## Numerical refocusing in digital holographic microscopy with extended-sources illumination |

Optics Express, Vol. 21, Issue 23, pp. 28258-28271 (2013)

http://dx.doi.org/10.1364/OE.21.028258

Acrobat PDF (4532 KB)

### Abstract

Numerical refocusing can be seen as a method of compensating the defocus aberration based on
deconvolution by inverse filtering [

© 2013 Optical Society of America

## 1. Introduction

3. L. Lovicar, J. Komrska, and R. Chmelík, “Quantitative-phase-contrast imaging of a two-level surface described as a 2D linear filtering process,” Opt. Express **18**, 20585–20594 (2010). [CrossRef] [PubMed]

4. R. Barer, “Interference microscopy and mass determination,” Nature (London) **169**, 366–367 (1952). [CrossRef]

5. Y. Cotte, F. Toy, C. Arfire, S. Kou, D. Boss, I. Bergoënd, and C. Depeursinge, “Realistic 3D coherent transfer function inverse filtering of complex fields,” Biomed. Opt. Express **2**, 2216–2230 (2011). [CrossRef] [PubMed]

2. F. Dubois, L. Joannes, and J. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. **38**, 7085–7094 (1999). [CrossRef]

2. F. Dubois, L. Joannes, and J. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. **38**, 7085–7094 (1999). [CrossRef]

*z*satisfies where NA is the numerical aperture of the objective and NA

_{i}is the effective numerical aperture of illumination — NA

_{i}=

*n*sin

*φ*

_{max}, where

*n*is the index of refraction of immersion and

*φ*

_{max}is the greatest angle of incidence of the illumination rays in the object space.

7. T. Kozacki and R. Jóźwicki, “Digital reconstruction of a hologram recorded using partially coherent illumination,” Opt Commun **252**, 188–201 (2005). [CrossRef]

8. T. Slabý, P. Kolman, Z. Dostál, M. Antoš, M. Lošt’ák, and R. Chmelík, “Off-axis setup taking full advantage of incoherent illumination in coherence-controlled holographic microscope,” Opt. Express **21**, 14747–14762 (2013). [CrossRef] [PubMed]

9. F. Dubois, M. Novella Requena, C. Minetti, O. Monnom, and E. Istasse, “Partial spatial coherence effects in digital holographic microscopy with a laser source,” Appl. Opt. **43**, 1131–1139 (2004). [CrossRef] [PubMed]

_{i}≪ NA: where

*c*is a constant depending on the experimental setup,

**Q**

_{t}is a particular spatial frequency vector and

*S*is the Fourier transform of the source intensity distribution. In the referenced case, the source was a laser beam focused onto a rotating ground glass, so the intensity distribution was assumed to have the Gaussian shape. The refocusing distance then ranges from tens to hundreds of micrometers for objectives with NA = 0.25.

2. F. Dubois, L. Joannes, and J. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. **38**, 7085–7094 (1999). [CrossRef]

10. F. Dubois, O. Monnom, C. Yourassowsky, and J–C. Legros, “Border processing in digital holography by extension of the digital hologram and reduction of the higher spatial frequencies,” Appl. Opt. **41**, 2621–2626 (2002). [CrossRef] [PubMed]

11. F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express **14**, 5895–5908 (2006). [CrossRef] [PubMed]

7. T. Kozacki and R. Jóźwicki, “Digital reconstruction of a hologram recorded using partially coherent illumination,” Opt Commun **252**, 188–201 (2005). [CrossRef]

_{i}> NA/2) that are based on 3D coherent transfer function of the system. These propagators significantly differ from those for coherent sources.

## 2. Imaging in DHM

*w*(

*x*,

*y*,

*z*) be a value of the reconstructed

*complex amplitude*[13

13. P. Kolman and R. Chmelík, “Coherence-controlled holographic microscope,” Opt. Express **18**, 21990–22003 (2010). [CrossRef] [PubMed]

*x*,

*y*,

*z*] that we can express alternatively as

*w*

_{t}(

**q**

_{t};

*z*),

**q**

_{t}= (

*x*,

*y*). The image point is determined by the

*x*,

*y*coordinates of the optically conjugated point of the object plane [14

14. R. Chmelík, “Three-dimensional scalar imaging in high-aperture low-coherence interference and holographic microscopes,” J. Mod. Opt. **53**, 2673–2689 (2006). [CrossRef]

*z*refers to the axial position of the sample relative to the object plane. Numerical refocusing of reconstructed complex amplitudes is a process of recovering

*w*

_{t}(

**q**

_{t};

*z*

_{0}) from

*w*

_{t}(

**q**

_{t};

*z*

_{0}+ Δ

*z*), where we call Δ

*z*the

*refocusing distance*.

