Spectral propagation-based x-ray phase-contrast computed tomography

Abstract.

Purpose

Propagation-based x-ray imaging (PBI) is a phase-contrast technique that is employed in high-resolution imaging by introducing some distance between sample and detector. PBI causes characteristic intensity fringes that have to be processed with appropriate phase-retrieval algorithms, which has historically been a difficult task for objects composed of several different materials. Spectral x-ray imaging has been introduced to PBI to overcome this issue and to potentially utilize the spectral nature of the data for material-specific imaging. We aim to explore the potential of spectral PBI in three-dimensional computed tomography (CT) imaging in this work.

Approach

We demonstrate phase-retrieval for experimental high-resolution spectral propagation-based CT data of a simple two-component sample, as well as a multimaterial capacitor test sample. Phase-retrieval was performed using an algorithm based on the Alvarez–Macovski model. Virtual monochromatic (VMI) and effective atomic number images were calculated after phase-retrieval.

Results

Phase-retrieval results from the spectral data set show a distinct gray-level for each material with no residual phase-contrast fringes. Several representations of the phase-retrieved data are provided. The VMI is used to display an attenuation-equivalent image at a chosen display energy of 80 keV, to provide good separation of materials with minimal noise. The effective atomic number image shows the material composition of the sample.

Conclusions

Spectral photon-counting detector technology has already been shown to be compatible with spectral PBI, and there is a foreseeable need for robust phase-retrieval in high-resolution, spectral x-ray CT in the future. Our results demonstrate the feasibility of phase-retrieval for spectral PBI CT.

1. Introduction

Propagation-based x-ray phase contrast imaging (PBI) utilizes the self interference of a wave-field during free-space propagation to obtain phase information of an object.¹ In contrast to other x-ray phase contrast methods, such as crystal interferometry,² speckle tracking,^3,4 coded aperture,⁵ grating-,⁶ and analyzer-based imaging,⁷ this does not require any additional optical elements nor monochromatic radiation.⁸ Provided an x-ray source with sufficient spatial coherence, free-space propagation causes phase differences introduced by an object to give rise to detectable intensity variations. Phase effects in PBI predominantly appear as pairs of narrow bright/dark fringes at interfaces between materials with different refractive indices. Due to the characteristic size of these fringes, typically a high-resolution detector has to be employed. Furthermore, phase-contrast fringes are superimposed onto the conventional attenuation image, which leads to the necessity of an image processing step called phase-retrieval for quantitative imaging. A prime motivation for the use of PBI is the potential of phase-retrieval to greatly improve signal quality, particularly in combination with computed tomography (CT).⁹ On the other hand, phase-retrieval may also be necessary to remove phase-contrast fringes that occur as an undesired consequence in high-resolution x-ray imaging.¹⁰ There exist a multitude of approaches to perform phase-retrieval, using either one or more input images. Single-image methods demand stringent approximations to be met, such as a weakly attenuating object,¹¹ an object with homogeneous composition,¹² or that attenuation is primarily caused by Compton scattering.¹³ Despite efforts in recent years, phase-retrieval of general, heterogeneous objects using a single image remains a challenge.^14,15 To overcome these limitations, multi-image methods seek to combine multiple measurements using either different propagation distances¹⁶ or illumination spectra.^17–21

The combination of PBI with spectral data is of particular interest, considering the intrinsic potential of spectral data for material-specific imaging, and the potential for spectral data to solve the problem of effective phase retrieval of heterogeneous objects. Figure 1 shows a basic spectral PBI experiment. The intensity profiles of a coherent wave-front after interaction with an object are sketched as a function of propagation distance $R$. These profiles result as a superposition of conventional attenuation, and the characteristic PBI fringes, which both depend on the x-ray energy. CT imaging is realized by rotating the object. Experimental demonstrations of spectral PBI so far have been on two-dimensional (2D) projection imaging.^18–21 We are unaware of any experimental demonstrations of three-dimensional (3D) spectral PBI-CT, which we will showcase in this paper. In the following, we highlight the feasibility of spectral phase-retrieval based on experimental high-resolution spectral PBI CT data taken at the SPring-8 synchrotron in Japan. We show that after successful phase-retrieval, the underlying spectral information allows us to utilize representation methods that are well-established in spectral CT, namely the virtual monochromatic image (VMI)²³ and effective atomic number image.²⁴ We also show that the order of phase-retrieval and CT reconstruction can be interchanged and discuss practical benefits arising from this.

Fig. 1.

