## Realistic 3D coherent transfer function inverse filtering of complex fields |

Biomedical Optics Express, Vol. 2, Issue 8, pp. 2216-2230 (2011)

http://dx.doi.org/10.1364/BOE.2.002216

Acrobat PDF (2629 KB)

### Abstract

We present a novel technique for three-dimensional (3D) image processing of complex fields. It consists in inverting the coherent image formation by filtering the complex spectrum with a realistic 3D coherent transfer function (CTF) of a high-NA digital holographic microscope. By combining scattering theory and signal processing, the method is demonstrated to yield the reconstruction of a scattering object field. Experimental reconstructions in phase and amplitude are presented under non-design imaging conditions. The suggested technique is best suited for an implementation in high-resolution diffraction tomography based on sample or illumination rotation.

© 2011 OSA

## 1. Introduction

1. Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express **17**(1), 266–277 (2009). [CrossRef] [PubMed]

2. C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical-system,” J. Mod. Opt. **40**, 1631–1651 (1993). [CrossRef]

3. S. Frisken Gibson and F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A **8**(10), 1601–1613 (1991). [CrossRef]

4. R. Arimoto and J. M. Murray, “A common aberration with water-immersion objective lenses,” J. Microsc. **216**, 49–51 (2004). [CrossRef] [PubMed]

6. A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging **4**(4), 336–350 (1982). [CrossRef] [PubMed]

7. Y. Cotte, M. F. Toy, E. Shaffer, N. Pavillon, and C. Depeursinge, “Sub-Rayleigh resolution by phase imaging,” Opt. Lett. **35**, 2176–2178 (2010). [CrossRef] [PubMed]

8. Y. Cotte, M. F. Toy, N. Pavillon, and C. Depeursinge, “Microscopy image resolution improvement by deconvolution of complex fields,” Opt. Express **18**(19), 19462–19478 (2010). [CrossRef] [PubMed]

10. A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. **225**, 156–169 (2007). [CrossRef] [PubMed]

12. F. Montfort, F. Charrire, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Purely numerical compensation for microscope objective phase curvature in digital holographic microscopy: influence of digital phase mask position,” J. Opt. Soc. Am. A **23**(11), 2944–2953 (2006). [CrossRef]

13. T. Latychevskaia, F. Gehri, and H.-W. Fink, “Depth-resolved holographic reconstructions by three-dimensional deconvolution,” Opt. Express **18**(21), 22527–22544 (2010). [CrossRef] [PubMed]

14. P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. **23**, 32–45 (2006). [CrossRef]

## 2. Description of method

### 2.1. Regularized 3D deconvolution of complex fields

*U*is expressed as the convolution of the complex object function, called

*o*, and the complex point spread function (APSF), called

*h*[9]: where

*r⃗*= (

*x,y,z*) denotes the location vector in the object space

*r⃗*

_{1}and the image space

*r⃗*

_{2}as shown in Fig. 1(a). Equation (1) can be recast into reciprocal space by a 3D Fourier transformation

*ℱ*defined as: The reciprocal space based on the free-space (index of refraction

*n*= 1) norm of wavenumber

*k*with wavelength

*λ*, relates to the spatial frequency

*ν*and wave vector

*k⃗*= (

*k*,

_{x}*k*,

_{y}*k*) by According to the convolution theorem, applying Eq. (2) to Eq. (1) results in: Conventionally, the 3D Fourier transforms of

_{z}*U*,

*o*, and

*h*are called

*G*, the complex image spectrum,

*O*, the complex object spectrum, and,

*c*, the coherent transfer function (CTF), as summarized in Eq. (4). The latter is bandpass limited through

*h*, with the maximal lateral wave vector, and the maximal longitudinal wave vector scaled by

*k*from Eq. (3). The angle

*α*indicates the half-angle of the maximum cone of light that can enter into the MO (cf. Fig. 1) and is given by its NA =

*n*sin

_{i}*α*(

*n*is the immersion’s index of refraction). Through Eq. (4), the complex image formation can be easily inverted: The 3D inverse filtering can be directly performed by dividing the two complex fields

_{i}*G*and

*c*. As known from intensity deconvolution [15

15. J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods **19**, 373–385 (1999). [CrossRef] [PubMed]

*G*(

*k⃗*)

*/c*(

*k⃗*), particularly at high spatial frequencies.

