Abstract

Projection methods have been shown to be extremely powerful for signal restoration and other signal and image processing tasks. However, they break down when some system parameters cannot be exactly defined. To mitigate this problem, we propose a new algorithm that is based on the extended parallel projection method combined with fuzzy set theory. The incompleteness of the available information is taken into account by considering the constraint sets and/or the iterates as fuzzy. Then some maximum membership is searched by using parallel projections. The introduction of the fuzzy sets formalism results in a flexible technique that improves substantially the results obtained by conventional methods. The conventional projection method is shown to be a special case of this fuzzy algorithm. Moreover, whereas in the conventional algorithm the projection weights were chosen arbitrarily, in the new algorithm they are related to the degree of uncertainty involved.

© 1999 Optical Society of America

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References

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  1. P. L. Combettes, H. J. Trussell, “The use of noise properties in set theoretic estimation,” IEEE Trans. Signal Process. 39, 1630–1641 (1991).
    [CrossRef]
  2. M. Goldburg, R. J. Marks, “Signal synthesis in the presence of an inconsistent set of constraints,” IEEE Trans. Circuits Syst. CAS-32, 647–663 (1985).
    [CrossRef]
  3. A. Levi, H. Stark, “Signal restoration from phase by projections onto convex sets,” J. Opt. Soc. Am. 73, 810–822 (1983).
    [CrossRef]
  4. A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from amplitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  5. M. I. Sezan, H. Stark, “Image restoration by convex projections in the presence of noise,” Appl. Opt. 22, 2781–2789 (1983).
    [CrossRef] [PubMed]
  6. D. C. Youla, V. Velasco, “Extensions of a result on the synthesis of signals in the presence of inconsistent constraints,” IEEE Trans. Circuits Syst. CAS-33, 465–468 (1986).
    [CrossRef]
  7. P. L. Combettes, H. J. Trussell, “Method of successive projections for finding a common point of sets in metric spaces,” J. Optim. Theory Appl. 67, 487–507 (1990).
    [CrossRef]
  8. T. Kotzer, N. Cohen, J. Shamir, “Generalized projection algorithms with applications to optics and signal restoration,” Opt. Commun. 156, 77–91 (1998).
    [CrossRef]
  9. T. Kotzer, N. Cohen, J. Shamir, “A projection-based algorithm for consistent and inconsistent constraints,” SIAM (Soc. Ind. Appl. Math.) J. Optim. 7, 527–546 (1997).
  10. R. Aharoni, Y. Censor, “Block-iterative methods for parallel computation of solutions to convex feasibility problems,” Linear Algebr. Appl. 120, 165–175 (1989).
    [CrossRef]
  11. P. L. Combettes, “Signal recovery by best feasible approximation,” IEEE Trans. Image Process. 2, 269–271 (1993).
    [CrossRef] [PubMed]
  12. P. L. Combettes, “Inconsistent signal feasibility problems: least-squares solutions in a product space,” IEEE Trans. Signal Process. 42, 2955–2966 (1994).
    [CrossRef]
  13. Y. Censor, T. Elfving, “A multiprojection algorithm us-ing Bregman projections in a product space,” Numer. Algorithms 8, 221–239 (1994).
    [CrossRef]
  14. T. Kotzer, N. Cohen, J. Shamir, “Image reconstruction by a novel parallel projection onto constraint set method,” Opt. Lett. 20, 1172–1174 (1995).
    [CrossRef] [PubMed]
  15. T. Kotzer, J. Rosen, J. Shamir, “Application of serial and parallel projection methods to correlation filter design,” Appl. Opt. 34, 3883–3895 (1995).
    [CrossRef] [PubMed]
  16. M. R. Civanlar, H. J. Trussell, “Digital signal restoration using fuzzy sets,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-34, 919–936 (1986).
    [CrossRef]
  17. S. Oh, R. J. Marks, “Alternating projection onto fuzzy convex sets,” in Proceedings of 1993 IEEE Conference on Fuzzy Systems, pp. 148–155.
  18. E. Cox, The Fuzzy Systems Handbook (Academic, San Diego, Calif., 1994).
  19. M. M. Gupta, R. K. Ragade, R. R. Yager, Advances in Fuzzy Set Theory and Applications (North-Holland, New York, 1979).
  20. A. Kandel, Fuzzy Mathematical Techniques with Applications (Addison-Wesley, Reading, Mass., 1986).
  21. A. Kaufmann, Introduction to the Theory of Fuzzy Subsets (Academic, New York, 1975), Vol. I.
  22. L. A. Zadeh, “From circuit theory to system theory,” Proc. IRE 50, 856–865 (1962).
    [CrossRef]
  23. L. A. Zadeh, “Fuzzy sets,” Inf. Control. 8, 338–353 (1965).
    [CrossRef]
  24. L. A. Zadeh, “Fuzzy sets and systems,” in System Theory, Microwave Research Institute Symposia Series XV, J. Fox, ed. (Polytechnic, Brooklyn, N.Y.1965), pp. 29–37.
  25. C. V. Negoita, “The current interest in fuzzy optimization,” Fuzzy Sets Syst. 6, 261–269 (1981).
    [CrossRef]
  26. R. Pearce, P. H. Cowley, “Use of fuzzy logic to describe constraints derived from engineering judgment in genetic algorithms,” IEEE Trans. Ind. Electron. 43, 535–540 (1996).
    [CrossRef]
  27. W. Pedrycz, Fuzzy Control and Fuzzy Systems (Wiley, New York, 1989).
  28. J. Ramı́k, “Extension principle in fuzzy optimization,” Fuzzy Sets Syst. 19, 29–35 (1986).
    [CrossRef]
  29. G. Pierra, “Decomposition through formalization in a product space,” Math. Program. 18, 96–115 (1984).
    [CrossRef]

