Abstract

The problem of retrieving a complex function when both its square modulus and the square modulus of its Fourier transform are known is considered. When these intensities are directly assumed to be data, it amounts to performing the inversion of a quadratic operator. The solution is found to be the global minimum of an appropriate functional. Moreover, inasmuch as the unknown function is modeled within a finite-dimensional set, the data are also consistently represented within finite-dimensional subspaces, and a coherent discretization of the problem results. Because the assumed formulation involves nonquadratic functionals, the crucial problem of the existence of local minima in the course of the minimization procedure is discussed. The main factors affecting these minima can be identified, such as the amount of available independent data. Furthermore, quadraticity makes it possible to define an efficient conjugate-gradient-based minimization procedure. The numerical results confirm the distinguishing feature of the proposed approach—its ability to obtain the solution starting from a completely random guess.

© 1996 Optical Society of America

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References

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  1. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
    [CrossRef]
  2. G. Poulton, “Antenna application of phase retrievial: a review,” in Proceedings of the 1992 URSI International Symposium on Electromagnetic Theory (Union Radio-Scientifique Internationale, Sydney, Australia, 1992).
  3. T. Isernia, G. Leone, R. Pierri, “Antenna testing from phaseless measurements: probe compensation and experimental results for the cylindrical case,” Proc. Inst. Electr. Eng. H 140, 395–400 (1993).
  4. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction planar pictures,” Opt. Lett. 35, 237–246 (1972).
  5. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2764 (1983).
    [CrossRef]
  6. R. Barakat, G. Newsman, “Algorithms for reconstruction of partially known band-limited Fourier transform pairs from noisy data,” J. Opt. Soc. Am. A 2, 2027–2039 (1985).
    [CrossRef]
  7. A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–942 (1984).
    [CrossRef]
  8. D. C. Dobson, “Phase reconstruction via nonlinear least squares,” Inv. Prob. 8, 541–557 (1992).
    [CrossRef]
  9. R. Barakat, B. Sandler, “Determination of the wavefront aberration function from measured values of the point-spread function: a two-dimensional phase retrieval problem,” J. Opt. Soc. Am. A 9, 1715–1723 (1992).
    [CrossRef]
  10. W. H. Southwell, “Wave-front analyzer using a maximum likelihood algorithm,”J. Opt. Soc. Am. 67, 396–399 (1977).
    [CrossRef]
  11. T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inv. Prob. 11, 183–203 (1995).
    [CrossRef]
  12. V. Yu. Ivanov, V. P. Sivokon, M. A. Vorontsov, “Phase retrieval from a set of intensity measurements: theory and experiment,” J. Opt. Soc. Am. A 9, 1515–1524 (1992).
    [CrossRef]
  13. I. Sabba Stefanescu, “On the phase retrieval problem in two dimensions,”J. Math. Phys. 26, 2141–2160 (1985).
    [CrossRef]
  14. A. M. J. Huiser, H. A. Fewerda, “On the problem of the phase retrieval in electron microscopy from image and diffraction pattern,” Optik 46, 407–420 (1976).
  15. D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-I,” Bell Syst. Tech. J. 40, 43–64 (1961).
  16. J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, San Diego, Calif., 1970).
  17. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).
  18. G. Chavent, “A new size × curvature condition for strict quasi convexity of sets,” SIAM J. Control Optim. 29, 1348–1372 (1991).
    [CrossRef]
  19. T. Isernia, G. Leone, R. Pierri, “Phaseless near field techniques: uniqueness condition and attainment of the solution,”J. Electromagn. Waves Appl. 8, 889–908 (1994).
  20. D. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1987).
  21. T. Isernia, G. Leone, V. Pascazio, R. Pierri, G. Schirinzi, F. Soldovieri, “Phase retrieval in antennas and remote sensing,” in Proceedings of the Third ESA European Workshop on Electromagnetic Compatibility and Computational Electromagnetics (European Space Agency, Noordwijk, The Netherlands, 1993).
  22. T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “Reflector diagnostics from aperture and far field intensities,” Proc. Fondazione Ronchi 40, 727–731 (1994).

