Research Article | Open Access
Mansi Butola, Sunaina Rajora, Kedar Khare, "Robust Phase Retrieval with Complexity-Guidance for Coherent X-Ray Imaging", Intelligent Computing, vol. 2022, Article ID 9819716, 12 pages, 2022. https://doi.org/10.34133/2022/9819716
Robust Phase Retrieval with Complexity-Guidance for Coherent X-Ray Imaging
Reconstruction of a stable and reliable solution from noisy and incomplete Fourier intensity data is a challenging problem for iterative phase retrieval algorithms. The typical methodology employed in the coherent X-ray imaging (CXI) literature involves thousands of iterations of well-known phase retrieval algorithms, e.g., hybrid input-output (HIO) or relaxed averaged alternating reflections (RAAR), that are concluded with a smaller number of error reduction (ER) iterations. Since the single run of this methodology may not provide a reliable solution, hundreds of trial solutions are first obtained by initializing the phase retrieval algorithm with independent random guesses. The resulting trial solutions are then averaged with appropriate phase adjustment, and resolution of the averaged reconstruction is assessed by plotting the phase retrieval transfer function (PRTF). In this work, we examine this commonly used RAAR-ER methodology from the perspective of the complexity parameter introduced by us in recent years. It is observed that the single run of the RAAR-ER algorithm provides a solution with undesirable grainy artifacts that persist to some extent even after averaging the multiple trial solutions. The grainy features are spurious in the sense that they are smaller in size compared to the resolution predicted by the PRTF curve. This inconsistency can be addressed by a novel methodology that we refer to as complexity-guided RAAR (CG-RAAR). The methodology is demonstrated with simulations and experimental data sets from the CXIDB database. In addition to providing consistent solution, CG-RAAR is also observed to require reduced number of independent trials for averaging.
Imaging of micro- and nano-sized objects remains one of the prime interests in structural biology, material science, chemistry, medical science, etc. With the rapid advancements in coherent diffraction imaging hardware, particularly in X-ray free electron lasers (XFELs) , it is possible to perform single particle imaging (SPI) [2, 3]. In order to determine the structure of objects of interest, the acquisition of Fourier diffraction data is a challenging task that is being carried out by a number of dedicated instruments [4–6]. On the other hand, reconstruction of robust and reliable solution from an incomplete noisy diffraction data acquired by a typical CXI experiment remains an equally important and fundamental task which is addressed by phase retrieval algorithms [7–12]. Since uniqueness of the solution from a single coherent X-ray diffraction data frame cannot be guaranteed by phase retrieval algorithms, the typical procedure for the solution recovery  requires running the algorithm over hundreds of times starting with independent random guesses to get multiple trial solutions that are finally averaged together. The resolution of this averaged solution is estimated using the phase retrieval transfer function (PRTF), which is the ratio between the Fourier magnitude of the average solution and the measured Fourier magnitude as a function of the spatial frequencies.
In recent works [14, 15], we have proposed a novel approach that we call “complexity-guided phase retrieval” (CGPR) that is meant to address the typical stagnation problems with phase retrieval algorithms. This methodology uses a complexity parameter which is computed directly from the Fourier intensity data and provides a measure of fluctuations in the desired phase retrieval solution. As observed in [14, 15], the complexity parameter can be used to guide the application of constraints in the object domain and provides artifact free phase retrieval solution which degrades benignly with increasing noise. In [14, 15], the CGPR methodology was mainly developed with simulated noisy data in combination with the Fienup’s hybrid input-output (HIO) algorithm. The complexity-guidance idea is put to test for the first time in this work for experimental data available from the CXIDB database. The complexity-guidance concept is used along with the relaxed averaged alternating reflection (RAAR) phase retrieval algorithm in this study as RAAR is now commonly used in the CXI literature.
Our first aim is to examine the behavior of the phase retrieval solution obtained using the popular RAAR-ER methodology (both single run and averaged solution) from the perspective of the complexity parameter. A single run of RAAR-ER uses a number of RAAR iterations  that are concluded with a smaller number of error-reduction (ER) iterations for the purpose of stabilizing the solution. We observe that both single run and averaged solutions from RAAR-ER methodology consist of grainy artifacts and features that are smaller in size compared to the resolution as predicted by the PRTF. Such high frequency features are difficult to interpret  and therefore likely to be considered as “spurious.” In order to address this inconsistency, we add the complexity-guidance component to the RAAR algorithm to present what we call as complexity-guided RAAR (CG-RAAR) algorithm. CG-RAAR is first tested with simulated noisy data (with two noise levels) that do not have missing pixels for providing a clear understanding of the proposed methodology. The methodology is then applied to the real cyanobacterium diffraction data from the CXIDB database  to further validate our results. The experimental data is noisy and also has missing pixels. We report two main observations based on our study. First of all, a single run of CG-RAAR provides a solution without the high frequency grainy artifacts that are typically present in the RAAR-ER solution. A further consequence of this is that the number of independent runs required for averaging is reduced significantly. CG-RAAR essentially provides a regularized solution that does not contain spurious grainy features. The regularization is controlled in this methodology by means of the complexity parameter, thus making the solution consistent with the data.
