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A new phase demodulation approach is proposed that uses windowed Fourier transforms to achieve high spatial resolution in fringe pattern analysis with a high signal-to-noise ratio for single-shot X-ray grating-based interferometry. Conventionally, Fourier transforms have been used to demodulate single-fringe patterns, but this requires a fringe pattern with a long period to obtain an acceptable signal-to-noise ratio among the demodulated parameters. However, by controlling the signal-to-noise ratio, the spatial resolution of demodulated parameters is degraded below that obtained from the phase-stepping method, which requires several images to obtain these parameters. In this paper, we introduce the use of a windowed Fourier transform with a process for analyzing the objective spectrum in isolation from other spectra on the Fourier space to overcome the limitations of the Fourier transform method. It is proved that with suitable assumptions the objective spectrum is isolated theoretically, and the spatial resolution is improved by practically accepting the limitations from the assumptions. We demonstrate the validity of the proposed method by comparing the modulation transfer function of a synthetic phantom with the conventional FT method. The proposed method is also valid on practical data obtained by an experimental setup, by which it is demonstrated that a high spatial resolution with high signal-to-noise ratio can be achieved by our proposed method.
© 2014 Optical Society of America
1. Introduction
Recently, X-ray phase imaging (XPI) has been studied as a means to overcome the limitations of conventional X-ray absorption images, which cannot be used to visualize soft matter such as biomaterials [1–5]. In XPI systems, the phase demodulation method, by which the captured pattern is analyzed to obtain the physical parameters of samples, is a key technique that depends highly on the phase contrast system and the concepts of that system.
From among several XPI methods, X-ray Talbot interferometry (XTI), a kind of differential interferometry, has been investigated by several groups as a promising method [2, 3]. A typical XTI system requires a set of gratings that produce a detectable interference fringe pattern; this pattern then includes information on the measured sample. The first grating is used to produce an interference pattern from the incident coherent X-ray, which is distorted by the sample. However, micrometer-scale interference patterns are not detectable by conventional X-ray detectors, which have a spatial resolution in the tens of micrometers. Thus, a second absorption grating is used to enhance the scale of the fringe pattern by generating the moiré fringe pattern, which is visible to conventional detectors.
Most studies of XTI systems adopt a set of one-dimensional (1D) gratings, and a phase-stepping (PS) method is used for demodulation [6, 7]. An advantage of this combination is the simultaneous achievement of high spatial resolution and a high signal-to-noise (SN) ratio. PS methods require several images shot with changing the relative position of the grating along the axis perpendicular to the grating pattern, and the physical parameters of the object are demodulated from each pixel. This offers the highest possible spatial resolution. Furthermore, the PS method can take a high modulation amplitude of a fringe pattern and still maintain a high SN ratio. If one-directional spatial resolution is the most important measure (as in, for example, computational X-ray phase tomography), then 1D XTI with a PS method is the best choice.
However, two-dimensional (2D) XTI systems have also been studied [8–10]. In a projection imaging system, 2D information is preferred for constructing the integrated phase information of the imaged object from the differential phase information. 2D XTI can measure the two-directional differential phase information of a sample object [9]. It has been shown that a noiseless integrated phase can be obtained by using 2D differential images because these images complement each other. However, the PS method for 2D XTI requires a larger number of shots than the PS method for 1D XTI and also requires a precise control system for two 2D gratings. To overcome these difficulties, we have studied a one-shot XTI system with 2D structured gratings [11–13]. We adopted the Fourier transform (FT) method first introduced by Takeda et al. [14] as the demodulation method for single-shot XTI.
However, the spatial resolution of the FT method is theoretically lower than that of the PS method. The spatial resolution of the FT method is, in principle, determined by the radius of the filter function used in the Fourier space. Therefore, to extend the radius of the filter function, the FT method requires a fringe pattern with a short period. Unfortunately, a shorter period in the fringe pattern reduces the modulation amplitude observed during actual use of the XTI because of properties of the modulation transfer function (MTF) of the detector; this means that the SN ratio of the demodulated parameters deteriorates with shortening the period. Therefore, setting the period for the FT method forces a trade-off between spatial resolution and SN ratio.
To overcome this dilemma, we proposed a demodulation approach that uses a windowed Fourier transform (WFT) with a Gaussian window [15]. We indicated that under appropriate assumptions the WFT with subsequent analysis could achieve a qualitatively higher spatial resolution than possible with the FT method. However, that research did not give a general formulation of this approach or quantitative results. The purpose of this paper is to describe the generalized formulation of the method and to demonstrate by simulation and experiment that the proposed method can achieve high spatial resolution with a SN ratio equivalent to that obtained by methods with lower spatial resolution.
