1. Introduction

Digital holographic microscopy (DHM) is a powerful, non-invasive light-microscopy technique that enables retrieval of not only the amplitude of light scattered by the observed sample, but also of its phase. The detected phase shift has physical interpretation: in reflected-light DHM (RDHM), it is proportional to the displacement of the observed surface [3], and in transmitted-light DHM (TDHM) it is, to a large extent, proportional to dry-mass density distribution of observed cells [4].

Apart from having this significance, it provides us with more complete knowledge of the light wave scattered by a sample, thus allowing application of advanced methods of image processing. These methods can, for instance, compensate optical aberrations. The aberrations compensation through inverse filtering has been successfully applied to highly aberrated images of samples acquired using objectives with high numerical aperture (NA) [5]. Another of these methods is numerical refocusing (see section 2 for the definition).

Numerical refocusing of reconstructed complex amplitudes acquired using a quasi-point, monochromatic (coherent) source has been demonstrated [2]. Additionally, limits for refocusing were derived for cases when images acquired with an extended-source are refocused using formulas that were derived for point-source illumination. According to [2] (after transformation of variables), the refocusing is accurate if the refocusing distance Δz satisfies

The benefit of the extended-source illumination is twofold: it removes high-coherence effects which give rise to speckles that degrade the image [6], [7] and it also increases the transversal resolution [8]. The image acquired using extended-source illumination has its frequency spectrum (see (3) for the definition and (5) for the meaning) filtered, which also affects numerical refocusing. The function that expresses the attenuation of the image frequency spectrum has been derived in [9] for the case when the source is assumed to be partially spatially coherent and NA_i ≪ NA:

Another limitation of numerical refocusing follows from the discrete nature of the data [2]. This limitation actually relates to whether the field of view is sufficiently large in the sense that surroundings of all points we want to refocus correctly is recorded, too. Typically, this limitation is not significant and it can be mitigated by using a border processing technique (see eg [10]).

Advanced applications of numerical refocusing are known and include focal plane detection [11] and 3D motion tracking of moving objects [12]. Numerical refocusing can be a step in the complex amplitude reconstruction process [7].

We have derived correct formulas (propagators) for numerical refocusing for the case of spatially incoherent large extended sources (ie NA_i > NA/2) that are based on 3D coherent transfer function of the system. These propagators significantly differ from those for coherent sources.

Our arrangement uses extended-source primarily to increase both the transversal and longitudinal resolution, the speckle reduction is regarded as a side-effect. Inseparable from that, the presence of out-of-focus sample features is suppressed even further, so refocusing distances are in our case lower.

We have experimentally proved that when using a large extended-source, propagators based on coherent source produce wrong results and our derived propagators produce accurate results. We present a comparison of the two propagators and evaluate their accuracy using statistical methods.

2. Imaging in DHM

Let w(x, y, z) be a value of the reconstructed complex amplitude [13] at point [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]. The coordinate 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; z₀) from w_t(q_t; z₀ + Δz), where we call Δz the refocusing distance.

We define the frequency spectrum W_t(Q_t; z) of the acquired 2D reconstructed complex amplitude w_t(q_t; z) in the form

Linear systems theory can be used to describe the image formation in DHM [14] if we restrict ourselves to the 1^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], the reconstructed complex amplitude w(q) = w_t(q_t; z) can be expressed [1] as

Means of calculating the H have been described in [14] for sources of arbitrary coherence state. Cross-sections of supports of CTFs for monochromatic point-source illumination and monochromatic large extended-source illumination are shown in Figs. 1(a) and 1(b) respectively. We assume rotational symmetry of H with respect to the optical axis meaning that Q_t1 = Q_t2 ⇒ H(Q_t1, Z) = H(Q_t2, Z).

 

Fig. 1 Cross-sections of support of CTFs for an objective with NA = 0.5 in the reflected-light (H_R) and transmitted-light mode (H_T). Calculated for (a) point-source illumination and (b) extended-source illumination with NA_i = NA, B_L denotes the longitudinal frequency bandwidth (see section 4.2). CTFs are rotationally symmetric with respect to the Z axis.