*G*(

*k⃗*) is physically band limited by the CTF, thus it can be low-pass filtered with the maximal frequency

*k*of Eq. (5) in order to suppress noise [8

_{xy,c}8. Y. Cotte, M. F. Toy, N. Pavillon, and C. Depeursinge, “Microscopy image resolution improvement by deconvolution of complex fields,” Opt. Express **18**(19), 19462–19478 (2010). [CrossRef] [PubMed]

*c*smaller than

*τ*, the CTF’s amplitude is set to unity, so that its noise amplification is eliminated while its phase value still acts for the deconvolution. By controlling

*τ*, truncated inverse complex filtering (

*τ*<< 1) or pure phase filtering (

*τ*=1) can be achieved. Therefore, the deconvolution result depends on the parameter

*τ*. Compared to standard regularization 3D intensity deconvolution [14

14. P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. **23**, 32–45 (2006). [CrossRef]

16. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express **17**(15), 13040–13049 (2009). [CrossRef] [PubMed]

### 2.2. 3D field reconstruction of 2D hologram

14. P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. **23**, 32–45 (2006). [CrossRef]

17. 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]

### 2.3. Experimental pseudo 3D APSF

*o*(

*r⃗*

_{1}) =

*δ*(

*r⃗*

_{1}), the image field

*U*(

_{δ}*r⃗*

_{2}) is the APSF

*h*(

*r⃗*

_{2}). This approximation yields aperture diameters ⌀<<

*d*(

_{min}*d*: limit of resolution), and its imaged amplitude and phase have been shown to be characteristic [10

_{min}10. A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. **225**, 156–169 (2007). [CrossRef] [PubMed]

18. X. Heng, X. Q. Cui, D. W. Knapp, J. G. Wu, Z. Yaqoob, E. J. McDowell, D. Psaltis, and C. H. Yang, “Characterization of light collection through a subwavelength aperture from a point source,” Opt. Express **14**, 10410–10425 (2006). [CrossRef] [PubMed]

*λ*= 680

*nm*and is equipped with a dry MO for long working distance of nominal NA = 0.7 and magnification

*M*= 100 ×. The sample rotation by

*θ*, as depicted in Fig. 2, introduces non-design MO conditions of imaging [4

4. R. Arimoto and J. M. Murray, “A common aberration with water-immersion objective lenses,” J. Microsc. **216**, 49–51 (2004). [CrossRef] [PubMed]

*n*, the defocus of the object in a mismatched mounting medium

_{i}*n*[3

_{m}3. S. Frisken Gibson and F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A **8**(10), 1601–1613 (1991). [CrossRef]

*θ*= 15°.

10. A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. **225**, 156–169 (2007). [CrossRef] [PubMed]

*J*

_{1}. The phase part oscillates at

*J*

_{1}’s roots with spacing

*λ*/NA from –

*π*to

*π*[7

7. Y. Cotte, M. F. Toy, E. Shaffer, N. Pavillon, and C. Depeursinge, “Sub-Rayleigh resolution by phase imaging,” Opt. Lett. **35**, 2176–2178 (2010). [CrossRef] [PubMed]

2. C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical-system,” J. Mod. Opt. **40**, 1631–1651 (1993). [CrossRef]

*θ*= 15°, the field in Fig. 3(b) features asymmetries of the diffraction pattern in amplitude and phase. The aberration can be especially well observed in a lateral phase distortion. Likewise, the asymmetric aberration is also expressed in the axial direction of the APSF. The introduction of coma-like aberration [2

2. C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical-system,” J. Mod. Opt. **40**, 1631–1651 (1993). [CrossRef]

3. S. Frisken Gibson and F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A **8**(10), 1601–1613 (1991). [CrossRef]

20. J. Braat, “Analytical expressions for the wave-front aberration coefficients of a tilted plane-parallel plate,” Appl. Opt. **36**(32), 8459–8467 (1997). [CrossRef]

*A*and phase Φ

_{δ}*on the sample rotation (cf. Fig. 2) is defined by with the illumination wavevector*

_{δ}*h*background illumination free. As a result, the SNR is advantageous and the required

*h*can be directly used for Eq. (2). The amplitude and phase of the recorded field

*A*

_{δ,k⃗0}(

*r⃗*

_{2}) and Φ

_{δ,k⃗0}(

*r⃗*

_{2}) then correspond to the scattered components

### 2.4. Experimental 3D CTF

*k⃗*

_{0}and the scattered field is measured in direction of

*k⃗*, the first-order Born approximation states that the 3D CTF is given by the cap of an Ewald sphere defined for as schematically shown in Fig. 1(b) for DHM [21

21. S. S. Kou and C. J. Sheppard, “Imaging in digital holographic microscopy,” Opt. Express **15**(21), 13,640–13,648 (2007). [CrossRef]

