Abstract

On-axis digital holography (DH) is becoming widely used for its time-resolved three-dimensional (3D) imaging capabilities. A 3D volume can be reconstructed from a single hologram. DH is applied as a metrological tool in experimental mechanics, biology, and fluid dynamics, and therefore the estimation and the improvement of the resolution are current challenges. However, the resolution depends on experimental parameters such as the recording distance, the sensor definition, the pixel size, and also on the location of the object in the field of view. This paper derives resolution bounds in DH by using estimation theory. The single point resolution expresses the standard deviations on the estimation of the spatial coordinates of a point source from its hologram. Cramér–Rao lower bounds give a lower limit for the resolution. The closed-form expressions of the Cramér–Rao lower bounds are obtained for a point source located on and out of the optical axis. The influences of the 3D location of the source, the numerical aperture, and the signal-to-noise ratio are studied.

© 2010 Optical Society of America

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2009 (4)

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. (Bellingham) 48, 095801 (2009).
[CrossRef]

L. Denis, D. Lorenz, and D. Trede, “Greedy solution of ill-posed problems: Error bounds and exact inversion,” Inverse Probl. 25, 115017 (2009).
[CrossRef]

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
[CrossRef] [PubMed]

L. Denis, D. Lorenz, E. Thiebaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Opt. Lett. 34, 3475–3477 (2009).
[CrossRef] [PubMed]

2008 (1)

2007 (5)

2006 (5)

2004 (2)

A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. (Bellingham) 43, 239–250 (2004).
[CrossRef]

S. Sotthivirat and J. A. Fessler, “Penalized-likelihood image reconstruction for digital holography,” J. Opt. Soc. Am. A 21, 737–750 (2004).
[CrossRef]

2001 (1)

M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87–94 (2001).
[CrossRef]

1997 (1)

1969 (1)

1945 (1)

C. R. Rao, “Information and the accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–89 (1945).

Bertaux, N.

Brady, D. J.

Chaumet, P. C.

Choi, K.

Cramér, H.

H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, 1946).

den Dekker, A. J.

Denis, L.

Drsek, F.

Fade, J.

Fessler, J. A.

Fournier, C.

Garcia-Sucerquia, J.

Giovannini, H.

Goepfert, C.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

Grier, D. G.

Guérin, C. A.

Helstrom, C. W.

Hennelly, B. M.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. (Bellingham) 48, 095801 (2009).
[CrossRef]

Holschneider, M.

Horisaki, R.

Jacquot, M.

M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87–94 (2001).
[CrossRef]

Javidi, B.

Jericho, M. H.

Jericho, S. K.

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 2005).

Kelly, D. P.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. (Bellingham) 48, 095801 (2009).
[CrossRef]

Kim, S. H.

Klages, P.

Kreis, T. M.

T. M. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, 2005).

Kreuzer, H. J.

Lee, S. H.

Lim, S.

Lorenz, D.

L. Denis, D. Lorenz, E. Thiebaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Opt. Lett. 34, 3475–3477 (2009).
[CrossRef] [PubMed]

L. Denis, D. Lorenz, and D. Trede, “Greedy solution of ill-posed problems: Error bounds and exact inversion,” Inverse Probl. 25, 115017 (2009).
[CrossRef]

Marks, D. L.

Milanfar, P.

M. Shahram and P. Milanfar, “Statistical and information-theoretic analysis of resolution in imaging,” IEEE Trans. Inf. Theory 52, 3411–3437 (2006).
[CrossRef]

Naughton, T. J.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. (Bellingham) 48, 095801 (2009).
[CrossRef]

Ober, R. J.

S. Ram, E. Sally Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidimens. Syst. Signal Process. 17, 27–57 (2006).
[CrossRef]

Pandey, N.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. (Bellingham) 48, 095801 (2009).
[CrossRef]

Ram, S.

S. Ram, E. Sally Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidimens. Syst. Signal Process. 17, 27–57 (2006).
[CrossRef]

Rao, C. R.

C. R. Rao, “Information and the accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–89 (1945).

Réfrégier, P.

Rhodes, W. T.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. (Bellingham) 48, 095801 (2009).
[CrossRef]

Roche, M.