*W*

_{t}(

**Q**

_{t};

*z*) of the acquired 2D reconstructed complex amplitude

*w*

_{t}(

**q**

_{t};

*z*) in the form ie

*W*

_{t}is the Fourier transform of

*w*

_{t}along

*x*and

*y*for a given

*z*.

**Q**= (

*X*,

*Y*,

*Z*) denotes a spatial frequency vector and

**Q**

_{t}= (

*X*,

*Y*) then analogously denotes a transverse spatial frequency vector.

14. R. Chmelík, “Three-dimensional scalar imaging in high-aperture low-coherence interference and holographic microscopes,” J. Mod. Opt. **53**, 2673–2689 (2006). [CrossRef]

^{st}Born approximation of the scattering theory, if we assume that the DHM consists of aplanatic imaging system and if we assume that the extended-source is spatially incoherent. Then, according to [14

14. R. Chmelík, “Three-dimensional scalar imaging in high-aperture low-coherence interference and holographic microscopes,” J. Mod. Opt. **53**, 2673–2689 (2006). [CrossRef]

*w*(

**q**) =

*w*

_{t}(

**q**

_{t};

*z*) can be expressed [1

1. M. Týč and R. Chmelík, “Numerical refocusing of planar samples unlimited,” Proc. SPIE **7746**774620 (2010). [CrossRef]

*H*×

*T*, where

*H*is the system’s 3D coherent transfer function (CTF) and

*T*is the 3D Fourier transform of the sample’s 3D scattering potential

*t*(

**q**) [14

**53**, 2673–2689 (2006). [CrossRef]

*H*have been described in [14

**53**, 2673–2689 (2006). [CrossRef]

*H*with respect to the optical axis meaning that

*Q*

_{t1}=

*Q*

_{t2}⇒

*H*(

**Q**

_{t1},

*Z*) =

*H*(

**Q**

_{t2},

*Z*).

*t*can be expressed [1

1. M. Týč and R. Chmelík, “Numerical refocusing of planar samples unlimited,” Proc. SPIE **7746**774620 (2010). [CrossRef]

*t*(

**q**) =

*t*

_{t}(

**q**

_{t})

*δ*(

*z*), where

*δ*refers to the Dirac distribution. Then we have

*T*(

**Q**

_{t},

*Z*) =

*T*

_{t}(

**Q**

_{t}) for all

*Z*, where

*T*

_{t}is the 2D Fourier transform of

*t*

_{t}. If we combine (4) with (3) while using the fact that

*T*is constant with respect to its

*Z*-axis, we obtain simpler expression: where

*H*

_{t}(

**Q**

_{t};

*z*) can be regarded as a 2D CTF for a planar sample defocused by

*z*:

## 3. Refocusing theory

*z*to

*z*+Δ

*z*can be expressed in terms of frequency spectra using (5) for DHM setups that satisfy (4): where we call the generally complex-valued function

*P propagator*and We will discuss cases when

*H*

_{t}(

**Q**

_{t};

*z*) equals zero (or it is very low) in the section 3.2. We use the notation

*P*(

**Q**

_{t}; Δ

*z*) further in the text for cases when

*z*= 0 (this corresponds to the case when recovery of focused sample is in question) and the function

*H*the propagator is based upon clearly follows from the context.

**53**, 2673–2689 (2006). [CrossRef]

*P*in a polar form as

*P*= |

*P*| exp(iarg

*P*), we denote

*P*

_{A}= |

*P*| the

*amplitude*and

*P*= arg

_{φ}*P*the

*phase*of the propagator

*P*. Then filtering the image frequency spectrum

*W*

_{t}using only

*P*is called

_{φ}*phase deconvolution*and the use of both the amplitude and the phase is referred to as to

*complex deconvolution*[5

5. Y. Cotte, F. Toy, C. Arfire, S. Kou, D. Boss, I. Bergoënd, and C. Depeursinge, “Realistic 3D coherent transfer function inverse filtering of complex fields,” Biomed. Opt. Express **2**, 2216–2230 (2011). [CrossRef] [PubMed]

### 3.1. Coherent source

*K*=

*n/λ*is the reduced wave vector in the object space and

*λ*is the vacuum wavelength. The exact form of the term

*P*weakly depends on the sample type and can be looked up in [14

_{o}**53**, 2673–2689 (2006). [CrossRef]

*H*

_{t}| corresponding to this CTF does not change with Δ

*z*. Also note that it is virtually constant (except the cut-off) for low numerical apertures and reflective conductive surfaces, as it is shown in Fig. 2 (dashed line).

