Abstract

A discrete diffraction transform (DDT) is a novel discrete wavefield propagation model that is aliasing free for a pixelwise invariant object distribution. For this class of distribution, the model is precise and has no typical discretization effects because it corresponds to accurate calculation of the diffraction integral. A spatial light modulator (SLM) is a good example of a system where a pixelwise invariant distribution appears. Frequency domain regularized inverse algorithms are developed for reconstruction of the object wavefield distribution from the distribution given in the sensor plane. The efficiency of developed frequency domain algorithms is demonstrated by simulation.

© 2008 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. 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]
  3. V. Katkovnik, J. Astola, and K. Egiazarian, “Wavefield reconstruction and design as discrete inverse problems,” Proceedings of 3D TV Conference, Istanbul (2008).
  4. G. S. Sherman, “Integral-transform formulation of diffraction theory,” J. Opt. Soc. Am. 57, 1490-1498 (1967).
    [CrossRef]
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996) 2nd edition.
  6. L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A 24, 359-367 (2007).
    [CrossRef]
  7. 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]
  8. J. M. Arnold, “Phase-space localization and discrete representations of wave fields,” J. Opt. Soc. Am. A 12, 111-123 (1995).
    [CrossRef]
  9. 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]
  10. 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]
  11. M. Liebling, Th. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 12, 29-43 (2003).
    [CrossRef]
  12. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP Publishing, 1998).
    [CrossRef]
  13. A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, 1977).
  14. V. Katkovnik, K. Egiazarian, J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006).
    [CrossRef]
  15. V. Katkovnik, J. Astola, and K. Egiazarian, “Numerical wavefield reconstruction in phase-shifting holography as inverse discrete problem,” submitted to EUSIPCO 2008(2008).

2007

2006

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]

2004

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

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

1995

1967

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]

Arnold, J. M.

Arsenin, V. Y.

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

Astola, J.

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

V. Katkovnik, J. Astola, and K. Egiazarian, “Wavefield reconstruction and design as discrete inverse problems,” Proceedings of 3D TV Conference, Istanbul (2008).

V. Katkovnik, J. Astola, and K. Egiazarian, “Numerical wavefield reconstruction in phase-shifting holography as inverse discrete problem,” submitted to EUSIPCO 2008(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]

Egiazarian, K.

V. Katkovnik, J. Astola, and K. Egiazarian, “Numerical wavefield reconstruction in phase-shifting holography as inverse discrete problem,” submitted to EUSIPCO 2008(2008).

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

V. Katkovnik, J. Astola, and K. Egiazarian, “Wavefield reconstruction and design as discrete inverse problems,” Proceedings of 3D TV Conference, Istanbul (2008).

Goodman, J. W.

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

Katkovnik, V.

V. Katkovnik, J. Astola, and K. Egiazarian, “Numerical wavefield reconstruction in phase-shifting holography as inverse discrete problem,” submitted to EUSIPCO 2008(2008).

V. Katkovnik, J. Astola, and K. Egiazarian, “Wavefield reconstruction and design as discrete inverse problems,” Proceedings of 3D TV Conference, Istanbul (2008).

V. Katkovnik, K. Egiazarian, 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]

Shen, F.

Sherman, G. S.

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.

Appl. Opt.

IEEE Trans. Image Process.

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. Signal Process.

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.

J. Opt. Soc. Am. A

Opt. Eng.

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]

Other

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

V. Katkovnik, J. Astola, and K. Egiazarian, “Wavefield reconstruction and design as discrete inverse problems,” Proceedings of 3D TV Conference, Istanbul (2008).

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

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

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

V. Katkovnik, K. Egiazarian, 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,” submitted to EUSIPCO 2008(2008).

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Figures (12)

Fig. 1
Fig. 1

Principal setup of wavefield propagation and reconstruction.

Fig. 2
Fig. 2

Conditioning number versus the distance z for the transfer function A ˜ z with N = 512 , N = 1024 and λ = 0.632 nm .

Fig. 3
Fig. 3

Module frequency characteristics of the discrete diffraction transfer function: (a)  Δ 0 ; (b)  Δ = 0.01 / 1024 , z = 0.75 m .