Download Full Size | PDF

We restrict ourselves to planar samples — typically reflective surfaces observed in RDHM, whose scattering potential t can be expressed [1] as 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:

3. Refocusing theory

Numerical refocusing for planar samples from z to z+Δz can be expressed in terms of frequency spectra using (5) for DHM setups that satisfy (4):

We can compute the 3D CTF according to [14], then the corresponding 2D CTFs according to (6) and the propagator according to (8). As a result, we obtain an invertible model of imaging for aberration-free imaging systems with extended non-monochromatic sources.

By expressing the complex function P in a polar form as P = |P| exp(iargP), we denote P_A = |P| the amplitude and P_φ = argP the phase of the propagator P. Then filtering the image frequency spectrum W_t using only P_φ is called phase deconvolution and the use of both the amplitude and the phase is referred to as to complex deconvolution [5].

Further in the text, we are going to focus only on imaging in RDHM, mainly because numerical refocusing of planar samples is more applicable in this type of microscopy.

3.1. Coherent source

When illumination by an axial monochromatic point-source is used and observation of reflective surfaces is assumed, the corresponding RDHM 3D CTF can be expressed in a closed-form:

This CTF support is shown in Fig. 1(a). The frequency transmission |H_t| corresponding to this CTF does not change with Δz. 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).

 

Fig. 2 Dependence of |H_t(0, Q_t; z)| (ie transmission of spatial frequencies) on defocus. Computed for RDHM, NA = NA_i = 0.25 (full lines) and NA_i = 0.05 (dashed lines). Narrow-band filter with central λ = 550nm, full width in the half maximum (FWHM) = 10 nm.

Download Full Size | PDF

Then, after calculating the respective H_t(Q_t; Δz) and using it in (8), we obtain the appropriate propagator:

We extend the definition of the propagator to $\text{exp}\left[2\pi \text{i}\mathrm{\Delta }z{\left(K^2-Q_{\text{t}}^2\right)}^{\frac{1}{2}}\right]$ for all spatial frequencies, because it is applied to W_t, which is equal to zero for Q_t > NA/λ nevertheless.

If we consider the inversion of the imaging process based on (5) and denote the result (ie the recovered frequency spectrum of the sample) by $T_{\text{f}}^{*}({\mathbf{Q}}_{\text{t}};\mathrm{\Delta }z)$, we get

This is another expression of the fact that phase numerical refocusing produces an accurate image of the sample in case of point-source illumination. Finally, if we take the 2D inverse Fourier transform of $T_{\text{f}}^{*}({\mathbf{Q}}_{\text{t}};\mathrm{\Delta }z)$, we obtain the corresponding sample’s scattering potential $t_{\text{f}}^{*}({\mathbf{q}}_{\text{t}};\mathrm{\Delta }z)$.

3.2. Extended source

In this section, we focus on the extended monochromatic source, when generally neither the 3D CTF 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; Δz) 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).

 

Fig. 3 Phase of the 2D CTF for defocus of 8 μm) for TDHM (a) and RDHM (b). Monochromatic extended-source illumination for selected values of NA_i. NA = 0.25, λ = 550nm. Shifted so that argH_t = 0 for Q_t = 0.

Download Full Size | PDF

We extend the definition of 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] or [5]. We introduce the operator G{·}:

The constant 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.

Given the defocus z, refocusing distance Δz the system’s CTF H and the constant c, we define the propagator as

The recovered sample’s scattering potential t^*(q_t; Δz; c₁) is again the inverse 2D Fourier transform of T^* (Q_t; Δz; c₁). Since further in the text, we use either the complex deconvolution (14) with c₁ = 0.4 or the phase deconvolution, in order to keep the notation simple, we define $t_{\text{p}}^{*}\left({\mathbf{q}}_{\text{t}};z\right)=t^{*}\left({\mathbf{q}}_{\text{t}};z;\infty \right)$ and $t_{\text{c}}^{*}\left({\mathbf{q}}_{\text{t}};z\right)=t^{*}\left({\mathbf{q}}_{\text{t}};z;0.4\right)$.