*H*

_{t}(

**Q**

_{t}; Δ

*z*) and using it in (8), we obtain the appropriate propagator:

*W*

_{t}, which is equal to zero for

*Q*

_{t}> NA/

*λ*nevertheless.

**Q**

_{t}and Δ

*z*.

### 3.2. Extended source

*H*(

**Q**) nor its corresponding

*H*

_{t}(

**Q**

_{t};

*z*) can be expressed in a closed-form. Considering the phase deconvolution,

*P*(

_{φ}**Q**

_{t}; Δ

*z*) is very sensitive to the size of the source. Behavior of some phase propagators that were obtained by numerical evaluation of their corresponding CTFs is shown in Fig. 3. Notice that in the TDHM case (Fig. 3(a)), the propagator’s phase variation decreases with the growing source size. In other words, phase refocusing is not relevant in TDHM with large extended-source (ie when NA

_{i}→ NA).

*P*to all spatial frequencies analogously to the point-source case. This time, |

*H*

_{t}| is not constant on its support as it can be observed in Fig. 2. In order to prevent spurious amplification of spatial frequencies in the frequency spectrum we deconvolve, we introduce measures similar to ones introduced in [16

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

5. Y. Cotte, F. Toy, C. Arfire, S. Kou, D. Boss, I. Bergoënd, and C. Depeursinge, “Realistic 3D coherent transfer function inverse filtering of complex fields,” Biomed. Opt. Express **2**, 2216–2230 (2011). [CrossRef] [PubMed]

*G*{·}: Selecting

*c*= 0 yields

*G*{

*H*

_{t}; 0} =

*H*

_{t}, conversely, we define

*G*{

*H*

_{t}; ∞} = exp(i arg

*H*

_{t}). As it is shown in Fig. 2, transmission for defocus

*z*= 10

*μ*m equals zero both for high and low spatial frequencies. The operator

*G*{·} prevents amplification of noise equally well when filtering the focused signal, where only the highest spatial frequencies are problematic, and the mentioned signal defocused by

*z*= 10

*μ*m, when possible division by zero becomes an issue for low spatial frequencies, too.

*c*depends on the noise, however the acquisition noise is not the main limiting factor. Artifacts present along the optical path cause disturbances that appear in reconstructed complex amplitudes. They are not band-limited and they are the main source of the image degradation when applying the deconvolution and they determine the minimum value of

*c*.

*z*, refocusing distance Δ

*z*the system’s CTF

*H*and the constant

*c*, we define the propagator as We now introduce the function

*T*

^{*}(

**Q**

_{t}; Δ

*z*) presenting the inversion of the imaging process of complex amplitudes defocused by Δ

*z*for extended-source: where

*H*

_{t}≡

*H*

_{t}(

**Q**

_{t}; Δ

*z*).

*t*

^{*}(

**q**

_{t}; Δ

*z*;

*c*

_{1}) is again the inverse 2D Fourier transform of

*T*

^{*}(

**Q**

_{t}; Δ

*z*;

*c*

_{1}). Since further in the text, we use either the complex deconvolution (14) with

*c*

_{1}= 0.4 or the phase deconvolution, in order to keep the notation simple, we define

## 4. Experiment

*amplitude image*to refer to the absolute value of reconstructed complex amplitude and we use

*intensity image*to refer to the classical microscope output signal.

### 4.1. Sample and methods

_{2}(Fig. 4). The optical setup referred to as RDHM is the one described in [17

17. R. Chmelík and Z. Harna, “Parallel-mode confocal microscope,” Opt. Eng. **38**, 1635–1639 (1999). [CrossRef]

*n*= 1.

