Abstract

We propose a Riesz transform approach to the demodulation of digital holograms. The Riesz transform is a higher-dimensional extension of the Hilbert transform and is steerable to a desired orientation. Accurate demodulation of the hologram requires a reliable methodology by which quadrature-phase functions (or simply, quadratures) can be constructed. The Riesz transform, by itself, does not yield quadratures. However, one can start with the Riesz transform and construct the so-called vortex operator by employing the notion of quasi-eigenfunctions, and this approach results in accurate quadratures. The key advantage of using the vortex operator is that it effectively handles nonplanar fringes (interference patterns) and has the ability to compensate for the local orientation. Therefore, this method results in aberration-free holographic imaging even in the case when the wavefronts are not planar. We calibrate the method by estimating the orientation from a reference hologram, measured with an empty field of view. Demodulation results on synthesized planar as well as nonplanar fringe patterns show that the accuracy of demodulation is high. We also perform validation on real experimental measurements of Caenorhabditis elegans acquired with a digital holographic microscope.

© 2012 Optical Society of America

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References

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  7. W. T. Freeman and E. H. Adelson, “The design and use of steerable filters,” IEEE Trans. Pattern Anal. Machine Intell. 13, 891–906 (1991).
    [CrossRef]
  8. M. Unser, and D. Van De Ville, “Wavelet steerability and the higher-order Riesz transform,” IEEE Trans. Image Process. 19, 636–652 (2010).
    [CrossRef]
  9. M. Unser, D. Sage, and D. Van De Ville, “Multiresolution monogenic signal analysis using the Riesz–Laplace wavelet transform,” IEEE Trans. Image Process. 18, 2402–2418 (2009).
    [CrossRef]
  10. T. Bülow, and G. Sommer, “Hypercomplex signals—a novel extension of the analytic signal to the multidimensional case,” IEEE Trans. Signal Process. 49, 2844–2852 (2001).
    [CrossRef]
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  13. M. Felsberg, and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136–3144 (2001).
    [CrossRef]
  14. T. Colomb, E. Cuche, F. Charriere, J. Kuhn, N. Aspert, F. Montfort, P. Marquet, and C. Depeursinge, “Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation,” Appl. Opt. 45, 851–863 (2006).
    [CrossRef]
  15. M. Liebling, T. Blu, and M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A 21, 367–377 (2004).
    [CrossRef]
  16. C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Exact complex-wave reconstruction in digital holography,” J. Opt. Soc. Am. A 28, 983–992 (2011).
    [CrossRef]
  17. J. P. Havlicek, D. S. Harding, and A. C. Bovik, “Multidimensional quasi-eigenfunction approximations and multicomponent AM–FM models,” IEEE Trans. Image Process. 9, 227–242(2000).
    [CrossRef]
  18. J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
    [CrossRef]
  19. E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38, 6994–7001 (1999).
    [CrossRef]
  20. E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
    [CrossRef]
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    [CrossRef]
  22. T. Colomb, J. Kühn, F. Charrière, C. Depeursinge, P. Marquet, and N. Aspert, “Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram,” Opt. Express 14, 4300–4306 (2006).
    [CrossRef]
  23. J. W. Head and C. G. Mayo, “The response of a network to a frequency-modulated input voltage,” Proc. IEE C 305R, 509–512 (1958).
    [CrossRef]
  24. E. Bedrosian, “A product theorem for Hilbert transforms,” Proc. IEEE 51, 868–869 (1963).
    [CrossRef]
  25. A. H. Nuttall and E. Bedrosian, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966).
    [CrossRef]
  26. H. Urkowitz, “Hilbert transforms of band-pass functions,” Proc. IRE 50, 2143 (1962).
  27. G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (Johns Hopkins University, 1996).