Roichman, Y.

Sally Ward, E.

S. Ram, E. Sally Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidimens. Syst. Signal Process. 17, 27–57 (2006).
[CrossRef]

Sandoz, P.

M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87–94 (2001).
[CrossRef]

Sentenac, A.

Shahram, M.

M. Shahram and P. Milanfar, “Statistical and information-theoretic analysis of resolution in imaging,” IEEE Trans. Inf. Theory 52, 3411–3437 (2006).
[CrossRef]

Sotthivirat, S.

Soulez, F.

Stern, A.

Thiebaut, E.

Trede, D.

L. Denis, D. Lorenz, and D. Trede, “Greedy solution of ill-posed problems: Error bounds and exact inversion,” Inverse Probl. 25, 115017 (2009).
[CrossRef]

L. Denis, D. Lorenz, E. Thiebaut, C. Fournier, and D. Trede, “Inline hologram reconstruction with sparsity constraints,” Opt. Lett. 34, 3475–3477 (2009).
[CrossRef] [PubMed]

Tribillon, G.

M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87–94 (2001).
[CrossRef]

Van Aert, S.

van Blaaderen, A.

Van den Bos, A.

Van Dirk, D.

van Oostrum, P.

Xu, W.

Yang, S. M.

Yi, G. R.

Appl. Opt. (1)

Bull. Calcutta Math. Soc. (1)

C. R. Rao, “Information and the accuracy attainable in the estimation of statistical parameters,” Bull. Calcutta Math. Soc. 37, 81–89 (1945).

IEEE Trans. Inf. Theory (1)

M. Shahram and P. Milanfar, “Statistical and information-theoretic analysis of resolution in imaging,” IEEE Trans. Inf. Theory 52, 3411–3437 (2006).
[CrossRef]

Inverse Probl. (1)

L. Denis, D. Lorenz, and D. Trede, “Greedy solution of ill-posed problems: Error bounds and exact inversion,” Inverse Probl. 25, 115017 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Multidimens. Syst. Signal Process. (1)

S. Ram, E. Sally Ward, and R. J. Ober, “A stochastic analysis of performance limits for optical microscopes,” Multidimens. Syst. Signal Process. 17, 27–57 (2006).
[CrossRef]

Opt. Commun. (1)

M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87–94 (2001).
[CrossRef]

Opt. Eng. (Bellingham) (2)

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. (Bellingham) 48, 095801 (2009).
[CrossRef]

A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. (Bellingham) 43, 239–250 (2004).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Other (5)

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 2005).

P. Réfrégier, Noise Theory and Application to Physics: From Fluctuations to Information (Springer Verlag, 2004).

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

T. M. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, 2005).

H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, 1946).

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Figures (3)

Fig. 1
Fig. 1

Illustration of the parametric model g θ ( x k , y k ) of an on-axis digital hologram of a point source ( x = 0.2 L , y = 0.4 L , z = 100   mm , λ = 0.532 μ m , Ω = 8.6 × 10 3 ).

Fig. 2
Fig. 2

Single point resolution in a transversal plane: (a) x-resolution map normalized by the value of x-resolution on the optical axis; (b) z-resolution map normalized: σ z / σ ̊ z ; (c) x-resolution for y ¯ = 0 ; (d) z-resolution for y ¯ = 0 ; for z = 100   mm , λ = 0.532 μ m , Ω = 8.6 × 10 3 , and SNR = 10 . The squares in the centers of the figures represent the sensor boundaries.

Fig. 3
Fig. 3

Relative differences (in percent) between resolution maps (Fig. 2) and resolution maps taking into account pixel integration for z = 100   mm , λ = 0.532 μ m , Ω = 8.6 × 10 3 , SNR = 10 , Δ x = Δ y = 6.7 μ m , κ x = κ y = 1 , and N = 256 . The squares in the centers of the figures represent the sensor boundaries.