**W**,

**P**and

**E**:

**W**Single reference observation was done on Nikon Eclipse L-150 classical reflected-light widefield microscope in white light with a 100 × / NA = NA_{i}= 0.95 objective.**P**z-stack (2*μ*m step) using the RDHM with a 10×/ NA = 0.25 objective, central*λ*= 550nm, filter FWHM 10nm using a quasi-point-source. Value of NA_{i}= 0.05 was determined by analysis of transmitted frequencies in the frequency spectra.**E**z-stack 1*μ*m step) using the RDHM with a 10 × / NA = 0.25 objective, central*λ*= 550nm, filter FWHM 10nm using an extended-source. The source image has filled condenser pupil so the theoretical NA_{i}= 0.25. Analysis of transmitted frequencies in the frequency spectrum of the defocused complex amplitude has revealed the effective NA_{i}we use for the processing is lower, NA_{i}= 0.15.

18. L. Lovicar, L. Kvasnica, and R. Chmelík, “Surface observation and measurement by means of digital holographic microscope with arbitrary degree of coherence,” Proc. SPIE , **7141**p. 71411S (2008). [CrossRef]

**|**Δ

*z*| < 8

*μ*m, so we analyze numerical refocusing for two extreme values of defocus ±8

*μ*m for both quasi-point-source and large extended-source illumination.

### 4.2. Sampling

_{i})/

*λ*≤ 2NA/

*λ*(see Fig. 1), it can be sampled. The minimal sampling distance is the inverse of the signal’s bandwidth

*B*[19].

*D*,

*N/B*=

*D*, where

*N*is the number of samples. We can express

*N*=

*B*×

*D*. The transversal frequency bandwidth of the reconstructed complex amplitude is twice the highest transmitted transversal spatial frequency, the longitudinal bandwidth

*B*

_{L}is shown in Fig. 1(b). So for the number of transversal and longitudinal samples,

*N*

_{t}and

*N*

_{L}respectively, we have where

*D*

_{t}refers to the length or width and

*D*

_{L}to the height of the area in which we want to be able to refocus. The conclusion is that in our experimental setup, the sufficiently sampled CTF 3D array contains 268×200×30 elements. Therefore it is feasible to perform calculations involving it on ordinary office PCs.

20. F. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE **66**, 51–83 (1978). [CrossRef]

*Q*

_{t}> 2NA/

*λ*.

### 4.3. Observations

*λ*/(NA + NA

_{i}), which corresponds to the Rayleigh resolution criterion in the sense that images of two points where one is located at the center of the circle and the other one outside of it are considered resolved [15].

*w*

_{t}alone is not an accurate representation of the sample because of the different transmission of spatial frequencies in the case of extended-source (shown in Fig. 2), corresponding

*w*

_{t}(

**q**

_{t}; 0)| and

*W*

_{t}of the two observations reveals that the frequency spectrum of the extended-source-illuminated amplitude image contains transmitted frequencies at least 1.6× higher than the other one (the theoretical increase is 2×, compare the difference in Fig. 5).

- Higher spatial frequencies transmitted from the sample may be present in the frequency spectra, but they are overpowered by noise due to the fact that their transmission is progressively reduced — see Fig. 2. This applies particularly to the defocused complex amplitude, where the signal-to-noise ratio is essentially lower.
- Certain degree of illumination misalignment is inevitable, rendering the effective source size is lower than the nominal one as it is described in [13].
13. P. Kolman and R. Chmelík, “Coherence-controlled holographic microscope,” Opt. Express

**18**, 21990–22003 (2010). [CrossRef] [PubMed]

### 4.4. Propagator measurement

*W*

_{t}(

**Q**

_{t}; 0)| ≠ 0,

*P*(

_{φ}**Q**

_{t}; Δ

*z*). The mapping is based on taking averages of the examined function over circles.

*P*(

_{φ}**Q**

_{t}) (we omit the parameter Δ

*z*now for simplicity), we focus on its domain, the space of spatial frequencies. We choose a set of spatial frequencies that lie on a ring of a given central radius

*Q*

_{t}and width equal to the distance between adjacent spatial frequencies. We denote this set as

*S*(

*Q*

_{t}). Finally, we define the set of all values of the function

*P*(

_{φ}**Q**

_{t}) on the ring as

*F*(

*P*,

_{φ}*Q*

_{t}) =

*P*(

_{φ}**Q**

_{t}) for all

**Q**

_{t}∈

*S*(

*Q*

_{t})

*P*(

_{φ}**Q**

_{t}) to be rotationally symmetric, we could expect elements of

*F*(

*P*,

_{φ}*Q*

_{t}) to be all the same. However, due to the noise or aberrations, it is not the case, so we define the function

*P*(

_{φ}′*Q*

_{t}) of the radial variable

*Q*

_{t}as the average of values of

*F*(

*P*,

_{φ}*Q*

_{t}).