2011

2010

M. Unser, and D. Van De Ville, “Wavelet steerability and the higher-order Riesz transform,” IEEE Trans. Image Process. 19, 636–652 (2010).
[CrossRef]

2009

M. Unser, D. Sage, and D. Van De Ville, “Multiresolution monogenic signal analysis using the Riesz–Laplace wavelet transform,” IEEE Trans. Image Process. 18, 2402–2418 (2009).
[CrossRef]

2006

2004

2003

2001

2000

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
[CrossRef]

J. P. Havlicek, D. S. Harding, and A. C. Bovik, “Multidimensional quasi-eigenfunction approximations and multicomponent AM–FM models,” IEEE Trans. Image Process. 9, 227–242(2000).
[CrossRef]

1999

1992

S. L. Hahn, “Multidimensional complex signals with single-orthant spectra,” Proc. IEEE 80, 1287–1300 (1992).
[CrossRef]

1991

W. T. Freeman and E. H. Adelson, “The design and use of steerable filters,” IEEE Trans. Pattern Anal. Machine Intell. 13, 891–906 (1991).
[CrossRef]

1967

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

1966

A. H. Nuttall and E. Bedrosian, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966).
[CrossRef]

1963

E. Bedrosian, “A product theorem for Hilbert transforms,” Proc. IEEE 51, 868–869 (1963).
[CrossRef]

1962

H. Urkowitz, “Hilbert transforms of band-pass functions,” Proc. IRE 50, 2143 (1962).

1958

J. W. Head and C. G. Mayo, “The response of a network to a frequency-modulated input voltage,” Proc. IEE C 305R, 509–512 (1958).
[CrossRef]

1946

D. Gabor, “Theory of communication,” J. IEE Part 3 93, 429–457 (1946).

Adelson, E. H.

W. T. Freeman and E. H. Adelson, “The design and use of steerable filters,” IEEE Trans. Pattern Anal. Machine Intell. 13, 891–906 (1991).
[CrossRef]

Aspert, N.

Bedrosian, E.

A. H. Nuttall and E. Bedrosian, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966).
[CrossRef]

E. Bedrosian, “A product theorem for Hilbert transforms,” Proc. IEEE 51, 868–869 (1963).
[CrossRef]

Blu, T.

Bone, D. J.

Bovik, A. C.

J. P. Havlicek, D. S. Harding, and A. C. Bovik, “Multidimensional quasi-eigenfunction approximations and multicomponent AM–FM models,” IEEE Trans. Image Process. 9, 227–242(2000).
[CrossRef]

Bülow, T.

T. Bülow, and G. Sommer, “Hypercomplex signals—a novel extension of the analytic signal to the multidimensional case,” IEEE Trans. Signal Process. 49, 2844–2852 (2001).
[CrossRef]

Charriere, F.

Charrière, F.

Colomb, T.

Coppola, G.

Cuche, E.

Depeursinge, C.

Felsberg, M.

M. Felsberg, and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136–3144 (2001).
[CrossRef]

Ferraro, P.

Finizio, A.

Freeman, W. T.

W. T. Freeman and E. H. Adelson, “The design and use of steerable filters,” IEEE Trans. Pattern Anal. Machine Intell. 13, 891–906 (1991).
[CrossRef]

Gabor, D.

D. Gabor, “Theory of communication,” J. IEE Part 3 93, 429–457 (1946).

Golub, G. H.

G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (Johns Hopkins University, 1996).

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

Hahn, S. L.

S. L. Hahn, “Multidimensional complex signals with single-orthant spectra,” Proc. IEEE 80, 1287–1300 (1992).
[CrossRef]

Harding, D. S.

J. P. Havlicek, D. S. Harding, and A. C. Bovik, “Multidimensional quasi-eigenfunction approximations and multicomponent AM–FM models,” IEEE Trans. Image Process. 9, 227–242(2000).
[CrossRef]

Havlicek, J. P.

J. P. Havlicek, D. S. Harding, and A. C. Bovik, “Multidimensional quasi-eigenfunction approximations and multicomponent AM–FM models,” IEEE Trans. Image Process. 9, 227–242(2000).
[CrossRef]

Haykin, S.

S. Haykin, Communication Systems, 4th ed. (Wiley, 2001).

Head, J. W.

J. W. Head and C. G. Mayo, “The response of a network to a frequency-modulated input voltage,” Proc. IEE C 305R, 509–512 (1958).
[CrossRef]

Kuhn, J.

Kühn, J.

Larkin, K. G.

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

Liebling, M.

Loan, C. F. V.