1998

T. Kotzer, N. Cohen, J. Shamir, “Generalized projection algorithms with applications to optics and signal restoration,” Opt. Commun. 156, 77–91 (1998).
[CrossRef]

1997

T. Kotzer, N. Cohen, J. Shamir, “A projection-based algorithm for consistent and inconsistent constraints,” SIAM (Soc. Ind. Appl. Math.) J. Optim. 7, 527–546 (1997).

1996

R. Pearce, P. H. Cowley, “Use of fuzzy logic to describe constraints derived from engineering judgment in genetic algorithms,” IEEE Trans. Ind. Electron. 43, 535–540 (1996).
[CrossRef]

1995

1994

P. L. Combettes, “Inconsistent signal feasibility problems: least-squares solutions in a product space,” IEEE Trans. Signal Process. 42, 2955–2966 (1994).
[CrossRef]

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

1993

P. L. Combettes, “Signal recovery by best feasible approximation,” IEEE Trans. Image Process. 2, 269–271 (1993).
[CrossRef] [PubMed]

1991

P. L. Combettes, H. J. Trussell, “The use of noise properties in set theoretic estimation,” IEEE Trans. Signal Process. 39, 1630–1641 (1991).
[CrossRef]

1990

P. L. Combettes, H. J. Trussell, “Method of successive projections for finding a common point of sets in metric spaces,” J. Optim. Theory Appl. 67, 487–507 (1990).
[CrossRef]

1989

R. Aharoni, Y. Censor, “Block-iterative methods for parallel computation of solutions to convex feasibility problems,” Linear Algebr. Appl. 120, 165–175 (1989).
[CrossRef]

1986

D. C. Youla, V. Velasco, “Extensions of a result on the synthesis of signals in the presence of inconsistent constraints,” IEEE Trans. Circuits Syst. CAS-33, 465–468 (1986).
[CrossRef]

M. R. Civanlar, H. J. Trussell, “Digital signal restoration using fuzzy sets,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-34, 919–936 (1986).
[CrossRef]

J. Ramı́k, “Extension principle in fuzzy optimization,” Fuzzy Sets Syst. 19, 29–35 (1986).
[CrossRef]

1985

M. Goldburg, R. J. Marks, “Signal synthesis in the presence of an inconsistent set of constraints,” IEEE Trans. Circuits Syst. CAS-32, 647–663 (1985).
[CrossRef]