2. Spatial resolution and phase sensitivity
2.1. Fourier transform method for one-shot 2D XTI
For 2D XTI, the selection of grating design is important in determining performance. Many grating designs are possible; Zanette et al. [10] gives a large selection. From among the grating designs, we choose a combination of a π-checker-shaped phase-shift grating and a mesh-shaped absorption grating for the first and second gratings, respectively, due to the high contrast of the resultant fringe patterns; these are the same grating patterns used in our previous studies.
With our chosen gratings, the fringe pattern is approximated by the following:
where the fringe pattern distortion caused by the object is described by five parameters: a(x, y), bx(x, y), by(x, y), Px(x, y), and Py(x, y). Parameter a(x, y) is related to the absorption of X-rays by the object. Parameters bx(x, y) and by(x, y), which are related to the amplitude of fringe modulation, express X-ray scattering from the microstructure of the object in the x and y directions. Parameters Px(x, y) and Py(x, y) are the amount of the phase shift of the fringe pattern in the x and y directions, respectively; they reflect the differential phases of the object along the x and y directions. These five parameters are affected by refraction from the sample object. The purpose of demodulation is to determine these parameters from the obtained fringe pattern.Here, we assume that the direction of the fringe pattern is aligned on the x and y axes. Parameters ωx and ωy are the fundamental angular frequencies of the carrier fringe pattern (hereafter, carrier frequency) in the x and y directions, respectively.
The carrier frequencies are artificially determined by optimizing the grating specification and system geometry. Because the spatial resolution of the FT method for one-shot XTI is determined by the carrier frequency, it is important to define the desired performance of the XTI system to decide on a period for the carrier frequency. The FT method was first introduced by Takeda et al., and the parameters mentioned above are demodulated by using the following expressions:
The demodulated parameters A(x, y), P(x, y), and S(x, y) are the absorption image, the differential phase, and the signal degradation from X-ray scattering, respectively. Ĩ(kx, ky) is the FT of the fringe pattern I(x, y); ℱ−1{···} indicates the inverse FT operation. G(kx, ky) is an even filter function for cropping the spectrum in Fourier space, and (qx, qy) is the peak position of the Fourier spectrum. The parameters are demodulated from the Fourier spectrum corresponding to the carrier frequency. Note that the spatial resolution, determined by the filter-function size, is limited to avoid overlap between neighboring spectra; for this, the carrier frequency should be as high as possible.
However, as mentioned above, during actual laboratory experiments, the carrier frequency should be as low as possible to compensate for effects of the MTF of the X-ray detector. In other words, the modulated fringe pattern becomes invisible because the pattern is blurred by scintillation from the detector. This seems to be the primary reason that the PS method is typically used for XTI study: the PS method does not use the fringe pattern and so the modulation amplitude can be higher.
2.2. Windowed Fourier transform method with carrier isolation process for one-shot 2D XTI
To overcome the trade-off between resolution and SN ratio, we newly introduce a demodulation method that uses WFT and a carrier spectrum isolation process (hereafter, CI process). First, we discuss demodulation by WFT. WFT is defined as
where f(x, y) and g(x, y) are the input function and the window function, respectively. The ordered pair (x, y) indicates the spatial coordinates, and the ordered pair (x′, y′) denotes the center of the window function. Phase demodulation (retrieval) using WFT has been already reported; for example, Kemao [16] used WFT to reduce noise in the demodulated phase, in which the phase is drastically changed.A simple demodulation process using WFT is achieved by using the following expressions, which we call simple-WFT:
It is known that Eqs. (6)–(8) are mathematically equivalent to the FT method, Eqs. (2)–(4), by the convolution theorem. The Fourier transform of the window function g(x, y) used in the simple-WFT method is mathematically equivalent to the filter function G(kx, ky) that is used in the FT method. Thus, the simple-WFT method has the same limitations as the FT method. Namely, the spatial resolution of the FT method is improved by widening the filter function G(kx, ky) while the spatial resolution of the WFT method is improved by narrowing the window function g(x, y). Therefore, the simple-WFT method offers no advantage over the FT method.