4. Experiment

From now on, we use the term 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

Experiments were conducted on a reflective amplitude planar sample — a golden film with small pits on SiO₂ (Fig. 4). The optical setup referred to as RDHM is the one described in [17]. No immersion was used for any observation, so the index of refraction n = 1.

 

Fig. 4 Visualized |w_t(q_t; 0)| of the field of view recorded by RDHM in the extended-source imaging setup (setup E, see the list in section 4.1). Edges of the field of view were apodized (see section 4.4). The stroke in the upper right corner is a diffraction grating artifact, the white rectangle points out the section of the sample shown further in the paper in more detail.

Download Full Size | PDF

Two fundamentally different devices were used for the following three series of observations marked as W, P and E:

Since the non-monochromatic RDHM with an extended-source has the optical sectioning property [18], the refocusing length has an upper bound — the reconstructed complex amplitude’s frequency spectrum gets progressively attenuated (see the relative transmission decrease in Fig. 2). We have found out that we can acquire data with acceptable signal-to-noise ratio for |Δz| < 8 μm, so we analyze numerical refocusing for two extreme values of defocus ±8 μm for both quasi-point-source and large extended-source illumination.

4.2. Sampling

Since the complex amplitude is a band-limited signal with the highest transmitted transverse spatial frequency equal to (NA + NA_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].

Given a coordinate axis and the length of field of view along that axis 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

Concerning the sample itself, spans beyond the lower edge of the field of view (see Fig. 4), so we have used the Hanning function [20] to apodize it. This measure suppresses spurious spatial frequencies in the sample’s frequency spectrum and its success is apparent in Fig. 5, since the visualized phase difference is pure noise for any frequency Q_t > 2NA/λ.

 

Fig. 5 The unwrapped phase difference P_φ (Q_t; 8 μm) between W_t(Q_t; 0) and W_t(Q_t; 8 μm). Both phase shifts are unwrapped in such way that the zero frequency has phase equal to 0 in both cases. (a) extended-source, setup E and (b) quasi-point-source P.

Download Full Size | PDF

4.3. Observations

Observations of the section of the in-focus sample (see Fig. 4) are shown in Fig. 6.

 

Fig. 6 Comparison of in-focus sample images (see the full sample in Fig. 4): (a) reference intensity image (setup W), (b) RDHM (setup E) filtered extended-source amplitude image $\left|t_{\text{c}}^{*}\left({\mathbf{q}}_{\text{t}};0\right)\right|$, (c) RDHM (setup E) extended-source amplitude image |w_t(q_t; 0)| and (d) RDHM (setup P) point-source amplitude image |w_t(q_t; 0)|. The black circle has radius of 1.22(NA + NA_i)/λ, see section 4.3. The colormap is linear and adjusted separately for each image to provide the highest contrast possible.

Download Full Size | PDF

The black circle • at the bottom left corner of images has radius of 1.22λ /(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].

Since the reconstructed complex amplitude 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 $t_{\text{c}}^{*}$ was calculated using (14). We can see the difference between the unprocessed |w_t(q_t; 0)| and $\left|t_{\text{c}}^{*}\left({\mathbf{q}}_{\text{t}};0\right)\right|$ in Figs. 6(c) and 6(b) respectively and Figs. 7(c) and 7(b) respectively. Visual comparison of Fig. 7 (a)–(c) entitles us to say that the imaging inversion (14) produces a higher-contrast and more accurate representation of the sample without adding any kind of artifacts.

 

Fig. 7 Comparison of close-ups corresponding to Fig. 6 (center of the upper edge): (a) reference intensity image (setup W), (b) RDHM (setup E) filtered extended-source amplitude image $\left|t_{\text{c}}^{*}\left({\mathbf{q}}_{\text{t}};0\right)\right|$, (c) RDHM (setup E) extended-source (raw) amplitude image |w_t(q_t; 0)|. The colormap is common for all three images. The cross-section at the bottom is taken along the path marked by the white bar in individual images above.