1995 (1)

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inv. Prob. 11, 183–203 (1995).
[CrossRef]

1994 (2)

T. Isernia, G. Leone, R. Pierri, “Phaseless near field techniques: uniqueness condition and attainment of the solution,”J. Electromagn. Waves Appl. 8, 889–908 (1994).

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “Reflector diagnostics from aperture and far field intensities,” Proc. Fondazione Ronchi 40, 727–731 (1994).

1993 (1)

T. Isernia, G. Leone, R. Pierri, “Antenna testing from phaseless measurements: probe compensation and experimental results for the cylindrical case,” Proc. Inst. Electr. Eng. H 140, 395–400 (1993).

1992 (3)

1991 (1)

G. Chavent, “A new size × curvature condition for strict quasi convexity of sets,” SIAM J. Control Optim. 29, 1348–1372 (1991).
[CrossRef]

1990 (1)

1985 (2)

1984 (1)

1983 (1)

1977 (1)

1976 (1)

A. M. J. Huiser, H. A. Fewerda, “On the problem of the phase retrieval in electron microscopy from image and diffraction pattern,” Optik 46, 407–420 (1976).

1972 (1)

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

1961 (1)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-I,” Bell Syst. Tech. J. 40, 43–64 (1961).

Barakat, R.

Chavent, G.

G. Chavent, “A new size × curvature condition for strict quasi convexity of sets,” SIAM J. Control Optim. 29, 1348–1372 (1991).
[CrossRef]

Dobson, D. C.

D. C. Dobson, “Phase reconstruction via nonlinear least squares,” Inv. Prob. 8, 541–557 (1992).
[CrossRef]

Fewerda, H. A.

A. M. J. Huiser, H. A. Fewerda, “On the problem of the phase retrieval in electron microscopy from image and diffraction pattern,” Optik 46, 407–420 (1976).

Fienup, J. R.

Gerchberg, R. W.

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

Huiser, A. M. J.

A. M. J. Huiser, H. A. Fewerda, “On the problem of the phase retrieval in electron microscopy from image and diffraction pattern,” Optik 46, 407–420 (1976).

Isernia, T.

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inv. Prob. 11, 183–203 (1995).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, “Phaseless near field techniques: uniqueness condition and attainment of the solution,”J. Electromagn. Waves Appl. 8, 889–908 (1994).

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “Reflector diagnostics from aperture and far field intensities,” Proc. Fondazione Ronchi 40, 727–731 (1994).

T. Isernia, G. Leone, R. Pierri, “Antenna testing from phaseless measurements: probe compensation and experimental results for the cylindrical case,” Proc. Inst. Electr. Eng. H 140, 395–400 (1993).

T. Isernia, G. Leone, V. Pascazio, R. Pierri, G. Schirinzi, F. Soldovieri, “Phase retrieval in antennas and remote sensing,” in Proceedings of the Third ESA European Workshop on Electromagnetic Compatibility and Computational Electromagnetics (European Space Agency, Noordwijk, The Netherlands, 1993).

Ivanov, V. Yu.

Leone, G.

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inv. Prob. 11, 183–203 (1995).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, “Phaseless near field techniques: uniqueness condition and attainment of the solution,”J. Electromagn. Waves Appl. 8, 889–908 (1994).

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “Reflector diagnostics from aperture and far field intensities,” Proc. Fondazione Ronchi 40, 727–731 (1994).

T. Isernia, G. Leone, R. Pierri, “Antenna testing from phaseless measurements: probe compensation and experimental results for the cylindrical case,” Proc. Inst. Electr. Eng. H 140, 395–400 (1993).

T. Isernia, G. Leone, V. Pascazio, R. Pierri, G. Schirinzi, F. Soldovieri, “Phase retrieval in antennas and remote sensing,” in Proceedings of the Third ESA European Workshop on Electromagnetic Compatibility and Computational Electromagnetics (European Space Agency, Noordwijk, The Netherlands, 1993).

Levi, A.

Luenberger, D.

D. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1987).

Millane, R. P.

Newsman, G.

Ortega, J. M.

J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, San Diego, Calif., 1970).

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

Pascazio, V.