Fig. 4
Fig. 4

Phase frequency characteristics of the discrete diffraction transfer function: (a)  Δ 0 ; (b)  Δ = 0.01 / 1024 , z = 0.75 m .

Fig. 5
Fig. 5

Image of the Baboon test-distribution.

Fig. 6
Fig. 6

Wavefield distribution at the sensor plane: (a) amplitude distribution; (b) phase distribution.

Fig. 7
Fig. 7

Object wavefield reconstruction (amplitude distribution), double pixel size Δ = 3.9 × 10 5 : (a) inverse regularized DDT reconstruction, RMSE = 0.108 ; (b) standard FFT algorithm. The standard algorithm fails with a pattern of clear aliasing effects. DDT shows a good quality aliasing free reconstruction.

Fig. 8
Fig. 8

Wavefield reconstruction (amplitude distribution): (a) the recursive regularized inverse DDT technique, RMSE = 0.051 after 10 iterations; (b) the standard FFT technique, RMSE = 0.086 . The DDT algorithm shows sharper and better resolution imaging.

Fig. 9
Fig. 9

Phase distribution reconstruction: (a) the recursive regularized inverse DDT technique, RMSE = 0.185 after 10 iterations; (b) the standard FFT technique, RMSE = 0.26 . The DDT algorithm shows sharper and better imaging with no boundary ringing effect seen in the FFT reconstruction.

Fig. 10
Fig. 10

Phase distribution design for the object plane to obtain a desirable amplitude distribution in the sensor plane: (a) designed phase distribution for the object plane; (b) the corresponding amplitude distribution in the sensor plane.

Fig. 11
Fig. 11

Convergence of the recursive regularized inverse phase design algorithm: RMSE versus the number of iterations.

Fig. 12
Fig. 12

Integration area in the original and transformed (rotated) coordinates.

Equations (56)

