1. Introduction

Quantitative phase imaging (QPI) has been used extensively for visualizing hidden features of biological and technical samples [1–4]. Moreover, QPI allows measurements on the sample without disturbing or destroying it. The advent of Interferometry and Holography provided the necessary tools to carry out QPI with high accuracy. These well-established techniques provide the integrated phase (averaged refractive index) of the light as it passes through the specimen. One implementation of these techniques is Digital Holographic Microscopy (DHM) [5–8] that is successfully applied in the areas of life sciences and technology giving remarkable results [7,8]. DHM is fast, accurate and with additional effort allows tracking in real time processes in life sciences and engineering. However, DHM and related Interferometry techniques have a number of weaknesses: they require a reference beam, high mechanical and environmental stability, and coherent illumination. Furthermore, the obtained phase needs to be unwrapped requiring an extra computational effort.

Single Beam Phase Retrieval Techniques (SBRTs) that seek to obtain the phase information from a sequence of through focus intensities [2,4,9] are an alternative solution to the phase problem. SBRTs based on iterative algorithms [1,9–11] recover the phase of the object by employing a backward-forward wave-propagation scheme of the captured intensities in a loop until certain numbers of iterations are accomplished or certain stagnation value is reached. However, iterative SBRTs may suffer from stagnation in local minima, giving a slow convergence and a solution that may not be guaranteed due to the non-convex nature of the problem [12]. Non-iterative SBRTs are referred as Deterministic Phase Retrieval Techniques (DPRTs) [13–15]. DPRTs based on the Transport of Intensity Equation (TIE) [2,3] have gained increased interest because they relate the transversal energy flux of the phase with axial energy flux of the intensity in the Fresnel region by means of a linear transformation [3,16]. Notably, TIE techniques give a unique solution [17], the retrieved phase does not need to be unwrapped [18] and it can be employed with partially coherent illumination [3,19,20]. Therefore, TIE based techniques have been applied in diverse areas ranging from optical microscopy to X-ray imaging [2,16,21]. However, major disadvantages of TIE methods are that they are strongly affected by Low Frequency Artifacts (LFAs) which are the effect of low frequency noise amplification [22]. The impact of these LFAs will depend on the plane separation between images [18]. Furthermore, TIE based algorithms have a poor efficiency for recovering high frequency components while the accuracy in the recovering of low frequency components is limited by the maximal plane separation [23]. Regularization Techniques (RTs) [21,24] have been proposed in order to suppress LFAs in the retrieved phase by TIE techniques. The aim of these RTs is to exclude low frequency components from the retrieved phase. Some RTs (for example the Tikhonov regularization [24]) are designed as High Pass Filters (HPFs) that can exclude low frequency components from the retrieved phase. Nevertheless, HPFs cannot distinguish between the low frequency components that belong to the LFAs and those that belong to the phase of the object [20]. Thus, RTs based on HPFs can therefore destroy important low frequency components corrupting the final phase reconstruction. More sophisticated RTs can be designed for suppressing LFAs. However, these techniques need a priori information of the object in order to distinguish between the low frequency components that belong to LFAs and those that belong to the phase of the object. In the particular case of the optical regime, it has been shown that the performance of TIE methods can be improved by choosing the proper plane separation strategy for capturing the intensities [15,22,25,26], and thus, the employment of RTs can be avoid. When selecting large plane separations, the effect of LFAs in the retrieved phase can be mitigated without employing RTs [18,26] but high frequency components of the phase cannot be recovered [18,27]. This problem can be overcome when capturing a large number of intensities obtained at equally spaced planes having small plane separations. This will mitigate LFAs while recovering high frequencies [22,23]. However, an efficient suppression of LFAs through this spacing technique does require a large amount of measurement planes with small plane separations. A different approach to this problem is an exponential plane selection strategy, which has been presented for the case of CTF based solvers [15]. That technique mitigates LFAs and recovers a wide frequency spectrum of the phase by employing a few captured intensities [10,15]. This strategy has been successfully implemented in the Gaussian Process regression TIE approach (GP-TIE) [23] giving high accuracy that outperforms current state of the art techniques. The exponential spacing strategy has been developed originally on the specific properties of CTF solvers [15], which did not consider the properties of TIE solvers. This is an important aspect, because the GP-TIE solver [23] seeks to find optimal estimation of the axial derivative. However, in [22], Martinez et al showed that the condition for an accurate estimation of the phase is different from the one for the axial derivative.