Basic spectral PBI-CT experiment. A coherent wavefront interacts with an object, and the resulting intensity is sketched as a function of propagation distance $R$. Different x-ray energies are indicated by the colors. Figure adapted from Ref. 22.

2. Spectral Phase-Retrieval

Existing phase-retrieval algorithms for spectral PBI can be classified into single-step Fourier methods^17–19 and iterative methods.^20,21 All methods share a common approach; transforming spectral data into another basis, spanned either by the coefficients of real materials,^17,18 or those of pseudomaterials according to the Alvarez–Macovski model.^18–21 Here, we use a phase-retrieval algorithm that builds on a single-step Fourier method initially introduced by Gursoy and Das.¹⁹ We will briefly summarize the main idea of the method and introduce the formalism used in the following, with more details on the implementation, advantages and optimum regimes provided in Ref. 18.

Underpinning the approach is the Alvarez–Macovski (AM) model,²⁵ which suggests that x-ray attenuation can be approximated as a linear combination of photoelectric absorption and Compton scattering, regardless of the materials involved. A major reason this approach is particularly useful for spectral PBI is the fact that both Compton scattering and phase-contrast effects are proportional to the electron density ${\rho }_{\mathrm{e}}$. Thus, a mathematical connection between phase and attenuation exists, similar to that used in the single-material approximation,¹² which in turn facilitates stable phase-retrieval.¹⁸ The second cornerstone of this phase-retrieval approach is the use of a linearized form of the transport of intensity equation (TIE).¹⁹ Assuming comparatively small intensity gradients, and moderate phase effects, the linear forward model for the intensity measured in spectral PBI, $I(E)$, is then given as

$$-\ln I(E)=\frac{P_{\perp }}{E^3}+[{\sigma }_{\mathrm{KN}}(E)-\frac{h^2{c}^2{r}_{\mathrm{e}}R}{2\pi E^2}{\nabla }_{\perp }^2]{\rho }_{\mathrm{e},\perp }.$$ (1)

According to the AM model, the dependencies of photoelectric attenuation and Compton scattering on the x-ray energy, $E$, are defined as a heuristic function $E^{-3}$, and the Klein–Nishina cross-section ${\sigma }_{\mathrm{KN}}(E)$, respectively.¹⁸ Phase effects are described by the ${\nabla }_{\perp }^2$-term, where $c$ is the speed of light, $h$ is Planck’s constant, and $r_{\mathrm{e}}$ is the classical electron radius. We also see that the magnitude of phase effects can easily be adjusted to meet the earlier assumption of moderate phase effects by changing the propagation distance $R$. Finally, structural information about the object is contained in the projected nonphysical coefficient for photoelectric absorption, $P_{\perp }$, and the quantitative measure of projected electron density ${\rho }_{\mathrm{e}\perp }$. Note that $I(E)$, $P_{\perp }$ and ${\rho }_{\mathrm{e},\perp }$ are 2D quantities, for which the $(x,y)$ indices in the plane perpendicular to the propagation direction, $z$, have been omitted in the interest of clarity here and in the following.

Given multiple intensity measurements at different energies, $I(E)$, spectral phase-retrieval then aims to recover phase-retrieved maps of $P_{\perp }$ and $\int {\rho }_{\mathrm{e},\perp }$ by solving a linear system of equations based on Eq. (1). Similar to the single-material phase-retrieval algorithm,¹² this is conveniently done in Fourier space to utilize Fourier integration by replacing ${\nabla }_{\perp }^2$ with ${\mathbf{k}}^2=k_x^2+k_y^2$.¹⁸ In case of more than two $I(E)$, the least-squares solution to the overdetermined problem is found.

The logical next step of spectral PBI is the extension of the method to the third dimension using CT. Consider a spectral PBI data set, $I(E,\theta )$, taken at all combinations of a given number of energies, $E$, and projection angles, $\theta$. For each set of images with fixed projection angle, $I_{\theta }(E)$, the phase-retrieved projections $P_{\theta }$ and ${\rho }_{\mathrm{e},\theta }$ are obtained by virtue of spectral phase-retrieval. The 3D quantities for phoelectric absorption, $P$, and electron density, ${\rho }_{\mathrm{e}}$, can then be computed using standard reconstruction techniques, e.g., filtered backprojection.

2.1. Spectral Phase-Retrieval Post-Reconstruction

It has been shown that the order of phase-retrieval and CT reconstruction can be reversed in single-image PBI, provided that minor adjustments are made that account for the third dimension after reconstruction.^10,26,27 Even though spectral PBI has more than a single set of input projections, and two output volumes, the same arguments can be applied to the case of spectral PBI CT.

