1. Introduction

X-ray phase imaging is of great interest for improving the contrast of low-density material, in complementarity with classical absorption imaging. Several X-ray phase imaging techniques are reported in the literature [1], but we focus here on those that use a phase modulator to form a reference intensity pattern on the detector. When adding a sample in the optical path, the reference pattern shifts and allows retrieving the phase gradient signal. Phase modulators can be chosen to produce either a randomly distributed intensity pattern, such as the ones used for the speckle-based techniques [2,3], or a regular intensity pattern, such as 1D or 2D grating based interferometry techniques [4–10]. Both modulators allow the measurement of at least two orthogonal phase gradients with one or multiple acquisitions.

The quality of the phase gradient measurement - which is the capability to estimate the phase shift induced by the sample - is crucial and has an impact on the retrieved phase image. One way to evaluate this point is to calculate a phase derivative closure map $C(x,y)$. This approach was used for the first time for phase measurements by Primot [11] and applied in the X-ray domain on synchrotron light source by Rizzi et al. [12]. The phase derivative closure map assumes that the wave front issued from the sample is a continuous surface. The surface is thus twice derivable at each point and the circulation around it should be equal to zero. In other words, for the phase $\phi (x,y)$ defined in the plan $(x,y)$ perpendicular to the wave propagation axis $(z)$, if we consider two orthogonal phase gradients $[\partial _x \phi (x,y),\partial _y \phi (x,y)]$ and apply a curl operator, the phase derivative closure map should be equal to zero:

But in real conditions, $C(x,y)$ can suffer from different contributions, and Eq. (1) can be rewritten as

We propose to use $C(x,y)$ to build a confidence map which indicates the trust level of the retrieved phase image. This article is organized as follow: section 2 introduces how to build the confidence map by estimating the parameters $\epsilon _d$, $\epsilon _u$ and $\epsilon _n$ from Eq. (2) using simulated images. Then, direct applications on experimental images - a canonical object and a carbon fiber reinforced polymer - are presented in section 3. Finally the confidence map is put into perspective in a discussion in section 4.

2. How to build a confidence map

To illustrate the estimation of the parameters $\epsilon _d$, $\epsilon _u$ and $\epsilon _n$ and build a confidence map, we propose to use a phase contrast simulation tool based on wave-front description of the multilateral shearing interferometry technique [13]. With this tool, the diffracted orders are directly simulated after the 2D-checkerboard grating (modulator) and propagate onto the detection plan. A simple object with analytical description can be imported into the simulated scene and placed either before or after the modulator. We have chosen here the second implementation, but the position of the object does not affect the basic idea of the confidence map. Here, we consider an ideal case with a monochromatic source at 17.48 keV (Molybdenum K$\alpha$ line). The simulated modulator is designed to have a 0-$\pi$ shift at this energy and an orthogonal periodicity $a$ of 12 $\mathrm{\mu}$m. The simulated object is a PMMA cylinder of diameter of 500 $\mathrm{\mu}$m with the inclusion of a defect in the center. The detection plane is set to have a pixel size $S_{pix}$ of 6 $\mathrm{\mu}$m sideways. The distances are set as follow: i) the source-detector distance is 60 cm; ii) the source-modulator distance is 14 cm and iii) the modulator-object distance is 4 cm. It leads to a magnification of $G_m = 4.3$ for the modulator and $G_s = 6$ for the sample. To complete the simulation a Gaussian noise is added, since it is a fair description of the experimental noise measured in our setup.

From this simulation setup, a phase gradient image is generated and a phase derivative closure map $C(x,y)$ is calculated [12] (see respectively Figs. 1(a), 1(b)). For each of them, we extract intensity profiles along the blue and red lines. The profiles are shown in Fig. 1(c). The phase gradient profile, in blue, refers to the left axis and the phase derivative closure map, in red, refers to the right axis.

 

Fig. 1. Simulation of PMMA cylinder with the inclusion of a defect in the center, for a monochromatic X-ray source at 17.48 keV. (a) Phase gradient with local extinction. (b) Phase derivative closure map. (c) Corresponding plot profile. (d) Histogram of the phase derivative closure map (absolute values).

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The phase gradient profile shows a slow phase variation across the cylinder with more abrupt variations at the center, where the defect causes a local phase extinction (see green rectangle), and at the edges (see orange rectangles), where weak fringes sampling leads to a non-accurate phase gradient measurement. When we look now at the phase derivative closure map profile, we see that $C(x,y)$ shows a strong signal at the defect and the edges locations, emphasizing the errors introduced by the phase extinction (green rectangle) and the weak fringes sampling (orange rectangles).