1984

1983

1981

C. V. Negoita, “The current interest in fuzzy optimization,” Fuzzy Sets Syst. 6, 261–269 (1981).
[CrossRef]

1965

L. A. Zadeh, “Fuzzy sets,” Inf. Control. 8, 338–353 (1965).
[CrossRef]

1962

L. A. Zadeh, “From circuit theory to system theory,” Proc. IRE 50, 856–865 (1962).
[CrossRef]

Aharoni, R.

R. Aharoni, Y. Censor, “Block-iterative methods for parallel computation of solutions to convex feasibility problems,” Linear Algebr. Appl. 120, 165–175 (1989).
[CrossRef]

Censor, Y.

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

R. Aharoni, Y. Censor, “Block-iterative methods for parallel computation of solutions to convex feasibility problems,” Linear Algebr. Appl. 120, 165–175 (1989).
[CrossRef]

Civanlar, M. R.

M. R. Civanlar, H. J. Trussell, “Digital signal restoration using fuzzy sets,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-34, 919–936 (1986).
[CrossRef]

Cohen, N.

T. Kotzer, N. Cohen, J. Shamir, “Generalized projection algorithms with applications to optics and signal restoration,” Opt. Commun. 156, 77–91 (1998).
[CrossRef]

T. Kotzer, N. Cohen, J. Shamir, “A projection-based algorithm for consistent and inconsistent constraints,” SIAM (Soc. Ind. Appl. Math.) J. Optim. 7, 527–546 (1997).

T. Kotzer, N. Cohen, J. Shamir, “Image reconstruction by a novel parallel projection onto constraint set method,” Opt. Lett. 20, 1172–1174 (1995).
[CrossRef] [PubMed]

Combettes, P. L.

P. L. Combettes, “Inconsistent signal feasibility problems: least-squares solutions in a product space,” IEEE Trans. Signal Process. 42, 2955–2966 (1994).
[CrossRef]

P. L. Combettes, “Signal recovery by best feasible approximation,” IEEE Trans. Image Process. 2, 269–271 (1993).
[CrossRef] [PubMed]

P. L. Combettes, H. J. Trussell, “The use of noise properties in set theoretic estimation,” IEEE Trans. Signal Process. 39, 1630–1641 (1991).
[CrossRef]

P. L. Combettes, H. J. Trussell, “Method of successive projections for finding a common point of sets in metric spaces,” J. Optim. Theory Appl. 67, 487–507 (1990).
[CrossRef]

Cowley, P. H.

R. Pearce, P. H. Cowley, “Use of fuzzy logic to describe constraints derived from engineering judgment in genetic algorithms,” IEEE Trans. Ind. Electron. 43, 535–540 (1996).
[CrossRef]

Cox, E.

E. Cox, The Fuzzy Systems Handbook (Academic, San Diego, Calif., 1994).

Elfving, T.

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

Goldburg, M.

M. Goldburg, R. J. Marks, “Signal synthesis in the presence of an inconsistent set of constraints,” IEEE Trans. Circuits Syst. CAS-32, 647–663 (1985).
[CrossRef]

Gupta, M. M.

M. M. Gupta, R. K. Ragade, R. R. Yager, Advances in Fuzzy Set Theory and Applications (North-Holland, New York, 1979).

Kandel, A.

A. Kandel, Fuzzy Mathematical Techniques with Applications (Addison-Wesley, Reading, Mass., 1986).

Kaufmann, A.

A. Kaufmann, Introduction to the Theory of Fuzzy Subsets (Academic, New York, 1975), Vol. I.

Kotzer, T.

T. Kotzer, N. Cohen, J. Shamir, “Generalized projection algorithms with applications to optics and signal restoration,” Opt. Commun. 156, 77–91 (1998).
[CrossRef]

T. Kotzer, N. Cohen, J. Shamir, “A projection-based algorithm for consistent and inconsistent constraints,” SIAM (Soc. Ind. Appl. Math.) J. Optim. 7, 527–546 (1997).

