Influence of Data Acquisition Algorithms on X-Ray Phase Contrast Imaging Computed Tomography

X-ray phase contrast imaging (XPCI) is a radiography technique that leverages the wave properties of X-rays to obtain phase contrast data in addition to traditional absorption contrast data. In the hard X-ray regime, the phase cross section can exceed the attenuation cross section by up to three orders of magnitude for elements with low atomic numbers [1,2]. Volumetric reconstructions of XPCI data can be reconstructed using computed tomography (CT) techniques [3,4], which include capturing numerous angular samples of a rotating object.

We implement a table-top grating-based XPCI system with a laboratory tube source using a Talbot-Lau interferometer [5] (Fig. 1). XPCI reconstruction results in three image products: tau, dark field, and differential phase (Fig. 2). The tau image is equivalent to traditional attenuation-contrast radiography, the dark field image is due to small-angle scatter, and the differential phase image reveals material interfaces. The differential phase image may be integrated to yield an absolute phase image.

The combination of interferometer motion required for grating-based XPCI and sample rotation mandated by CT dictates at least two distinct image acquisition techniques: capture an entire phase image set before rotating the object to a new angle and capture a full rotational projection set before phase-stepping the analyzer grating. We compare two simple data acquisition techniques and their influence on resulting XPCI CT reconstructions.

The sample object is a paper cup containing numerous 1/2 in. (12.7 mm) diameter plastic spheres (Fig. 2). The spheres are composed of six different materials: nylon, clear acrylic, polyamide-imide Torlon, Delrin acetyl resin, and polypropylene.

Current X-ray detector technology cannot directly measure phase. The Talbot-Lau interferometer instead transforms phase information into a periodic intensity fringe pattern that the detector can measure, which enables phase reconstruction.

The source grating G0 divides a finite incoherent X-ray source into a set of smaller individually coherent (yet mutually incoherent) sources. The phase grating G1 imposes a periodic phase shift of magnitude π on the wavefront, creating an interference pattern across the detector. The analyzer grating G2 samples the interference pattern and produces an observable moiré fringe pattern on the detector [6,7].

Phase reconstruction with this apparatus requires the acquisition of a sequence of images captured as the analyzer grating G2 is stepped transverse to the optical axis. The movement of G2 causes lateral movement of the fringe pattern. We capture the requisite images for XPCI reconstruction by translating the grating over a full grating period at regular intervals and capturing a radiograph at each grating position. This process, known as phase stepping [8], samples the fringe pattern and must be completed for both reference and sample image sets. The number of phase steps is dictated by the Shannon–Nyquist Theorem [9].

In these equations, the subscript indicates whether the sinusoid parameter value is derived from the sample (S) or reference (R) image. With the FFT method, XPCI reconstruction quality heavily depends on the fidelity of the sinusoidal fit, which in turn depends on the data sampling period and uniformity.

We employ the traditional Feldkamp–Davis–Kress analytic algorithm [10] to reconstruct the CT data.

All gratings used in the Sandia National Laboratories (SNL) laboratory grating-based three-dimensional X-ray phase contrast imaging system were fabricated at the Microsystems Engineering, Science and Applications (MESA) facility at SNL [11]. The grating features are summarized in Table 1.

The X-ray source is a Philips MG 225 kV laboratory bench-top mini-focus tube source operated at tube potential V = 41 kV_p and current I = 15 mA. The detector is a PaxScan^® 1313DX digital image receptor from Varex Imaging Corporation with 1024 × 1024 pixels and pixel pitch p = 127 μm operated in 2 × 2 binning mode. The entire system is mounted on a floating optical table to minimize vibrations. A detailed description of the grating-based XPCI CT system used in this investigation is available in the study by Thompson et al. [12].

X-ray phase contrast imaging CT requires the acquisition of radiographs that satisfy both XPCI and CT reconstruction requirements. The image data set must include sufficient projection image sets for CT reconstruction, and each projection set must contain radiographs adequate for XPCI reconstruction of that projection. Therefore, data acquisition algorithms must coordinate two mechanical stages: the analyzer grating G2 horizontal translation stage and the sample rotational mount.