Performing phase-retrieval with subsequent CT reconstruction is the conventional approach to spectral PBI CT. As explained above, this involves individually performing phase-retrieval on all subsets $I_{\theta }(E)$ to obtain the 2D projections $P_{\theta }$ and ${\rho }_{\mathrm{e},\theta }$, from which the volumes $P$ and ${\rho }_{\mathrm{e}}$ are then reconstructed. As an alternative, there exists the possibility to reverse the order of phase-retrieval and CT reconstruction. For this, CT reconstruction is performed on all subsets $I_E(\theta )$, prior to any phase-retrieval step. Several 3D reconstructions, $V_E$, are obtained this way. These volumes still contain phase-contrast fringes that have to be removed in a subsequent phase-retrieval step. Thompson et al.²⁷ have shown how single-material phase-retrieval can be extended to the third dimension, based on a linear TIE model similar to the one used here. Following these arguments, the algorithm for spectral phase retrieval in three dimensions is obtained by replacing all 2D quantities in Eq. (1), including ${\nabla }_{\perp }^2$, with their respective 3D counterparts:

$$V(E)=\frac{P}{E^3}+[{\sigma }_{\mathrm{KN}}(E)-\frac{h^2{c}^2{r}_{\mathrm{e}}R}{2\pi E^2}{\nabla }^2]{\rho }_{\mathrm{e}}.$$ (2)

Analogous to the 2D case, $P$ and ${\rho }_{\mathrm{e}}$ are obtained by solving a linear system of equations based on Eq. (2). Finding the solution by Fourier integration requires using 3D coordinates ${\mathbf{k}}^{\mathbf{2}}=k_x^2+k_y^2+k_z^2$, given the third dimension of ${\nabla }^2$. We have previously shown that spectral phase-retrieval can be recognized as a linear combination of low-/high-pass filtered input images.¹⁸ Phase retrieval after CT reconstruction is equivalent to 3D Fourier filtering of input images $V_E$.

Even though the results do not differ significantly between the two possible approaches, performing phase-retrieval after CT reconstruction has practical advantages. A commonly used approach in conventional PBI CT is to iteratively adjust a filter parameter ($\delta /\beta$) until an optimal reconstruction result is achieved. This is an effective way to account for unknown experimental parameters, such as detector point-spread function, source blurring, and spectrum.²⁸ However, the process can be very time-consuming, considering that the CT reconstruction has to be performed during each step of the parameter search. Given the increase in free parameters, e.g., propagation distance and effective spectrum, this issue is exacerbated in spectral PBI-CT. By moving the phase-retrieval step to the end of the processing chain, changes of the phase-retrieval parameters do not require recalculation of the CT reconstructions. In addition, phase-retrieval can be restricted to smaller subvolumes to provide rapid feedback on parameter changes or to overcome memory limitations.²⁷ Together, this allows for greatly accelerated parameter optimization. All results shown here were generated using phase-retrieval post-CT reconstruction.

3. Experimental Results

We performed spectral PBI CT experiments at the Medical and Imaging Beamline, BL20B2, of the SPring-8 Synchrotron in Hyōgo, Japan. We measured a simple two-component sample made from two well-known materials—aluminum foil and poly(methyl methacrylate) (PMMA) spheres—as well as a more complex sample in the form of an aluminum electrolytic capacitor, of the type that uses an etched foil. The capacitor sample is a well-suited object for spectral PBI CT due to its size, material composition, and internal structure. The various different interfaces within both samples prevent the use of a single-material phase-retrieval algorithm.

Images were recorded using a Hamamatsu C11440-52U series sCMOS camera with $6.5\mu \mathrm{m}$ pixel size and a $10\mu \mathrm{m}$ thick Gadox (${\mathrm{Gd}}_2{\mathrm{O}}_2\mathrm{S}:{\mathrm{Tb}}^{2+}$; P43) scintillator. The position of the detector determines the propagation distance and was set to 12.5 cm behind the Al-foil/PMMA and 20 cm behind the capacitor sample during the measurements. These relatively short propagation distances allowed the requirements for a linearization of the TIE to be met. We sequentially recorded three CT data sets with monochromatic x-ray beams of 24, 29, and 34 keV for the Al-foil/PMMA sample, and 25, 30, and 35 keV for the capactior sample. For each individual CT measurement, 2572 images were captured over 360 deg with an exposure time of 100 ms each.