T. Kotzer, J. Rosen, J. Shamir, “Application of serial and parallel projection methods to correlation filter design,” Appl. Opt. 34, 3883–3895 (1995).
[CrossRef] [PubMed]

T. Kotzer, N. Cohen, J. Shamir, “Image reconstruction by a novel parallel projection onto constraint set method,” Opt. Lett. 20, 1172–1174 (1995).
[CrossRef] [PubMed]

Levi, A.

Marks, R. J.

M. Goldburg, R. J. Marks, “Signal synthesis in the presence of an inconsistent set of constraints,” IEEE Trans. Circuits Syst. CAS-32, 647–663 (1985).
[CrossRef]

S. Oh, R. J. Marks, “Alternating projection onto fuzzy convex sets,” in Proceedings of 1993 IEEE Conference on Fuzzy Systems, pp. 148–155.

Negoita, C. V.

C. V. Negoita, “The current interest in fuzzy optimization,” Fuzzy Sets Syst. 6, 261–269 (1981).
[CrossRef]

Oh, S.

S. Oh, R. J. Marks, “Alternating projection onto fuzzy convex sets,” in Proceedings of 1993 IEEE Conference on Fuzzy Systems, pp. 148–155.

Pearce, R.

R. Pearce, P. H. Cowley, “Use of fuzzy logic to describe constraints derived from engineering judgment in genetic algorithms,” IEEE Trans. Ind. Electron. 43, 535–540 (1996).
[CrossRef]

Pedrycz, W.

W. Pedrycz, Fuzzy Control and Fuzzy Systems (Wiley, New York, 1989).

Pierra, G.

G. Pierra, “Decomposition through formalization in a product space,” Math. Program. 18, 96–115 (1984).
[CrossRef]

Ragade, R. K.

M. M. Gupta, R. K. Ragade, R. R. Yager, Advances in Fuzzy Set Theory and Applications (North-Holland, New York, 1979).

Rami´k, J.

J. Ramı́k, “Extension principle in fuzzy optimization,” Fuzzy Sets Syst. 19, 29–35 (1986).
[CrossRef]

Rosen, J.

Sezan, M. I.

Shamir, J.

T. Kotzer, N. Cohen, J. Shamir, “Generalized projection algorithms with applications to optics and signal restoration,” Opt. Commun. 156, 77–91 (1998).
[CrossRef]

T. Kotzer, N. Cohen, J. Shamir, “A projection-based algorithm for consistent and inconsistent constraints,” SIAM (Soc. Ind. Appl. Math.) J. Optim. 7, 527–546 (1997).

T. Kotzer, J. Rosen, J. Shamir, “Application of serial and parallel projection methods to correlation filter design,” Appl. Opt. 34, 3883–3895 (1995).
[CrossRef] [PubMed]

T. Kotzer, N. Cohen, J. Shamir, “Image reconstruction by a novel parallel projection onto constraint set method,” Opt. Lett. 20, 1172–1174 (1995).
[CrossRef] [PubMed]

Stark, H.

Trussell, H. J.

P. L. Combettes, H. J. Trussell, “The use of noise properties in set theoretic estimation,” IEEE Trans. Signal Process. 39, 1630–1641 (1991).
[CrossRef]

P. L. Combettes, H. J. Trussell, “Method of successive projections for finding a common point of sets in metric spaces,” J. Optim. Theory Appl. 67, 487–507 (1990).
[CrossRef]

M. R. Civanlar, H. J. Trussell, “Digital signal restoration using fuzzy sets,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-34, 919–936 (1986).
[CrossRef]

Velasco, V.

D. C. Youla, V. Velasco, “Extensions of a result on the synthesis of signals in the presence of inconsistent constraints,” IEEE Trans. Circuits Syst. CAS-33, 465–468 (1986).
[CrossRef]

Yager, R. R.

M. M. Gupta, R. K. Ragade, R. R. Yager, Advances in Fuzzy Set Theory and Applications (North-Holland, New York, 1979).

Youla, D. C.