The motion requirements for XPCI CT indicate that each stage may be considered as a motion loop, a set of movements that could occur repeatedly. The organization of potential data acquisition algorithms then follow a nested loop form in which one motion loop executes within the other. Two acquisition algorithms are then readily apparent: in one, sample rotation is the outer loop and grating translation is the inner loop, and vice versa in the other. Other data acquisition algorithms for XPCI CT have been explored, including interlaced grating translation [13,14], helical XPCI CT [15], and no grating movement at all [16]. We consider the two simple nested loop algorithms, which we call Step First and Rotate First. In our implementation, the angular positions of the sample stage and the phase step positions of the analyzer grating G2 for both motion loops are calculated with an arithmetic progression rather than relative to the previous position to prevent accumulation of position errors.

Theoretically, the two data acquisition techniques capture the same images but in a different order. Nevertheless, there are distinct advantages and disadvantages to each algorithm. If a Step First scan fails and does not capture a complete XPCI CT image set, the incomplete data include a subset of projections with complete data for XPCI reconstruction. In the event that a sufficient number of complete projections were captured, a limited-angle CT scan of that data set can be reconstructed. Conversely, a failed Rotate First scan results in an incomplete data set that cannot be reconstructed into XPCI image products since no individual projection contains a complete fringe profile for XPCI reconstruction. A traditional absorption CT scan may be reconstructed from a single sample rotation. Only the absorption image is available, and the visible fringes introduce significant reconstruction artifacts. However, Rotate First offers the potential for shorter scan times compared to Step First. If the scan runs at a high frame rate, Rotate First may rotate the sample continuously, with G2 translating to the next position at the conclusion of each sample revolution.

In practice, the data captured by the two acquisition algorithms differ slightly due to the system error. Numerous potential hardware challenges that introduce acquisition error exist in a laboratory grating-based XPCI system, such as X-ray source fluctuation, detector electronic noise, system thermal drift, mechanical and acoustic disturbances, and phase stepping errors [17]. Acquisition algorithm choice may exacerbate or mitigate certain error sources.

We refer to the acquisition algorithm that rotates the sample in the outer loop and translates G2 in the inner loop as Step First. In Step First, a complete image set required for XPCI reconstruction for a given projection is acquired before rotating to the next projection. When the object is rotated, G2 is reset to its initial position. The process repeats for all projections required for CT reconstruction (Fig. 3(a)). Pseudocode describing the Step First algorithm is presented in Algorithm 1.

The algorithm that translates G2 in the outer loop and rotates the sample in the inner loop is named Rotate First. All the projections required for a CT scan are acquired for a given grating position. G2 then translates to the next position, repeating until the entire image set is captured (Fig. 3(b)). Algorithm 2 outlines Rotate First in pseudocode.

As described in Sec. 2.1, XPCI reconstruction with the FFT method depends heavily on the fidelity of the underlying signal. If the signal is sinusoidal, the phase steps are evenly spaced and the steps occur over exactly one G2 grating period, then the FFT method is reasonably accurate. Figure 4 displays the sampled fringe data for a reference XPCI image set, a set acquired during a Step First scan and a set acquired during a Rotate First scan. The sampled pixel is located away from the sample toward the edge of the field of view for direct comparison between reference and sample sinusoids. Therefore, in the absence of system errors, all three fringe profiles should look similar and have roughly the same amplitude, phase, and offset. The fringe profile from the Rotate First scan is noticeably degraded compared to the reference and Step First profiles. Both fringe profiles also exhibit the significant phase shift relative to the reference signal. The degradation and the aberrant phase shift are likely due to a combination of the hardware errors articulated previously.

Degraded fringe profiles result in reconstruction errors in the resulting XPCI images. The errors are compounded when the images are used for CT reconstruction.

The fringe degradation effect is more pronounced for longer scans. Potential causes include mechanical and/or thermal drift, imprecise, unrepeatable G2 stage motion and flat-field calibrations expiring toward the end of a long scan.

Sinusoidal purity is simple to calculate during reconstruction because the FFT reconstruction calculates and subsequently discards the higher-order coefficients. High purity indicates low error in G2 position and translation during data acquisition. Resampling the raw data may improve signal purity by correcting for phase-stepping errors [18].

