Abstract

We consider reconstruction of a wave field distribution in an input/object plane from data in an output/diffraction (sensor) plane. We provide digital modeling both for the forward and backward wave field propagation. A novel algebraic matrix form of the discrete diffraction transform (DDT) originated in Katkovnik et al. [Appl. Opt. 47, 3481 (2008)] is proposed for the forward modeling that is aliasing free and precise for pixelwise invariant object and sensor plane distributions. This “matrix DDT” is a base for formalization of the object wave field reconstruction (backward propagation) as an inverse problem. The transfer matrices of the matrix DDT are used for calculations as well as for the analysis of conditions when the perfect reconstruction of the object wave field distribution is possible. We show by simulation that the developed inverse propagation algorithm demonstrates an improved accuracy as compared with the standard convolutional and discrete Fresnel transform algorithms.

© 2009 Optical Society of America

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References

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  1. Th. Kreis, Handbook of Holographic Interferometry (Optical and Digital Methods) (Wiley-VCH, 2005).
  2. V. Katkovnik, J. Astola, and K. Egiazarian, “Discrete diffraction transform for propagation, reconstruction, and design of wave field distributions,” Appl. Opt. 47, 3481-3493 (2008).
    [CrossRef] [PubMed]
  3. G. S. Sherman, “Integral-transform formulation of diffraction theory,” J. Opt. Soc. Am. 57, 1490-1498 (1967).
    [CrossRef]
  4. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  5. L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A 24, 359-367 (2007).
    [CrossRef]
  6. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh--Sommerfeld diffraction formula,” Appl. Opt. 45, 1102-1110 (2006).
    [CrossRef] [PubMed]
  7. L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. 43, 2557-2563 (2004).
    [CrossRef]
  8. I. Aizenberg and J. Astola, “Discrete generalized Fresnel functions and transforms in an arbitrary discrete basis, “ IEEE Trans. Signal Process. 54, 4261-4270 (2006).
    [CrossRef]
  9. M. Liebling, Th. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29-43 (2003).
    [CrossRef]
  10. B. M. Hennelly and J. T. Sheridan, “Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms,” J. Opt. Soc. Am. A 22, 917-927 (2005).
    [CrossRef]
  11. L. Yaroslavsky, “Discrete transforms, fast algorithms and point spread functions of numerical reconstruction of digitally recorded holograms,” in Advances in Signal Transforms: Theory and Applications, J. Astola and L. Yaroslavsky, eds., Vol. 7 of EURASIP Book Series on Signal Processing and Communications (Hindawi Publishing, 2007), pp. 93-141.
  12. I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt. 40, 6177-6186 (2001).
    [CrossRef]
  13. A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 2nd ed., Signal Processing Series (Prentice-Hall, 1999).
  14. H. Theil, “Linear algebra and matrix methods in econometrics, in Handbook of Econometrics, Z. Griliches and M. D. Intriligator, eds. (North-Holland, 1983), Vol. 1.
    [CrossRef]
  15. A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, 1977).
  16. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP Publishing, 1998).
    [CrossRef]
  17. L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D (Amsterdam) 60, 259-268 (1992).
    [CrossRef]
  18. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
    [CrossRef]
  19. E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
    [CrossRef]
  20. V. Katkovnik, J. Astola, and K. Egiazarian, “Numerical wavefield reconstruction in phase-shifting holography as inverse discrete problem,” Proceedings of the 2008 European Signal Processing Conference (EUSIPCO 2008) (Wiley, 2008).
  21. V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006).
    [CrossRef]
  22. L. L. Scharf, Statistical Signal Processing (Prentice-Hall, 1991).

2008 (1)

2007 (1)

2006 (4)

F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh--Sommerfeld diffraction formula,” Appl. Opt. 45, 1102-1110 (2006).
[CrossRef] [PubMed]

I. Aizenberg and J. Astola, “Discrete generalized Fresnel functions and transforms in an arbitrary discrete basis, “ IEEE Trans. Signal Process. 54, 4261-4270 (2006).
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

2005 (1)

2004 (1)

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. 43, 2557-2563 (2004).
[CrossRef]

2003 (1)

M. Liebling, Th. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29-43 (2003).
[CrossRef]

2001 (1)

1992 (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D (Amsterdam) 60, 259-268 (1992).
[CrossRef]

1967 (1)

Aizenberg, I.

