T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Multi-distance, Multi-projection, parallel projection method,” presented at the International Conference on Optical Computing-OC94, Edinburgh, Scotland, August, 1994.

T. Kotzer, N. Cohen, J. Shamir, “Signal synthesis and reconstruction by projection methods in a hyper space,” presented at the Annual Meeting of the Optical Society of America, Dallas, Tex., October 1994.

T. Kotzer, N. Cohen, J. Shamir, “Extended and alternative projections onto convex constraint sets: theory and applications,” EE Pub. 900, (Dept. of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, November1993).

T. Kotzer, N. Cohen, J. Shamir, “A projection algorithm for consistent and inconsistent constraints,” EE Pub. 920 (Dept. of Electrical Engineering, Technion Israel Institute of Technology, Haifa, August1994).

T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Summed distance error reduction of simultaneous multiprojections and applications,” EE Pub. 909 (Dept. of Electrical Engineering, Technion—Israel Institute of Technology, Haifa, Israel, August1994).

N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space (Ungar, New York, 1961), Vol. I.

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Figures (9)

Fig. 1
Fig. 1

Input distribution.

Fig. 2
Fig. 2

Correlation results with a LC. The input is Fig. 1, where the filter is (a) a POF matched to letter F, (b) generated by algorithm 2 for the LC.

Fig. 3
Fig. 3

Correlation results with the PEC. The input is Fig. 1, where the filter is (a) a POF matched to letter F, (b) generated by algorithm 2 for the PEC.

Fig. 4
Fig. 4

(a) Impulse response of the filter. (b) As Fig. 3b, but with the letters of the input (from Fig. 1) F and E interchanged.

Fig. 5
Fig. 5

Block diagram of the phase-extraction rotation-invariant correlator: IFT, inverse FT; f(x, y), h(x, y) the input and the filter functions, respectively; CHCF, CHC filter.

Fig. 6
Fig. 6

Three input distributions.

Fig. 7
Fig. 7

Output correlation distributions for the LC corresponding to the input patterns of Figs. 6(a) and 6(b), with the appropriate CHC POF matched to the letter P of the order of N = 1.

Fig. 8
Fig. 8

As Fig. 7, but for the PEC.

Fig. 9
Fig. 9

(a)–(c) Output correlation distributions for the GPEC that correspond to the input patterns of Figs. 6(a)6(c), respectively, but with the filter generated by the serial POCS algorithm, for the GPEC.

Tables (2)

Tables Icon

Table 1 PCE and Rejection Measurements for the LC and the PEC for Various Orders of CHC’sa

Tables Icon

Table 2 EM for the LC and the PEC for Various Orders of the CHC’s

Equations (81)