In view of the rapid advancements in coherent X-ray imaging (CXI) experiments, the phase retrieval algorithms are critical for obtaining a reliable solution. Recent development of open-source software called Hawk  provides the platform for the entire process of solution reconstruction from the oversampled diffraction data and creating high performance algorithms. The Hawk software implements hybrid input-output (HIO) , difference map (DM) , and RAAR as phasing algorithms. From our prior work in  and from our experimental and simulation results demonstrated here, it is evident that complexity-guidance when combined with traditional phase retrieval algorithms such as HIO and RAAR can offer a better noise-robust estimate of the object. We believe that complexity-guidance as an idea may potentially be integrated into existing software tools and can improve the performance of existing phasing algorithms in coherent X-ray imaging.
2. Methods: Fourier Phase Retrieval
In a typical coherent X-ray imaging experiment, where X-ray beam illuminates the object or sample, the Fourier intensity measured at the detector can be mathematically expressed as where and are the spatial and Fourier space co-ordinates. is the electron density of the object, and is the Fourier transform of . Throughout the paper, the hat notation will be the representative of Fourier domain. The aim of iterative phase retrieval algorithms is to find from the measured intensity . Starting with a random guess , the Fourier magnitude constraint can be defined  as
Here, is the Fourier modulus projection operator in the Fourier domain, and is the set of pixels in that have zero photon counts (either due to Poisson noise or missing pixels). The above equation can be translated in the real space as where is the Fourier projection operator in the object domain. The second projection operator in object domain is the support projection which signifies the extent of the object contained within the support and is defined as
Commonly used phase retrieval algorithms aim to reach a solution that satisfies the Fourier magnitude and a priori object domain constraints in an iterative fashion. Inspired by the work of Gerchberg and Saxton  from electron microscopy for two measurements, the error reduction (ER) algorithm  tries to solve the phase retrieval problem for single Fourier measurement via following solution update:
The Fourier constraint being non-convex in nature makes the problem harder and the ER algorithm is known to suffer from stagnation in the sense that the quality of the solution does not improve even after a large number of iterations. The hybrid input-output (HIO) algorithm  addresses the stagnation issue by introducing a negative feedback outside the object support. The HIO feedback step can be stated as
where is the feedback parameter in the range . For noiseless data case, in a wide variety of images, the HIO algorithm shows a significantly improved reconstruction that is close to an ideal solution . However, for the noisy data case, HIO solution also shows stagnation artifacts  that corrupt the quality of the recovered solution. Phase retrieval with noisy data is an ill-posed problem, and therefore a unique solution to the problem cannot be estimated by satisfying both the Fourier magnitude and object support constraints. The difference map (DM) proposed by Elser  is an interesting approach which avoids local minima by using relaxed projections. The -th iteration of the DM is given as
Here, and are the relaxed generalized projections of and , respectively , and is the identity operator. is the relaxation parameter which needs to be optimized depending on the problem at hand, while the other two parameters and are generally chosen to be and . Finally, in DM the final solution is obtained as . Another approach that is also known to surpass the stagnation problems is the relaxed averaged alternating reflections (RAAR)  algorithm, which is a descendant of the hybrid projection reflection (HPR) algorithm . The RAAR algorithm shows good convergence to a stable solution even when an exact solution satisfying both the Fourier magnitude and the support constraints does not exist. The RAAR update is given as where and are the reflections of and that are obtained by setting . Apart from this, some other variants of HIO algorithm have also been reported for the noisy data case [24–26]. At this point, we remark that among all the phase retrieval algorithms HIO, DM, and RAAR are the most widely used ones in CXI. They are included in the Hawk open-source software and have been tested for a numerous coherent X-ray imaging data sets. From application point of view where there is a wide range of CXI data, it is not easy to compare these algorithms. One of the reasons is that the behavior of phase retrieval solution is not clearly understood for the various phasing schemes. In phase retrieval literature, the performance of the iterative algorithms is commonly evaluated by means of the error function which calculates the L2-norm of the error between the Fourier magnitude data and the Fourier magnitude of the iterative solution. While this measure is natural, its use can pose problems when the data is corrupted with high degree of noise and missing pixels. In such cases, matching the Fourier magnitude of the solution and the raw data may lead to overfitting the noise, resulting in appearance of grainy artifacts in the solution. This practical problem of examining the quality of phase retrieval solutions from noisy data is a challenge that needs further work.