Therefore, some CI process is required for analyzing the WFT spectrum. As mentioned above, the effective area of window function should be to be narrow to achieve high spatial resolution, and it is efficiently narrow in the region where CI process is required. Therefore, we can rely on the assumption that the demodulated parameters— differential phase, absorption, and scattering—are constant within the narrow area of the window function:
This assumption is deemed reasonable if there are no notable changes in fringe frequency in most of the cropped area considered by the window function. Furthermore, by considering that an XTI system is a differential interferometer, the differential phase is not notably changed within a short distance. Therefore, this assumption is reasonable for most of the area of the captured fringe image.We obtain the following expression by performing WFT under this assumption:
where Bx and By are defined as and The function g̃(kx, ky) is the Fourier transform of window function g(x, y). Generally, the WFT of the fringe pattern expressed by Eq. (1) consists of nine peaks whose shapes are determined by the FT of the window function g(x, y). Parameters can be demodulated similarly by using Eqs. (6)–(8).This also implies that the function g̃(kx, ky) is extended by narrowing the window function g(x, y). Specifically, the effective radius of the window function g(x, y) should be controlled because the FT of the window function g̃(kx, ky) does not affect the neighboring spectra. In the limit, this is equivalent to the simple-WFT method. Therefore, we add a CI process to analytically isolate the desired spectrum from the neighboring spectra.
Although several candidate isolation processes exist, here we use a simple process on the WFT spectrum data at (ωx, 0), (0, ωy), and (0, 0) at the beginning of the CI process. An example of this isolation process, specified for the Gaussian window function, has been discussed in our previous research. Thus, we introduce the general formulation of our idea for the detailed understanding in this paper.
Here, we focus on the specific position (x′, y′), so coordinate expressions are omitted to simplify the expressions. We have the following expression.
and Furthermore, the window function is a product of one-directional even functions, H̃(kx, ky) = H̃(kx)H̃(ky), so Eqs. (18) and (19) can be factorized as follows: and Because the second factors of Eqs. (20) and (21) share a common expression, we can replace this factor by Ty. By calculating 𝒮I(0, 0)/H̃(0) − 𝒮I(ωx, 0)/H̃(+ωx) we obtain where ℜ{...} and ℑ{...} indicate real and imaginary parts, respectively, and and are factors defined as follows: From these expressions, we can calculate the factor Rx ≡ aBxTy as Furthermore, by subtracting Rx and from (20), we obtain Finally, we isolate B by dividing RxH̃(0) by (26) as We obtain by(x′, y′)exp{−iPy(x′, y′)} in a similar manner. Thus, the modulation amplitude bx(x′, y′), by(x′, y′) and the differential phase Px(x′, y′), Py(x′, y′) are calculated by taking the absolute values and angular values of Bx and By, respectively. The absorption value a(x, y) can also obtained by inserting Bx and By into Eq. (18), for example. Consequently, the chosen spectrum can be isolated, and the parameters can be demodulated without interference from other spectra under the given assumption by the proposed process.In principle, this isolation process does not depend on the period of the fringe pattern. This indicates that the limitations of the FT method and the simple-WFT method are overcome and the window function can used to obtain higher spatial resolution by means of the CI-WFT method.
3. Demodulation results
3.1. Gaussian window function
In this section, we use the FT method and the proposed CI-WFT method to process results. In the previous subsection, we mentioned that any even window function could be used. For the sake of simplicity we assume that the Gaussian function is used as the window function. Because the Gaussian window function maintains its shape in the Fourier space, it is convenient to discuss quantitative differences between the CI-WFT method and the FT method. The window function is defined as
where σ is the standard deviation of the Gaussian function. Because a Gaussian is Fourier transformed into a Gaussian, g̃(kx, ky) is expressed as Therefore, H̃(kx, ky) and H̃(k) in Eq. (17) are defined as In this case, the relation between the width of the window function g(x, y) in Eqs. (6)–(8) and the filter function G(kx, ky) in Eqs. (2)–(4) is clarified by comparing these equations. Plainly stated, the simple-WFT method with Gaussian window function having standard deviation σ is equivalent to the FT method with Gaussian filter function having standard deviation 1/σ.3.2. Comparison using synthetic objects
To quantify the effectiveness of our proposed method, we compare the demodulation results of the proposed method on a synthetic sample with those obtained by the FT method. Figure 1(a) shows a sample of the synthetic fringe pattern used in this subsection. The period of the fringe pattern is taken as 6 pixels per fringe. Figure 1(b) shows the spectrum intensity pattern of Fig. 1(a). As is shown, the spectrum intensity map includes nine peaks, as expected from Eq. (14). To be more precise, higher-order harmonics also appeared in Fig. 1(b) but they were ignored because they are relatively small and assumed to have negligible effects on the demodulation results.