I. Aizenberg and J. Astola, “Discrete generalized Fresnel functions and transforms in an arbitrary discrete basis, “ IEEE Trans. Signal Process. 54, 4261-4270 (2006).
[CrossRef]

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, 1977).

Astola, J.

V. Katkovnik, J. Astola, and K. Egiazarian, “Discrete diffraction transform for propagation, reconstruction, and design of wave field distributions,” Appl. Opt. 47, 3481-3493 (2008).
[CrossRef] [PubMed]

I. Aizenberg and J. Astola, “Discrete generalized Fresnel functions and transforms in an arbitrary discrete basis, “ IEEE Trans. Signal Process. 54, 4261-4270 (2006).
[CrossRef]

V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006).
[CrossRef]

V. Katkovnik, J. Astola, and K. Egiazarian, “Numerical wavefield reconstruction in phase-shifting holography as inverse discrete problem,” Proceedings of the 2008 European Signal Processing Conference (EUSIPCO 2008) (Wiley, 2008).

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP Publishing, 1998).
[CrossRef]

Blu, Th.

M. Liebling, Th. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29-43 (2003).
[CrossRef]

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP Publishing, 1998).
[CrossRef]

Candès, E.

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
[CrossRef]

Egiazarian, K.

V. Katkovnik, J. Astola, and K. Egiazarian, “Discrete diffraction transform for propagation, reconstruction, and design of wave field distributions,” Appl. Opt. 47, 3481-3493 (2008).
[CrossRef] [PubMed]

V. Katkovnik, J. Astola, and K. Egiazarian, “Numerical wavefield reconstruction in phase-shifting holography as inverse discrete problem,” Proceedings of the 2008 European Signal Processing Conference (EUSIPCO 2008) (Wiley, 2008).

V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006).
[CrossRef]

Fatemi, E.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D (Amsterdam) 60, 259-268 (1992).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Hennelly, B. M.

Katkovnik, V.

V. Katkovnik, J. Astola, and K. Egiazarian, “Discrete diffraction transform for propagation, reconstruction, and design of wave field distributions,” Appl. Opt. 47, 3481-3493 (2008).
[CrossRef] [PubMed]

V. Katkovnik, J. Astola, and K. Egiazarian, “Numerical wavefield reconstruction in phase-shifting holography as inverse discrete problem,” Proceedings of the 2008 European Signal Processing Conference (EUSIPCO 2008) (Wiley, 2008).

V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006).
[CrossRef]

Kato, J.

Kreis, Th.

Th. Kreis, Handbook of Holographic Interferometry (Optical and Digital Methods) (Wiley-VCH, 2005).

Liebling, M.

M. Liebling, Th. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29-43 (2003).
[CrossRef]

Mizuno, J.

Ohta, S.

Onural, L.

L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A 24, 359-367 (2007).
[CrossRef]

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. 43, 2557-2563 (2004).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 2nd ed., Signal Processing Series (Prentice-Hall, 1999).

Osher, S.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D (Amsterdam) 60, 259-268 (1992).
[CrossRef]

Romberg, J.

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

Rudin, L.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D (Amsterdam) 60, 259-268 (1992).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 2nd ed., Signal Processing Series (Prentice-Hall, 1999).

Scharf, L. L.

L. L. Scharf, Statistical Signal Processing (Prentice-Hall, 1991).

Shen, F.

Sheridan, J. T.

Sherman, G. S.

Tao, T.

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

Theil, H.

H. Theil, “Linear algebra and matrix methods in econometrics, in Handbook of Econometrics, Z. Griliches and M. D. Intriligator, eds. (North-Holland, 1983), Vol. 1.
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, 1977).

Unser, M.

M. Liebling, Th. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29-43 (2003).
[CrossRef]

Wang, A.

Yamaguchi, I.

Yaroslavsky, L.