In this paper, our effort is to understand the nature of phase retrieval solution from a new perspective of the complexity parameter. The complexity information which can be computed a priori and provides a measure of fluctuations present in the ground-truth object is utilized to evaluate the characteristics of the phase retrieval solution. For the present work, we examine the behavior of the solution obtained by the RAAR algorithm which is popularly used by the CXI community. It is a common practice to conclude the standard algorithms like RAAR (or HIO) with ER iterations to further stabilize/refine the RAAR solution and improve the error performance in Fourier domain [16, 27]. We use the combination of RAAR algorithm followed by 100 iterations of ER throughout this paper which will be referred by the name “RAAR-ER” algorithm.
For the sake of completeness, we start by describing the complexity parameter which was introduced in . The parameter is a measure of fluctuations present in pixel values of an object , where the coordinate r is expanded as . Mathematically, the complexity is defined as follows :
Here, and are the partial derivatives with respect to the and coordinates, respectively. Using Parseval’s theorem and employing the derivative property of Fourier transform along with the notion of modified wave numbers, the above equation can be equivalently written as where and are the grid spacing in spatial domain and denote the spatial frequency samples as per the FFT convention. Note that the usual continuous case of this result is evident by setting . Interestingly, the complexity parameter is a real space constraint but derived directly from the raw Fourier intensity data . Moreover, the complexity information in the form of as defined in Equation (10) is readily available a priori to all the phase retrieval algorithms. We observe empirically that the complexity parameter is not too sensitive to the typical Poisson noise in the Fourier magnitude data. As discussed in Section 3, we will explain how the complexity information may be utilized when applying object domain constraints to obtain a noise robust phase retrieval solution.
3. Complexity Behavior of the RAAR-ER Solution with Noisy Data
In this section, we investigate the nature of a single run of RAAR-ER solution recovered from simulated noisy Fourier intensity data from the complexity viewpoint. The practical CXI data is noisy and has missing information due detector beam stop, hot pixels, and gaps between the detector segments, etc. For simplicity of understanding the complexity behavior, we restrict our initial simulation study to noisy data without any missing pixels. The missing pixels in the data can add one more factor of uncertainty since the complexity calculation as per Equation (10) requires the Fourier magnitude information at all pixels. As we will describe later in Section 6, we handle the missing pixels in the real data with the standard methodology prevalent in the literature.
In order to illustrate the effect of noise on the phase retrieval solution, we simulate two Fourier intensity data corresponding to a red blood cell (RBC) object in a computational window of size pixels. The amplitude of this test object (included within the central pixels) is shown in Figure 1(a). The complex-valued RBC object for simulating noisy diffraction patterns was generated using a digital holographic microscope system (make: Holmarc) and corresponds to image plane field for the cell recovered using an off-axis hologram. The noisy Fourier intensity data corresponding to this RBC object have been simulated by assuming the Poisson noise corresponding to two different average photon count levels of 1000 and 500 photons/pixels. The noisy data frames have ~36% and 51% pixels with zero photon counts as shown in Figures 1(b) and 1(c), respectively. These noise levels are comparable to those in the experimental CXI data to be discussed in Section 6. The Fourier intensities are oversampled by a factor of approximately 3 in both directions.
We ran 1100 iterations of RAAR-ER algorithm comprising of 1000 RAAR iterations, as per Equation (8), that were concluded by 100 ER iterations starting with the data in Figures 1(b) and 1(c). The relaxation parameter for RAAR update was selected to be . Since the object is complex-valued, only support constraint has been used. Figures 1(d) and 1(e) are the absolute magnitudes of the solution after 1000 RAAR iterations for the noisy data in Figures 1(b) and 1(c), respectively, whereas the same solutions after additional 100 ER iterations are shown in Figures 1(f), and 1(g), respectively. From the solution recoveries, we clearly see that for the higher noise data, both the RAAR and RAAR-ER solutions contain grainy features that were not present in the ground truth in Figure 1(a).