*c*

_{1}in (14) as high as 0.4. Setting the right value of

*c*

_{1}is based on preventing accentuation of these disturbances, so they don’t distort their surroundings. Unfortunately, this type of noise can’t be easily separated neither from the image nor from the spectrum.

*P*

_{A}(

**Q**

_{t}; Δ

*z*) = |

*W*

_{t}(

**Q**

_{t}; Δ

*z*)/

*W*

_{t}(

**Q**

_{t}; 0)| differs significantly from the theoretical

*P*

_{A}= |

*H*

_{t}(

**Q**

_{t}; Δ

*z*)/

*H*

_{t}(

**Q**

_{t};0)|. As a result, we don’t use complex deconvolution (that involves it) for other purposes than to filter in-focus complex amplitudes.

*P*(

_{φ}**Q**

_{t}; 8

*μ*m) — ie the change of phase in the frequency spectra corresponding to the defocus of 8

*μ*m for the two illumination source sizes.

*P*(

_{φ}**Q**

_{t};8

*μ*m) for extended-source is less symmetric, but the area where it is well-defined (ie not noisy) spans beyond NA/

*λ*. Conversely, in the case of coherent illumination,

*P*(

_{φ}**Q**

_{t}; Δ

*z*) is well-defined only for

*Q*

_{t}< NA/

*λ*.

*P*(

_{φ}′*Q*

_{t}; 8

*μ*m) corresponding to Fig. 5 is visualized in Fig. 8(a) (solid lines). The experimental data are noisy, therefore 95% confidence intervals for the mean value of

*P*(

_{φ}′*Q*

_{t}) were added. Since the examined variable is phase, a statistical model that can work with directional data had to be used. Therefore we use statistical tools based on von Mises statistical distribution, which itself can be described as being very close to wrapped Gaussian distribution [21]. The 95% confidence intervals were obtained by fitting the von Mises distribution using [22

22. C. Agostinelli and U. Lund, R package circular: Circular Statistics (version 0.4-3), CA: Department of Environmental Sciences, Informatics and Statistics, Ca’ Foscari University, Venice, Italy. UL: Department of Statistics, California Polytechnic State University, San Luis Obispo, California, USA (2011).

*F*(

*P*,

_{φ}*Q*

_{t}) for each given

*Q*

_{t}.

*F*(

*P*,

_{φ}*Q*)| > 50 for

*Q*

_{t}> 0.16NA/

*λ*. In other words, the data sample for any spatial frequency greater than the 0.08 multiple of the theoretical highest transmitted spatial frequency has more than 50 elements and therefore is large enough to draw conclusions from it.

*P*(

_{φ}′*Q*

_{t}) is visualized in Fig. 8 (dashed lines). It has been computed using numerically evaluated CTF and it is described in more detail in section 3.2. Figure 8 shows that the theory is most inaccurate for low spatial frequencies. Otherwise the theoretically computed

*P*mostly falls in the confidence interval of the mean of experimental data. This indicates numerical refocusing using the theoretical values should yield good results.

_{φ}′### 4.5. Refocusing

**53**, 2673–2689 (2006). [CrossRef]

*w*

_{t}(

**q**

_{t}; 8

*μ*m) from the z-stack, and computed the properly refocused

*w*

_{t}(

**q**

_{t}; 8

*μ*m)|.

**P**, we obtain the expression 8 ≪ 40 that indicates that coherent propagation should yield acceptable results. Actual results presented in Fig. 9 confirm this. While the top view in Fig. 9 illustrates that it is possible recover the focused amplitude image of the sample even if the defocused amplitude image is severely distorted, the cross-section confirms that the recovered function is accurate in the sense that it differs very little from the reference (see Fig. 9(c) and 9(d)).

**E**as we did in the previous case, we obtain the expression 8 ≪ 14, which is not likely to hold. Figure 10(d) illustrates that phase refocusing using the coherent propagator expectedly produces wrong results, while the correctly derived phase propagator is able to invert the imaging process accurately.