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

P ( h ) = h if and only if h C and inf y C d ( y , h ) = d ( h , h ) ;
d ( h , h ) = h h H : = | h ( x ) h ( x ) | 2 d x .
P λ ( h ) = P ( h ) + λ [ P ( h ) h ] ,
T = P N , λ N P N 1 , λ N 1 P 1 , λ 1 ,
h k + 1 = T ( h k ) , k 0 .
d i ( h 1 , h 2 ) : = H 1 H 2 W i 2 : = | H 1 ( u ) H 2 ( u ) | 2 W i ( u ) d u h 1 , h 2 H ,
J ˆ ( h ) 2 : = i = 1 N β i d i [ P C i d i ( h ) , h ] = i = 1 N β i F { P C i d i ( h ) } F { h } W i 2 ,
P C i d i ( h ) = h if and only if inf h 1 C i d i ( h 1 , h ) = d i ( h , h ) , h C i .
υ i k + 1 ( x ) : = P i , λ [ h k ( x ) ] , for all i = 1 , 2 , , N ,
h k + 1 ( x ) = F 1 { i = 1 N β i W i ( u ) F { υ i k + 1 } ( u ) i = 1 N β i W i ( u ) } ,
Φ ( x ) = F 1 { N l { F { f ( x ) } } N l { F { h ( x ) } } }
N l { R ( u ) } = | R ( u ) | l exp [ i φ ( u ) ] R ( u ) = | R ( u ) | exp [ i φ ( u ) ] , 0 l 1 .
Φ ( x ) = h ( x ) f ( x ) .
Φ ( x ) = h p ( x ) f p ( x ) .
Φ ( x ) = h ( x ) f p ( x ) ,
C 1 : = { h | ( h f ) ( j ) C ˆ 1 , j } ,
C ˆ 1 : = { Φ ( j ) | | Φ ( j ) | T 2 , for j R 2 ; Φ re ( j ) T 1 and Φ im ( j ) = 0 , for j R 1 } ,
C 2 : = { h | | F { h ( j ) } | 1 } ,
C 3 : = { h | h ( j ) = 0 , for j [ a , a ] ; a > 0 } ,
Φ ( j ) : = ( h f ) ( j ) , Φ ( j ) : = Φ re ( j ) + i Φ im ( j ) ; Φ re ( j ) = Re { Φ ( j ) } , Φ im ( j ) = Im { Φ ( j ) } .
P C 2 d 2 { h ( x ) } = F 1 { H ( u ) } ,
H ( u ) = { H ( u ) if | H ( u ) | 1 exp [ i φ H ( u ) ] otherwise , P C 3 d 3 { h ( x ) } = { h ( x ) if x ( a , a ) 0 otherwise ,
d 1 ( H 1 , H 2 ) = m W 1 ( m ) | H 1 ( m ) H 2 ( m ) | 2 = j | F 1 { [ W 1 ( m ) ] 1 / 2 } ( j ) [ h 1 ( j ) h 2 ( j ) ] | 2 .
{ F 1 { V 1 } = } υ 1 : = P C 1 d 1 ( h ) , where V 1 ( m ) = F { Φ ( j ) } ( m ) F ( m ) , Φ ( j ) = { T 2 exp [ i φ Φ ( j ) ] , if j R 2 and | Φ ( x ) | > T 2 Φ ( j ) , if j R 2 and | Φ ( j ) | T 2 T 1 , if j R 1 and Φ re ( j ) < T 1 , Φ re ( j ) , if j R 1 and C re ( j ) T 1 Φ ( j ) , otherwise
Φ ( j ) = | Φ ( j ) | exp [ + i φ Φ ( j ) ] = F 1 { H ( m ) F ( m ) } = h ( j ) f ( j ) .
H k + 1 ( m ) = F { P C 1 d 1 { F 1 { H k } } } ( m ) W 1 ( m ) + F { P C 2 d 2 { F 1 { H k } } } ( m ) + F { P C 3 d 3 { F 1 { H k } } } ( m ) W 1 ( m ) + 1 + 1 ,
C 2 nc : = { h | | F { h ( j ) } | = 1 } .
F ( ρ , φ ) | A ( ρ , φ ) | exp [ i γ ( ρ , φ ) ] .
exp [ i γ ( ρ , φ ) ] = N = A N ( ρ ) exp ( i N φ ) ,
A N ( ρ ) = | A N ( ρ ) | exp { i arg [ A N ( ρ ) ] } = 1 2 π 0 2 π exp [ i γ ( ρ , φ ) ] exp ( i N φ ) d φ .
PCE = | Φ ( 0 , 0 ) | 2 S 0 | Φ ( x , y ) | 2 d x d y ,
H N ( ρ , φ ) = exp { i arg [ H N ( ρ , φ ) ] } = exp ( i { N φ + arg [ A N ( ρ ) ] } ) .
Φ ( 0 , 0 ) = ( 1 2 π ) 2 0 2 π 0 R 0 F ( ρ , α + φ ) H N ( ρ , φ ) ρ d ρ d φ = exp ( i N α ) 2 π 0 R 0 | A N ( ρ ) | ρ d ρ ,
S 0 | Φ ( x , y ) | 2 d x d y = 1 2 π M = 0 ρ 0 | A M ( ρ ) | 2 ρ d ρ .
PCE N P = 1 2 π [ 0 R 0 | A N ( ρ ) | ρ d ρ ] 2 M = 0 ρ 0 | A M ( ρ ) | 2 ρ d ρ ,
B N ( ρ ) = | B N ( ρ ) | exp { i arg [ B ( ρ ) ] } = 1 2 π 0 2 π | A ( ρ , θ ) | exp [ i γ ( ρ , θ ) ] exp ( i N θ ) d θ ,