Next, we examine the complexity behavior of RAAR and RAAR-ER solutions shown in Figures 1(d)–1(g). To do that in each RAAR or ER iteration, the complexity of the solution is calculated using Equation (9) and is denoted by . In a similar vein, complexity inside and outside the support can be evaluated by summing over corresponding pixels. Clearly, we have . Figures 2(a) and 2(b) show the graphs of complexities , , and along with the ground-truth complexity against the iteration number for the Fourier intensities with Poisson noise corresponding to average light levels of 1000 photons/pixel and 500 photons/pixel, respectively. The complexities and for both the cases stabilize to values higher and lower than the ground-truth complexity . In terms of complexity behaviour of solution, we observe that the solution’s complexity within the support stabilizes at a higher level for data with higher noise. The outside support complexity is also seen to stabilize to a slightly higher value for the higher noise data. The manifestation of higher complexity or fluctuations inside the support region with increasing noise can be clearly seen in solutions in Figures 1(e) and 1(g) when compared to those in Figures 1(d) and 1(f). When ER iterations are applied after 1000 RAAR iterations, stabilizes to a lower value than that in the RAAR case. On the other hand increases as compared to that at the end of RAAR iterations and also contributes most to the total complexity . The reduction of during ER iterations is easy to understand as the pixels outside the support region are set to zero during the ER iterations so that only contribution to comes from the pixels adjacent to the support boundary. From the complexity curves and in Figures 2(a) and 2(b) and solutions in Figures 1(f) and 2(g), we observe that in the process of applying the ER iterations (for “stabilizing” the solution), the complexity of the RAAR solution within the support region has actually increased, which causes additional grainy artifacts that in our opinion are due to over-fitting of noise. The application of ER iterations that typically reduce the data domain error is thus not necessarily desirable in our opinion, as this is accompanied by appearance of increased grainy artifacts within the support region. The high frequency nature of the grainy artifacts may also lead to a slightly exaggerated estimate of the resolution of the solution based on the PRTF curve as we will see later in Section 5.
The prior literature [12, 24, 25] has stated the significance of outside support region in relation to noise in the Fourier intensity data. However, the procedures for suppressing outside support information in the literature are somewhat ad hoc in our opinion. In the complexity-guided phase retrieval approach described next, the a priori knowledge about the ground-truth complexity plays an important role in application of constraints in the object domain (both within and outside the support).
4. Complexity-Guided RAAR (CG-RAAR) Algorithm
From the complexity curves in Figures 2(a) and 2(b), we observed that the solution complexity for RAAR as well as RAAR-ER algorithms stabilizes to a higher numerical value compared to the ground-truth complexity computed from the raw data. The concept of complexity-guidance is based on the strategy of matching with . Instead of concluding the RAAR algorithm with ER iterations (which leads to undesirable increase in solution complexity within the support), we match the total complexity of the guess solution with the ground-truth complexity in each RAAR iteration. To bring down to the level of , the values of and are decreased via complexity reduction steps in an adaptive manner. In our previous work , while studying the behavior of the HIO solution, we observed that for the increasing noise in the data, the showed a monotonic increase with iteration number, and for that reason, was required to be suppressed actively in each HIO iteration. Unlike HIO solution, the RAAR solution does not show continous increase in complexity outside the support region with iterations. This observation has not been reported before in our opinion in the context of the RAAR algorithm. Therefore, most complexity reduction in the present case needs to be done within the support region. It is interesting to note that an improvement over the native RAAR algorithm has been suggested previously by introducing a generalized feedback term inside the support region . Based on our observations, we control the total complexity of the RAAR solution primarily by suppressing the complexity within the support which is essentially like providing a complexity feedback within the support.
We remark that complexity-guidance is essentially a regularization scheme where complexity (or degree of fluctuations) in the desired solution is predetermined from the raw data. In order to implement complexity-guidance, the recursive iterations for complexity reduction are applied to the RAAR solution in Equation (8) in the outside and inside the support region in each outer RAAR iteration. For brevity of notation, we denote the outside and inside part of the solution as and , respectively. Firstly, recursive sub-iterations are applied outside the support as represented by
Next, further recursive complexity reduction iterations similar to above are also applied to the :
For both the above equations, is the step size which was chosen to be so that the solution changes by a small amount in each sub-iteration and is the normalized functional gradient of complexity defined as where is the complex derivative  of complexity parameter . As explained in , the number of sub-iterations for reduction are adjusted by actively tracking the trend of . Particularly, the value is nudged towards the mean minus the standard deviation of of the previous iterates in order to control any growth in . After that, the complexity reduction sub-iterations within the support are applied until the numerical value of the total complexity is close to (within ) the ground-truth complexity . The number of sub-iterations in this step typically depends on the noise level (higher noise data requires more complexity reduction sub-iterations inside the support). The (n +1)-th iterate of CG-RAAR is updated as
Here, and refer to the inside and outside support solutions after applying the complexity-guidance sub-iterations. The recovered solutions using a single run of CG-RAAR from the same noisy data in Figures 1(b) and 1(c) are shown in Figures 1(h) and 1(i), respectively. Visually, the CG-RAAR solution in Figures 1(h) and 1(i) are free from severe grainy artifacts that were seen in RAAR and RAAR-ER solutions in Figures 1(d)–1(g), respectively.