*α*according to the cut-off frequencies of Eqs. (5) and (6). The experimental CTF includes experimental conditions, i. e. aberrations, as well. By comparing Fig. 4(a) and Fig. 4(b) the impact of illumination becomes apparent. Due to the sample rotation, the MO can accept higher frequencies from one side of the aperture, while frequencies from the opposed side are cut off. As a result, the CTF is displaced along the Ewald’s sphere [22

22. S. S. Kou and C. J. R. Sheppard, “Image formation in holographic tomography: high-aperture imaging conditions,” Appl. Opt. **48**(34), H168–H175 (2009). [CrossRef] [PubMed]

*K⃗*: where a hat indicates the Fourier component in amplitude

*Â*and phase Φ̂ as summarized in Fig. 5(a). Similarly to the 3D CTF reconstruction, the three-dimensional complex spectrum

*G*(

*k⃗*–

*k⃗*

_{0}) is calculated from experimental configurations shown in Fig. 2(c).

### 2.5. Scattered field retrieval

*o*. According to diffraction theory [5], the total field

*o*can be expressed as the sum of the incident field

*o*

^{(i)}in direction of

*k⃗*

_{0}and the scattered field

*o*

^{(s)}, where with the scattered field amplitude

23. 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**(10), 4300–4306 (2006). [CrossRef] [PubMed]

*A*

^{(i)}, propagating in the direction specified by

*k⃗*

_{0}. The time-independent part of the incident field is then given by the expression and Eqs. (15) and (16) yield On the other hand, according to Eqs. (13), (14), and (16) the image spectrum may be expressed as

*Â*

^{(s)}can be normalized to

*o*

^{(s)}can be obtained for any illumination by Eq. (15) if an experimental reference field is provided. Alternatively, under the assumption of plane wave illumination, it can be calculated by Eqs. (19) or (20). The latter is used for processing in section 3.

*F*(

*r⃗*

_{1}): It states that the scattered field

*z*

^{±}, is filtered by an ideal Ewald half sphere

*n*: refractive index of mounting medium), and propagated by the latest term as known by the filtered back propagation algorithm of conventional diffraction tomography [6

_{m}6. A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging **4**(4), 336–350 (1982). [CrossRef] [PubMed]

8. Y. Cotte, M. F. Toy, N. Pavillon, and C. Depeursinge, “Microscopy image resolution improvement by deconvolution of complex fields,” Opt. Express **18**(19), 19462–19478 (2010). [CrossRef] [PubMed]

*F̂*(

*K⃗*) of the scattering potential within the full Ewald limiting sphere of 2

*k*= 4

*π*/

*λ*. One could synthesize all Fourier components to obtain the approximation called the low-pass filtered approximation of the scattering potential [5] that gives rise to diffraction tomography. Opposed to this full approach, the approximation of the partial low-pass filtered scattering potential

*F*in only one direction of

*k⃗*

_{0}gives rise to optical sectioning, as discussed in section 3.3.

## 3. Applications

### 3.1. Coherent imaging inversion

*n*= 1.59, ⌀ ≈ 5.8

_{sph}*μm*) in water (

*n*

_{m,H2O}= 1.33) are recorded at a tilt angle of

*θ*= 15°, shown in Fig 6(a) for raw data and in Fig. 6(b) after deconvolution.

**Background extinction:**The transparent sample images are recorded with the incident light

*o*

^{(i)}in direction of

*k⃗*

_{0}. According to Eq. (20), a DC value is added to the APSF to compensate for

*o*

^{(i)}well seen by reduced background haze in the amplitude (cf. Fig. 6 ROI-1). Similarly, the removed background results in full 2

*π*-dynamic range of phase oscillation (cf. Fig. 6 ROI-3). Finally, the object ROI-2 in Fig. 6 displays improved contrast since the objects’ edges are sharpened by the complex deconvolution.

**Diffraction pattern suppression:**A second motivation consists in correcting the diffraction pattern of the MO’s APSF. This correction is in particular required for high-NA imaging systems since the APSF diffraction pattern may result in incorrect tomographic reconstruction in the near resolution limit range. The diffraction pattern can be observed to be well suppressed by comparing ROI-1 in Fig. 6. As a result, the diffraction pattern of the refractive index mismatched sphere [24

24. J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. II. transmission and cross-polarization effects,” Appl. Opt. **35**(3), 515–531 (1996). [CrossRef] [PubMed]

25. D. Q. Chowdhury, P. W. Barber, and S. C. Hill, “Energy-density distribution inside large nonabsorbing spheres by using Mie theory and geometrical optics,” Appl. Opt. **31**(18), 3518–3523 (1992). [CrossRef] [PubMed]