As is shown in Eqs. (2)–(4), the spectrum corresponding to the carrier frequency is cropped by a filter function in the FT method. The effective area of the filter function must be determined so as not to include the neighboring spectra; this is shown by the solid line in Fig. 1(b). In this Gaussian case, we choose π/9 as the standard deviation of the Gaussian filter, so that three standard deviations is π/3 and the effective area is included. Therefore, for example, the standard deviation of the window function equivalent to the filter function is .
Figure 2 is a comparison of the demodulation results with changing σ on a synthetic sphere. Figures 2(a)–2(c) are the demodulated differential phase images obtained by using the FT method, where the standard deviations of filter function are taken as 3/π, , and 6/π, respectively. The effective areas in each case are also illustrated in Fig. 2(b): The areas surrounded by the dashed line and the dash-dotted line represent standard deviations and 6/π, respectively. An artifact from unchosen spectra occurs because these neighboring spectra are included within the effective filtering area in Figs. 2(b) and 2(c).
Fig. 2 Differential phase images demodulated along x direction with changing the filter radius. Demodulation results of the FT method are shown in the top and bottom row with filter radius (a) 3/π, (b) , and (c) 6/π, respectively. (d)–(f) Each image shows the results of the CI-WFT method with a window radius equivalent to the filter radius of the image directly above it.
Figures 2(d)–2(f) are the demodulated results obtained by using the proposed CI-WFT process with standard deviations equivalent to the filter function radii in the FT method of 3/π, , and 6/π, respectively. It is clear that the artifact has vanished and the differential phase images have become clearer. This indicates that the CI process effectively removes the effect of neighboring spectra and so the equivalent filter can be expanded outward. This also indicate the spatial resolution of obtained parameters can be improved by an isolation process. This fact is of practical importance for experiments, as shown in the next subsection.
Figure 3 shows the improvement of the spatial resolution obtained by calculating the MTF curve of the differential phase for the tilted pyramid synthetic samples. The calculation process of the differential phase MTF is schematically illustrated in Fig. 3(a). First, we calculate the differential phase image of a tilted synthetic pyramid by each method, as shown in the top-left image of the figure. Second, because the differential phase of the pyramid shape includes a plateau area, the step region of the image (128 × 128 pixels) is cropped after correcting the tilt, as shown in top-right image of the figure. The tilt is applied to avoid position dependency in the demodulation results. Third, the MTF curve is obtained by Fourier transformation of the derivative of the step image. The results are shown in Fig. 3(b): The open rhobus and open circle show the MTF obtained by using the FT method with a 6 pixel fringe pattern and a 3 pixel fringe pattern, respectively. The closed triangle and closed square show the MTF obtained by using the CI-WFT method with the 6 pixel fringe pattern. The MTF of the CI-WFT method becomes wider than that of the FT method at the 6 pixel fringe pattern case, and the MTF of the CI-WFT method at σ = 1.4 achieves to the results of the FT method with 3 pixel fringe pattern. This indicates that we can achieve a spatial resolution equivalent to a 3 pixel period fringe pattern with using a 6 pixel period fringe pattern.
Fig. 3 (a) A schematic diagram of MTF calculation. (b) MTF curves obtained by using the FT method, the simple-WFT method, and the CI-WFT method.
3.3. Experimental results
We also applied the proposed method to non-synthetic images. Experimental images are captured by using the 2D XTI system in our laboratory [12]. The experiments are performed using a conventional X-ray source (Rigaku ultraX 18) and a silver target. The incident X-rays are assumed to be 22 keV, and the gratings used in the system are designed for that effective energy. The phase grating is fabricated with a 4.3 μm checkerboard pattern. The moiré fringe pattern is captured by the detector (Rad-icon Shad-o-Box; pixel size: 48 μm), and the fringe pattern is captured at the third self-image distance. The magnification of the system is set to 1.41×. The fringe period, which is controlled by rotating the grating, as explained by Momose [6], is set to 6 pixels wide in both the x and y directions. The demodulated data presented here were normalized by the reference data without the object because practical experimental datasets are nonuniform due to grating distortion and other effects.