L. Yaroslavsky, “Discrete transforms, fast algorithms and point spread functions of numerical reconstruction of digitally recorded holograms,” in Advances in Signal Transforms: Theory and Applications, J. Astola and L. Yaroslavsky, eds., Vol. 7 of EURASIP Book Series on Signal Processing and Communications (Hindawi Publishing, 2007), pp. 93-141.

Appl. Opt. (3)

IEEE Trans. Image Process. (1)

M. Liebling, Th. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29-43 (2003).
[CrossRef]

IEEE Trans. Inf. Theory (2)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

IEEE Trans. Signal Process. (1)

I. Aizenberg and J. Astola, “Discrete generalized Fresnel functions and transforms in an arbitrary discrete basis, “ IEEE Trans. Signal Process. 54, 4261-4270 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

L. Onural, “Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations,” Opt. Eng. 43, 2557-2563 (2004).
[CrossRef]

Physica D (Amsterdam) (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D (Amsterdam) 60, 259-268 (1992).
[CrossRef]

Other (10)

L. Yaroslavsky, “Discrete transforms, fast algorithms and point spread functions of numerical reconstruction of digitally recorded holograms,” in Advances in Signal Transforms: Theory and Applications, J. Astola and L. Yaroslavsky, eds., Vol. 7 of EURASIP Book Series on Signal Processing and Communications (Hindawi Publishing, 2007), pp. 93-141.

Th. Kreis, Handbook of Holographic Interferometry (Optical and Digital Methods) (Wiley-VCH, 2005).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

V. Katkovnik, J. Astola, and K. Egiazarian, “Numerical wavefield reconstruction in phase-shifting holography as inverse discrete problem,” Proceedings of the 2008 European Signal Processing Conference (EUSIPCO 2008) (Wiley, 2008).

V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006).
[CrossRef]

L. L. Scharf, Statistical Signal Processing (Prentice-Hall, 1991).

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 2nd ed., Signal Processing Series (Prentice-Hall, 1999).

H. Theil, “Linear algebra and matrix methods in econometrics, in Handbook of Econometrics, Z. Griliches and M. D. Intriligator, eds. (North-Holland, 1983), Vol. 1.
[CrossRef]

A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, 1977).

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP Publishing, 1998).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Principal setup of wave field propagation and reconstruction.

Fig. 2
Fig. 2

Image reconstruction: (a) by the backward Fresnel transform for z = d f (perfect quality), (b) by the backward Fresnel transform for z = 3 d f (average quality), and (c) by the regularized inverse M-DDT algorithm (quality is improved with respect to (b)).

Fig. 3
Fig. 3

Rank of the matrix A y H A y versus the distance z = d for averaged (av) A y and nonaveraged (non-av) matrices B y for different sizes of the square sensor defined by the parameter q.

Fig. 4
Fig. 4

RMSE versus the distance z = d for q = 1 , q = 2 , and q = 4 averaged matrices. A nearly perfect reconstruction is obtained for all d d f | q = 2 if q = 2 and for all d d f | q = 4 if q = 4 .

Fig. 5
Fig. 5

Accuracy of the image restoration (RMSE) versus the distance d for different algorithms: M-DDT, the convolutional inverse using the transfer functions of the image size (conv-1) and the double-size zero-padding image (conv-2). M-IDFrT, and the recursive regularized inverse F-DDT , d f = 0.02 m , q = 1 .

Fig. 6
Fig. 6

Comparative imaging by M-DDT and M-IDFrT algorithms, for various distances between the object and sensor planes: (a)  d = d f , (b)  d = 3 d f , and (c)  d = 6 d f , where d f is the in-focus distance for q = 1 , d f = d f | q = 1 . The images in the object and sensor planes are of equal size.

Fig. 7
Fig. 7

Comparative imaging by M-DDT and M-IDFrT algorithms, for various distances between the object and sensor planes: (a)  d = d f , (b)  d = 3 d f , and (c)  d = 6 d f , where d f is the in-focus distance for q = 1 , d f = d f | q = 1 . The image in the sensor plane is of double the size of the image in the object plane, q = 2 .

Fig. 8
Fig. 8

Object wave field reconstruction (amplitude distribution) with the distance z = 0 . 5 d f : (left) standard M-IDFrT fails with a pattern of clear aliasing effects; (right) M-DDT with the averaged matrices gives a good quality aliasing free reconstruction.