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u z ( x , y ) = g z ( x ξ , y η ) u 0 ( ξ , η ) d ξ d η ,
g z = z exp ( j 2 π r / λ ) j λ r 2 , r = x 2 + y 2 + z 2 .
G z ( f x , f y ) = F { g z ( x , y ) } = { exp ( j 2 π z λ 1 ( λ f x ) 2 ( λ f y ) 2 ) , ( f x , f y ) D λ 0 , otherwise .
U z ( f x , f y ) = G z ( f x , f y ) U 0 ( f x , f y ) ,
U 0 ( f x , f y ) = G z * ( f x , f y ) U z ( f x , f y ) ,
u z ( x , y ) = U z ( f x , f y ) exp ( j 2 π ( f x x + f y y ) ) d f x d f y .
u 0 ( x , y ) = D z 1 { u z ( x , y ) } = g z ( x ξ , y η ) u z ( ξ , η ) d ξ d η = D z { u z ( x , y ) } .
U ¯ z ( v x , v y ) = G ¯ z ( v x , v y ) U ¯ 0 ( v x , v y ) ,
U ¯ 0 ( v x , v y ) = G ¯ z * ( v x , v y ) U ¯ z ( v x , v y ) .
u z ( x , y ) = s , t = N 0 / 2 N 0 / 2 1 u 0 [ s , t ] Δ / 2 Δ / 2 Δ / 2 Δ / 2 g z ( x s Δ + ξ , y t Δ + η ) d ξ d η , u 0 [ s , t ] = u 0 ( s Δ + ξ , t Δ + η ) , Δ / 2 ξ , η < Δ / 2 ,
u z [ k , l ] = 1 Δ 2 Δ / 2 Δ / 2 Δ / 2 Δ / 2 u z ( k Δ + ξ , l Δ + η ) d ξ d η .
u z [ k , l ] = s , t = N 0 / 2 N 0 / 2 1 a z [ k s , l t ] u 0 [ s , t ] , k , l = N z / 2 + 1 , , N z / 2 1 ,
a z [ k , l ] = 1 Δ 2 Δ / 2 Δ / 2 Δ / 2 Δ / 2 Δ / 2 Δ / 2 Δ / 2 Δ / 2 g z ( k Δ + ξ + ξ , l Δ + η + η ) d ξ d η d ξ d η , k , l = N a / 2 + 1 , , N a / 2 1 , N a = N 0 + N z .
a z [ k , l ] Δ 0 Δ 2 g z ( k Δ , l Δ ) .
u z = A z · u 0 ,
u 0 = ( A z H A z ) 1 A z H u z ,
U ˜ 0 ( v x , v y ) = FFT { u ˜ 0 } = k , l = N 0 / 2 N 0 / 2 1 u 0 [ k , l ] W v x k W v y l = k , l = N a / 2 N a / 2 1 u ˜ 0 [ k , l ] W v x k W v y l , W = exp ( j 2 π / ( 2 N ) ) , v x , v y = N a / 2 , , N a / 2 1.
A ˜ z ( v x , v y ) = FFT { a ˜ z } = u , v = N a / 2 + 1 N a / 2 1 a z [ u , v ] W v x u W v y v = u , v = N a / 2 N a / 2 1 a ˜ z [ u , v ] W v x u W v y v .
U ˜ z ( v x , v y ) = A ˜ z ( v x , v y ) U ˜ 0 ( v x , v y ) , u ˜ z [ k , l ] = FFT 1 { U ˜ z ( v x , v y ) } , k , l = N a / 2 , , N a / 2 1.
u z [ k , l ] = u ˜ z [ k , l ] , k , l = N z / 2 , , N z / 2 1 .
cond A ˜ = max v x , v y | A ˜ z ( v x , v y ) | / min v x , v y | A ˜ z ( v x , v y ) | .
g z 1 j λ z exp [ j ( 2 π z λ + π λ z ( x 2 + y 2 ) ) ] .
a z [ k , l ] exp ( j 2 π z / λ ) j λ z ρ z , λ [ k ] ρ z , λ [ l ] ,
ρ z , λ [ k ] = 1 Δ Δ / 2 Δ / 2 Δ / 2 Δ / 2 exp ( j π λ z ( k Δ + ξ + ξ ) 2 ) d ξ d ξ = Δ 1 / 2 1 / 2 1 / 2 1 / 2 exp ( j π λ z ( k Δ + ξ Δ + ξ Δ ) 2 ) d ξ d ξ = 2 Δ 1 1 ( 1 | v | ) exp ( j π λ z ( k Δ + Δ v ) 2 ) d v .
A ˜ z ( v x , v y ) = exp ( j 2 π z / λ ) j λ z Λ z , λ ( v x , N a ) × Λ z , λ ( v y , N a ) ,
Λ z , λ ( v , N ) = k , l = N / 2 N / 2 1 ρ z , λ [ k ] W v k , W = exp ( j 2 π / N a ) , v = N a / 2 , , N a / 2 1 .
u ˜ z = A ˜ z · u ˜ 0 ,
Λ z , λ ( f , N a ) 0 ,     f = N a / 2 , , N a / 2 1 ,
( u z # u z ) = A ˜ z ( u 0 0 ) ,
( u 0 0 ) = A ˜ z 1 ( u z # u z ) .