D. C. Youla, V. Velasco, “Extensions of a result on the synthesis of signals in the presence of inconsistent constraints,” IEEE Trans. Circuits Syst. CAS-33, 465–468 (1986).
[CrossRef]

Zadeh, L. A.

L. A. Zadeh, “Fuzzy sets,” Inf. Control. 8, 338–353 (1965).
[CrossRef]

L. A. Zadeh, “From circuit theory to system theory,” Proc. IRE 50, 856–865 (1962).
[CrossRef]

L. A. Zadeh, “Fuzzy sets and systems,” in System Theory, Microwave Research Institute Symposia Series XV, J. Fox, ed. (Polytechnic, Brooklyn, N.Y.1965), pp. 29–37.

Appl. Opt.

Fuzzy Sets Syst.

J. Ramı́k, “Extension principle in fuzzy optimization,” Fuzzy Sets Syst. 19, 29–35 (1986).
[CrossRef]

C. V. Negoita, “The current interest in fuzzy optimization,” Fuzzy Sets Syst. 6, 261–269 (1981).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process.

M. R. Civanlar, H. J. Trussell, “Digital signal restoration using fuzzy sets,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-34, 919–936 (1986).
[CrossRef]

IEEE Trans. Circuits Syst.

D. C. Youla, V. Velasco, “Extensions of a result on the synthesis of signals in the presence of inconsistent constraints,” IEEE Trans. Circuits Syst. CAS-33, 465–468 (1986).
[CrossRef]

M. Goldburg, R. J. Marks, “Signal synthesis in the presence of an inconsistent set of constraints,” IEEE Trans. Circuits Syst. CAS-32, 647–663 (1985).
[CrossRef]

IEEE Trans. Image Process.

P. L. Combettes, “Signal recovery by best feasible approximation,” IEEE Trans. Image Process. 2, 269–271 (1993).
[CrossRef] [PubMed]

IEEE Trans. Ind. Electron.

R. Pearce, P. H. Cowley, “Use of fuzzy logic to describe constraints derived from engineering judgment in genetic algorithms,” IEEE Trans. Ind. Electron. 43, 535–540 (1996).
[CrossRef]

IEEE Trans. Signal Process.

P. L. Combettes, “Inconsistent signal feasibility problems: least-squares solutions in a product space,” IEEE Trans. Signal Process. 42, 2955–2966 (1994).
[CrossRef]

P. L. Combettes, H. J. Trussell, “The use of noise properties in set theoretic estimation,” IEEE Trans. Signal Process. 39, 1630–1641 (1991).
[CrossRef]

Inf. Control.

L. A. Zadeh, “Fuzzy sets,” Inf. Control. 8, 338–353 (1965).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Optim. Theory Appl.

P. L. Combettes, H. J. Trussell, “Method of successive projections for finding a common point of sets in metric spaces,” J. Optim. Theory Appl. 67, 487–507 (1990).
[CrossRef]

Linear Algebr. Appl.

R. Aharoni, Y. Censor, “Block-iterative methods for parallel computation of solutions to convex feasibility problems,” Linear Algebr. Appl. 120, 165–175 (1989).
[CrossRef]

Math. Program.

G. Pierra, “Decomposition through formalization in a product space,” Math. Program. 18, 96–115 (1984).
[CrossRef]

Numer. Algorithms

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

Opt. Commun.

T. Kotzer, N. Cohen, J. Shamir, “Generalized projection algorithms with applications to optics and signal restoration,” Opt. Commun. 156, 77–91 (1998).
[CrossRef]

Opt. Lett.

Proc. IRE

L. A. Zadeh, “From circuit theory to system theory,” Proc. IRE 50, 856–865 (1962).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Optim.

T. Kotzer, N. Cohen, J. Shamir, “A projection-based algorithm for consistent and inconsistent constraints,” SIAM (Soc. Ind. Appl. Math.) J. Optim. 7, 527–546 (1997).