To evidence the contribution of $\epsilon _d$, $\epsilon _u$ and $\epsilon _n$ in the phase derivative closure map (Fig. 1(b)), we calculate the histogram of the absolute gray values (Fig. 1(d)). It shows three areas, that we label $A, B$ and $C$. $A$ has few counts and high values of $C(x,y)$ (a few hundreds of rad/m$^2$). It estimates the phase dislocation parameter $\epsilon _d$. Indeed, phase dislocations are very intense and local, therefore $C(x,y)$ values correlated with this phenomenon are intense and few, precisely located in the map. $B$ has more counts, but weak $C(x,y)$ values (a few tens of rad/m$^2$). This is related to the under-sampling $\epsilon _u$ phenomenon. The high counts reveal that a significant portion of the image is impacted by an under-sampling issue. Note that under-sampling dominates at location where the object encounters strong phase variations, for instance at the edge of the cylinder. Finally, $C$ has very high-counts at low $C(x,y)$ values (more than zero but less than 1 rad/m$^2$). It corresponds to the pixels whose values are related to $\epsilon _n$, the simulated numerical noise. To build the confidence map, we need to locate on the $C(x,y)$ image the pixels impacted by the $\epsilon _d, \epsilon _u, \epsilon _n$ contributions ($C(x,y) \ne 0$).

For the phase dislocation parameter $\epsilon _d$, the intense values of $C(x,y)$ can be filtered by threshold methods. For instance, in this work, we have chosen to define a threshold $T_{hd}$ based on a maximum entropy method [14]. It consists of measuring the uncertainty of an event taking place, in this case a phase dislocation. After defining the threshold, every pixels of the $C(x,y)$ image are evaluated, and an alert is generated by coloring the pixel - in red in this work - when the pixel value $(x_i,y_j)>T_{hd}$ where $T_{hd}$ = 114 rad/m$^2$ in the simulation example presented in Fig. 1.

After filtering $\epsilon _d$, we can perform a moving average evaluation of the phase derivative closure map related to $\epsilon _u$ and $\epsilon _n$. To do so, we evaluate $C(x,y)$ in a square region of interest $\Omega$ of size of $S_\Omega = (aG_g)/(2S_{pix})$ with the following odd condition $\{2S_\Omega +1~|~S_\Omega \in \mathbb {N}\}$. $S_\Omega$ corresponds to the phase gradient measurement sensitivity that can be reached when using a fringe interference pattern of periodicity $aG_g$, sampled by the detection plan of pixel size $S_{pix}$. In order to distinguish the contribution of $\epsilon _n$ from $\epsilon _u$ a threshold $T_{hu}$ is used to perform a binarization of the image. $T_{hu}$ is determined by the isodata method [15] which is achieved in three steps: i) the pixels are separated into two groups, object and background, by taking an initial threshold; ii) the average of each group is calculated; iii) finally, the average of those two values (called composite average) are computed, the threshold is incremented and the process is repeated until the threshold is larger than the composite average. Here the threshold value is estimated to $T_{hu} = 4$ rad/m$^2$. All pixel values $(x_i,y_j) \ge T_{hu}$ are related to $\epsilon _u$ and colored in blue, and all pixel values $(x_i,y_j)<T_{hu}$ are related to $\epsilon _n$ and colored in cyan.

In this textbook case, we can also distinguish the contribution of $\epsilon _n$ from $\epsilon _u$ simply by visual interpretation: we expect $\epsilon _u$ alerts to be intense and precisely located on the image, where phase variations are not well described, such as at the edge of the simulated PMMA cylinder. In comparison, we expect $\epsilon _n$ alerts to be less intense and uniformly distributed on the image.

Figure 2 presents the computed confidence map from the derivative closure map presented in Fig. 1(b). Red pixel alerts (dislocation issue) are located in the center of the PMMA cylinder due to the local extinction caused by the defect. Blue pixels alerts are present on the edge of the cylinder due to the under-sampling issue. Cyan alerts (noise issue) are distributed on the image, but are more intense inside the sample than outside (see the zoomed area on Fig. 2). The confidence map presented on Fig. 2 emphasizes the visual aid provided by this tool in detecting possible artifacts induced by the phase treatment. It will be used in the next section on real X-ray phase contrast images.