Our implementation used initial 500 RAAR iterations followed by 500 RAAR iterations with complexity-guidance. In Figures 2(c) and 2(d), we observe that after iterations when complexity-guidance sub-iterations are applied, by design, the complexity of the solution matches well with . In particular, the high frequency artifacts within the support are now suppressed by the complexity-reduction steps. Therefore, the CG-RAAR solution is smoother in nature and is much closer to the ground-truth shown in Figure 1(a). We want to emphasize here that all the results shown in Figure 1 correspond to a single run of the corresponding algorithms (RAAR or RAAR-ER or CG-RAAR). The use of complexity constraint with noisy data is observed to provide a better estimate for single run of the phase retrieval algorithm. In turn, this is beneficial for the averaging process as well as we will see in the next section. The trial solutions selected for the averaging process can be improved further by methods suggested by  such as voting, patching, and use of non-centrosymmetric support  as well as by use of sparsity enhancing functionals like total variation, Huber, or other penalties .
5. Evaluating the Resolution of Reconstructed Solution Using Phase Retrieval Transfer Function (PRTF)
The reconstruction of solution from the experimental Fourier transform intensity data consisting of noise and missing pixels due to detector blocking poses challenges. First of all, generally just one realization of the data is often available as the sample may get destroyed after exposure to high energy X-ray pulse. Secondly, a single run of the algorithm starting with a random guess cannot be fully relied upon when the data is corrupted by noise and missing pixels. It is therefore a standard practice to arrive at a better estimate of the solution by averaging a large number of trial solutions  starting with the same data. These independent trial solutions are obtained by running the phase retrieval algorithm over hundreds of runs that are initialized with different random guesses. Among a number of trial reconstructions, only some following a certain criterion are selected for the averaging process. The averaging criterion can be decided on the basis of least errors in real space or Fourier space. Here, we have selected correlation coefficient between trial solution and a reference solution with least error as an averaging criterion. Since the final reconstruction is based on the averaging process, the goodness of reconstruction can be assessed using phase retrieval transfer function (PRTF) which compares the magnitude of Fourier transform of averaged complex reconstruction with the measured Fourier modulus and defined as follows [27, 32]:
Here, the average of complex Fourier estimates can be simplified as where denotes the trial number index and represents the phase adjusted trial reconstructions following the correlation criterion for averaging. In order to add the trial solutions coherently, the phases of trial solutions are adjusted to a common value so that the variation in phases does not reduce the average value. The phase adjustment of all the trial reconstructions can be done as per . The PRTF curve can provide a measure of the resolution of the recovered solution by selecting the point where PRTF falls to 1/e . In this section, we first present the PRTF curves for the simulated data followed by similar analysis of the experimental data in the next section.
Figures 3(a) and 3(b) are the solutions obtained after averaging over trials of RAAR-ER and CG-RAAR algorithms, respectively, for noisy data in Figure 1(b). The correlation coefficient with value 0.96 is chosen as the criterion for selecting a trial solution. For the simulated data, we have not assumed any missing pixels due to detector blocking and assumed that the data is corrupted by Poisson noise alone. In this case, we observe that an average over a small number of trial solutions is sufficient to stabilize the PRTF curve. One can clearly see that the averaging process has reduced the grainy appearance of the RAAR-ER solution. However, some of the faint artifacts are still persistent which may possibly get misinterpreted as the real features, particularly in the case of experimental Fourier intensity data where the ground-truth object is unknown. Since the individual CG-RAAR solution was already free from grainy artifacts, the averaged CG-RAAR solution closely resembles the ground-truth object. The performance of the CG-RAAR and RAAR-ER solutions are tabulated in Table 1 in terms of error in Fourier and real domain. The real space error metric is defined as follows : where
Here, represents the solution estimate (individual trial or average solution), is the ground-truth object, and is the correlation between and . Next, the relative Fourier domain error for the Fourier modulus estimate with respect to the measured Fourier modulus can be written as
Both RAAR-ER and CG-RAAR solutions seem to have errors of similar orders in terms of the above metrics despite the fact that CG-RAAR solution has much reduced grainy artifacts. Figure 3(c) shows the PRTF curves for the solutions in Figures 3(a) and 3(b) obtained by both the algorithms. The spatial resolution for the averaged RAAR-ER and CG-RAAR solutions is ~3 pixels. The resolution estimate for the RAAR-ER solution is slightly higher compared to the CG-RAAR solution. This higher resolution estimate is however spurious and arises mainly due to the remaining grainy artifacts. From the zoomed-in portions of Figures 3(a) and 3(b) and the profile plot in Figure 3(d) across the yellow dotted lines in both the solutions, we see that the CG-RAAR solution is smooth on a 3-pixel scale unlike the RAAR-ER solution. Therefore, it is quite evident from the PRTF plot that the proposed CG-RAAR method leads to a self-consistent solution compared to that obtained by the traditional RAAR-ER methodology for noisy Fourier intensity data.