Equations (22)

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δ x = δ y = λ Ω ,     δ z = λ Ω 2 .
g θ ( x k , y k ) = A   sin ( π ( x x k ) 2 + ( y y k ) 2 λ z ) ,
var ( θ ̂ i ) [ I 1 ( θ ) ] i , i ,
[ I ( θ ) ] i , j = def E [ 2 ln   p ( d ; θ ) θ i θ j ] .
ln   p ( d ; θ ) = 1 2 σ b 2 x k y k [ d ( x k , y k ) g θ ( x k , y k ) ] 2 + C ,
[ I ( θ ) ] i , j = 1 σ b 2 1 L 2 sensor ( g θ ( x k , y k ) θ i g θ ( x k , y k ) θ j ) d x k d y k .
[ I ( θ ) ] i , j = 1 σ b 2 1 L 2 L / 2 L / 2 | L / 2 L / 2 ( g θ ( x k , y k ) θ i g θ ( x k , y k ) θ j ) | x = 0 y = 0 d x k d y k .
L 2 / λ z 1.
I x x = I y y π 2 A 2 σ b 2 L 2 6 λ 2 z 2 ,     I z z π 2 A 2 σ b 2 7 L 4 360 λ 2 z 4 .
σ ̊ x = σ ̊ y = λ Ω c 1 SNR ,     σ ̊ z = λ Ω 2 c 2 SNR ,
[ I ( θ ) ] i , j = 1 σ b 2 1 L 2 L / 2 x L / 2 x | L / 2 y L / 2 y ( g θ ( x k , y k ) θ i g θ ( x k , y k ) θ j ) | x = 0 y = 0 d x k d y k .
I x x π 2 A 2 σ b 2 L 2 λ 2 z 2 ( 1 + 12 x ¯ 2 ) 6 ,
I y y π 2 A 2 σ b 2 L 2 λ 2 z 2 ( 1 + 12 y ¯ 2 ) 6 ,
I x y = I y x π 2 A 2 σ b 2 L 2 λ 2 z 2 2 x ¯ y ¯ ,
I z z π 2 A 2 σ b 2 L 4 λ 2 z 4 ( 7 + 120 ( x ¯ 2 + y ¯ 2 ) + 180 ( x ¯ 2 + y ¯ 2 ) 2 ) 360 ,
I x z = I z x π 2 A 2 σ b 2 L 3 λ 2 z 3 x ¯ ( 1 / 3 + x ¯ 2 + y ¯ 2 ) ,
I y z = I z x π 2 A 2 σ b 2 L 3 λ 2 z 3 y ¯ ( 1 / 3 + x ¯ 2 + y ¯ 2 ) ,
( ( 1 + 156 x ¯ 2 K ) σ ̊ x 2 156 x ¯ y ¯ K σ ̊ x σ ̊ y 60 7 15 x ¯ 1 + 3 ρ ¯ 2 K σ ̊ x σ ̊ z 156 x ¯ y ¯ K σ ̊ x σ ̊ y ( 1 + 156 y ¯ 2 K ) σ ̊ y 2 60 7 15 y ¯ 1 + 3 ρ ¯ 2 K σ ̊ y σ ̊ z 60 7 15 x ¯ 1 + 3 ρ ¯ 2 K σ ̊ x σ ̊ z 60 7 15 y ¯ 1 + 3 ρ ¯ 2 K σ ̊ y σ ̊ z ( 1 + 60 ρ ¯ 2 2 3 ρ ¯ 2 K ) σ ̊ z 2 ) ,
σ x = σ ̊ x 1 + 156 x ¯ 2 7 36 ρ ¯ 2 + 180 ρ ¯ 4 , σ y = σ ̊ y 1 + 156 y ¯ 2 7 36 ρ ¯ 2 + 180 ρ ¯ 4 ,
σ z = σ ̊ z 1 + 60 ρ ¯ 2 2 3 ρ ¯ 2 7 36 ρ ¯ 2 + 180 ρ ¯ 4 .
g θ p ( x k , y k ) = g θ ( x k , y k ) sinc ( π κ x Δ x ( x x k ) λ z ) sinc ( π κ y Δ y ( y y k ) λ z ) ,
[ I ( θ ) ] i , j = 1 σ b 2 1 N M i = 1 N j = 1 M ( g θ p ( i Δ x , j Δ y ) θ i g θ p ( i Δ x , j Δ y ) θ j ) ,

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