 

Fig. 2. Confidence map computed from the derivative closure map presented in Fig. 1(b). The pixels colored in red indicate a dislocation issue; in blue, an under-sampling issue; and in cyan, a noise issue.

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3. Experimental results

This section presents experimental results on a canonical object – in order to illustrate the proposed approach – and on a complex and mastered sample, a carbon fiber reinforced polymer test piece, highlighting the fact that the method performs equally well on more representative object.

3.1 Canonical object

We will now focus on experimental results obtained from a optical fiber made of PMMA material with a diameter of 500 $\mathrm{\mu}$m. Multilateral shearing interferometry is used as gradient-based phase contrast technique [13]. The source is a microfocus X-ray tube (Feinfocus FXE-160.51) with a solid transmitted Tungsten anode and a measured spot size of 5.5 $\mathrm{\mu}$m [16]. It is used at a tube intensity of 60 $\mathrm{\mu}$A and a voltage of 75 kV. The detector is a Hamamatsu C12849-102U high resolution sensor made of a 20 $\mathrm{\mu}$m layer of Gadox scintillator deposited on fiber plate coupled with a sCMOS sensor. The detector pixel size is $S_{det} = 6.5$ $\mathrm{\mu}$m sideways. This imaging setup can reach a spatial resolution limit of 4.6 $\mathrm{\mu}$m, according to the Rayleigh criterion estimated with an image quality indicator (X-radia type X500-200-30) magnified by a factor of 19 [17]. The modulator is a single 2D-checkerboard phase grating (made by the Microworks company) with available periodicity of $a = [24, 20, 16, 12]$ $\mathrm{\mu}$m. It consists of square Gold platforms of size $a/2$ and thickness $3.49 \pm 0.16$ $\mathrm{\mu}$m, deposited on a polymer substrate, and induces a $[$0-$\pi]$ phase shift at 17.48 keV. The source-detector distance is $d_{sd} = 57$ cm and the source-object and source-grating distances are, respectively $d_{so} = 11$ cm and $d_{sg} = 14$ cm, implying a magnification factor of roughly $G_s = 5$ for the fiber and $G_g = 4$ for the grating. The following experimental images are the result of an average of 15 images with an exposure time of 20 seconds per image.

Figure 3(a) presents the experimental raw image obtained with the orthogonal periodicity of $a = 24$ $\mathrm{\mu}$m. We measure then the phase gradient image (Fig. 3(b)) and retrieve the phase image (Fig. 3(c)) by applying the Fourier derivative theorem [18] pre-treated by an anti-symmetric derivative integration proposed by Bon et al. [19]. Multilateral shearing interferometry technique allows a direct measurement of phase gradient in multiple spatial directions. Therefore, we can use a derivative closure map (Eq. (1)) to get direct feedback of the measurement as presented in Fig. 3(d).

 

Fig. 3. Top row: raw image made with grating orthogonal periodicity of 24 $\mathrm{\mu}$m (a); phase gradient (b); phase image (c). Bottom row: phase derivative closure map (d); confidence map merged with the phase gradient (e) and phase image (f) emphasizing especially the under-sampling at the edge of the fiber.

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This image highlights different areas, especially at the fiber edges and some pixels with extreme gray values. As described above, treatments on the phase derivative closure map are applied in order to get the confidence map. This confidence map is merged with the gradient and phase images as presented in Figs. 3(e) and 3(f) respectively. For a better visualization of the merged images, cyan alerts, related to the noise issue, are not displayed. One can see local red alerts revealing the presence of phase dislocations (orange arrows in Fig. 3(e) related to the yellow arrows in Fig. 3(d) on the phase derivative closure map image), as well as blue alerts revealing under-sampling on the edges of the fiber and they are not limited to this area. Indeed, in Fig. 3(f), some blue alerts are present inside the fiber (see inset, red arrow) as well as outside.