T. Isernia, G. Leone, V. Pascazio, R. Pierri, G. Schirinzi, F. Soldovieri, “Phase retrieval in antennas and remote sensing,” in Proceedings of the Third ESA European Workshop on Electromagnetic Compatibility and Computational Electromagnetics (European Space Agency, Noordwijk, The Netherlands, 1993).

Pierri, R.

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inv. Prob. 11, 183–203 (1995).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, “Phaseless near field techniques: uniqueness condition and attainment of the solution,”J. Electromagn. Waves Appl. 8, 889–908 (1994).

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “Reflector diagnostics from aperture and far field intensities,” Proc. Fondazione Ronchi 40, 727–731 (1994).

T. Isernia, G. Leone, R. Pierri, “Antenna testing from phaseless measurements: probe compensation and experimental results for the cylindrical case,” Proc. Inst. Electr. Eng. H 140, 395–400 (1993).

T. Isernia, G. Leone, V. Pascazio, R. Pierri, G. Schirinzi, F. Soldovieri, “Phase retrieval in antennas and remote sensing,” in Proceedings of the Third ESA European Workshop on Electromagnetic Compatibility and Computational Electromagnetics (European Space Agency, Noordwijk, The Netherlands, 1993).

Pollak, H. O.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-I,” Bell Syst. Tech. J. 40, 43–64 (1961).

Poulton, G.

G. Poulton, “Antenna application of phase retrievial: a review,” in Proceedings of the 1992 URSI International Symposium on Electromagnetic Theory (Union Radio-Scientifique Internationale, Sydney, Australia, 1992).

Rheinboldt, W. C.

J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, San Diego, Calif., 1970).

Sabba Stefanescu, I.

I. Sabba Stefanescu, “On the phase retrieval problem in two dimensions,”J. Math. Phys. 26, 2141–2160 (1985).
[CrossRef]

Sandler, B.

Saxton, W. O.

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

Schirinzi, G.

T. Isernia, G. Leone, V. Pascazio, R. Pierri, G. Schirinzi, F. Soldovieri, “Phase retrieval in antennas and remote sensing,” in Proceedings of the Third ESA European Workshop on Electromagnetic Compatibility and Computational Electromagnetics (European Space Agency, Noordwijk, The Netherlands, 1993).

Sivokon, V. P.

Slepian, D.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-I,” Bell Syst. Tech. J. 40, 43–64 (1961).

Soldovieri, F.

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “Reflector diagnostics from aperture and far field intensities,” Proc. Fondazione Ronchi 40, 727–731 (1994).

T. Isernia, G. Leone, V. Pascazio, R. Pierri, G. Schirinzi, F. Soldovieri, “Phase retrieval in antennas and remote sensing,” in Proceedings of the Third ESA European Workshop on Electromagnetic Compatibility and Computational Electromagnetics (European Space Agency, Noordwijk, The Netherlands, 1993).

Southwell, W. H.

Stark, H.

Vorontsov, M. A.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-I,” Bell Syst. Tech. J. 40, 43–64 (1961).

Inv. Prob. (2)

T. Isernia, G. Leone, R. Pierri, “Phase retrieval of radiated fields,” Inv. Prob. 11, 183–203 (1995).
[CrossRef]

D. C. Dobson, “Phase reconstruction via nonlinear least squares,” Inv. Prob. 8, 541–557 (1992).
[CrossRef]

J. Electromagn. Waves Appl. (1)

T. Isernia, G. Leone, R. Pierri, “Phaseless near field techniques: uniqueness condition and attainment of the solution,”J. Electromagn. Waves Appl. 8, 889–908 (1994).

J. Math. Phys. (1)

I. Sabba Stefanescu, “On the phase retrieval problem in two dimensions,”J. Math. Phys. 26, 2141–2160 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Lett. (1)

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

Optik (1)

A. M. J. Huiser, H. A. Fewerda, “On the problem of the phase retrieval in electron microscopy from image and diffraction pattern,” Optik 46, 407–420 (1976).

Proc. Fondazione Ronchi (1)

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “Reflector diagnostics from aperture and far field intensities,” Proc. Fondazione Ronchi 40, 727–731 (1994).