Other

S. Oh, R. J. Marks, “Alternating projection onto fuzzy convex sets,” in Proceedings of 1993 IEEE Conference on Fuzzy Systems, pp. 148–155.

E. Cox, The Fuzzy Systems Handbook (Academic, San Diego, Calif., 1994).

M. M. Gupta, R. K. Ragade, R. R. Yager, Advances in Fuzzy Set Theory and Applications (North-Holland, New York, 1979).

A. Kandel, Fuzzy Mathematical Techniques with Applications (Addison-Wesley, Reading, Mass., 1986).

A. Kaufmann, Introduction to the Theory of Fuzzy Subsets (Academic, New York, 1975), Vol. I.

L. A. Zadeh, “Fuzzy sets and systems,” in System Theory, Microwave Research Institute Symposia Series XV, J. Fox, ed. (Polytechnic, Brooklyn, N.Y.1965), pp. 29–37.

W. Pedrycz, Fuzzy Control and Fuzzy Systems (Wiley, New York, 1989).

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Figures (6)

Fig. 1
Fig. 1

Original signal.

Fig. 2
Fig. 2

Distorted signal.

Fig. 3
Fig. 3

Typical evolution of the reconstruction errors: A, weighted EPPOCS; B, nonuniformly relaxed, weighted EPPOCS; C, serial POCS; D, fuzzy EPPOCS.

Fig. 4
Fig. 4

Signals restored by the different algorithms: (a) crisp EPPOCS, (b) crisp nonuniformly relaxed EPPOCS, (c) crisp serial POCS, (d) fuzzy EPPOCS.

Fig. 5
Fig. 5

Image presented in the input plane of the correlator.

Fig. 6
Fig. 6

Correlation plane intensities and discrimination ratios: (a) crisp method without distortions or noise (r=3.0632), (b) crisp method with distortions and noise (r=1.6731), (c) fuzzy method without distortions or noise (r=2.1773), (d) fuzzy method with distortions and noise (r=2.7793).

Equations (73)