To highlight the utility of the confidence map and present the evolution of the alerts as a function of the experimental parameters variations, three additional acquisitions were made in the same experimental conditions but for a grating periodicity a of 20 $\mathrm{\mu}$m, 16 $\mathrm{\mu}$m and 12 $\mathrm{\mu}$m. The retrieved phase images and the merge with their respective confidence map are shown in Figs. 4(a) and 4(b) for a = 20 $\mathrm{\mu}$m and Figs. 4(c) and 4(d) for a = 12 $\mathrm{\mu}$m (the result for a = 16 $\mathrm{\mu}$m is not shown). By comparing the phase images, we can see that the sample edges sharpness improves when the grating periodicity is reduced. This is confirmed by a decrease of blue and red alerts among the associated confidence maps (see the left histogram in Fig. 5). More precisely, the number of blue alerts drops from 6.15 % of the total pixel image when using an orthogonal grating periodicity of $a = 24$ $\mathrm{\mu}$m to 1.83 % when using $a = 12$ $\mathrm{\mu}$m (3.12 % for $a = 20$ $\mathrm{\mu}$m and 2.32 % for $a = 16$ $\mathrm{\mu}$m). At the same time, the number of red alerts decreases continuously when going from $a = 24$ $\mathrm{\mu}$m to 16 $\mathrm{\mu}$m, before increasing again for $a = 12$ $\mathrm{\mu}$m. The noise value related to the phase measurement (cyan alerts, not displayed on Figs. 3 and 4) is quite stable for all grating configurations tested (see the right histogram on Fig. 5).

 

Fig. 4. Phase images made with a grating orthogonal periodicity of 20 $\mathrm{\mu}$m (a) and 12 $\mathrm{\mu}$m (c) with the corresponding confidence map merging (b,d). One can see the diminution of the dislocation (red) and under-sampling (blue) alerts when the sampling is improved.

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Fig. 5. Histogram of the evolution of the under-sampling and phase dislocation alerts as a function of the orthogonal grating periodicity (left). Histogram of the evolution of the noise alert calculation by taking the ratio standard deviation over the mean of the noise alert value, as in function of the orthogonal grating periodicity (right)

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Another way to show the variation of the confidence map alerts could have been to acquire images at low and high exposure time in order to induce two different noise levels on the raw image.

3.2 Carbon fiber reinforced polymer – CFRP

As final illustration, an acquisition was made on a carbon fiber reinforced polymer (CFRP) sample. CFRP is widely used in aerospace industries and its production is mastered. For the purpose of this work, the studied piece was manufactured in a controlled laboratory environment and was especially made of sixteen plys of align carbon fibers with a diameter of 12 $\pm$3 $\mathrm{\mu}$m (successive plys are oriented at -45° ; +45°) surrounded by Epoxy resin. This kind of sample serves as model for laboratory lightning damage experiment [20]. The considered sample is a pristine one which has not suffered from lightning tests. Figure 6 presents the CFRP sample dimension as well as our laboratory X-ray bench. Here, $d_{so} = 25$ cm and $d_{sg} = 17$ cm with no change for $d_{sd} = 57$ cm giving roughly a grating magnification of $G_g = 3$ and an object magnification of $G_s = 2$. The grating orthogonal periodicity is $a = 16$ $\mathrm{\mu}$m. Other experimental parameters are the sames as before.

 

Fig. 6. CFRP sample with the imaged region indicated by the red dash line rectangle (left). Experimental bench with a microfocus X-ray source, 2D-gratings, CFRP sample and a high-resolution detector (right).

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Figure 7(a) presents the phase retrieval of a CFRP sample region (indicated on Fig. 6 by a red dash line rectangle). We can see two areas (see yellow arrows) with what seems to be strong artifacts. The outputs of the associated confidence map are displayed separately for a better visualization in Figs. 7(b), 7(c), and 7(d). Both red and blue alerts confirmed that these two areas are indeed artifacts. It can be also noted that these artifacts are not linked to a local increase of the noise, as illustrated by the cyan alerts (see yellow arrows on Fig. 7(d)) but linked to a local phase variation more abrupt than the sensitivity range of the imaging system.

 

Fig. 7. Phase image of the CFRP sample (a) and the associate confidence map with the dislocation alerts (b), the under-sampling alerts (c) and noise alerts (d) indicating artifact on the edge of the object (yellow arrows) and an over estimation of the phase gray values linked to the carbon fibers (area indicated by the red brace).

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Another interesting comment can be done on the core of the sample, spotted by a red brace on Fig. 7(a). Indeed, on the phase image the carbon fibers alignment seems to be highlighted locally by dark gray level lines. However, it appears clearly, thank to the dislocation and under-sampling alerts, that these gray contrast levels are not phase information but artifacts. In addition, the gray levels of the sample are not uniform locally. It could have been interpreted as a contribution of the noise but as we can see on the Fig. 7(d), the noise is quite uniform and the heterogeneity of the phase image is mainly due to phase dislocation and under-sampling of the carbon fibers of diameter size of 12 $\pm$3  $\mathrm{\mu}$m which is close to the phase sensitivity limit.