The goal of this paper is to introduce a new TIE technique that is able to recover the phase with minimal error when employing unequally spaced plane strategies. The principle of this technique is derived by exploiting the properties of the Classical TIE solver [18] with a set of three images. Initially, we study the frequency response of three plane TIE solver [3,18] with respect to LFAs, nonlinear error components, plane separation, and the Inverse Laplacian (IL). From this analysis, it is found that only a limited range of spatial frequencies can be retrieved accurately for a given plane separation. Thus, we show that the frequency components retrieved by the three plane TIE solver can be enlarged when combining the phase from a series of band pass filtered three plane TIE reconstructions at different defocused distances. It is shown that the noise sensitivity and accuracy of each TIE reconstruction depends on the size of the band pass filter. This method gives a robust and accurate solution for the phase retrieval problem over a wide frequency range. Moreover, we develop a new separation strategy that equalizes the noise sensitivity of all band pass filtered TIE reconstructions. This allows controlling the final error of the retrieved phase without a priori information of the phase distribution as it is the case for other optimum TIE techniques [21–23,25].

2. Frequency response of a three plane TIE solver

Consider a paraxial wave field [3,18] where the intensity variations along z are calculated from two images placed at ± z and are related to the transversal energy flux of the phase as

Figure 1(a) shows the IPTF_TIE for three defocus distances; the positions of the COFs for ε = 0.01 are denoted by the vertical dash lines. Figure 1(b) shows the RMSE of the TIE solver at a given spatial frequency that is simulated using a single frequency sine grating φ_j = A₀sin(2πx/T_j) as a pure phase object, where T_j is the grating period and A₀ = 0.5. It should be noted that for these simulation, the regularization parameter is β = 0, the zero frequency sample is excluded, and the DC term is removed from the phase error in order to estimate the RMSE. The simulations in Fig. 1(b) are carried out using Δx = 4μm, λ = 0.633μm, N_p = 256, SNR = 40dB (σ = 0.01), and various grating periods T_j. The two intensities have a distance z to the either side of the in-focus plane. The selected separations for these simulations were z₁ = 10μm (green line), z₂ = 100μm (red line) and z₃ = 1000μm (blue line). In Fig. 1(b), the solid lines represent the analytical RMSE (denoted as RMSE^T, using Eq. (7)), while the lines labeled with RMSE^S correspond to the RMSE obtained by simulation. These results were obtained by making an average of 25 simulations. The analytical result for Eq. (11) is shown in Fig. 1(b) with the horizontal dash lines and the positions of the COFs that are denoted by the vertical dash lines. Notably, the analytical predictions of Eq. (7) (solid line) and Eq. (11) follow accurately the simulation results (RMSE^S) for spatial frequencies that do not exceed ζ. Hence, larger defocus distances give a smaller RMSE but allow recovering reliably less frequency components.

 

Fig. 1 a) IPTF_TIE for three different defocus distances and b) RMSE frequency response of the retrieved phases. The vertical lines are placed in its corresponding COF. The lines labeled with the legend RMSE^S correspond to the RMSE of the retrieved phase through simulations, and the theoretical RMSE (RMSE^T) are denoted with the solid lines. Moreover, the analytical results of Eq. (11) and the position of the COFs are plotted with the horizontal and the vertical dashed lines respectively. The RMSE^S presented in this figure are the average of 25 simulations.

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3. Improving the 3 plane TIE solution: The Multi-Filter TIE algorithm

From Fig. 1(b), it can be seen that the frequency components of the phase below the COF can be recovered with the smallest error possible. Following this idea, the performance of the three plane TIE solver can be improved, if for each defocus distance we are able to select and extract the range of frequencies for those where the phase retrieval is done accurately. For instance, for a large defocus distance ± z₃ (see Fig. 1 comparing z₁ and z₂), the retrieved phase with Eq. (2) will have the low frequency well resolved until the COF ζ(z₃); this information can be extracted when suppressing higher frequencies with a LPF. By adding two more planes with a smaller separation distance ± z₂, the frequencies in the radial range ζ(z₃)<f ≤ζ(z₂) could be well-resolved. The set of recovered frequencies in this region can be extracted using a Band Pass Filter (BPF) with COFs (ζ(z₃), ζ(z₂)), and can be added to the other recovered frequencies so that the range of frequencies 0 < f ≤ ζ(z₃) can be recovered accurately. The same process can be applied to the shortest defocus distance z₁, and thus, the set of frequencies in the region (ζ(z₂), f_max), where f_max = 1/(2Δx), could be extracted by a High Pass Filter (HPF) and added to the other data. Thus, the initial set of recovered frequencies (0, ζ(z₃)) has been enlarged to the range (0, f_max). This methodology can be generalized to the case of several distances ± z₁,…, ± z_N with z₁< z₂<…< z_N-1< z_N and several COFs given as