G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (Johns Hopkins University, 1996).

Marquet, P.

Mayo, C. G.

J. W. Head and C. G. Mayo, “The response of a network to a frequency-modulated input voltage,” Proc. IEE C 305R, 509–512 (1958).
[CrossRef]

Montfort, F.

Nicola, S. D.

Nuttall, A. H.

A. H. Nuttall and E. Bedrosian, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966).
[CrossRef]

Oldfield, M. A.

Papoulis, A.

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

A. Papoulis, Systems and Transforms With Applications in Optics (Krieger, 1981).

Pavillon, N.

Pierattini, G.

Sage, D.

M. Unser, D. Sage, and D. Van De Ville, “Multiresolution monogenic signal analysis using the Riesz–Laplace wavelet transform,” IEEE Trans. Image Process. 18, 2402–2418 (2009).
[CrossRef]

Seelamantula, C. S.

Sommer, G.

M. Felsberg, and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136–3144 (2001).
[CrossRef]

T. Bülow, and G. Sommer, “Hypercomplex signals—a novel extension of the analytic signal to the multidimensional case,” IEEE Trans. Signal Process. 49, 2844–2852 (2001).
[CrossRef]

Stein, E.

E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton University, 1971).

Unser, M.

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Exact complex-wave reconstruction in digital holography,” J. Opt. Soc. Am. A 28, 983–992 (2011).
[CrossRef]

M. Unser, and D. Van De Ville, “Wavelet steerability and the higher-order Riesz transform,” IEEE Trans. Image Process. 19, 636–652 (2010).
[CrossRef]

M. Unser, D. Sage, and D. Van De Ville, “Multiresolution monogenic signal analysis using the Riesz–Laplace wavelet transform,” IEEE Trans. Image Process. 18, 2402–2418 (2009).
[CrossRef]

M. Liebling, T. Blu, and M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A 21, 367–377 (2004).
[CrossRef]

Urkowitz, H.

H. Urkowitz, “Hilbert transforms of band-pass functions,” Proc. IRE 50, 2143 (1962).

Van De Ville, D.

M. Unser, and D. Van De Ville, “Wavelet steerability and the higher-order Riesz transform,” IEEE Trans. Image Process. 19, 636–652 (2010).
[CrossRef]

M. Unser, D. Sage, and D. Van De Ville, “Multiresolution monogenic signal analysis using the Riesz–Laplace wavelet transform,” IEEE Trans. Image Process. 18, 2402–2418 (2009).
[CrossRef]

Weiss, G.

E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton University, 1971).

Appl. Opt.

Appl. Phys. Lett.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[CrossRef]

IEEE Trans. Image Process.

J. P. Havlicek, D. S. Harding, and A. C. Bovik, “Multidimensional quasi-eigenfunction approximations and multicomponent AM–FM models,” IEEE Trans. Image Process. 9, 227–242(2000).
[CrossRef]

M. Unser, and D. Van De Ville, “Wavelet steerability and the higher-order Riesz transform,” IEEE Trans. Image Process. 19, 636–652 (2010).
[CrossRef]

M. Unser, D. Sage, and D. Van De Ville, “Multiresolution monogenic signal analysis using the Riesz–Laplace wavelet transform,” IEEE Trans. Image Process. 18, 2402–2418 (2009).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell.

W. T. Freeman and E. H. Adelson, “The design and use of steerable filters,” IEEE Trans. Pattern Anal. Machine Intell. 13, 891–906 (1991).
[CrossRef]

IEEE Trans. Signal Process.