Fig. 9
Fig. 9

M-DDT reconstructions, the distance d = 1.01 × d f : image (a) obtained with α = 0 is completely destroyed; image (b) obtained with α = 10 7 gives a very good accuracy of RMSE = 0.0049 .

Fig. 10
Fig. 10

Ranks of the matrices A y H A y and A x H A x for the rectangular pixels ( 5 × 8 ) μm and RMSE for the Baboon image reconstruction by M-DDT and M-IDFrT algorithms versus the distance d.

Fig. 11
Fig. 11

Comparative imaging by M-DDT and M-IDFrT algorithms for the distances d f , y = 0.0202 m (first row of images) and d f , x = 0.0518 m (second row of images: (a) M-DDT reconstruction, RMSE = 0.0115 ; (b) M-IDFrT reconstruction, RMSE = 0.874 ; (c) M-DDT reconstruction, RMSE = 0.0115 ; (d) M-IDFrT reconstruction, RMSE = 0.0809 .

Fig. 12
Fig. 12

Integration areas for calculation of the integral (A4).

Tables (3)

Tables Icon

Table 1 Computational Time (in Seconds) for Matrices A y and B y

Tables Icon

Table 2 Computational Time (in Seconds) for Matrices Q y

Tables Icon

Table 3 Computational Time (in Seconds) for Calculation of Estimates

Equations (80)