Figure 2 shows the shortcomings of a single-material phase-retrieval algorithm when dealing with multimaterial samples, based on the Al-foil/PMMA sample. Phase-contrast fringes can clearly be seen in a typical reconstructed slice without phase-retrieval, shown in Fig. 2(a) for the 24 keV data set. Single-material phase-retrieval¹² was applied to this data set in Figs. 2(b) and 2(c), aimed at the air/Al interface in Fig. 2(b), and the air/PMMA interface in Fig. 2(c), respectively. It is well-understood that the single-material algorithm only accurately addresses one material at a time (or one material interface).¹⁴ Phase-effects of the secondary material or material interface are either underestimated, as can be seen by the residual fringes around the PMMA spheres in Fig. 2(b), or overestimated, as indicated by the visible blurring of the Al-foil in Fig. 2(c). This clearly demonstrates the need for a phase-retrieval approach able to handle multimaterial objects. Spectral phase retrieval using the full spectral data set allows us to remove phase-contrast fringes at all interfaces and generate a quantitative ${\rho }_{\mathrm{e}}$ map, shown in Fig. 2(d). Despite the increased level of noise compared with the the single-material images, in particular at intermediate spatial frequencies, phase-contrast fringes are fully removed in this case. We note that this increase in noise is a consequence of the quantitative decomposition into ${\rho }_{\mathrm{e}}$, and we have shown that spectral phase retrieval, in fact, reduces this type of noise compared with spectral decomposition without phase-retrieval.¹⁸ We estimate the electron density ${\rho }_{\mathrm{e}}$ of PMMA from an average over the red box in Fig. 2(d) as $3.51\times {10}^{2}4{\mathrm{cm}}^{-1}$, which is close to the theoretical value of $3.86\times {10}^{2}4{\mathrm{cm}}^{-1}$. A likely explanation for this slight mismatch is a noticeably reduced apparent attenuation in our sample, caused, e.g., by Compton scattered radiation reaching the detector, or effects caused by the PSF of the scintillator-based detector.

Fig. 2.

Axial slices through CT reconstructions of an aluminium foil/PMMA sphere test sample. (a) 24 keV reconstruction prior to any phase retrieval. (b) Single-material phase retrieval using ${\delta }_{\mathrm{Al}}/{\beta }_{\mathrm{Al}}$. (c) Single-material phase retrieval using ${\delta }_{\mathrm{PMMA}}/{\beta }_{\mathrm{PMMA}}$. (d) Quantitative spectral PBI phase-retrieval. A quantitative colorbar is only given for (d).

Figure 3 shows slices through the reconstructions without phase-retrieval at 25, 30, and 35 keV. Several of the different internal components can be recognized in the reconstructions, marked with the following letters in Fig. 3(e): Two thin aluminum foils (f) make up the capacitor plates, one of which has been chemically etched (e) for an increased surface area. These foils are spaced by an electrolyte (s) and connected to two solid aluminium contacts (c). Despite the seemingly artifact-free CT reconstructions, there exist phase-contrast effects at interfaces between the different materials, some of which are indicated by the arrows in (d). These characteristic bright-dark fringes prevent quantitative analysis of the images.

Fig. 3.

Axial slices through CT reconstructions of a capacitor sample, measured at (a) 25 keV, (b) 30 keV, and (c) 35 keV. (d)–(f) Show a magnified region of (a)–(c), according to the box in (a). No phase-retrieval was performed on these CT reconstructions.

3.1. Virtual Monochromatic Image

$P$ and ${\rho }_{\mathrm{e}}$ were obtained by performing spectral phase-retrieval using the reconstructed data shown in Fig. 3. Rather than displaying $P$ and ${\rho }_{\mathrm{e}}$ directly, we employ the method of the virtual monochromatic image (VMI) here. The VMI is a technique commonly used for clinical dual-energy CT that allows the radiologists to display the CT scan as it would appear at an arbitrarily chosen, monochromatic energy.²³ The VMI is created by recombining the constituent parts of the AM model at a chosen energy, i.e.,

$$\mathrm{VMI}(E)=\frac{P}{E^3}+{\sigma }_{\mathrm{KN}}(E){\rho }_{\mathrm{e}}.$$ (3)