Download Full Size | PDF

Figure 6(d) shows the quasi-point-source amplitude image, that has clearly lower resolution than the extended-source one. The difference between Figs. 6(d) and 6(c) reveals the increase of resolution due to the extended-source illumination. Examining 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).

There are two main reasons that can explain why the observed resolution increase 1.6× is lower than the one predicted by the theory 2×:

4.4. Propagator measurement

Having a z-stack allows us to measure the propagator introduced in (8) directly using the frequency spectra and relate it to the model. Specifically, assuming |W_t(Q_t; 0)| ≠ 0,

The experimental data are discrete and affected by noise, so we introduce a mapping that maps function of two discrete variables to a function of one (radial) continuous variable. This facilitates analysis of data that are assumed to be rotationally symmetric, such as P_φ(Q_t; Δz). The mapping is based on taking averages of the examined function over circles.

Having a discrete function of two-dimensional variable such as the phase difference P_φ(Q_t) (we omit the parameter Δz 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)

Since we assume the function 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).

As it was explained below (12), artifacts present along the optical path disturb the frequency spectrum most. In our case, those disturbances were caused by dust particles present on a glass just in front of the camera’s CCD chip. Such disturbances are not band-limited in the way the signal is and they are the main reason we have to set c₁ in (14) as high as 0.4. Setting the right value of c₁ 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.

The experimentally measured P_A(Q_t; Δz) = |W_t(Q_t; Δz)/W_t(Q_t; 0)| differs significantly from the theoretical P_A = |H_t(Q_t; Δz)/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.

Analysis of the spectrum was not conducted on the part of the sample shown for example in Fig. 6, but on the whole sample — see Fig. 4. Figure 5 shows the measured P_φ(Q_t; 8 μm) — ie the change of phase in the frequency spectra corresponding to the defocus of 8 μm for the two illumination source sizes.

The measured P_φ(Q_t;8 μm) 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; Δz) is well-defined only for Q_t < NA/λ.

The experimental P_φ′(Q_t; 8 μm) 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] on the sample of values F(P_φ, Q_t) for each given Q_t.

 

Fig. 8 Phase difference P_φ′(Q_t) corresponding to defocus (a) 8 μm (see also Fig. 5) and (b) −8 μm. Solid lines correspond to experimental data, dashed lines to the theory and the bands show 95% confidence interval for the experimental data. The experimental data has been smoothed out to reduce the quantization error by applying convolution with a Gaussian kernel of σ = 0.05NA/λ.

Download Full Size | PDF

The number of values in the data sample |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.

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

4.5. Refocusing

It is clear that there are features on the sample that are imaged in utterly different way between the classical and holographic microscopes (such as the features by the center of the upper edge in Fig. 6 or the feature above the cross-section line in Fig. 7 that yield much higher contrast in DHM images). This is caused by the fact that opposed to classical microscopes, the DHM imaging process itself is coherent even if an incoherent illumination source is used [14]. Consequently, we are going to compare cross-sections of data acquired only by RDHM.

We have applied the filtering (14) to obtain $\left|t_{\text{c}}^{*}({\mathbf{q}}_{\text{t}};0)\right|$ from in-focus complex amplitudes first. Then we have taken the defocused complex amplitude w_t(q_t; 8 μm) from the z-stack, and computed the properly refocused $\left|t_{\text{p}}^{*}({\mathbf{q}}_{\text{t}};8\hspace{0.17em}\mu \text{m})\right|$ and the coherent approximation $\left|t_{\text{f}}^{*}({\mathbf{q}}_{\text{t}};8\hspace{0.17em}\mu \text{m})\right|$. Then we compare the corresponding RDHM reference $\left|t_{\text{c}}^{*}({\mathbf{q}}_{\text{t}};0)\right|$ with $\left|t_{\text{p}}^{*}({\mathbf{q}}_{\text{t}};8\hspace{0.17em}\mu \text{m})\right|$, $\left|t_{\text{f}}^{*}({\mathbf{q}}_{\text{t}};8\hspace{0.17em}\mu \text{m})\right|$ and the originally acquired defocused amplitude image |w_t(q_t; 8 μm)|.