 

Fig. 2 Flow chart of the MF-TIE based solver.

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3.1 Error response of the MF-TIE

Given that the filtered phases are not correlated, we can estimate the corresponding RMSE for each filtered phase. This RMSE will be denoted as RMSE_i and can be calculated using similar considerations leading to Eq. (7), but in contrary to the case of Eq. (7) it is necessary to include the effect of the filter, as

4. Plane selection strategy for the MF-TIE

As it was discussed in Section 2, the selection of the plane separation strategy is crucial for SBPRTs because this allows mitigate the effects of LFAs due to noise and determines the range of spatial frequencies to be recovered [15,23,26]. Recently reported, the exponential plane selection strategy [15,23] allows such phase reconstructions with small error. The defocus distances in that strategy are calculated as

By using Eq. (26) the total error of the reconstructed phase can be estimated without specific object information. It should further be noted, that in contrary to other separation strategies (e.g. the exponential spacing [15,23]), the noise sensitive spacing strategy allows to include the information of an acceptable level of RMSE in the plane selection methodology. This important finding has not been reported by previous works [15,22,23]. The result given in Eq. (26) allows us to retrieve the phase of the object with a specific error for a fixed number of planes provided that a maximum value for ε is not exceeded. The maximum value for ε is object dependent and is related to the validity of TIE, i.e. cases where non-linear error components are negligible. This property is visualized by simulations. Consider a pure phase object in form of a Shepp-Logan phantom with a phase difference of π/4, π/8 and π/16 employing 11 intensities (N = 5) and different values of RMSE_T. These simulation were carried out for different levels of noise given as SNR = 30,40 and 50dB.

The results of these simulations are shown in Fig. 3. The solid line in this plot is the noise boundary limit given by Eq. (26) (no nonlinear error components). This line can be obtained by plotting the corresponding values of ε and RMSE_T for which Eq. (25) generates only five defocus distances. The red, green and black circles dots represent the actual RMSE_T of the retrieved phase (obtained by simulation) when employing the MF-TIE for the corresponding values of ε and different phase amplitudes (π/4, π/8 and π/16 respectively). When comparing the solid line and the circles, one can observe that the simulations results follow the noise boundary limit until certain value ε = ε_MAX is reached. In other words: the nonlinear error components can be neglected for ε < ε_MAX. The value of this ε_MAX depends of the noise and phase amplitude. The errors present for ε>ε_MAX result from the fact that at larger ε the conditions discussed in Sections 2 and 3 are not fulfilled, and thus, nonlinear error components are not suppressed. In consequence, the retrieved phase with the MF-TIE will be inaccurate. From Fig. 3, we can observe that the smaller is the phase amplitude of the object the larger is the range of ε where the error of the retrieved phase is similar to the noise boundary limit. In Fig. 3(a), ε_MAX≈0.035, 0.25, and 0.47 for the phase amplitudes of π/4, π/8 and π/16 respectively. Notably, for the case of weak phase objects the MF-TIE allows large values of ε without introduce non-linear errors. For the case of Fig. 3(b), we have that ε_MAX≈0.006, 0.09, and 0.17 for the phase amplitudes of π/4, π/8 and π/16 respectively. In these simulations, the value of ε_MAX has decreased at least in a factor of five, which is expected. The reason of these results comes from the fact that the noise amplitude has been decreased, and thus, the non-linear component error become more dominant (relative to noise) as ε increased. In Fig. 3(c) it is observed that ε_MAX≈0.002, 0.006, and 0.08 for π/4, π/8 and π/16 respectively. Similarly to Fig. 3(b), the value of ε_MAX are lower than in Fig. 3(b), because the noise in the system lower. Thus, the nonlinear error becomes more dominant in the RMSE_T at larger values of ε (see for example the red circles in Fig. 3).