Proc. Inst. Electr. Eng. H (1)

T. Isernia, G. Leone, R. Pierri, “Antenna testing from phaseless measurements: probe compensation and experimental results for the cylindrical case,” Proc. Inst. Electr. Eng. H 140, 395–400 (1993).

SIAM J. Control Optim. (1)

G. Chavent, “A new size × curvature condition for strict quasi convexity of sets,” SIAM J. Control Optim. 29, 1348–1372 (1991).
[CrossRef]

Other (5)

J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, San Diego, Calif., 1970).

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

G. Poulton, “Antenna application of phase retrievial: a review,” in Proceedings of the 1992 URSI International Symposium on Electromagnetic Theory (Union Radio-Scientifique Internationale, Sydney, Australia, 1992).

D. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1987).

T. Isernia, G. Leone, V. Pascazio, R. Pierri, G. Schirinzi, F. Soldovieri, “Phase retrieval in antennas and remote sensing,” in Proceedings of the Third ESA European Workshop on Electromagnetic Compatibility and Computational Electromagnetics (European Space Agency, Noordwijk, The Netherlands, 1993).

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Figures (14)

Fig. 1
Fig. 1

Set of admissible data is made up of arcs of parabolas, and local minima do not exist if the curvature radius of the parabola is large enough and if the data point is properly located.

Fig. 2
Fig. 2

Pictorial view of the influence of additional independent information on the conditions of the existence of local minima.

Fig. 3
Fig. 3

Flow chart of the solution algorithm.

Fig. 4
Fig. 4

Normalized contour plot of the amplitude of the reference function [contour levels (in dB): 1, −3; 2, −6; 3, −10; 4, −20].

Fig. 5
Fig. 5

Normalized contour plot of the phase of the reference function (nine equispaced contour levels from −0.1 to 0.7 rad).

Fig. 6
Fig. 6

Normalized contour plot of the amplitude of the reconstructed function in the local minimum (contour levels are the same as in Fig. 4).

Fig. 7
Fig. 7

Normalized contour plot of the phase of the reconstructed function in the local minimum (contour levels are the same as in Fig. 5).

Fig. 8
Fig. 8

Three-dimensional plot of the difference between the reconstructed phase and the reference phase.

Fig. 9
Fig. 9

Normalized contour plot of the amplitude of the reconstructed function when the square amplitude of the function is also considered (contour levels are the same as in Fig. 4).

Fig. 10
Fig. 10

Normalized contour plot of the phase of the reconstructed function when the square amplitude of the function is also considered (contour levels are the same as in Fig. 5).

Fig. 11
Fig. 11

Three-dimensional plot of the difference between the reconstructed phase and the reference phase.

Fig. 12
Fig. 12

Behaviors of the functionals Φ and Φ1 along the direction f = f0 + λΔf from the solution (λ = 0) to the local minimum (λ = 1) [solid (lower) curve, functional Φ1; dotted (top) curve, functional Φ].

Fig. 13
Fig. 13

Three-dimensional plot of the phase of the reference function.

Fig. 14
Fig. 14

Three-dimensional plot of the phase of the reconstructed function.

Equations (46)