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u z ( y , x ) = D z { u 0 } g z ( y η , x ξ ) u 0 ( η , ξ ) d η d ξ , ( y , x ) R 2 ,
g z ( y , x ) = z exp ( j 2 π r / λ ) j λ · r 2 , r = x 2 + y 2 + z 2 , z λ ,
u 0 ( η , ξ ) = D z 1 { u z } g z ( η y , ξ x ) u z ( y , x ) d x d y .
u z ( Δ y , z s , Δ x , z t ) = k , l u 0 ( Δ y , 0 k , Δ x , 0 l ) g z ( Δ y , z s Δ y , 0 k , Δ x , z t Δ x , 0 l ) · Δ y , 0 Δ x , 0 ,
u ^ 0 ( Δ y , 0 k , Δ x , 0 l ) = s , t u z ( Δ y , z s , Δ x , z t ) g z ( Δ y , 0 k Δ y , z s , Δ x , 0 l Δ x , z t ) · Δ y , z Δ x , z ,
U z ( f y , f x ) = G z ( f y , f x ) U 0 ( f y , f x ) ,
U ^ 0 ( f y , f x ) = G z ( f y , f x ) U z ( f y , f x ) ,
g z ( y , x ) exp ( j 2 π z / λ ) j λ z exp [ j π λ z ( x 2 + y 2 ) ] .
u z ( Δ y , z s , Δ x , z t ) j exp [ j ( 2 π z / λ + π ( ( Δ y , z s ) 2 + ( Δ x , z t ) 2 ) / ( λ z ) ) ] λ · z × k , l { exp [ j π λ z ( ( Δ y , 0 k ) 2 + ( Δ x , 0 l ) 2 ) ] u 0 ( Δ y , 0 k , Δ x , 0 l ) } × exp [ j 2 π Δ y , 0 Δ y , z s k + Δ x , 0 Δ x , z t l λ z ] · Δ y , 0 Δ x , 0
u ^ 0 ( Δ y , 0 k , Δ x , 0 l ) j exp [ j ( 2 π z / λ + π ( ( Δ y , 0 k ) 2 + ( Δ x , 0 l ) 2 ) / ( λ z ) ) ] λ · z × s , t exp [ j π λ z ( ( Δ y , z s ) 2 + ( Δ x , z t ) 2 ) ] u z ( Δ y , z s , Δ x , z t ) × exp [ j 2 π ( Δ y , 0 Δ y , z s k + Δ x , 0 Δ x , z t l ) λ z ] · Δ x , z Δ y , z .
N y = λ · z Δ y , 0 Δ y , z , N x = λ · z Δ x , 0 Δ x , z .
u z ( Δ y , z s , Δ x , z t ) j λ · z exp [ j ( 2 π z / λ + π ( ( Δ y , z s ) 2 + ( Δ x , z t ) 2 ) / ( λ z ) ) ] × D F T k , l { exp [ j π λ z ( ( Δ y , 0 k ) 2 + ( Δ x , 0 l ) 2 ) ] u 0 ( Δ y , 0 k , Δ x , 0 l ) } [ s , t ] · Δ y , 0 Δ x , 0 ,
u ^ 0 ( Δ y , 0 k , Δ x , 0 l ) j λ · z exp [ j ( 2 π d / λ + π ( ( Δ y , 0 k ) 2 + ( Δ x , 0 l ) 2 ) / ( λ z ) ) ] × D F T s , t 1 { exp [ j π λ z ( ( Δ y , z s ) 2 + ( Δ x , z t ) 2 ) ] u z ( Δ y , z s , Δ x , z t ) } [ k , l ] × N y N x Δ x , z Δ y , z ,
u ^ 0 = u 0 .
s , t exp [ j 2 π ( Δ y , 0 Δ y , z · s k + Δ x , 0 Δ x , z · t l ) λ z ] · w ( s , t ) ,
u z = μ z · C y , z · u 0 · C x , z T ,
u ^ 0 = μ 0 · C y , z T · u z · C x , z ,
C y , z = ( C y , z [ s , s ] ) N y , z × N y , 0 , C x , z = ( C x , z [ s , s ] ) N x , z × N x , 0
C y , z [ s , s ] = exp ( j π λ z ( s Δ y , z s Δ y , 0 ) 2 ) , C x , z [ s , s ] = exp ( j π λ z ( s Δ x , z s Δ x , 0 ) 2 ) , μ z = μ · Δ x , 0 Δ y , 0 , μ 0 = μ * · Δ x , z Δ y , z , μ = exp ( j 2 π z / λ ) j λ · z .
u z = ( u z [ s , t ] ) N y , z × N x , z , u ^ 0 = ( u ^ 0 [ k , l ] ) N y , 0 × N x , 0
