## Microscopy image resolution improvement by deconvolution of complex fields |

Optics Express, Vol. 18, Issue 19, pp. 19462-19478 (2010)

http://dx.doi.org/10.1364/OE.18.019462

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### Abstract

Based on truncated inverse filtering, a theory for deconvolution of complex fields is studied. The validity of the theory is verified by comparing with experimental data from digital holographic microscopy (DHM) using a high-NA system (NA=0.95). Comparison with standard intensity deconvolution reveals that only complex deconvolution deals correctly with coherent cross-talk. With improved image resolution, complex deconvolution is demonstrated to exceed the Rayleigh limit. Gain in resolution arises by accessing the objects complex field - containing the information encoded in the phase - and deconvolving it with the reconstructed complex transfer function (CTF). Synthetic (based on Debye theory modeled with experimental parameters of MO) and experimental amplitude point spread functions (APSF) are used for the CTF reconstruction and compared. Thus, the optical system used for microscopy is characterized quantitatively by its APSF. The role of noise is discussed in the context of complex field deconvolution. As further results, we demonstrate that complex deconvolution does not require any additional optics in the DHM setup while extending the limit of resolution with coherent illumination by a factor of at least 1.64.

© 2010 Optical Society of America

## 1. Introduction

1. S. V. Aert, D. V. Dyck, and A. J. den Dekker, “Resolution of coherent and incoherent imaging systems reconsidered—classical criteria and a statistical alternative,” Opt. Express **14**, 3830–3839 (2006). [CrossRef] [PubMed]

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

4. W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A workingperson’s guide to deconvolution in light microscopy.” Biotechniques31 (2001). [PubMed]

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

5. B. Colicchio, O. Haeberl, C. Xu, A. Dieterlen, and G. Jung, “Improvement of the lls and map deconvolution algorithms by automatic determination of optimal regularization parameters and pre-filtering of original data,” Opt. Commun. **244**, 37 – 49 (2005). [CrossRef]

6. F. Aguet, S. Geissbühler, I. Märki, T. Lasser, and M. Unser, “Super-resolution orientation estimation and localization of fluorescent dipoles using 3-d steerable filters,” Opt. Express **17**, 6829–6848 (2009). [CrossRef] [PubMed]

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

8. 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]

10. C. J. Sheppard, “Fundamentals of superresolution,” Micron **38**, 165–169 (2007). [CrossRef]

11. D. Mendlovic, A. W. Lohmann, N. Konforti, I. Kiryuschev, and Z. Zalevsky, “One-dimensional superresolution optical system for temporally restricted objects,” Appl. Opt. **36**, 2353–2359 (1997). [CrossRef] [PubMed]

12. A. Shemer, D. Mendlovic, Z. Zalevsky, J. Garcia, and P. G. Martinez, “Superresolving optical system with time multiplexing and computer decoding,” Appl. Opt. **38**, 7245–7251 (1999). [CrossRef]

14. E. N. Leith, D. Angell, and C. P. Kuei, “Superresolution by incoherent-to-coherent conversion,” J. Opt. Soc. Am. A **4**, 1050–1054 (1987). [CrossRef]

*A*as well as the phase Φ from the reconstructed complex field

*U*. Time multiplexing methods combined with DHM methods have been demonstrated to work with low-NA systems [16

16. V. Mico, Z. Zalevsky, C. Ferreira, and J. García, “Superresolution digital holographic microscopy for three-dimensional samples,” Opt. Express **16**, 19260–19270 (2008). [CrossRef]

17. G. Indebetouw, Y. Tada, J. Rosen, and G. Brooker, “Scanning holographic microscopy with resolution exceeding the rayleigh limit of the objective by superposition of off-axis holograms,” Appl. Opt. **46**, 993–1000 (2007). [CrossRef] [PubMed]

18. V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. **205**, 165–176 (2002). [CrossRef] [PubMed]

19. M. Debailleul, V. Georges, B. Simon, R. Morin, and O. Haeberle, “High-resolution three-dimensional tomographic diffractive microscopy of transparent inorganic and biological samples,” Opt. Lett. **34**, 79–81 (2009). [CrossRef]

20. M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. **198**, 82–87 (2000). [CrossRef] [PubMed]

*λ*/(2NA)] by phase structuring, the complex detection is only partially used in excitation.