4. Discussion

All gradient based X-ray phase contrast techniques require a post treatment of the image acquisitions in order to extract the phase information. These treatments can cause artifacts and induce incorrect interpretations. The example of the canonical PMMA sample presented in Figs. 2 and 3, gives a first indication of what level of confidence we can have at the edge of the phase object. Indeed, in this area, a saw-tooth shape appears, which can lead to a confusing interpretation of the intrinsic shape of the sample for more complex geometries. The phase derivative closure map gives a first warning, but in order to connect gray intensity values to physical information, the confidence map is built and displayed on the phase measurement images (see Figs. 3 and 4).

As expected, the number of phase dislocation and under-sampling alerts decreases together with the grating orthogonal periodicity $a$, except for the case of phase dislocation alerts when $a=12$ $\mathrm{\mu}$m. This effect arises due to the appearance of a large number of red pixel alerts outside the sample (see green arrows in Fig. 4(d)) compared to those inside the sample (see red arrow). This is most likely due to local grating flaws which can induce abrupt local phase variations leading to a red pixel alert made by the confidence map algorithm. So the confidence map can provide alerts not only related to the quality of the image sample produced but also on the quality of the material used to perform the measurement (here the 2D-checkerboard grating). Finally, the stable value of the noise emphasize that $\epsilon _n$ is not dependent on grating periodicity variations but more on the Fourier demodulation used to obtain the gradients and phase images.

At last, the confidence map is applied to a real object, with a mastered design from the aeronautic domain, a CFRP sample. Again, on this more realistic example, the confidence map helps to interpret the phase image. Indeed, the dark gray level lines could be interpreted as strong phase variations induced by the carbon fibers, but as evidenced by the confidence map, they correspond mainly to dislocation alerts (red). Also a very large part of the bulk of the sample is branded by under-sampling alerts (blue). Indeed, we know that the carbon fibers (about 12 $\pm$3 $\mathrm{\mu}$m in diameter) are sampled between 3 to 5 pixels. However, the phase sensitivity limit of the system $(a G_g) / S_{det}$, which ensures phase variations to be properly sampled, is 7 pixels. Therefore, the phase sensitivity limit is overtaken and the number of under-sampling and dislocation alerts is important. On the other hand the contribution of the noise alerts is uniform in this sample area which tells us that the major artifacts contribution is linked to the carbon fibers. A small nuance can be noted on Fig. 7(d): with the actual threshold method, the noise alert map can report small residual values (see red arrows) of a strong artifact highlighted by $\epsilon _d$ and $\epsilon _u$ (see yellow arrow on the right Figs. 7(b), 7(c)). However, this approximation does not affect the interpretation of the artifacts since they are correctly pointed out by red and blues alerts.

The confidence map is really a qualification of the results given by the $C(x,y)$ calculation, not linked to an a priori evaluation of the performance of the set-up, but directly evaluated from the data themselves. Therefore, an image interpreter can be directly alerted to the fact that the saw-tooth shape on the edges of the canonical sample or the dark gray level lines on the CFRP sample, are truly artifacts. This is very interesting especially when the shape and texturing of an object of interest give important information. From the point of view of an experimenter, the confidence map is a very useful tool to improve the acquisition configuration. As we can see on the grating periodicity variation (Fig. 4) if the goal is to make a measurement of the phase variation at the center of the PMMA fiber, edges are less important and a grating periodicity of 24 $\mathrm{\mu}$m can be considered. On the other hand, if the experimenter wants to optimize the edge PMMA fiber measurement, the best choice is a grating periodicity of 12 $\mathrm{\mu}$m. The confidence map is a means of attaining the best compromise in terms of acquisition parameter and image quality.

5. Conclusion

Our aim in this communication has been to present a tool, that we call the confidence map, applicable to gradient-based X-ray phase contrast techniques. The confidence map provides a means for optimizing an experimental set-up, but is also an important tool for an end-user who must interpret the final image and make decisions based on it. As an optimization tool the confidence map can highlight the phase measurement sensitivity required and the possible flaws linked to the experimental set-up. As an aid for image interpretation, the confidence map has clear and simple color alerts, with a possible merging with the phase images produce.

This approach can be adapted to any type of gradient-based X-ray phase contrast technique. It was applied here to a microfocus X-ray tube using multi-lateral shearing interferometry allowing a multiple direction phase derivatives evaluation from one measurement. For future work in phase contrast tomography, confidence maps can be used as prior input to minimize the artifact propagation in the iterative tomography process.