 

Fig. 3 RMSET vs ε using several phase difference amplitudes and different levels of noise. The red, black and green circles correspond to the phase difference amplitudes of π/4, π/8 and π/16 respectively. a) SNR = 30dB. b) SNR = 40dB c) SNR = 50dB. The solid line is the noise boundary limit given by Eq. (26). The circles are the actual RMSET when employing the MF-TIE. The triangle, pentagon, star and square markers indicate the number of planes necessary to retrieve the phase with a constant value of the RMSET (dash line).

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Furthermore, Fig. 3 shows four additional additional markers (triangle, pentagon, star and square) for which the RMSE_T has been fixed to RMSE_T = 0.021 (Fig. 3(a)), 0.011 (Fig. 3(b)), and 0.008rad (Fig. 3(c)) for the level of noise SNR = 30,40 and 50dB respectively. The fixed value for the RMSE_T is depicted by the dash-line in each picture. Table 1 shows the numerical conditions for which the markers were obtained in Fig. 3 by employing Eq. (20) and (25). Notably, for generating Table 1 no phase information about the object is necessary. Moreover, in Table 1 can be observed that the number of planes increases as ε decreases. This table also highlights that a high noise robustness can be achieved using only fewer measurement planes by increasing the value of ε. Similarly, the smaller the value for ε, the higher the ability to reconstruct stronger phase objects. Hence, MF-TIE together with the noise sensitivity separation strategy described here allows controlling the mount of error down to an acceptable level, i.e. RMSE_T of the retrieved the phase.

Table 1. Defocus distances for various SNR, ε and RMSE_T

View Table

5. Numerical experiments and comparison to other TIE techniques

The performance and frequency response of the MF-TIE solver is investigated by simulation for the case of pure sine gratings having various periods and defocus distances. The accuracy in the reconstructed phase with the MF-TIE is compared to the results obtained by the GP-TIE [23].

Figure 4(a) shows the obtained simulation results for both the MF-TIE solver (solid line) and the GP-TIE (dash line) when using the exponential separation strategy. For this simulation we have employed 11 planes with exponential spacing given by z_i = ± 6; ± 23; ± 89; ± 350; ± 1375μm, and various levels of noise SNR = 25,30, and 35dB. Figure 4(b) shows the effectiveness of MF-TIE and GP-TIE solvers under same simulation conditions as in Fig. 4(a), but using the plane selection strategy defined in Eq. (25). The plane separation for this simulation is given as z_i = ± 6; ± 10; ± 24, ± 98, ± 1375μm. For a fair comparison, we carried out these simulations using z₁ = ± 6μm (same maximum spatial frequency) and z₄ = ± 1375μm (same low frequency artifacts) and ε = 0.01. Note that z_2,3,4 are smaller than their GOMF counterpart. As it was the case with previous simulations, the zero frequency sample is excluded from the calculations and the DC term is removed from the phase error in order to estimate the RMSE_T. All simulation have been carried out for β = 0, because the unequal separation strategies mitigate LFAs without using regularization parameters [15,23]. When comparing Fig. 4(a) and Fig. 4(b), we see that despite the high accuracy that can be obtained by the GP-TIE algorithm, the MF-TIE solver gives, under same conditions, a lower RMSE (by a factor of two). Notably, the MF-TIE solver reported here, gives the more accurate result compared to the GP-TIE technique. The performance loss of the GP-TIE solver [23] originates from the fact that this approach only seeks an optimal estimation of the axial derivative; however, as Martinez et at shown in [22], this not imply a retrieved phase with minimal error. The MF-TIE algorithm suppresses LFAs through the BPFs but without affecting the retrieved frequencies of the object. A solver can adapt to this TIE specific behavior by either choosing appropriate measurement planes (as Martinez et at shown in [22]) or by adapting the corner frequencies of the filters (as it is the case here).

 

Fig. 4 RMSE vs for various grating frequencies when using different plane selection strategies for the GP-TIE and MF-TIE for several levels of noise. a) Plane separation strategy defined in Eq. (17). b) Separation strategy defined in Eq. (25). The vertical lines are placed on its corresponding COF. The final results of these plots are the averages of 15 simulations.