*λ*/(2NA)] with high-NA (NA=0.95) by directly using the information content available from amplitude as well as from phase in DHM. By adapting mentioned standard deconvolution post processing methods to coherent illumination imaging conditions, the phase imaging process does not need to be compromised. No additional optical components nor scanning procedures are required since the method is applied at a step posterior to the experiment itself. We also show that the usual trade-off between precision in object localization and Rayleigh’s resolution criterion [1

1. S. V. Aert, D. V. Dyck, and A. J. den Dekker, “Resolution of coherent and incoherent imaging systems reconsidered—classical criteria and a statistical alternative,” Opt. Express **14**, 3830–3839 (2006). [CrossRef] [PubMed]

^{nd}section. In the 3

^{rd}section experimental details are provided and the 4

^{th}section shows how to treat the test target’s and the APSF’s data. Furthermore, the adaption of the synthetic CTF is outlined and used for the determination of the influence of noise in the 5

^{th}section. Also in this section, the final results are presented and compared to intensity deconvolution. A discussion of the implementation of complex deconvolution concludes in the 6

^{th}section.

## 2. Theory

*ν*

### 2.1. Inverse filter deconvolution of intensity fields

*M*, the intensity function

*I*(

*x*

_{2},

*y*

_{2}) in the image plane is presented as a convolution integral (following notation is based on [23])

*o*(

_{i}*x*

_{1},

*y*

_{1}) is the intensity function in the object plane and

*h*(

*x*,

*y*) is the complex point spread function (APSF). Because of the use of the intensity point spread function (IPSF) ∣

*h*(

*x*,

*y*)∣

^{2}, no phase term is included. One can express this in

*k*-space as

*J*,

*O*, and

_{i}*C*are the 2-D Fourier transform of

*I*,

*o*, and ∣

_{i}*h*∣

^{2}, respectively, such as

*C*is called the optical transfer function (OTF),

*J*the intensity image spectrum, and

*O*the intensity object spectrum. For incoherently illuminated imaging systems, the standard deconvolution approach, namely inverse filtering [7

_{i}7. P. Sarder and A. Nehorai, “Deconvolution methods for 3-d fluorescence microscopy images,” IEEE Signal Process. Mag. **23**, 32–45 (2006). [CrossRef]

*k*,

_{c}*k*, and

_{max}*k*is discussed in detail in section 5.2. The basic idea of the function

_{s}*f*is to linearly decrease the frequency values from

_{s}*f*(

_{s}*k*) = 1 to

_{s}*f*(

_{s}*k*) = 0 within a small smoothing kernel

_{max}*k*−

_{max}*k*. The purpose is to smooth the mask’s rim borders in order to suppress aliasing effects.

_{s}7. P. Sarder and A. Nehorai, “Deconvolution methods for 3-d fluorescence microscopy images,” IEEE Signal Process. Mag. **23**, 32–45 (2006). [CrossRef]

**23**, 32–45 (2006). [CrossRef]

### 2.2. Inverse filter deconvolution of complex fields

*U*(

*x*

_{2},

*y*

_{2}) can be expressed as the convolution of the complex object function

*o*(

*x*

_{1},

*y*

_{1}) and the APSF [23]:

*G*,

*O*, and c are the 2-D Fourier transform of

*U*,

*o*, and

*h*, respectively, such as

*G*the complex image spectrum and

*O*the complex object spectrum.

*G*and

*c*. However, just as the intensity based approach [2

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

*G̃*(

*k*,

_{x}*k*)/

_{y}*c*(

*k*,

_{x}*k*), especially at high spatial frequencies. Consequently,

_{y}*G̃*is defined accordingly to Eq. (9):

## 3. Experiment

*=80nm) are drilled with focused ion beam (FIB) milling in the coating and are placed at very close pitches*

_{nominal}*η*. The fabricated pitch is controlled and measured by scanning electron microscopy (SEM), as shown in Fig. 1.

*≈90nm) than the nominal ones due to their slightly conical shapes. The real pitch*

_{real}*η*, however, varies only within ±5nm from the nominal specifications.