## 5. Conclusion

_{i}

*>*NA/2) extended, spatially incoherent light sources, introducing the image inversion concept that is specific to the case of large extended sources. We have described the refocusing theory in terms of spatial frequencies spectra and developed methods how to verify the theory on experimental data.

*t*

^{*}(

**q**

_{t}; 0) representing the sample using solely the phase deconvolution. The refocused images retain high spatial frequencies that are not present in images acquired using the quasi-point-source.

## Acknowledgments

## References and links

1. | M. Týč and R. Chmelík, “Numerical refocusing of planar samples unlimited,” Proc. SPIE |

2. | F. Dubois, L. Joannes, and J. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. |

3. | L. Lovicar, J. Komrska, and R. Chmelík, “Quantitative-phase-contrast imaging of a two-level surface described as a 2D linear filtering process,” Opt. Express |

4. | R. Barer, “Interference microscopy and mass determination,” Nature (London) |

5. | Y. Cotte, F. Toy, C. Arfire, S. Kou, D. Boss, I. Bergoënd, and C. Depeursinge, “Realistic 3D coherent transfer function inverse filtering of complex fields,” Biomed. Opt. Express |

6. | F. Dubois, C. Yourassowsky, N. Callens, C. Minetti, and P. Queeckers, “Applications of digital holographic microscopes with partially spatial coherence sources,” JPCS, 139(IOP Publishing, 2008), p. 012027. |

7. | T. Kozacki and R. Jóźwicki, “Digital reconstruction of a hologram recorded using partially coherent illumination,” Opt Commun |

8. | T. Slabý, P. Kolman, Z. Dostál, M. Antoš, M. Lošt’ák, and R. Chmelík, “Off-axis setup taking full advantage of incoherent illumination in coherence-controlled holographic microscope,” Opt. Express |

9. | F. Dubois, M. Novella Requena, C. Minetti, O. Monnom, and E. Istasse, “Partial spatial coherence effects in digital holographic microscopy with a laser source,” Appl. Opt. |

10. | F. Dubois, O. Monnom, C. Yourassowsky, and J–C. Legros, “Border processing in digital holography by extension of the digital hologram and reduction of the higher spatial frequencies,” Appl. Opt. |

11. | F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express |

12. | O. Carriere, J. Hermand, and F. Dubois, “Underwater microorganisms observation with off-axis digital holography microscopy using partially coherent illumination,” in “ |

13. | P. Kolman and R. Chmelík, “Coherence-controlled holographic microscope,” Opt. Express |

14. | R. Chmelík, “Three-dimensional scalar imaging in high-aperture low-coherence interference and holographic microscopes,” J. Mod. Opt. |

15. | J. W. Goodman, |

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

17. | R. Chmelík and Z. Harna, “Parallel-mode confocal microscope,” Opt. Eng. |

18. | L. Lovicar, L. Kvasnica, and R. Chmelík, “Surface observation and measurement by means of digital holographic microscope with arbitrary degree of coherence,” Proc. SPIE , |

19. | Y. Geerts, M. Steyaert, and W. Sansen, |

20. | F. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE |

21. | E. Olson, “On computing the average orientation of vectors and lines,” in “Robotics and Automation (ICRA)”, Proc. IEEE, pp. 3869–3874, (2011). |

22. | C. Agostinelli and U. Lund, R package circular: Circular Statistics (version 0.4-3), CA: Department of Environmental Sciences, Informatics and Statistics, Ca’ Foscari University, Venice, Italy. UL: Department of Statistics, California Polytechnic State University, San Luis Obispo, California, USA (2011). |

**OCIS Codes**

(100.1830) Image processing : Deconvolution

(110.0180) Imaging systems : Microscopy

(110.4980) Imaging systems : Partial coherence in imaging

(170.6900) Medical optics and biotechnology : Three-dimensional microscopy

(090.1995) Holography : Digital holography

**ToC Category:**

Image Processing

**History**

Original Manuscript: August 13, 2013

Revised Manuscript: October 18, 2013

Manuscript Accepted: October 23, 2013

Published: November 11, 2013

**Virtual Issues**

Vol. 9, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Matěj Týč, Lukáš Kvasnica, Michala Slabá, and Radim Chmelík, "Numerical refocusing in digital holographic microscopy with extended-sources illumination," Opt. Express **21**, 28258-28271 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-28258

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### References

- M. Týč and R. Chmelík, “Numerical refocusing of planar samples unlimited,” Proc. SPIE7746774620 (2010). [CrossRef]
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