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

xX,μC(x)=μA(x) t μB(x).
a t b
=g-1[g(a)+g(b)]ifg(a)+g(b)[0, g(0)]0otherwise.
Fhˆ(h)  limδ0 F(h+δhˆ)-F(h)δ.
h1, h2X,
di2(h1, h2)=Wi(u)|H1(u)-H2(u)|2 du.
h1, h2i=Wi(u)H1(u)H2*(u)du,
J(h)=i=1Nβidi2[h, Pi(h)],βi0,i=1Nβi=1,
hX,
QF(h)=F-1i=1NηiWi(u)Pi(u)i=1NηiWi(u),
ηi=fi(di[h, Pi(h)])di[h, Pi(h)]=1di (βidi2)di=2βi
QF(h)  Qβ=F-1i=1NβiWi(u)Pi(u)i=1NβiWi(u),
hX,μCi(h)=μi(di[h,Pi(h)]).
find h¯X/μC1(h¯) t  t μCN(h¯)ismaximizedoverX.
G(h)=μC1(h)t  tμCN(h)=g-1i=1Ng[μCi(h)].
QF(h)=F-1i=1NηiWi(u)F{Pi(h)}(u)i=1NηiWi(u),
ηi=g[μCi(h)]μi(di[h,Pi(h)])di[h,Pi(h)].
μi(di)=exp-di2wi2,
g(m)=-ln m  g[μi(di)]=di2wi2
 ηi=2wi2,βi=1wi2,
J(h)=i=1N 1wi2 di2[h, Pi(h)].
C(u)={H/H(u)=P(u)};
dW2[h, Pu(h)]=W(u)|H(u)-P(u)|2.
uβ(u)dW2[h, Pu(h)]du
=uβ(u)W(u)|H(u)-P(u)|2 du
=dβW2[h, PCdW(h)];
C={H/|H(u)-HC(u)|(u)},
F{PC(h)}=H(u)|H(u)-HC(u)|(u)HC(u)+(u) H(u)-HC(u)|H(u)-HC(u)|otherwise,
F{PC(h)}=H(u)+λ(u)[HC(u)-H(u)],
λ(u)=max0, 1-(u)|H(u)-HC(u)|.
PC(h)=h(x)+λ(x)[hC(x)-h(x)],
QF(h)=F-1i=1Nηi(u)Wi(u)Pi(u)i=1Nηi(u)Wi(u).
HD=F{h0}exp(jϕ˜)+n˜ exp(jθ˜),
Cb={h|h(x, y)R, 0h(x, y)255x, y},
C0={HD},
C0={H|H(u)-HD(u)|  (u)}.
Pb(h)=0Re(h)<0Re(h)0  Re(h)  255255Re(h)>255,
F{P0(h)}=HD,
F{P0(h)}=H(u)+λ(u)[HD(u)-H(u)],
λ(u)=max0, 1-(u)|H(u)-HD(u)|.
hk+1=F-1wF{Pb,λ1(hk)}+P0,λ2(F{hk})w+1,
Pb,λ1=λ1Pb+(1-λ1)I,
P0,λ2=λ2P0+(1-λ2)I,
hk+1=F-1wbF{Pb(hk)}+F{P0(hk)}wb+1.
hk+1=Pbλ1(P0λ2(hk)),
hk+1=F-1η(u)HD(u)+ηbF{Pb(hk)}η(u)+ηb.
CSL={h/h(x)=0, xD1},
Cpof={H/|H(u)|=1},
CF={XF/XF(u)T1andXF(u) real, uD2;
|XF(u)|T2, uD2},
CE={XE/|XE(u)|  T3u},
PSL(h)=h(x)xD10xD1,Ppof(H)=H(u)|H(u)|,
PF(H)=F-1{XF}(u)F(u),
XF(u)
=T1uD2and Re[XF(u)]<T1Re[XF(u)]uD2and Re[XF(u)]T1T2 exp[jXF(u)]uD2and|XF(u)|T2XF(u)otherwise .
PE(H)=F-1{XE}(u)E(u),
XE(u)=T3 exp[jXE(u)]|XE(u)|T3XE(u)otherwise.
Fhˆ(h)  limδ0 F(h+δhˆ)-F(h)δ.
Fhˆ(h)=limδ0 G(h+δhˆ), K(h+δhˆ)-G(h), K(h)δ=limδ0 G(h+δhˆ), K(h+δhˆ)-G(h), K(h+δhˆ)+G(h), K(h+δhˆ)-G(h), K(h)δ=limδ0G(h+δhˆ), K(h+δhˆ)-G(h), K(h+δhˆ)δ+G(h), K(h+δhˆ)-G(h), K(h)δ=Ghˆ(h), K(h)+G(h), Khˆ(h).
Thˆ(h)  limδ0 T(h+δhˆ)-T(h)δ.
[F2(h)]hˆ=[T(h), T(h)]hˆ=Thˆ(h), T(h)+T(h), Thˆ(h)=2 ReThˆ(h), T(h),
[F2(h)]hˆ=2F(h)Fhˆ(h)  Fhˆ(h)=ReThˆ(h), T(h)F(h).
Fhˆ(h)=Rehˆ, h-pWdW(h, p).
Fhˆ(h)=i=1N Fdi dihˆ=i=1N Rehˆ, Fdi 1di (h-pi)i=ReHˆ(u)i=1NηiWi(u)[H(u)-Pi(u)]*du,
ηi=1di Fdi,
i=1NηiWi(u)[H(u)-Pi(u)]0
H(u)=i=1NηiWi(u)Pi(u)i=1NηiWi(u).
hX,F(h)=i=1Nfi(di[h, Pi(h)]),
hX,ψh(h)=i=1Nfi(di[h, Pi(h)]).
h=F-1i=1NηiWi(u)Pi(u)i=1NηiWi(u),
ηi=fi(di[h, Pi(h)])di[h, Pi(h)],
QF(h)=F-1i=1NηiWi(u)Pi(u)i=1NηiWi(u),
ηi=fi(di[h, Pi(h)])di[h, Pi(h)].

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