M. Felsberg, and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136–3144 (2001).
[CrossRef]

T. Bülow, and G. Sommer, “Hypercomplex signals—a novel extension of the analytic signal to the multidimensional case,” IEEE Trans. Signal Process. 49, 2844–2852 (2001).
[CrossRef]

J. IEE Part 3

D. Gabor, “Theory of communication,” J. IEE Part 3 93, 429–457 (1946).

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Proc. IEE C

J. W. Head and C. G. Mayo, “The response of a network to a frequency-modulated input voltage,” Proc. IEE C 305R, 509–512 (1958).
[CrossRef]

Proc. IEEE

E. Bedrosian, “A product theorem for Hilbert transforms,” Proc. IEEE 51, 868–869 (1963).
[CrossRef]

A. H. Nuttall and E. Bedrosian, “On the quadrature approximation to the Hilbert transform of modulated signals,” Proc. IEEE 54, 1458–1459 (1966).
[CrossRef]

S. L. Hahn, “Multidimensional complex signals with single-orthant spectra,” Proc. IEEE 80, 1287–1300 (1992).
[CrossRef]

Proc. IRE

H. Urkowitz, “Hilbert transforms of band-pass functions,” Proc. IRE 50, 2143 (1962).

Other

G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed. (Johns Hopkins University, 1996).

S. Haykin, Communication Systems, 4th ed. (Wiley, 2001).

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

A. Papoulis, Systems and Transforms With Applications in Optics (Krieger, 1981).

E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton University, 1971).

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Figures (13)

Fig. 1.
Fig. 1.

Off-axis digital holography—recording.

Fig. 2.
Fig. 2.

Off-axis digital holography—reconstruction.

Fig. 3.
Fig. 3.

Illustration of the (a) eigenfunction property and (b) quasi-eigenfunction approximation for LSI systems. The error ϵ(x) can be bounded in some suitable norm for well-behaved h(x).

Fig. 4.
Fig. 4.

Characterization of the Hilbert transformer: (a) impulse response, (b) magnitude spectrum, and (c) phase spectrum.

Fig. 5.
Fig. 5.

(a) Spectrum of o(x) and (b) spectra of various functions in Eq. (6). The bandpass filter is used to select the desired imaging order.

Fig. 6.
Fig. 6.

Performance of the Hilbert-transform-based demodulation algorithm. The AM is complex. The FM is linear, sweeping from 0.4π radians to 0.5π radians over the observation duration of 1024 points. The subplots (a) and (c) show the real and imaginary parts of the object wave, respectively, in comparison with the reconstructed ones. The subplots (b) and (d) show the associated estimation error. The signal-to-estimation-error ratio (SEER) values are also indicated. Note that the demodulation accuracy is sufficient for most practical applications.

Fig. 7.
Fig. 7.

Phase response of the complex Riesz operator over the domain [π,π]×[π,π]. The units for all three axes are radians.

Fig. 8.
Fig. 8.

(a) Spectral support of o(x); (b) Spectral support of various functions in Eq. (26). The bandpass filter is used to suppress the zero-order terms.

Fig. 9.
Fig. 9.

Analogy between 1D and 2D demodulation: (a) Hilbert-transform-based demodulation of 1D signals and (b) Riesz-transform-based demodulation of 2D fringes. I, in-phase channel; Q, quadrature-phase channel (standard terminology in communication/modulation literature).

Fig. 10.
Fig. 10.

Simulation results for linear-frequency modulation. (a) Original image, (b) carrier wave, (c) modulated image, (d) reconstructed image, (e) orientation map, and (f) reconstruction error. The orientation map is estimated with a Gaussian kernel of standard deviation σ=5.

Fig. 11.
Fig. 11.

Simulation results for circular-frequency modulation. (a) Original image, (b) carrier wave, (c) modulated image, (d) reconstructed image, (e) orientation map, and (f) reconstruction error. The orientation map is estimated with a Gaussian kernel of standard deviation σ=5.

Fig. 12.
Fig. 12.

Riesz demodulation performance on C. elegans holograms. (a) Hologram of C. elegans, (b) reference hologram, (c) zoomed-in part of the hologram (approximately 1/16th the size of the hologram from the lower left corner), (d) zoomed-in part of the reference hologram (approximately 1/16th the size of the reference hologram from the lower left corner, (e) amplitude-contrast image, and (f) phase-contrast image (wrapped phase). The orientation map is estimated with a Gaussian kernel of standard deviation σ=5.

Fig. 13.
Fig. 13.

Riesz demodulation performance on C. elegans holograms. (a) Hologram of C. elegans, (b) reference hologram, (c) zoomed-in part of the hologram (approximately 1/16th the size of the hologram from the lower left corner), (d) zoomed-in part of the reference hologram (approximately 1/16 the size of the reference hologram from the lower left corner), (e) amplitude-contrast image , and (f) phase-contrast image (wrapped phase). The orientation map is estimated with a Gaussian kernel of standard deviation σ=5.