6. Tests with Experimental CXI Data
In order to further demonstrate the performance of CG-RAAR algorithm, we take an experimental CXI data submitted to the coherent X-ray imaging data bank (CXIDB) . The experiment was carried out in Linac coherent light source (LCLS), and the extensive details about the experimental set-up are described in . The data set contains ten Fourier intensity frames corresponding to live cyanobacteria at ten different stages. For the illustration purpose, we take two of these Fourier intensity data (Figures 2(i) and 2(j) of ), which contain a total of and photon counts and are shown in Figures 4(a) and 4(b), respectively. The Fourier intensities in Figures 4(a) and 4(b) have ~51% and 46% of missing pixels due to detector blocking, gaps between the detector segments, Poisson noise, etc. The only modification required to handle all the missing pixels is to replace them with the Fourier magnitude corresponding to the current iterate. The masks provided in the “.cxi” file for the corresponding intensities were used as the known object supports.
We ran both RAAR-ER and CG-RAAR algorithms 400 and 200 times independently in a MATLAB environment. In the first case, 2000 RAAR iterations were concluded by 100 ER steps. In CG-RAAR initial 1500 RAAR iterations were followed by the 500 outer RAAR iterations with complexity-guidance sub-iterations. Since we are filling the missing pixels with the Fourier magnitude of the current iterate, the ground-truth complexity estimate also needs to be revised in each CG-RAAR iteration. We however note that by 1500 RAAR iterations (after which we use the complexity-guidance), the numerical values in the missing pixels are seen to stabilize and the estimated does not fluctuate significantly. All these cases with multiple independent initial random guesses ran on the high power computing (HPC) cluster at Indian Institute of Technology Delhi.
Figures 4(c) and 4(e) show one of the trial solutions obtained using the RAAR-ER and CG-RAAR algorithms, respectively. Visually, we observe that the single instance of the CG-RAAR solution is smoother compared to that obtained using the RAAR-ER algorithm. The averaged solution for both the respective algorithms is shown in Figures 4(d) and 4(f). The averaging procedure is the same as followed in the simulations. Based on the correlation criterion as before, 261 out of 400 RAAR-ER solutions were selected for the averaging while in CG-RAAR 94 out of 200 solutions followed the criterion. Further, the performance of RAAR-ER and CG-RAAR algorithms for both the Fourier intensity data shown in Figures 4(a) and 4(b) has comparable order-of-magnitude error performance as shown in Table 2.
We observe that the PRTF curve is stabilized for both RAAR-ER and CG-RAAR cases and additional trial runs will not have any perceptible effect on the PRTF curve. In order to assess the quality of averaged solutions, we plot the PRTF as a function of spatial frequency which is shown in Figure 4(k). The PRTF curve is vanishing near zero frequency since there are blocked pixels in this region where the Fourier intensity data is missing. The spatial resolution (reciprocal of spatial frequency) at which the PRTF falls to is 4.54 and 4.76 for the RAAR-ER and CG-RAAR solution, respectively, which can be rounded off to ≈5 pixels. The white circles in Figures 4(d) and 4(f) represent resolution scale as estimated from the PRTF curve. From the zoomed-in portion of the averaged RAAR-ER solution in Figure 4(l), we clearly see features smaller than 5 pixels that are spurious and difficult to interpret. On the other hand from Figure 4(m), the CG-RAAR solution is clearly observed to be smoother and consistent with the spatial resolution reflected in PRTF curve.
We remark that the spatial resolution given here is in terms of pixels, however, one can always express it in physical units by the relation , where is the wavelength of the X-rays, is the distance between specimen and detector, is the pixel size of the detector, and is the half pixel number. For the Fourier intensity data shown in Figure 4(b), one can make similar observations from Figures 4(g)–4(j) and 4(n)–4(p). The final solutions shown in Figures 4(h) and 4(j) are obtained after averaging over 385 and 188 RAAR-ER and CG-RAAR trial solutions, respectively. Once again, the averaged RAAR-ER solution still seems to retain some grainy artifacts that are not present in the CG-RAAR solution. The grainy artifacts in the RAAR-ER solution lead to a slightly better resolution estimate which is however spurious in our opinion. CG-RAAR solution however seems to have features consistent with the resolution estimate provided by the PRTF curve.