H N ( ρ , φ ) = exp ( i { N φ + arg [ B N ( ρ ) ] } ) ,
PCE N L = 1 2 π [ 0 ρ 0 | B N ( ρ ) | ρ d ρ ] 2 m = 0 ρ 0 | B m ( ρ ) | 2 ρ d ρ .
0 2 π 0 R 0 | A ( ρ , φ ) exp [ i γ ( ρ , φ ) ] | 2 ρ d ρ d φ = 2 π N = 0 R 0 | B N ( ρ ) | 2 ρ d ρ < 0 2 π 0 R 0 | exp [ i γ ( ρ , φ ) ] | 2 ρ d ρ d φ = 2 π N = 0 R 0 | A N ( ρ ) | 2 ρ d ρ .
EM = the energy in the N th order the total energy .
EM N L = 0 ρ 0 | B N ( ρ ) | 2 ρ d ρ M = 0 ρ 0 | B M ( ρ ) | 2 ρ d ρ ,
EM N P = 0 ρ 0 | A N ( ρ ) | 2 ρ d ρ M = 0 ρ 0 | A M ( ρ ) | 2 ρ d ρ .
C p det : = { H ( ρ ) | H ( ρ ) α p ( ρ ) ρ d ρ = const , const IR } ,
C p f rej : = { H ( ρ ) | H ( ρ ) α p f ( ρ ) ρ d ρ = 0 } ,
C p x rej : = { H ( ρ ) | H ( ρ ) α p x ( ρ ) ρ d ρ = 0 } ,
C p det : = { H ( ρ ) | H ( ρ ) α p ( ρ ) ρ d ρ = const } ,
C f det : = { H ( ρ ) | H ( ρ ) α f ( ρ ) ρ d ρ = const } ,
C x det : = { H ( ρ ) | H ( ρ ) α x ( ρ ) ρ d ρ = const } ,
T { H ( ρ ) } : = P C pas P C p det P C p f rej P C p x rej { H ( ρ ) } ,
H : = i = 1 N H i ,
τ ( h ) = : ( h , h , , h ) . N times
D ( h 1 , h 2 ) = i = 1 N β i d i ( h 1 i , h 2 i ) ; h 1 i , h 2 i H i ; h j : = ( h j 1 , h j 2 , , h j N ) , j = 1 , 2 ,
P S D ( h ) = h if and only if inf h 1 S D ( h 1 , h ) = D ( h , h ) .
inf h 1 C D [ h 1 , τ ( h ) ] = i = 1 N β i [ inf h 1 i C i d i ( h 1 i , h ) ] .
P C D [ τ ( h ) ] = [ P C 1 d 1 ( h ) , P C 2 d 2 ( h ) , , P C N d N ( h ) ] .
h = P Δ D ( h ) if and only if h = τ ( h ) where h 1 D [ τ ( h 1 ) , h ] | h 1 = h = 0 .
h = τ ( h ) , where h : = F 1 { i = 1 N β i W i ( u ) V i ( u ) i = 1 N β i W i ( u ) } .
P C , λ ( h ) : = P C ( h ) + λ [ h P C ( h ) ] .
h k + 1 : = T 1 [ T 2 ( h k ) ] ; k 0 ,
J ( h ) : = i = 1 2 h P C i ( h ) ,
v k + 1 = P C , λ D ( h k ) ,
h k + 1 = P Δ D ( v k ) ,
J ( h ) : = { D [ P C D ( h ) , h ] } 1 / 2 + { D [ P Δ D ( h ) , h ] } 1 / 2 .
d 2 [ H ( ρ ) , H ( ρ ) ] = 0 { [ H r ( ρ ) H r ( ρ ) ] 2 + [ H i ( ρ ) H i ( ρ ) ] 2 } ρ d ρ ,
A = 0 [ H r ( ρ ) f r ( ρ ) H i ( ρ ) f i ( ρ ) ] ρ d ρ ,
B = 0 [ H r ( ρ ) f i ( ρ ) + H i ( ρ ) f r ( ρ ) ] ρ d ρ ,
H ( ρ ) = H r ( ρ ) + i H i ( ρ ) ; H ( ρ ) = H r ( ρ ) + i H i ( ρ ) , f ( ρ ) = f r ( ρ ) + i f i ( ρ ) .
0 [ H r ( ρ ) f r ( ρ ) H i ( ρ ) f i ( ρ ) ] ρ d ρ = 0 , 0 [ H r ( ρ ) f i ( ρ ) + H i ( ρ ) f r ( ρ ) ] ρ d ρ = 0
0 [ H r ( ρ ) f r ( ρ ) H i ( ρ ) f i ( ρ ) ] ρ d ρ T 1 , 0 [ H r ( ρ ) f i ( ρ ) + H i ( ρ ) f r ( ρ ) ] ρ d ρ = 0
g ( ρ ) = 0 [ H r ( ρ ) f r ( ρ ) H i ( ρ ) f i ( ρ ) ] ρ d ρ ( T 1 + ξ ) = 0 ,
q ( ρ ) = 0 [ H r ( ρ ) f r ( ρ ) + H r ( ρ ) f i ( ρ ) ] ρ d ρ = 0 ,
d 2 ( H , H ) = λ g + μ q .
H r ( ρ ) = H r ( ρ ) + λ f r ( ρ ) + μ f i ( ρ ) 2 ,
H i ( ρ ) = H i ( ρ ) + λ f i ( ρ ) + μ f r ( ρ ) 2 .
λ = 2 ( A T 1 ξ ) E ,
μ = 2 B E , E = 0 | f ( ρ ) | 2 ρ d ρ .
d ( H , H ) = min λ 2 + μ 2 4 E .
λ = 0 , if A > T 1 λ = + 2 ( T 1 A ) E if A T 1 ,
μ = 2 B E .
λ = 2 A E ,
μ = 2 B E .

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