If we apply the refocusing formula (1) to our quasi-point-source setup 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)).

 

Fig. 9 Refocusing in the setup P (quasi-monochromatic, quasi-point-source) amplitude images: (a) in-focus deconvolved $\left|t_{\text{c}}^{*}({\mathbf{q}}_{\text{t}};0)\right|$, (b) defocused by 8 μm |w_t(q_t; 8 μm)|, (c) phase refocused $\left|t_{\text{p}}^{*}({\mathbf{q}}_{\text{t}};8\hspace{0.17em}\mu \text{m})\right|$ using the correct propagator and (d) phase refocused $\left|t_{\text{f}}^{*}({\mathbf{q}}_{\text{t}};8\hspace{0.17em}\mu \text{m})\right|$ using the approximative (coherent) propagator. See Fig. 4 for the colorbar. The cross-section on the right is took along the path marked by the white bar in individual images on the left.

Download Full Size | PDF

Results for the extended-source are shown in Fig. 10. If we apply the refocusing formula (1) the setup 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.

 

Fig. 10 Refocusing in the setup E (quasi-monochromatic, extended-source) amplitude images: (a) in-focus deconvolved $\left|t_{\text{c}}^{*}({\mathbf{q}}_{\text{t}};0)\right|$, (b) defocused by 8 μm |w_t(q_t; 8 μm)|, (c) phase refocused $\left|t_{\text{p}}^{*}({\mathbf{q}}_{\text{t}};8\hspace{0.17em}\mu \text{m})\right|$ using the correct propagator and (d) phase refocused $\left|t_{\text{f}}^{*}({\mathbf{q}}_{\text{t}};8\hspace{0.17em}\mu \text{m})\right|$ using inappropriate (coherent) propagator. See Fig. 4 for the colorbar. The cross-section on the right is took along the path marked by the white bar in individual images on the left.

Download Full Size | PDF

Although the difference between the focused and defocused images is not as apparent as in the previous case, now there are much finer details present and also the contrast is higher (compare Fig. 9 with Fig. 10). Consequently, the cross-section plot in Fig. 10 (right) becomes much more important because it shows that that the pit the cross-section traverses is not localized very well in the defocused amplitude image, while it is accurately recovered by the refocusing process.

5. Conclusion

We have described the problem of numerical refocusing for large (NA_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.

We have conducted a refocusing experiment with two quasi-monochromatic RDHM arrangements — one with quasi-point-source and one with large extended-source. The processed data have shown that the theory corresponds to the experiment in terms of phase of spatial frequency spectra (see Fig. 8) as well as in terms of complex amplitudes.

Specifically, we have confirmed that the quasi-point-source setup can be accurately approximated by a point-source monochromatic setup (see Fig. 9) and the coherent propagation can be used for refocusing. We have demonstrated that this is not the case when the large extended-source is used (see Fig. 10(d)). We have confirmed that when the condition (1) doesn’t hold, usage of the coherent propagation for refocusing is inadequate. We have shown that the phase numerical refocusing can yield accurate results (see Fig. 10(c)), although it is only an approximation of the complex description of the phenomenon, formally similar to the coherent propagation. Moreover, this simplified approach described in section 4.5 is numerically stable since it doesn’t amplify the amplitude of spatial frequency spectra, which is often affected by disturbances that can disrupt the imaging process inversion.

The usage of extended source of illumination makes the setup more susceptible to various aberrations and the assumption of spatial incoherence of the extended-source may be incorrect. Despite these issues, we have demonstrated that it is possible to accurately recover the function 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.