It is worth noting from our results that the CG-RAAR solution is artifact-free despite the fact that the number of trial solutions used for the averaging process in CG-RAAR case is much less than half the number of trial solutions used for RAAR-ER case. This is expected since even the single run of CG-RAAR solution has much reduced artifacts compared to that of the RAAR-ER solution.
In conclusion, we have examined the nature of phase retrieval solutions for noisy Fourier intensity data obtained from simulations as well as CXI experiments from a new perspective of solution complexity. The ground-truth complexity is an object domain constraint which provides a measure of fluctuations in the desired solution and can be calculated a priori from the raw Fourier intensity data. With increasing noise in the data, the complexity of RAAR solution stabilized to a higher level compared to the ground-truth complexity. When the concluding 100 ER steps were applied for the purpose of refining the RAAR solution, the resulting solution showed further undesirable increase in the complexity of the solution inside the support region. This was manifested in the form of severe grainy artifacts in the solution. In order to address this issue, we proposed and tested a complexity-guided RAAR (CG-RAAR) algorithm. The goodness of the algorithm and the recovered resolution were judged using the PRTF plot.
The traditional RAAR-ER solution seems to retain some grainy artifacts (even after averaging over a number of trial solutions), whose feature size is smaller than the resolution estimated by the PRTF. On the other hand, the solution from the proposed CG-RAAR algorithm has smallest features consistent with the resolution estimate provided by the PRTF. The main idea behind complexity-guidance is to match the complexity of RAAR solution with the desired ground-truth complexity. The performance of CG-RAAR algorithm is demonstrated for the experimental CXI data from Coherent X-ray Imaging Data Bank (CXIDB). It is worth highlighting that the single run of CG-RAAR produces solutions with much reduced artifacts, and as a result, the number of trial solutions required for the averaging process with this methodology is less than the half number that are needed for the traditional RAAR-ER method. We believe that the complexity-guidance methodology can benefit the field of X-ray coherent diffraction imaging by providing reliable and self-consistent phase retrieval solution estimates.
8. Limitations of the Present Study
The computational time taken by the 1100 RAAR-ER iterations is ~15 seconds, while 1000 iterations of CG-RAAR algorithm takes ~3 minutes due to additional complexity reduction sub-iterations in a nominal MATLAB implementation. Despite more computational efforts, the phase retrieval with complexity-guidance provides much reliable and robust phase retrieval solution. The additional complexity reducing sub-iterations required within the traditional RAAR loop make the individual run of the algorithm computationally more expensive. It is up to a user to decide whether this additional computation provides sufficient benefit for a particular application. The work has noted a distinctly different behavior of the popular HIO and RAAR phase retrieval algorithms from the perspective of the complexity parameter. We are however not in a position at present to provide a reasoning behind this observation. The CG-RAAR methodology also needs to be further tested for objects with more structural details and for variable relative object support sizes.
Data used in the paper is freely available from Coherent X-ray Imaging Data Bank (CXIDB) [https://www.cxidb.org/, data ID:26]. The pseudo-codes used in this work can be made available upon reasonable request to the corresponding author.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this article.
All authors contributed equally to the conceptual design, simulations, and writing of the manuscript.
MB and SR acknowledge PhD Fellowship provided by Indian Institute of Technology Delhi.
- C. Bostedt, S. Boutet, D. M. Fritz et al., “Linac coherent light source: the first five years,” Reviews of Modern Physics, vol. 88, no. 1, p. 015 007, 2016.
- H. N. Chapman, A. Barty, M. J. Bogan et al., “Femtosecond diffractive imaging with a soft-X-ray free-electron laser,” Nature Physics, vol. 2, no. 12, pp. 839–843, 2006.
- H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nature Photonics, vol. 4, no. 12, pp. 833–839, 2010.
- L. Strüder, S. Epp, D. Rolles et al., “Large-format, high-speed, X-ray pnCCDs combined with electron and ion imaging spectrometers in a multipurpose chamber for experiments at 4th generation light sources,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, vol. 614, no. 3, pp. 483–496, 2010.
- P. Emma, R. Akre, J. Arthur et al., “First lasing and operation of an angstrom-wavelength free-electron laser,” Nature Photonics, vol. 4, no. 9, pp. 641–647, 2010.
- M. Liang, G. J. Williams, M. Messerschmidt et al., “The coherent x-ray imaging instrument at the Linac coherent light source,” Journal of Synchrotron Radiation, vol. 22, no. 3, pp. 514–519, 2015.