**Complex aberration correction:**Aberrations are intrinsic to MO, especially for high-NA objectives [2

**40**, 1631–1651 (1993). [CrossRef]

**8**(10), 1601–1613 (1991). [CrossRef]

*n*, mismatch of

_{i}*n*and

_{i}*n*, defocus of object in

_{m}*n*, and non-design coverslip thickness or refraction index. Also, axially asymmetric aberrations are introduced by the sample rotation [4

_{m}4. R. Arimoto and J. M. Murray, “A common aberration with water-immersion objective lenses,” J. Microsc. **216**, 49–51 (2004). [CrossRef] [PubMed]

20. J. Braat, “Analytical expressions for the wave-front aberration coefficients of a tilted plane-parallel plate,” Appl. Opt. **36**(32), 8459–8467 (1997). [CrossRef]

15. J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods **19**, 373–385 (1999). [CrossRef] [PubMed]

### 3.2. The impact of phase deconvolution

*τ*= 1 while the phase part still leads to image correction according to the previous section. As seen by Eq. (19), the phase deconvolution effectively acts as a subtraction of the diffraction pattern in phase. Strictly speaking, the recorded phase is not the phase difference between object and reference beam, but also includes the MO’s diffraction due to frequency cutoff. The coherent system is seen to exhibit rather pronounced coherent imaging edges, known as ‘ringing’ [26]. For biological samples [27

27. Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. U.S.A. **105**(37), 13730–13735 (2008). [CrossRef] [PubMed]

*n*= 1.334 at

_{m,sol}*λ*= 682

*nm*). This preparation method allows fixing the RBC directly on the coverslip surface to avoid defocus aberrations, and a space invariant APSF may be assumed within the field of view [28

28. Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. **98**(7), 3790–3795 (2001). [CrossRef] [PubMed]

27. Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. U.S.A. **105**(37), 13730–13735 (2008). [CrossRef] [PubMed]

*n*= 1.39. It shows that the deconvolved phase profile follows basically the raw trend. However, similar to coherent imaging ringing, edges are less prone of oscillations after phase deconvolution, in particular on the RBC’s edges. Thus the impact of the complex deconvolution is to de-blur the phase signal.

_{RBC}*τ*, implies a decreased SNR through amplitude noise amplification. As a consequence, random phasers can be introduced that degrade the phase signal. Figures 7(e) and 7(f) show that signal degradation affects most prominently RBC’s central and border regions. For a pure retrieval of phase that is unaffected by noise amplification,

*τ*should not be smaller than −1dB.

*θ*since projection surface along y-axis decreases, as well observed in Figs. 8(g)–8(l).

### 3.3. Scattered object field retrieval and optical sectioning

**23**, 32–45 (2006). [CrossRef]

13. T. Latychevskaia, F. Gehri, and H.-W. Fink, “Depth-resolved holographic reconstructions by three-dimensional deconvolution,” Opt. Express **18**(21), 22527–22544 (2010). [CrossRef] [PubMed]

16. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express **17**(15), 13040–13049 (2009). [CrossRef] [PubMed]

*τ*<< 1 and amplitude information is not discarded. From the inverse filtering point of view, the z-confinement arises from the filtering by the 3D CTF as shown in Fig. 4. To demonstrate that scattered object field retrieval features this effect, complex deconvolution was performed on the RBC sample of the previous section 3.2 with

*τ*determined by CTF’s noise level. For optimal amplitude contrast, the value of

*τ*was calculated by a histogram based method, Otsu’s rule, well known in intensity deconvolution [29

29. N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man Cybern. **9**(1), 62–66 (1979). [CrossRef]

*τ*≈ −3dB, lies beneath the pure phase deconvolution criterion of −1dB, but it allows effective diffraction pattern suppression. Therefore, high amplitude contrast compromises phase signal. The results are shown in Fig. 9.

*U*| in Fig. 9(a) shows the object spread along the axial direction according to Eq. (9). As expected, the reconstruction features no axial confinement and the RBCs cannot be recognized.

*o*

^{(s)}| after truncated inverse filtering according to Eq. (20) is depicted in Fig. 9(b). It shows that the background field and out-of-focus haze are successfully removed. The RBCs’ edges can be identified as strong scattering objects and the scattered field can be recognized to match in size and position the anticipated RBC values. Although the image quality is affected by artifacts in axial elongation [30

30. J. G. McNally, C. Preza, J.-A. Conchello, and L. J. Thomas, “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A **11**(3), 1056–1067 (1994). [CrossRef]

*n*> 1.4) and mounting medium (

_{lipid}*n*= 1.334 at

_{m,sol}*λ*= 682

*nm*) result in strong scattering well visible in the xz sections.