*h*[24, 25

_{exp}25. 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]

26. 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]

*U*. The light source is a YAG laser at

*λ*=532nm. The used microscope objective is a Zeiss ×63 NA=0.95 in air (refractive index

*n*= 1) in combination with a relay magnification to reach a lateral sampling of

_{m}*δx*= 56nm.

## 4. Processing

8. 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]

*A*(

*x,y*) as well as the phase Φ(

*x,y*) of the complex field emitted by nano-holes can be extracted by following the methods of [8

8. 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]

*a*is a real normalization constant.

_{n}### 4.1. Experimental APSF

*c*and is illustrated in Figs. 2(a) and 2(c).

_{exp}*k*-space results in the phase of the focal spot image. Vice versa, defocusing results in de-phasing of the transmitted wavefront. The effects of aberration on the system can be seen e.g. the coma as a deformation of the wavefront phase in Fig. 2(c).

27. N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt.48, H186–H195 (2009). [CrossRef] [PubMed]

*n*, and uniform lateral sampling

_{m}*δx*. The discrete spatial bandwidth of the microscope objective corresponds to

*m*. By combining Eq. (3) with Eq. (16), the axes of Fig. 2 are recast into

_{px}*k*-space and the effective NA can be directly read out to be in accordance with the nominal one.

### 4.2. Synthetic APSF

*c*ought to be compared to a reference system. This system is based on a synthetic CTF:

_{exp}*c*transformed by Eq. (12) from a synthetic APSF. The APSF represents a synthesis since the following scalar Debye theory is computed with experimentally assessed parameters of the optical imaging system.

_{syn}*o*(

*x*

_{1},

*y*

_{1}) =

*δ*(

*x*

_{1},

*y*

_{1}), the image field

*U*(

_{δ}*x*

_{2},

*x*

_{2}) is the APSF

*h*(

*Mx*

_{2},

*My*

_{2}). Therefore, the APSF can be experimentally measured with a sufficiently small object diameter (∅ ≪

*d*) or synthesized by a theoretical description. A synthetic

_{min}*h*for high-aperture systems can be approximated by the scalar Debye theory expressed in a spherical coordinate system in Eq. (17) of

*θ*and Φ within the object space

*P*(

*θ*,

*ϕ*) is the apodization function according to Eq. (20) and Φ(

*θ*,

*ϕ*) the aberration function. Generally, the sine condition holds for an aplanatic imaging system within the field of view

*θ*,

*ϕ*) may be developed as spherical harmonics in a complete orthogonal set [28

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

*z*

_{2}relative to the focal plane

*A*of the aberration function Φ have to be adapted in a fitting process. In this optimization process, each calculation of the synthetic APSF is performed by FFT of the pupil function as presented in [29

_{n,m}29. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express **14**, 11277–11291 (2006). [CrossRef] [PubMed]

29. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express **14**, 11277–11291 (2006). [CrossRef] [PubMed]

*f*:

*c*is a function of the spherical harmonics with amplitude factor

_{syn}*A*according to [28

_{n,m}28. C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical-system,” J. Mod. Opt. **40**, 1631–1651 (1993). [CrossRef]

*f*is found by a genetic algorithm [30] and the fine fitting is performed by pattern research [31

31. V. Torczon, “On the convergence of pattern search algorithms,” SIAM J. Optim. **7**, 125 (1997). [CrossRef]

*k*wavenumbers and therefore a direction dependent effect. The model does not include direction dependent intensity responses which would be suggested by a vectorial influence.

_{x}*A*are likely to be overestimated.

_{n,m}*U*and

_{syn}*c*serve as a reference system to determine the influence of noise on the deconvolution process. Without loss of generality, noise can be added to the synthetic APSF as a gaussian probability distribution

_{syn}*n*and yields the estimation of

_{g}*U*by computing Eq. (23).

_{noise}### 4.3. Test target

*η*of Fig. 1 and processed according to the procedure described in section 4.1. As an illustration, Figs. 3(a) and 3(b) (

*η*=400nm) show the complex field spectrum

*G̃*calculated by Fourier transforming the reconstructed complex image field

*U*.