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In the previous simulation, we have tested the response error of the GP-TIE and the MF-TIE for a wide range of frequencies showing an optimal performance for the MF-TIE by using the exponential or equal noise sensitivity plane selection strategy. For the following simulation, we want to compare the error response of the MF-TIE with two different TIE solvers: the classical TIE solver (CL-TIE) [18] with the optimized equally spaced plane separation presented in reference [22], and the GP-TIE solver with the exponential spacing [23]. For the exponential spacing without optimization, we have selected Δz = 10μm (in order to recover high frequency components for low noise conditions) and g₀ = 5 (to suppress LFAs in high noise conditions). Moreover, we carry out an additional simulation to show that the exponential spacing can be optimized. The optimization regards the choice of the shortest and largest plane separation. This is not trivial when one considers the effect of noise and non-linear error components. Previous approaches [15,18,23], choose z₁ according to the diffraction limit, and a value of z_N that suppresses visible effects of LFAs in the reconstructed phase. In this work, we show that the exponential plane selection strategy can be optimized when using the values for z₁ and z_N that one would obtain for the plane selection strategy with equal noise sensitivity of this work, i.e. Equations (20) and (25). We select the distance z₁ and z_N from this strategy (Eq. (20)) in order to generate the exponential spacing, and thus, g₀ = (z_N /z₁)^1/(N-1) [15]. We consider the same phase object as in Fig. 3 with a maximum phase difference of π/8 and 11 planes. The noise level in this simulation is ranging from 20 to 50dB. It should be noted that for each level of noise the value of ε and the defocus distances will change (see Table 1). A further comparison is made with optimum plane selection strategies for equidistant measurement planes. Notably, the optimal separation dz for equally spaced planes depends on the level of noise. In order to estimate the corresponding plane separation, we applied Eq. (17) from reference [22].

The simulation results are shown in Fig. 5. As expected, the RMSE_T of the equidistant separation strategy is large for all levels of noise. Notably, the exponential spacing does not give a minimal RMSE_T for strong noise conditions (SNR<34dB). In fact, the performance of this technique is a slightly better than the optimized even separation strategy for low noise conditions (SNR>34dB). However, when the exponential spacing is optimized with the help of the noise sensitivity formula (Eq. (25)), it can be observed that the RMSE_T in the retrieved phase is improved in at least a factor of four in relation to the green markers. Thus, the exponential spacing and the GP-TIE can benefit from the spacing strategy developed in this contribution. Finally, Fig. 5 shows that the technique with minimal error is given by the MF-TIE solver combined with the plane selection strategy of this work (Eq. (25)). For this solver, it is observed that the RMSE_T stagnates for SNR>40dB; a similar effect can be seen in Fig. 3 due to the presence of nonlinear components errors.

 

Fig. 5 RMSE of the retrieved phase for various levels of noise. Three different TIE solvers have been employed: the Classical TIE solver (CL-TIE) [27] with optimal equally spaced planes (triangles) [22], the GP-TIE solver [23] with exponential spacing (cross) and optimized exponential spacing (asterisk), and the MF-TIE with equal noise sensitivity strategy of Eq. (25) (circle). These simulations were carried out with eleven planes. Each marker is the average of 20 simulations.

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Figure 6 shows a further example of the retrieved phase for the case when the SNR = 40dB and eleven planes. The defocus distances were selected according to the values showed in Table 1. Figure 6(a), (b), and (c), shows the case of the classical TIE (CL-TIE) [18], GP-TIE [23], and the MF-TIE solver, respectively. Similar to previous simulations β = 0 in order to visualize the ability of the techniques to suppress LFAs. The retrieved phase with CL-TIE (Fig. 6a) is not affected by LFAs due to the large plane separation; however, all the sharp features of the phase have been lost. The retrieved phase in Fig. 6(a) has been obtained using the algorithm in [21,31]. As expected, GP-TIE (Fig. 6(b)) provides a sharp phase reconstruction but this result is affected by LFAs that are result of no considering the effect of the IL. Finally, the result obtained by the MF-TIE is not affected by LFAs. This is because the filtering process in the MF-TIE excludes all the noisy and redundant information from other measured planes, which is not the case for the GP-TIE approach. Moreover, the MF-TIE provides the best quantitative results having an RMSE_T that is four times lower than the RMSE_T of the GP-TIE and five times lower than the RMSE_T of CL-TIE. When comparing the corresponding RMSE_T of Table 1 one can see that these measurements planes have been chosen in order to obtain a RMSE_T = 0.01rad; when comparing this value with the error of the retrieved phase of Fig. 4(c) it can be seen that this anticipated results could be met.