- J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Optics Letters, vol. 3, no. 1, pp. 27–29, 1978.
- J. R. Fienup, “Phase retrieval algorithms: a comparison,” Applied Optics, vol. 21, no. 15, pp. 2758–2769, 1982.
- V. Elser, “Phase retrieval by iterated projections,” JOSA A, vol. 20, no. 1, pp. 40–55, 2003.
- S. Marchesini, H. He, H. N. Chapman et al., “X-ray image reconstruction from a diffraction pattern alone,” Physical Review B, vol. 68, no. 14, p. 140 101, 2003.
- D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Problems, vol. 21, no. 1, p. 37, 2005.
- C. Guo, S. Liu, and J. T. Sheridan, “Iterative phase retrieval algorithms I: optimization,” Applied Optics, vol. 54, no. 15, pp. 4698–4708, 2015.
- D. Shapiro, P. Thibault, T. Beetz et al., “Biological imaging by soft x-ray diffraction microscopy,” Proceedings of the National Academy of Sciences, vol. 102, no. 43, pp. 15 343–15 346, 2005.
- M. Butola, S. Rajora, and K. Khare, “Phase retrieval with complexity guidance,” JOSA A, vol. 36, no. 2, pp. 202–211, 2019.
- M. Butola, S. Rajora, and K. Khare, “Complexity-guided fourier phase retrieval from noisy data,” JOSA A, vol. 38, no. 4, pp. 488–497, 2021.
- G. Van Der Schot, M. Svenda, F. R. Maia et al., “Imaging single cells in a beam of live cyanobacteria with an X-ray laser,” Nature Communications, vol. 6, no. 1, pp. 1–9, 2015.
- F. R. Maia, T. Ekeberg, D. Van Der Spoel, and J. Hajdu, “Hawk: the image reconstruction package for coherent x-ray diffractive imaging,” Journal of Applied Crystallography, vol. 43, no. 6, pp. 1535–1539, 2010.
- H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” JOSA A, vol. 19, no. 7, pp. 1334–1345, 2002.
- R. W. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik, vol. 35, pp. 237–246, 1972.
- T. Ekeberg, M. Svenda, C. Abergel et al., “Three-dimensional reconstruction of the giant mimivirus particle with an x-ray free-electron laser,” Physical Review Letters, vol. 114, no. 9, p. 098 102, 2015.
- H. Stark, Y. Yang, and Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics, John Wiley & Sons, Inc., 1998.
- V. Elser, “Random projections and the optimization of an algorithm for phase retrieval,” Journal of Physics A: Mathematical and General, vol. 36, no. 12, pp. 2995–3007, 2003.
- H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Hybrid projection–reflection method for phase retrieval,” JOSA A, vol. 20, no. 6, pp. 1025–1034, 2003.
- A. V. Martin, F. Wang, N. T. Loh et al., “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Optics Express, vol. 20, no. 15, pp. 16 650–16 661, 2012.
- J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” Journal of Applied Crystallography, vol. 46, no. 2, pp. 312–318, 2013.
- R. Trahan and D. Hyland, “Mitigating the effect of noise in the hybrid input–output method of phase retrieval,” Applied Optics, vol. 52, no. 13, pp. 3031–3037, 2013.
- H. N. Chapman, A. Barty, S. Marchesini et al., “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” JOSA A, vol. 23, no. 5, pp. 1179–1200, 2006.
- D. E. Adams, L. S. Martin, M. D. Seaberg, D. F. Gardner, H. C. Kapteyn, and M. M. Murnane, “A generalization for optimized phase retrieval algorithms,” Optics Express, vol. 20, no. 22, pp. 24 778–24 790, 2012.
- D. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proceedings H-Microwaves, Optics and Antennas, IET, vol. 130, pp. 11–16, 1983.
- J. Fienup and C. Wackerman, “Phase-retrieval stagnation problems and solutions,” JOSA A, vol. 3, no. 11, pp. 1897–1907, 1986.
- C. Gaur, B. Mohan, and K. Khare, “Sparsity-assisted solution to the twin image problem in phase retrieval,” JOSA A, vol. 32, no. 11, pp. 1922–1927, 2015.
- T. Latychevskaia, “Iterative phase retrieval in coherent diffractive imaging: practical issues,” Applied Optics, vol. 57, no. 25, pp. 7187–7197, 2018.
- J. R. Fienup, “Invariant error metrics for image reconstruction,” Applied Optics, vol. 36, no. 32, pp. 8352–8357, 1997.
- F. R. Maia, “The coherent X-ray imaging data bank,” Nature Methods, vol. 9, no. 9, pp. 854-855, 2012.
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