Equations (39)

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

i(x)=|r(x)+o(x)|2=|r(x)|2+|o(x)|2+r*(x)o(x)+r(x)o*(x).
ψo(x)=u(x)i(x)=u(x)+u(x)|o(x)|2zero-order terms+u(x)r*(x)o(x)+u(x)r(x)o*(x)imaging terms.
H{cos(ωc·+θ)}(x)=sin(ωcx+θ).
fas(x)=f(x)+jHf(x),=f2(x)+(Hf)2(x)amplitude modulation (AM)exp(jtan1(Hf(x)f(x))phase modulation (PM))(polar form).
f(x)=|1+o(x)ejϕ(x)|2,
f(x)=1+|o(x)|2+2Re{o(x)ejωcx},
g(x)=2+2cos(ωcx)r(x).
ras(x)=r(x)+jHr(x)=2ejωcx.
f˜as(x)=f˜(x)+jHf˜(x),=2Re{o(x)ejωcx}+jH{Re{2o(x)ejωcx}},=2o(x)ejωcx
o(x)=f˜as(x)ras(x),
H{o(x)ej(ωcx+φ(x))}h^(ωc+dφ(x)dx)o(x)ej(ωcx+φ(x))=jo(x)ej(ωcx+φ(x)).
H{o(x)ej(ωcx+φ(x))}h^(ωcdφ(x)dx)o(x)ej(ωcx+φ(x))=jo(x)ej(ωcx+φ(x)).
H{o(x)cos(ωcx+φ(x))}o(x)sin(ωcx+φ(x)).
fR(x)=(f1(x)f2(x))=((hx*f)(x)(hy*f)(x)),
h^x(ω)=jωxω,h^y(ω)=jωyω.
hx(x)=x2πx3,hy(x)=y2πx3.
Rf(x)Fjωx+ωyωh^R(ω)f^(ω),
R*f(x)Fjωx+ωyωf^(ω).
Rcos(ωc,.)(x)=ejθ0sin(ωc,x).
Hθf(x)=defcosθf1(x)+sinθf2(x)=Re{ejθRf(x)},
Hθ0cos(ωc,.)(x)=sin(ωc,x).
Hθcos(ωc,.)(x)=cos(θθ0)roll-offsin(ωc,x).
θ(x0)=argmaxα[π,π](v*|Hαf|2)(x0),
(v*|Hαf|2)(x0)=v*(cosθf1(x)+sinθf2(x))2=cos2θ(v*f12)+sin2θ(v*f22)+2cosθsinθv*(f1f2)=uTJ(x0)u,
J(x0)=(v*f12v*(f1f2)v*(f1f2)v*f22).
θ(x0)=12tan1(2v*(f1f2)v*f22v*f12).
f(x)=1+|o(x)|2zero order+2Re{o(x)ejϕ(x)}.
rdas(x)=r(x)+jHθ^(x){r(x)},
rdas(x)=r(x)+jRe{ejθ^(x)Rr(x)}2ejωc,x,
f˜das(x)=f˜(x)+jRe{ejθ^(x)Rf˜(x)}2aejωc,x.
af˜das(x)rdas(x).
R{o(·)ejϕ(·)}(x)o(x)ejϕ(x)h^R(ϕ)(x).
R{o(·)ejϕ(·)}(x)o(x)ejϕ(x)h^R(ϕ)(x).
R{o(·)cosϕ(·)}(x)jejψ(ϕ(x))o(x)sinϕ(x),
jejψ(ϕ(x))RV{o(·)cosϕ(·)}(x)o(x)sinϕ(x).
V{o(·)cosϕ(·)}(x)o(x)sinϕ(x).
f˜das(x)=o(x)cosϕ(x)+jV{o(·)cosϕ(·)}(x)o(x)ejϕ(x)
rdas(x)=cosϕ(x)+jV{cosϕ(·)}(x)ejϕ(x).
o(x)f˜das(x)rdas(x).

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