u ^ 0 = 1 μ z · C y , z 1 · u z · C x , z T ,
u z ( y , x ) = k = N y , 0 / 2 N y , 0 / 2 1 l = N x , 0 / 2 N x , 0 / 2 1 u 0 [ k , l ] Δ x , 0 / 2 Δ x , 0 / 2 d ξ Δ y , 0 / 2 Δ y , 0 / 2 d η g z ( y k Δ y , 0 + η , x l Δ x , 0 + ξ ) , u 0 [ k , t ] = u 0 ( k Δ y , 0 + η , l Δ x , 0 + ξ ) , | η | Δ y , 0 / 2 , | ξ | Δ x , 0 / 2 ,
u z [ s , t ] = 1 Δ y , z Δ x , z Δ y , z / 2 Δ y , z / 2 Δ x , z / 2 Δ x , z / 2 u z ( s Δ y , z + η , t Δ x , z + ξ ) d ξ d η .
u z [ s , t ] = s = N y , 0 / 2 N y , 0 / 2 1 t = N x , 0 / 2 N x , 0 / 2 1 a z [ s , k ; t , l ] · u 0 [ k , l ] , s = N y , z / 2 , ... , N y , z / 2 1 , t = N x , z / 2 , ... , N x , z / 2 1 ,
a z [ s , k ; t , l ] = 1 Δ y , z Δ x , z Δ x , z / 2 Δ x , z / 2 Δ x , 0 / 2 Δ x , 0 / 2 d ξ d ξ × Δ y , z / 2 Δ y , z / 2 Δ y , 0 / 2 Δ y , 0 / 2 d η d η g z ( s Δ y , z k Δ y , 0 + η + η , t Δ x , z l Δ x , 0 + ξ + ξ ) , s = N y , z / 2 , ... , N y , z / 2 1 , k = N y , 0 / 2 , ... , N y , 0 / 2 1 , t = N x , z / 2 , ... , N x , z / 2 1 , l = N x , 0 / 2 , ... , N x , 0 / 2 - 1.
a z [ s , k ; t , l ] = μ A y [ s , k ] A x [ t , l ] , μ = exp ( j 2 π z / λ ) j λ · z ,
A y [ s , k ] = 1 Δ y , z Δ y , z / 2 Δ y , z / 2 Δ y , 0 / 2 Δ y , 0 / 2 exp ( j π λ z ( s Δ y , z k Δ y , 0 + η + η ) 2 ) d η d η , A x [ t , l ] = 1 Δ x , z Δ x , z / 2 Δ x , z / 2 Δ x , 0 / 2 Δ x , 0 / 2 exp ( j π λ z ( t Δ x , z l Δ x , 0 + ξ + ξ ) 2 ) d ξ d ξ .
u z [ s , t ] = μ k = N y , 0 / 2 N y , 0 / 2 1 l = N x , 0 / 2 N x , 0 / 2 1 A y [ s , k ] u 0 [ k , l ] A x [ t , l ] ,
u z = μ · A y · u 0 · A x T .
u z col = μ · G · u 0 col , G = A x A y ,
G H u z col = μ · G H G · u ^ 0 col .
G H · G = ( A x A y ) H · ( A x A y ) = ( A x H A y H ) · ( A x A y ) = A x H A x A y H A y .
cond G = cond A x H A x · cond A y H A y , rank G = rank A x H A x · rank A y H A y .
u ^ 0 = 1 μ A y 1 u z A x T ,
N x , z N x , 0 , N y , z N y , 0 ;
rank ( A y ) = N y , 0 , rank ( A x ) = N x , 0 .
u ^ 0 = 1 μ ( A y H A y ) 1 A y H u z A x * ( A x T A x * ) 1 , u ^ 0 = u 0 ,
u ^ 0 = argmin u 0 L ,
L = u z μ A y u 0 A x T F 2 + α 2 u 0 F 2 ,
( | μ | A y H A y + α I ) u ^ 0 ( | μ | A x T A x * + α I ) α | μ | u ^ 0 A x T A x * α | μ | A y H A y u ^ 0 = μ * A y H u z A x * , α 0 .
u ^ 0 = 1 μ ( A y H A y + α | μ | I ) 1 A y H u z A x * ( A x T A x * + α | μ | I ) 1 .
rank = max j ( j ρ 1 / ρ j < 10 12 ) .