 

Fig. 6 Phase retrieval for three different solvers and employing the unequal phase separation defined by Eq. (25)

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5. Conclusions and discussion

In this work, a so called MF-TIE solver is reported that uses a series of low-pass, band-pass and high-pass filters in order to extract the particular object frequencies from a series of defocused intensities. The idea behind this approach is to deliberately exclude spatial frequencies from the measurement data if they do not correspond to the plane that provides minimal error at those frequencies. In this way, MF-TIE obtains phase reconstructions with higher noise robustness than the GP-TIE solver (that is currently the most accurate TIE technique but does not exclude less accurate data sets). The success of this method can be shown for the case of equidistant or non-equidistant measurement planes. Recent efforts in TIE have shown the advantageous use of non-equidistant plane selection strategies, i.e. exponential spacing [15,23]. However, exponential strategies have been originally developed for CTF based solvers and do not account for the properties of the TIE solver, i.e. the noise amplification of the IL. In particular, many TIE techniques as e.g. GP-TIE solver [23] seek to minimize the error in the axial derivative; however, in [22] Martinez et al shown that this not imply a minimal error in the retrieved phase. Hence, for the case of non-equidistant measurement planes there is room for improvement. Motivated by that, this paper reports a novel non-equidistant plane selection strategy that takes into account the properties of the TIE solver and thereby improves further the phase reconstruction. The idea behind this is based on the criterion to equalize the noise contribution to the RMSE_T of all measurement planes by appropriate selection of the measurement planes and the corner frequencies of the filters. Notably, the developed plane separation technique provides analytical expressions for the smallest and largest plane separation giving high LFA suppression with no need to employ RTs. The choice of the smallest and largest measurement planes for various levels of noise are not explicitly determined by exponential schemes [15,23]. Interestingly, as shown in Fig. 4 and 5, the exponential plane selection strategies may benefit from this information in terms of reconstruction accuracy. In conclusion, this works reports both a new TIE solver and a novel TIE plane selection methodology for high accuracy phase reconstructions. The so-called MF-TIE solver uses series of filter in order to extract the measurement data that gives higher accuracy, where optimal corner frequencies are chosen according to Eq. (6). The reported optimum non-equidistant plane selection strategy allows increasing further the accuracy without employing RTs. This strategy provides a minimum number of planes for a given level of noise and acceptable level of RMSE_T.

Appendix A

The MF-TIE algorithm of this work assumed the object to be a pure phase object, so that the in-focus intensity I₀≈constant [32]. In practice, this may be a good approximation [28,33], but for absorbing objects the use of Eq. (2) is not adequate. For these cases, one has to introduce the potential ∇ψ=I₀∇φ [34] into Eq. (1), which yields

(27)$$\nabla \cdot \nabla \psi =-k\frac{I(r_{\perp },z)-I(r_{\perp },-z)}{2z}.$$

When solving Eq. (27), the solution of Eq. (1) can be found as [35,36]

(28)$$\phi ={\nabla }^{-2}\left\{\nabla \cdot [I^{-1}\nabla \psi ]\right\},$$

where notably, the solution of Eq. (27) exists when ∇I₀x∇φ = 0 [32,36], which is fulfilled for φ = constant, I₀≈constant, or if the gradients of I₀ and φ are parallel [32]. The last case occurs when the transmitted light is in function on the thickness or density of the sample [32,36]. Hence, the accuracy of the solution of Eq. (28) is given by the degree of correlation between the intensity and the phase distributions.