Figure 5 shows the example of sinusoidal purity images calculated for example slices of data acquired with the Step First and Rotate First algorithms. The value P(x) = 1 is coded white, so brighter pixels indicate higher purity data. The Step First image is noticeably brighter than the Rotate First image, with a mean Purity of $P¯step=0.8167$ compared to $P¯rotate=0.6969$⁠. A lower purity value and dimmer image indicate that the Rotate First algorithm captures lower quality XPCI data. The Rotate First image also contains a residual background fringe pattern, an indication of misalignment with the reference image set. Misalignment is likely due to the accumulation of phase-stepping errors over the protracted period of time required to acquire a full phase projection set.

The Step First algorithm consistently yields higher sinusoidal purity values than the Rotate First algorithm. We examined the purity images reconstructed from 4 XPCI CT scans (2 Step First, 2 Rotate First) acquired of the same plastic spheres sample, with the same acquisition parameters. We acquired 451 angular projects with 16 phase steps per projection for each scan. The reconstructed field of view is 380 × 380 pixels. The pixels from the images acquired with the Step First have a mean purity of μ_S = 0.8144 with a standard deviation of σ_S = 0.0676. The Rotate First scans result in a mean pixel purity of μ_R = 0.6986 with a standard deviation of σ_R = 0.0687. Thus, Step First reliably captures higher quality grating-based XPCI data in aggregate.

The influence of acquisition algorithm on reconstructed data is most apparent in the differential phase modality. Differential phase images reconstructed from the data acquired with Step First vary in intensity over the course of the scan, while the images reconstructed from corresponding Rotate First data have relatively constant intensity. The intensity variation effect is most easily observed by examining the sinograms of the respective differential phase data sets (Fig. 6). A profile plot reveals a net background shift of about π/2 over the course of the Step First scan, compared to almost no phase shift during the Rotate First scan (Fig. 7). Recall that the differential phase value is due to the phase difference between reference and sample fringe profiles (e.g., see Fig. 4).

Projection intensity variability is a common problem in traditional CT. Fluctuations in the output of the X-ray source during a CT scan can cause projections to differ in brightness. We observe a similar phenomenon in Step First data, but not Rotate First data. While individual projections may have a more regular phase profile using Step First, the shift in the background phase between projections may be substantial. During a Rotate First scan, phase-stepping images in each fringe profile are acquired over the duration of the scan. The temporal spread effectively averages background phase, resulting in a more stable value across projections.

Existing CT reconstruction algorithms normalize projections to account for source intensity fluctuations. When applied to the differential phase data, the traditional normalization produces reconstructions that appear correct. However, the cause of apparent intensity variation in differential phase images is actually background phase shift, not X-ray beam intensity. The mechanism causing the error is located in G2 translation repeatability rather than X-ray source instability. Therefore, the traditional projection normalization used in CT reconstruction may be inappropriate for correcting differential phase data because it targets a different physical phenomenon. Further investigation is required to determine whether the background phase shift between projections is best addressed through hardware improvements or software postprocessing techniques.

X-ray phase contrast imaging CT using a grating-based Talbot-Lau interferometer system requires multiple moving stages to acquire the requisite data for successful reconstruction. Coordination of those stages demands deliberate attention to the design of data acquisition algorithms and their effects on the captured data.

While the choice of whether to place grating translation or sample rotation in the inner or outer acquisition loops primarily determines the order in which scan images are captured, it also influences the XPCI CT reconstruction. The Step First algorithm acquires higher fidelity fringe profiles as evidenced by the consistently higher purity of Step First data. However, the Step First technique is more vulnerable to background phase shifts than the Rotate First algorithm.

Step First results in XPCI CT data sets that can be salvaged if a scan fails and delivers superior sinusoidal profiles and resulting XPCI reconstructions. We therefore prefer Step First when pursuing high-quality volume data.

An investigation of the causes of fringe profile degradation may help mitigate observed artifacts. Potential system error sources include G2 phase-stepping error, system thermal drift, and vibration sensitivity. Fringe profile degradation may also be addressed with postprocessing techniques such as phase-stepping position resampling and background phase shift normalization or minimization.

This article describes objective technical results and analysis. Any subjective views or opinions that might be expressed do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

This work was supported by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. SAND2020-8573 J.

There are no conflicts of interest.