B y [ s , k ] lim Δ y , z , Δ y , 0 0 A y [ s , k ] = Δ y , 0 · exp ( j π λ z ( s Δ y , z k Δ y , 0 ) 2 ) = Δ y , 0 · C y , z [ s , k ] , B x [ t , l ] lim Δ x , z , Δ x , z 0 A x [ l , t ] = Δ x , 0 · exp ( j π λ z ( t Δ x , z l Δ x , 0 ) 2 ) = Δ x , 0 · C x , z [ t , l ] , s = N y , z / 2 , ... , N y , z / 2 1 , k = N y , 0 / 2 , ... , N y , 0 / 2 1 , t = N x , z / 2 , ... , N x , z / 2 1 , l = N x , 0 / 2 , ... , N x , 0 / 2 1 ,
B y = Δ y , 0 · C y , z , B x = Δ x , 0 · C x , z ,
1 Δ y , 0 2 N y B y H B y = I N y × N y , 1 Δ x , 0 2 N x B x T B x * = I N x × N x ,
1 N y C y , z H C y , z = I N y × N y , 1 N x C x , z T C x , z * = I N x × N x .
u ^ 0 = 1 μ ( B y H B y ) 1 B y H u z B x * ( B x T B x * ) 1 = 1 μ Δ y , 0 2 Δ x , 0 2 N x N y B y H u z B x * = 1 μ Δ y , 0 Δ x , 0 N x N y C y , z H u z C x , z * .
u ^ 0 = 1 μ · γ y · γ x ( B y H B y ˜ + α ˜ y · I ) 1 B y H u z B x * ( B x T B x * ˜ + α ˜ x · I ) 1 , B y H B y ˜ = B y H B y / γ y , B x T B x * ˜ = B x T B x * / γ x , α ˜ y = α / ( | μ | · γ y ) , α ˜ x = α / ( | μ | · γ x ) , γ y = Δ y , 0 2 N y , γ x = Δ x , 0 2 N x ,
0 < α 0 α | μ | min ( γ y , γ x ) .
| μ | min ( γ y , γ x ) = Δ 2 N λ d = Δ 2 N d f λ d d f = d f d ,
d f = N z Δ 2 / λ = N 0 q Δ 2 / λ .
N y , 0 = λ · d Δ y , 0 Δ y , z , N y , z = λ · d Δ y , 0 Δ y , z , N x , 0 = λ · d Δ x , 0 Δ x , z , N x , z = λ · d Δ x , 0 Δ x , z .
N y , 0 = λ · d Δ y , 0 Δ y , z , N y , z = λ · d Δ y , 0 Δ y , z ,
N x , 0 = λ · d Δ x , 0 Δ x , z , N x , z = λ · d Δ x , 0 Δ x , z .
d f , y = Δ y , 0 Δ y , z N y , z / λ , d f , x = Δ x , 0 Δ x , z N x , z / λ .
d f = min ( d f , y , d f , x ) .
I ϕ ( y , x ) = | u z ( y , x ) + u ref ( y , x ) e j ϕ ( y , x ) | 2 = | u z ( y , x ) | 2 + | u ref ( y , x ) | 2 + u z ( y , x ) u ref * ( y , x ) e j ϕ ( y , x ) + u z * ( y , x ) u ref ( y , x ) e j ϕ ( y , x ) .
I ϕ [ k , l ] = I z [ k , l ] + | u ref | 2 + u z [ k , l ] u ref * e j ϕ + u z * [ k , l ] u ref e j ϕ ,
I ϕ [ k , l ] = 1 Δ y , z Δ x , z Δ y , z / 2 Δ y , z / 2 Δ x , z / 2 Δ x , z / 2 I ϕ ( k Δ y , z + η , l Δ x , z + ξ ) d ξ d η , I z [ k , l ] = 1 Δ y , z Δ x , z Δ y , z / 2 Δ y , z / 2 Δ x , z / 2 Δ x , z / 2 | u z ( k Δ y , z + η , l Δ x , z + ξ ) | 2 d ξ d η ,
u z [ k , l ] = 1 4 u ref * ( I 0 [ k , l ] I π [ k , l ] j ( 2 I π / 2 [ k , l ] I 0 [ k , l ] I π [ k , l ] ) ) ,
u ^ 0 = 1 μ Q y u z Q x ,
( x y ) = ( cos φ sin φ sin φ cos φ ) ( Δ 1 ξ 1 Δ 2 ξ 2 ) .
( u v ) = 1 2 ( Δ 1 Δ 2 Δ 1 Δ 2 ) ( ξ 1 ξ 2 ) = 1 2 ( Δ 1 ξ 1 + Δ 2 ξ 2 Δ 1 ξ 1 + Δ 2 ξ 2 ) .
u 1 = 1 2 2 ( Δ 1 + Δ 2 ) , v 1 = 1 2 2 ( Δ 1 + Δ 2 ) , u 2 = 1 2 2 ( Δ 1 Δ 2 ) , v 2 = 1 2 2 ( Δ 1 Δ 2 ) .