( u ^ 0 ( r ) u ^ 0 ( r ) ) = A ˜ z 1 ( u z # u ^ z ( r 1 ) ) ,
( u ^ z ( r ) u ^ z ( r ) ) = A ˜ z ( u ^ 0 ( r ) 0 ) , r = 1 , .
J = U ˜ z A ˜ z U ˜ 0 2 + α U ˜ 0 2 .
U ^ 0 = A ˜ z * · U ˜ z / ( | A ˜ z | 2 + α 2 ) .
u ˜ 0 = FFT 1 { U ^ 0 } , u ^ 0 [ k , l ] = u ˜ 0 [ k , l ] , k , l = N 0 / 2 , , N 0 / 2 1 .
u ˜ z ( r 1 ) [ k , l ] = u z # [ k , l ] , k , l = N z / 2 , , N z / 2 1 , U ˜ z ( r ) [ v x , v y ] = FFT { u ˜ z ( r 1 ) [ k , l ] } , U ˜ 0 ( r ) [ v x , v y ] = A ˜ z * / ( | A ˜ z | 2 + α 2 ) · U ˜ z ( r ) [ v x , v y ] ;
u ˜ 0 ( r ) [ s , t ] = FFT 1 { U ˜ 0 ( r ) [ v x , v y ] } ;
u ˜ 0 ( r ) [ s , t ] = 0 , for all     s , t [ N 0 / 2 , , N 0 / 2 1 ] .
U ˜ 0 ( r ) [ v x , v y ] = FFT { u ˜ 0 ( r ) [ s , t ] } , U ˜ z ( r ) [ v x , v y ] = A ˜ z [ v x , v y ] U ˜ 0 ( r ) [ v x , v y ] , u ˜ z ( r 1 ) [ k , l ] = FFT 1 { U ˜ z ( r 1 ) [ v x , v y ] } .
u ^ 0 ( r ) [ s , t ] = u ˜ 0 ( r ) [ s , t ] , s , t = N 0 / 2 , , N 0 / 2 1 .
u ˜ 0 ( r ) [ s , t ] = | FFT 1 { U ˜ 0 ( r ) [ v x , v y ] } | .
u ˜ 0 ( r ) [ s , t ] = exp [ j × angle ( FFT 1 { U ˜ 0 ( r ) [ v x , v y ] } ) ] .
u z [ k , l ] = 1 N a 2 v x , v y = N a / 2 N a / 2 1 W v x k W v y l A ˜ z ( v x , v y ) U ˜ 0 ( v x , v y ) , k , l = N z / 2 , , N z / 2 1 .
u z [ k , l ] = 1 N a 2 u , v = N a / 2 + 1 N a / 2 1 u , v = N 0 / 2 N 0 / 2 1 a z [ u , v ] u 0 [ u , v ] × v y , v x = N a / 2 N a / 2 1 W v x ( k u u ) W v y ( l v v ) .
1 N a 2 v y , v x = N a / 2 N a / 2 1 W v x ( k u u ) W v y ( l v v ) = δ ( k u u ) δ ( l v v ) ,
u z [ k , l ] = u , v = N a / 2 + 1 N a / 2 1 u , v = N 0 / 2 N 0 / 2 1 a z [ u , v ] u 0 [ u , v ] δ ( k u u ) δ ( l v v ) .
N z / 2 ( N 0 / 2 1 ) k u N z / 2 1 + N 0 / 2 ,
N a / 2 + 1 k u N a / 2 1 .
N a / 2 + 1 l v N a / 2 1 ,
u z [ k , l ] = u , v = N 0 / 2 N 0 / 2 1 a z [ k u , l v ] u 0 [ u , v ] , k , l = N z / 2 , , N z / 2 1 .
1 Δ Δ / 2 Δ / 2 Δ / 2 Δ / 2 f ( k Δ + ξ + ξ ) d ξ d ξ = 2 Δ 1 1 f ( k Δ + Δ x ) ( 1 | x | ) d x .
J = 1 Δ Δ / 2 Δ / 2 Δ / 2 Δ / 2 f ( k Δ + ξ + ξ ) d ξ d ξ = Δ 1 / 2 1 / 2 1 / 2 1 / 2 f ( k Δ + Δ u + Δ u ) d u d u ,
( x y ) = ( cos φ sin φ sin φ cos φ ) ( u u ) .
( x y ) = 1 2 ( 1 1 1 1 ) ( u u ) ,
J = Δ 1 / 2 1 / 2 1 / 2 1 / 2 f ( k Δ + Δ u + Δ u ) d u d u = Δ 0 1 / 2 f ( k Δ + Δ 2 x ) 1 / 2 + x 1 / 2 x d y d x + Δ 1 / 2 0 f ( k Δ + Δ 2 x ) 1 / 2 x 1 / 2 + x d y d x = Δ 0 1 / 2 f ( k Δ + Δ 2 x ) 2 ( 1 / 2 x ) d x + Δ 1 / 2 0 f ( k Δ + Δ 2 x ) 2 ( 1 / 2 + x ) d x = 2 Δ 1 / 2 1 / 2 f ( k Δ + Δ 2 x ) 2 ( 1 / 2 | x | ) d x = 2 Δ 1 1 f ( k Δ + Δ x ) ( 1 | x | ) d x .
U ˜ z ( v x , v y ) = A ˜ z ( v x , v y ) U ˜ 0 ( v x , v y ) , U ˜ 0 ( v x , v y ) = A ˜ z 1 ( v x , v y ) U ˜ z ( v x , v y ) , k , l = N a / 2 , , N a / 2 1 .

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