*G̃*∣ in Fig. 3(a) shows the image spectrum accompanied by two frequency filters (seen as minimum transmittance) in

*k*direction. These minimum transmittance filters can be understood as spectral presentation of the destructive interference between the waves emitted by the two holes, reported as phase singularities in [32

_{y}32. 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]

*arg*[

*G̃*] [cf. Fig. 3(b)] occurs at spatial positions where the spherical waves emitted from each hole are out of phase. As reported in [32

32. 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]

*θ*of those lines of singularities varies systematically with the pitch

*η*of the two holes. Consequently, the

*η*−

*θ*relationship corresponds closely to a

*η*−

*k*relationship meaning that the position of the transmittances minima varies again as a function of the pitch. Figure 3(c) illustrates the behavior within the bandwidth. It can be seen that the minimum transmittance position of the filter shifts to higher frequencies as

_{y}*η*decreases. Finally, for

*η*=300nm, the minimum almost reaches

*k*and ∣

_{y,c}*G̃*∣ barely features higher frequency content. The exact maximally possible shift of the minimum transmittance in Fig. 3(c) matches with the largest observable angle

*θ*of the phase singularities. The corresponding limit of resolution is derived to be [32

32. 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]

*ϕ*indicates an offset phase difference for instance through a longitudinal displacement Δ

*z*= Δ

*ϕλ*/(2

*π*). In the case of the used test target of Fig. 1 Δ

*ϕ*yields 0. As a consequence, the relevant

*d*corresponds to a minimal distance 64% smaller than suggested by Eq. (2) for the coherent case and still 24% superior to the corresponding equation for the incoherently illuminated case. In the event of Δ

^{cd}_{min}*ϕ*≠ 0, asymmetric phase singularities would appear in the spatial phase map [32

**35**, 2176–2178 (2010). [CrossRef] [PubMed]

*arg*[

*G̃*]) shifts on one spectral side to lower and on the other spectral side to higher wavenumbers.

*J̃*∣ are compared with the experimental OTF in Fig. 3(d). It is important to note that ∣

*J̃*∣ is calculated by Fourier transforming the intensity fields

*I*of Eq. (25),

*J̃*∣ the data are processed as if they originated from an incoherent imaging system as in Eq. (5). The purpose of this approximation is to compare the performance of this ‘improper’ intensity deconvolution to proper complex deconvolution, and in particular to erode the phase’s role.

## 5. Results

### 5.1. Complex and intensity deconvolution of test targets

*I*of two PSF above [cf. Figs. 4(a)−4(b), inserts ‘rw’] and beneath [cf. Figs. 4(c)−4(d), inserts ‘rw’] the coherent limit of resolution

*d*=460nm. It can be seen that the PSF pairs beneath

_{min,coh}*d*converge and cannot be distinguished anymore by the contrast criterion. The inserts labeled ‘cd’, though, show the intensity images ∣

_{min,coh}*o*∣

^{2}complex deconvolved with

*k*(deduced in section 5.2). In the profile plot,

_{max}*I*and ∣

*o*∣

^{2}are compared to intensity deconvolved images

*o*along the

_{i}*y*cross-sections through the pitch centers. The exact results are listed in Table 2.

*η*=500nm and

*η*=600nm image. The contrast is higher for complex deconvolution whilst holding a more accurate match on the actual pitch

*η*. For the case of

*η*=400nm the intensity deconvolution fails to resolve individual peaks. Despite being beyond

*d*=460nm, the coherent deconvolution method results in a correct localization within 25nm while holding a contrast of 69%. However, the deconvolved image suffers from a residual artifact which is caused by a mismatch during the DHM reconstruction procedure [8

_{min,coh}**38**, 6994–7001 (1999). [CrossRef]

### 5.2. Determination of kmax and noise influence

*k*[expressed as the according minimal resolvable distance

_{max}*d*by Eq. (4)]. Note that the smoothing is fixed to a small value

_{min}*k*= 2

_{s}*π*/(

*d*−60

_{min}*nm*). For the results analysis, the deconvolved amplitude fields (cf. Fig. 4) are interpolated and fitted by two gaussian curves:

*µ*provide the peak-to-peak (p-t-p) distance of the holes’ images. Assuming equivalent transmittance of the imaged holes’ pairs, the effective full width at half maximum (FWHM) is averaged for

_{i}*b*

_{1}and

*b*

_{2}and determined from Eq. (26) as Eq. (27):

*a*

_{1}and

*a*

_{2}.