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B 1 ( f ) = M 2 .
B 2 ( f ) = m 2 .
M 2 ( M 2 m 2 ) T ,
B ( f ) = M 2 .
Φ ( f ) = B ( f ) - M 2 2 ,
Φ 1 ( f ) = B 1 ( f ) - M 2 2 ,
f ( x , y ) = 1 4 a 2 n = - N N m = - M M F n m exp [ j ( n π x + m π y ) / a ] ,
f ( x , y ) = n = - N N m = - M M f ( x n , y m ) S N ( x - x n ) S M ( y - y m ) ,
m 2 ( x , y ) = i = - 2 N 2 N j = - 2 M 2 M m 2 ( x i , y j ) S 2 N ( x - x i ) S 2 M ( y - y j ) ,
F ( u , v ) = n = - M N m = - M M F n m sinc ( a u - n π ) sinc ( a v - m π ) .
M 2 ( u , v ) = k l M k l 2 sinc ( 2 a u - k π ) sinc ( 2 a v - l π ) ,
M 2 ( u , v ) = n , n = - N N m , m = - M M F n m F n m * sinc ( a u - n π ) × sinc ( a v - m π ) sinc ( a u - n π ) × sinc ( a v - m π ) ,
M 2 ( u , v ) = p = 1 K 0 a p ϕ p ( u , v ) ,
Φ ( f ) = f 2 - m 2 2 + F f 2 - M 2 2 .
Φ 1 ( f 0 + λ n ) = λ 2 ( a 1 λ 2 + b 1 λ + c 1 ) ,
a 1 = F n 2 2 ,             b 1 = 4 Re { F f 0 ( F n ) * } , F n 2 , c 1 = 4 Re { F f 0 ( F n ) * } 2 ,
b 1 2 a 1 c 1 < 32 9 .
α { ( F n ) k l 2 } ,             γ { Re { ( F f 0 ) k l ( F n ) k l * } } ,
α , γ 2 α 2 γ 2 < 8 9 ,
Φ ( f 0 + λ n ) = λ 2 ( a λ 2 + b λ + c ) ,
a = a 1 + n 2 2 ,             b = b 1 + 4 Re { f 0 n * } , n 2 , c = c 1 + 4 Re { f 0 n * } 2 .
b 2 a c < 32 9
Φ 1 ( f 0 + λ n ¯ ) = a 1 λ 2 ( λ - 1 ) 2 .
Φ ( f ) = i = - 2 N 2 N j = - 2 M 2 M ( f i j 2 - m i j 2 ) 2 + k = - 2 N 2 N l = - 2 M 2 M ( F k l 2 - M k l 2 ) 2 ,
g n = { 0 , N < n 2 N f n , n N ,
f n = g n ,             n N .
g n = m = - I I f m exp [ - j 2 π m n / ( 2 I + 1 ) ] ,
f m = 1 2 I + 1 n = - I I g n exp [ j 2 π m n / ( 2 I + 1 ) ] .
f _ 1 = 4 N + 1 2 N + 1 DFT 2 N - 1 { Z DFT N { f _ } } .
f 2 n = f 1 n 2 - m n 2 ,
f _ 4 = DFT N - 1 { Π ( DFT 2 N { f _ 3 } ) } .
F _ 1 = DFT 2 N { Z ( f ) } .
F 2 k = F 1 k 2 - M k 2 ,
f _ 5 = Π ( DFT 2 N - 1 { F _ 3 } ) .
f _ ( k + 1 ) = f _ ( k ) + λ k G ̳ ( k ) [ f _ 4 ( k ) + f _ 5 ( k ) ] ,
Ψ ( f ) = k l ( F k l 2 - M k l 2 ) 2 M k l 2 + i j ( f i j 2 - m i j 2 ) 2 m i j 2 .
Ψ ( f 0 + λ n ) = λ 2 ( a ˜ λ 2 + b ˜ λ + c ˜ ) ,
c ˜ = 4 k l [ Re { ( F f 0 ) k l ( F n ) k l * } ] 2 M k l 2 + 4 i j [ Re { ( f 0 ) i j ( n ) i j * } ] 2 m i j 2 .
f ( x ) = 1 2 a n = - N N F n exp ( j n π x / a ) ,
F ( u ) = n = - N N F n sinc ( a u - n π ) .
F ( u ) 2 = n = - N N m = - N N F n F m * sinc ( a u - n π ) sinc ( a u - m π ) .
g n n ( u ) = sinc 2 ( a u - n π ) .
g k n n = { 1 , k = 2 n 0 , k is even and k 2 n 1 [ ( k - 2 n ) ( π / 2 ) ] 2 , k is odd ,
g n n ( u ) = sinc ( a u - n π ) sinc ( a u - m π )
g k n m = { 0 , k is even sin [ ( k - 2 n ) π / 2 ] sin [ ( k - 2 m ) π / 2 ] ( π / 2 ) 2 ( k - 2 n ) ( k - 2 m ) , k is odd ,
g k n m = ( - 1 ) k - n - m 2 ( n - m ) ( π / 2 ) 2 [ 1 ( k - 2 n ) - 1 ( k - 2 m ) ] .

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