## 4. Concluding remarks

## Acknowledgments

## References and links

1. | Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express |

2. | C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical-system,” J. Mod. Opt. |

3. | S. Frisken Gibson and F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A |

4. | R. Arimoto and J. M. Murray, “A common aberration with water-immersion objective lenses,” J. Microsc. |

5. | M. Born and E. Wolf, |

6. | A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging |

7. | Y. Cotte, M. F. Toy, E. Shaffer, N. Pavillon, and C. Depeursinge, “Sub-Rayleigh resolution by phase imaging,” Opt. Lett. |

8. | Y. Cotte, M. F. Toy, N. Pavillon, and C. Depeursinge, “Microscopy image resolution improvement by deconvolution of complex fields,” Opt. Express |

9. | M. Gu, |

10. | A. Marian, F. Charrière, T. Colomb, F. Montfort, J. Kühn, P. Marquet, and C. Depeursinge, “On the complex three-dimensional amplitude point spread function of lenses and microscope objectives: theoretical aspects, simulations and measurements by digital holography,” J. Microsc. |

11. | Y. Cotte and C. Depeursinge, “Measurement of the complex amplitude point spread function by a diffracting circular aperture,” in Focus on Microscopy, Advanced linear and non-linear imaging, pp. TU–AF2–PAR–D (2009). |

12. | F. Montfort, F. Charrire, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Purely numerical compensation for microscope objective phase curvature in digital holographic microscopy: influence of digital phase mask position,” J. Opt. Soc. Am. A |

13. | T. Latychevskaia, F. Gehri, and H.-W. Fink, “Depth-resolved holographic reconstructions by three-dimensional deconvolution,” Opt. Express |

14. | P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. |

15. | J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods |

16. | D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express |

17. | 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. |

18. | X. Heng, X. Q. Cui, D. W. Knapp, J. G. Wu, Z. Yaqoob, E. J. McDowell, D. Psaltis, and C. H. Yang, “Characterization of light collection through a subwavelength aperture from a point source,” Opt. Express |

19. | I. Bergoend, C. Arfire, N. Pavillon, and C. Depeursinge, “Diffraction tomography for biological cells imaging using digital holographic microscopy,” in Laser Applications in Life Sciences, SPIE vol. 7376 (2010). |

20. | J. Braat, “Analytical expressions for the wave-front aberration coefficients of a tilted plane-parallel plate,” Appl. Opt. |

21. | S. S. Kou and C. J. Sheppard, “Imaging in digital holographic microscopy,” Opt. Express |

22. | S. S. Kou and C. J. R. Sheppard, “Image formation in holographic tomography: high-aperture imaging conditions,” Appl. Opt. |

23. | 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 |

24. | J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. II. transmission and cross-polarization effects,” Appl. Opt. |

25. | D. Q. Chowdhury, P. W. Barber, and S. C. Hill, “Energy-density distribution inside large nonabsorbing spheres by using Mie theory and geometrical optics,” Appl. Opt. |

26. | J. W. Goodman, |

27. | Y. Park, M. Diez-Silva, G. Popescu, G. Lykotrafitis, W. Choi, M. S. Feld, and S. Suresh, “Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum,” Proc. Natl. Acad. Sci. U.S.A. |

28. | Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. |

29. | N. Otsu, “A threshold selection method from gray-level histograms,” IEEE Trans. Syst. Man Cybern. |

30. | J. G. McNally, C. Preza, J.-A. Conchello, and L. J. Thomas, “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A |

**OCIS Codes**

(100.1830) Image processing : Deconvolution

(100.5070) Image processing : Phase retrieval

(100.6890) Image processing : Three-dimensional image processing

(110.0180) Imaging systems : Microscopy

(180.6900) Microscopy : Three-dimensional microscopy

(090.1995) Holography : Digital holography

**ToC Category:**

Image Reconstruction and Inverse Problems

**History**

Original Manuscript: April 4, 2011

Revised Manuscript: June 29, 2011

Manuscript Accepted: June 30, 2011

Published: July 8, 2011

**Citation**

Yann Cotte, Fatih M. Toy, Cristian Arfire, Shan Shan Kou, Daniel Boss, Isabelle Bergoënd, and Christian Depeursinge, "Realistic 3D coherent transfer function inverse filtering of complex fields," Biomed. Opt. Express **2**, 2216-2230 (2011)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-2-8-2216

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### References

- Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009). [CrossRef] [PubMed]
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