*η*=400nm. The notation is as following: The legend ‘experimental’ indicates complex deconvolution of

*U*with

_{exp}*c*. Contrarily, the legend ‘synthetic(…)’ indicates the usage of synthesized fields and CTFs according to section 4.2. For indicator ‘…(no noise)’ Eqs. (19) and (23) are free of noise, whereas gaussian noise was added successively for the indicator ‘…(SNR=35)’ with the according signal-to-noise ration (SNR). Finally, the case ‘experimental-synthetic’ represents a hybrid, the complex deconvolution of the experimental fields by the synthetic noise free CTF.

_{exp}*k*(

_{max}*d*). This trend is strongest for the ‘experimental’ plot. On the contrary, the fully ‘synthetic (noise free)’ deconvolution shows a weak dependence which suggests noise as a source of the dependence trend. A stronger bending of the p-t-p curve for smaller

_{min}*d*can be created by adding noise to the ‘synthetic(SNR=35)’ deconvolution. Vice versa, the ‘experimental’ dependence becomes weaker for the ‘experimental-synthetic’ deconvolution but suffers from a vertical upward shift which may result from a modeling mismatch of

_{min}*h*. Finally, the filter’s radius dependence can be partially decoupled by using a noise free synthetic CTF, as expected for the use of synthetic OTF in intensity deconvolution [2

_{syn}**19**, 373–385 (1999). [CrossRef] [PubMed]

**23**, 32–45 (2006). [CrossRef]

*d*≈ 380

_{min}*nm*, the trend of increasing contrast is damped [cf. Fig. 5(b)] since the FWHM’s trend of narrowing stagnates [cf. Fig. 5(c)]. It is most likely that artifacts caused by the modeling mismatch lead to the (trend opposed) broadening.

*nm*(about half

*δx*) of the p-t-p distance as acceptable in order to minimize the FWHM or, in other words, to maximize the contrast. This error margin is indicated in Fig. 5(d), which shows the p-t-p distances for the 4 ‘experimental’ cases of the test target. According to the defined criterion, it can be seen that an unique filter diameter can be assessed. Moreover, a trend of

*k*is clearly observable: the optimal filter diameter corresponds to wavenumbers corresponding roughly to (150±30)

_{max}*nm*beneath the minimal structure

*η*to be observed. This trend responds to the hypothesis

*k*≈

_{max}*k*(

*η*−150

*nm*) if

*η*<

*d*.

_{min}## 6. Discussion

*k*direction indicates an aberration correction. The mean deconvolved spectrum transmittance drops down to ~0.6, therefore appearing to wane. Division, in the frequency domain, by very small values of the CTF gives rise to large transmission values [cf. Fig. 6(a)]. Those ‘bad’ pixels can also be seen in the phase spectrum [cf. Fig. 6(b)] as local pixel phase jumps. Since these local pixel phase jumps occur randomly or close to weak signal strengths, they can be considered as an effect of noise on the CTF. Just as it is known from intensity deconvolution [2

_{x}**19**, 373–385 (1999). [CrossRef] [PubMed]

*k*is chosen, in order to resolve smaller

_{max}*d*, the better the SNR must be.

_{min}*π*. This shift extends the spectrum continuously to higher frequencies.

*η*. In Fig. 7(d) it appears that the FWHMs tend to be very slim but artifacts cause a broadening and worse contrast as confirmed by Figs. 5(b) and 5(c). Thus, no real image improvement is achieved since the model mismatch causes severe artifacts. Finally, complex deconvolution by the experimental CTF holds the best match on

*η*.

*ϕ*= 0, the limit of resolution is shown to be extended by a factor of 1.64 as anticipated by Eq. (24).