J = 1 / 2 1 / 2 1 / 2 1 / 2 f ( x + Δ 1 ξ 1 + Δ 2 ξ 2 ) d ξ 1 d ξ 2 = 1 Δ 1 Δ 2 u , v S f ( x + 2 u ) d u d v ,
J = J 1 + J 2 + J 3 , J 1 = 1 Δ 1 Δ 2 u 2 u 2 f ( x + 2 u ) ( Δ 2 2 ) d u , J 2 = 1 Δ 1 Δ 2 u 2 u 1 f ( x + 2 u ) 2 ( u 1 u ) d u , J 3 = 1 Δ 1 Δ 2 - u 1 - u 2 f ( x + 2 u ) 2 ( u 1 + u ) d u .
J 1 = 1 Δ 1 ( Δ 1 Δ 2 ) / 2 ( Δ 1 Δ 2 ) / 2 f ( x + u ) d u , J 2 = 1 Δ 1 Δ 2 ( Δ 1 Δ 2 ) / 2 ( Δ 1 + Δ 2 ) / 2 f ( x + u ) ( Δ 1 + Δ 2 2 u ) d u , J 3 = 1 Δ 1 Δ 2 ( Δ 1 Δ 2 ) / 2 ( Δ 1 + Δ 2 ) / 2 f ( x u ) ( Δ 1 + Δ 2 2 u ) d u .
J 1 = 1 max ( Δ 1 , Δ 2 ) D 2 D 2 f ( x + u ) d u , J 2 = 1 Δ 1 Δ 2 D 2 D 1 f ( x + u ) ( D 1 u ) d u , J 3 = 1 Δ 1 Δ 2 D 2 D 1 f ( x u ) ( D 1 u ) d u ,
A y [ k , s ] = 1 Δ y , z Δ y , z / 2 Δ y , z / 2 Δ y , 0 / 2 Δ y , 0 / 2 exp ( j π λ d ( k Δ y , z s Δ y , 0 + ξ + ξ ) 2 ) d ξ d ξ = Δ y , 0 1 / 2 1 / 2 1 / 2 1 / 2 exp ( j π λ d ( k Δ y , z s Δ y , 0 + ξ 1 Δ y , 0 + ξ 2 Δ y , z ) 2 ) d ξ 1 d ξ 2 .
A y [ k , s ] = Δ y , 0 · ( J 1 ( k , s ) + J ˜ 2 ( k , s ) ) ,
J 1 ( k , s ) = 1 max ( Δ y , 0 , Δ y , z ) D 2 D 2 exp ( j π λ d ( k Δ y , z s Δ y , 0 + u ) 2 ) d u , J ˜ 2 ( k , s ) = 1 Δ y , 0 Δ y , d D 2 D 1 [ exp ( j π λ d ( k Δ y , d s Δ y , 0 + u ) 2 ) + exp ( j π λ d ( k Δ y , d s Δ y , 0 u ) 2 ) ] ( D 1 u ) d u .
A x [ l , t ] = Δ x , 0 · ( J 1 ( l , t ) + J ˜ 2 ( l , t ) ) ,
J 1 ( l , t ) = 1 max ( Δ x , 0 , Δ x , z ) D 2 D 2 exp ( j π λ d ( l Δ x , z t Δ x , 0 + u ) 2 ) d u , J ˜ 2 ( l , t ) = 1 Δ x , 0 Δ x , z D 2 D 1 [ exp ( j π λ d ( l Δ x , z t Δ x , 0 + u ) 2 ) + exp ( j π λ d ( l Δ x , z t Δ x , 0 u ) 2 ) ] ( D 1 u ) d u .
J = 1 1 f ( x + Δ u ) ( 1 | u | ) d u .
L = t r ( ( u z μ A y u 0 A x T ) H ( u z μ A y u 0 A x T ) + α 2 u 0 H u 0 ) = t r ( u z H u z μ u z H A y u 0 A x T μ * A x * u 0 H A y H u z + | μ | 2 A x * u 0 H A y H A y u 0 A x T + α 2 u 0 H u 0 ) .
L u 0 H = 0.
R t r ( QR ) = Q T , R t r ( QR T ) = Q ,
μ * A x H u z T A y * + | μ | 2 A x H A x u 0 T A y T A y * + α 2 u 0 T = 0 ,
μ * A y H u z A x * = ( | μ | A y H A y + α I ) u 0 ( | μ | A x T A x * + α I ) α | μ | u 0 A x T A x * α | μ | A y H A y u 0 ,
u 0 ( k + 1 ) = ( | μ | A y H A y + α I ) 1 μ * A y H u z A x * ( | μ | A x T A x * + α I ) 1 + | μ | α ( A y H A y | μ | + α I ) 1 u 0 ( k ) A x T A x * ( A x T A x * | μ | + α I ) 1 + | μ | α ( | μ | A y H A y + α I ) 1 A y H A y u 0 ( k ) ( | μ | A x T A x * + α I ) 1 , k = 0 , 1 , ... , u 0 ( 1 ) = 0 .

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