As was shown in the previous subsection, demodulation results typical of a given fringe pattern can be achieved with a fringe pattern that has a longer period by the CI-WFT method. Figure 4(b) shows the relation between the visibility V of a reference fringe pattern and the period of that pattern in this experimental setup. The visibility V for the fringe pattern expressed by Eq. (1) is calculated as the average of the local visibility v(x, y) ≡ 2b(x, y)/(a(x, y) + b(x, y)), where b(x, y) = (bx(x, y) + by(x, y))/2. It is known that visibility is drastically reduced with decreasing the fringe period. Generally, the modulation amplitude of fringe pattern decreases as the period of fringe is made shorter; this occurs because of the detector’s MTF, and it is easily understood that modulation amplitude strongly affects the SN ratio of modulated parameters. From this viewpoint, a longer period is preferred to maintain a tolerable SN ratio for demodulated parameters; it has been shown that the modulation amplitude is saturated at a fringe period of about 10 pixels. However, longer periods cause some problems. First, fringe patterns with long period tend to create location dependency in the phase sensitivity; this dependency is related to the phase of the fringe pattern. The phase sensitivity at the peak of fringe pattern, in particular, becomes worse than that at the slope of the fringe pattern; thus, line artifacts tend to appear in the demodulated parameters. Second, the Fourier spectra pattern of longer periods have higher order spectra, which are not assumed in Eq. (1). These higher-order spectra also cause other artifacts. A 6 pixel fringe was chosen as a balanced value that minimizes artifacts in the demodulated phase for our current system.
Fig. 4 (a) Fringe pattern captured by our experimental setup. (b) Relation between visibility and the period of the fringe pattern.
Figures 5 and 6 show the demodulation results of Fig. 4(a) from the FT method and the CI-WFT method, respectively. Here, the standard deviation σ used in the window function in Eq. (28) for Fig. 6 is set to 1.43, where the effective filter radius on the FT method is double that in Fig. 6 and is equal to the FT method results for a 3 pixel fringe period. Although five parameter images were successfully demodulated in both cases, the sharpness of the demodulated phase is improved in Figs. 6(a)–6(f). This means that the spatial resolution corresponding to the FT method results for a 3 pixel fringe period pattern can be experimentally achieved by using our proposed method. Additionally, Fig. 4(b) indicates that the modulation amplitude for a 3 pixel fringe period was one-fourth the modulation amplitude for a 6 pixel fringe period. Directly stated: the CI-WFT method can achieve the same spatial resolution as an FT method that uses a shorter fringe period pattern and can do so with the high SN ratio as in the case of 6 pixel fringe period pattern. From a different point of view, the proposed method helps to reduce X-ray dosage from XTI without sacrificing spatial resolution; this is important for practical use of XTI systems.
Fig. 5 Demodulation results obtained by the FT method. (a) Absorption image. Differential phase images along the (b) x and (c) y directions, and (d) integrated phase image. Modulation amplitude along the (e) x and (f) y directions.
Fig. 6 Demodulation results obtained by the proposed CI-WFT method. (a) Absorption image. Differential phase images along the (b) x and (c) y directions, and (d) integrated phase image. Modulation amplitude along the (e) x and (f) y directions.
4. Conclusion
In this paper, we proposed a WFT-based demodulation method with a carrier-spectrum isolation process for an FT-based demodulation process. One-shot XTI systems force a choice between high SN ratio and high spatial resolution for demodulated parameters due to visibility characteristics of the fringe pattern. In a conventional FT method, the period of the fringe pattern determines the spatial resolution of the demodulation parameters because of limitations in the filter function. However, our proposed method can overcome these limitations in the filter function by using WFTs and an isolation process on the objective spectrum. The results of simulation with synthetic objects indicate that the proposed method extends the filter function under some suitable assumptions, and we confirmed that the spatial resolution of the demodulated parameters was improved by calculating the MTF.
We also applied the proposed method to experimental data obtained from a laboratory 2D XTI system. The experimental results showed that the spatial resolution of demodulated parameters was successfully improved without a corresponding decrease in SN ratio. These results show that the proposed method can achieve demodulated parameters with high spatial resolution and high SN ratio.
Acknowledgments
The authors thank Prof. Mitsuo Takeda of Utsunomiya University for helpful discussions and advice. We also thank Mr. Takayuki Teshima and Mr. Yutaka Setomoto (Canon Inc.) for the fabrication of gratings. We are grateful to Mr. Naoki Kohara, Mr. Chidane Ouchi, and Mr. Masanobu Hasegawa (Canon Inc.) for helpful discussions. Finally, we express our gratitude to the members of the Frontier Research Center of Canon Inc.
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