*z*, an arbitrary phase offset difference Δ

*ϕ*≠ 0 between point-scatterers would be created. As underlined in test target specific section 4.3, any Δ

*ϕ*would result in at least one spectrally lower shifted phase discontinuity. We suspect that it could give rise to even higher resolution, scaling accordingly to Eq. (24). Similarly, for the case of phase objects, the occurrence of phase singularities has been reported [33

33. M. Totzeck and H. J. Tiziani, “Phase-singularities in 2d diffraction fields and interference microscopy,” Opt. Commun. **138**, 365–382 (1997). [CrossRef]

*η*= 300

*nm*sample’s complex deconvolution suggests that the use of a noise free synthetic CTF should be advantageous since the SNR becomes much more crucial in the spectral sub-discontinuity range. As pointed out in section 4.2, a vectorial CTF [34] may be more suitable to effectively avoid the discussed artifacts. A direct calculation of its vectorial components [35

35. H. Guo, S. Zhuang, J. Chen, and Z. Liang, “Imaging theory of an aplanatic system with a stratified medium based on the method for a vector coherent transfer function,” Opt. Lett. **31**, 2978–2980 (2006). [CrossRef] [PubMed]

## 7. Conclusion

## Acknowledgements

## References and links

1. | S. V. Aert, D. V. Dyck, and A. J. den Dekker, “Resolution of coherent and incoherent imaging systems reconsidered—classical criteria and a statistical alternative,” Opt. Express |

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

3. | C. Vonesch, “Fast and automated wavelet-regularized image restoration in fluorescence microscopy,” Ph.D. thesis, EPFL, LIB Laboratoire d’imagerie biomédicale (2009). |

4. | W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A workingperson’s guide to deconvolution in light microscopy.” Biotechniques31 (2001). [PubMed] |

5. | B. Colicchio, O. Haeberl, C. Xu, A. Dieterlen, and G. Jung, “Improvement of the lls and map deconvolution algorithms by automatic determination of optimal regularization parameters and pre-filtering of original data,” Opt. Commun. |

6. | F. Aguet, S. Geissbühler, I. Märki, T. Lasser, and M. Unser, “Super-resolution orientation estimation and localization of fluorescent dipoles using 3-d steerable filters,” Opt. Express |

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

8. | 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. |

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13. | Z. Zalevsky and D. Mendlovic, |

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17. | G. Indebetouw, Y. Tada, J. Rosen, and G. Brooker, “Scanning holographic microscopy with resolution exceeding the rayleigh limit of the objective by superposition of off-axis holograms,” Appl. Opt. |

18. | V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. |

19. | M. Debailleul, V. Georges, B. Simon, R. Morin, and O. Haeberle, “High-resolution three-dimensional tomographic diffractive microscopy of transparent inorganic and biological samples,” Opt. Lett. |

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21. | J. W. Goodman, |

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

23. | M. Gu, |

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

25. | 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 |

26. | 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. |

27. | N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt.48, H186–H195 (2009). [CrossRef] [PubMed] |

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

29. | M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express |

30. | D. E. Goldberg, |

31. | V. Torczon, “On the convergence of pattern search algorithms,” SIAM J. Optim. |

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

33. | M. Totzeck and H. J. Tiziani, “Phase-singularities in 2d diffraction fields and interference microscopy,” Opt. Commun. |

34. | C. J. Sheppard and K. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) |

35. | H. Guo, S. Zhuang, J. Chen, and Z. Liang, “Imaging theory of an aplanatic system with a stratified medium based on the method for a vector coherent transfer function,” Opt. Lett. |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(100.1830) Image processing : Deconvolution

(100.5070) Image processing : Phase retrieval

(100.6640) Image processing : Superresolution

(110.0180) Imaging systems : Microscopy

(090.1995) Holography : Digital holography

**ToC Category:**

Image Processing

**History**

Original Manuscript: June 11, 2010

Revised Manuscript: July 26, 2010

Manuscript Accepted: August 25, 2010

Published: August 30, 2010

**Virtual Issues**

Vol. 5, Iss. 13 *Virtual Journal for Biomedical Optics*

**Citation**

Yann Cotte, M. Fatih Toy, Nicolas Pavillon, and Christian Depeursinge, "Microscopy image resolution improvement by deconvolution of complex fields," Opt. Express **18**, 19462-19478 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-19462

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