In order to retrieve the phase with the MF-TIE for the absorbing case, the solution presented by Eq. (27) and (28) must be adopted as well. Firstly, we apply the same COFs for the BPFs as for the case of no absorption for calculating the filtered phases ψ_i. In the next step, the potential function ψ is estimated using a superposition of filtered functions ψ_i similar to Eq. (14), as

(29)$$\psi ={\displaystyle \sum _{i=1}^N{\psi }_i}.$$

By substituting Eq. (29) in Eq. (28), the phase can be estimated as

(30)$$\phi ={\displaystyle \sum _{i=1}^N{\phi }_i}={\displaystyle \sum _{i=1}^N{\nabla }^{-2}\left\{\nabla \cdot [I^{-1}\nabla {\psi }_i]\right\}}$$

For the absorbing case, the diffracted intensities for estimating ψ_i will contain information about the phase and absorption. In consequence, the BPFs will not be able to filter out the nonlinear errors from the retrieved phase. A practical solution to this problem is to employ the iterative TIE schemes proposed in references [35,36]. Such techniques allow removing the TIE solver errors from the retrieved phase caused by the absorption distribution. Hence, when applying the iterative technique to the retrieved phase by the MF-TIE, we will remove successfully the errors caused by absorption without the necessity of recalculating the COFs.

The applicability of this approach is tested by simulation for phase object of Fig. 6 but having an absorption distribution μ as it is shown in Fig 7(a). The maximum value for μ is set to μ_MAX=0.5. Simulations have been carried out using the 11 measurement planes of Table 1 for SNR=50dB. The reconstructed phase is shown in Figs. 7(b) and 7(c) for the MF-TIE and GP-TIE algorithm, respectively. Fig. 7(b) indicates that MF-TIE is less sensitive to absorbing samples than the GP-TIE algorithm. Given that the MF-TIE is based in the three plane TIE approach, and not by the weak absorption approximation (WAA) [23], the results showed in Fig. 7(b) are not strongly affected by the absorption of the object. Notably, the loss of accuracy in GP-TIE is because the fitting model only considers small variations due to the absorption of the object. Nevertheless, when having a closer look, the effect absorption does not remain unnoticed for the MF-TIE solver, and similar error may also occur for other TIE techniques [32,35–37]. Figure 8(a) shows the effect of the absorption in the RMSE, where the maximum absorption of Fig. 7(a) is varied from μ=0 (pure phase object) to μ=1 (strong absorption). Despite the resulting error, recently reported technique overcomes this problem within an iterative post-processing scheme [35,36]. The pseudo code of the iterative TIE approach is: (i) Retrieve the phase φ₀ with the MF-TIE; (ii)Estimate the axial intensity derivative ∂_zI_E from the experimental set of images with the GP Regression [23]; (iii)Initialize φ_i= φ₀.; (iv)For i=1 to n do: (iv.a) Calculate numerically the axial derivative ∂_zI_N with Eq. (29) of reference [23] when having I₀ and φ_i as inputs; (iv.b) Calculate the residual Δ∂_zI=∂_zI_N-∂_zI_E ; (iv.c)Use I₀, and Δ∂_zI as inputs to Eqs. (27) and (28) to obtain the phase Δφ_i; (iv.d) Correct initial phase estimate as φ_i+1= φ_i+ Δφ_{i .}

 

Fig. 7 Phase retrieval in case of absorption for SNR = 50dB. a) Absorption distribution μ. b) Retrieved phase with MF-TIE. c) Retrieved phase with GP-TIE.

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Fig. 8 RMSE of the retrieved phase for various values of μ. a) RMSE of the retrieved phase when the in-focus intensity is not constant. b) RMSE of the retrieved phase after employing the iterative TIE algorithm showed of reference [36]. These simulations were made with SNR = 60dB, n = 5. Each marker is the average of 25 simulations.

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Figure 8(b) shows the RMSE for the retrieved phase when applying the iterative scheme proposed by Zuo et al in [36]. In that simulations only five iterations have been employed (n=5) and each marker is the average of 25 simulations. Fig. 8(b) indicates that the previous iterative procedure reduces the RMSET for both the MF-TIE and GP-TIE solver, giving an increased accuracy by at least a factor of three. It is noticed that MF-TIE converges faster to the exact solution than the GP-TIE. The remaining error at larger values of μ may be reduced for both solvers by increasing the number of iterations; this however requires more computational effort.

Acknowledgments

The research leading to the described results are realized within program TEAM/2011-7/7 of Foundation for Polish Science, co-financed from European Funds of Regional Development. We would like to acknowledge the support of the statutory funds of Warsaw University of Technology. K. Falaggis also acknowledges the partial support of this work by the Outstanding Young Scientists Stipend Award 564/2014 of the Polish Ministry of Science and Higher Education. We also thank the authors of reference [23] for providing the Matlab implementation of the GP TIE algorithm.

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