Our previous theoretical work on spectral phase-retrieval¹⁸ has shown that for the present parameters, $P$ is the result of high-pass filtered x-ray intensity images, and ${\rho }_{\mathrm{e}}$ is the result of low-pass filtered images. Consequently, ${\rho }_{\mathrm{e}}$ has superior image quality in terms of image noise, so it is best to choose an energy where the contribution from ${\rho }_{\mathrm{e}}$ dominates the VMI. As Compton scattering is naturally more prominent at higher energies, we chose a display energy $E=80\mathrm{keV}$. The resulting VMI (80 keV) is shown in Fig. 4. We note that phase-contrast fringes are almost completely removed from the image. In addition, the attenuation values of the two Al foils are very similar after this phase-retrieval, whereas there was a noticeable difference prior to phase-retrieval. The plain foil, i.e., nonetched, in Fig. 3, appears to exhibit much stronger attenuation than the other Al parts, which we believe is caused in part by phase-contrast fringes from both sides of the foil elevating the local intensity and hence apparent attenuation. We also note that the VMI allows us to display the scan at energies otherwise unobtainable with the monochromator configuration at the beamline, which is also the reason we cannot provide a comparative direct measurement at 80 keV.

Fig. 4.

(a) Phase-retrieved virtual monochromatic reconstruction at 80 keV, generated from the spectral data shown in Fig. 3. (b) Magnification of the rectangular region marked in (a).

3.2. Effective Atomic Number Image

In addition to the VMI, we can also leverage the spectral data of a spectral PBI measurement to identify different materials within an object. To this end, an effective atomic number ${\mathrm{Z}}_{\mathrm{Eff}}$ is calculated for each pixel.

First, the relation between the atomic mass number $A$ and the atomic number $Z$ is approximated as $A\approx 2Z$. This is valid for low $Z$-elements up to around $Z=27$ (calcium). From this follows that the material density $\rho =2m_{\mathrm{u}}{\rho }_{\mathrm{e}}$, with the atomic mass $m_{\mathrm{u}}$. If we further recognize that $P\approx \rho Z^3$, we arrive at an expression for $Z_{\mathrm{Eff}}$:

$$Z_{\mathrm{Eff}}={\left(\frac{P}{2m_{\mathrm{u}}{\rho }_{\mathrm{e}}}\right)}^{1/3}.$$ (4)

The applicability of Eq. (4) is severely impaired by the high noise present in $P$, as is evident from Fig. 5(a). Strong denoising techniques are necessary to combat the high noise inevitably present in effective atomic number images and regularly used in this type of imaging. Here we employed a combination of total-variation (TV) smoothing and $k$-means clustering.

Fig. 5.

Effective atomic number mapping overlaid on the VMI shown in Fig. 4, (a) without, and (b), (c) with denoising applied.

We applied TV smoothing to the raw reconstructed slice at 25 keV, and subsequently used $k$-means clustering to partition the pixels into 100 different clusters. Within each cluster, $Z_{\mathrm{Eff}}$ was calculated from the per-cluster averaged $P$ and ${\rho }_{\mathrm{e}}$ and assigned to all pixels within that cluster. The resulting $Z_{\mathrm{Eff}}$ map is overlaid on VMI(80 keV) and shown in Fig. 5(b). As expected, the Al parts have a $Z_{\mathrm{Eff}}$ value close to the expected value $Z=13$. The etched Al layer has a slightly lower $Z_{\mathrm{Eff}}$ that can be explained by the presence of microscopic pores. Although we do not know the exact material composition of the electrolyte layer used in this specific capacitor, electrolyte layers are often on a polymer or water basis, for which the measured $Z_{\mathrm{Eff}}$ values are reasonable.

4. Conclusion

We have demonstrated phase-retrieval for spectral propagation-based phase-contrast CT. Our results illustrate how spectral phase-retrieval is able to simultaneously remove phase-contrast fringes between multiple different interfaces, which is one major limitation of the single-material approximation. Due to the use of universal basis functions, spectral phase-retrieval based on the Alvarez–Macovski model is well-suited for objects of which the exact material composition is unknown or for those that consist of a large number of different materials. In contrast to other approaches, no a priori knowledge about the investigated object is required. The virtual monochromatic and effective atomic number images were showcased as ways to fully utilize the spectral nature of this type of CT experiment. We have also discussed how the phase-retrieval and CT-reconstruction steps can be inverted, which has proven to be of great value during data processing.

In the future, simultaneous acquisition of spectral data is inevitable, to minimize measurement time and the associated movement and alignment artifacts. We foresee that the most promising approach to capturing multiple complementary x-ray measurements lies with energy discriminating detection. Spectral photon-counting detectors typically have relatively coarse resolution compared to scintillator-based detectors to avoid charge-sharing at smaller pixel sizes, which can limit their ability to capture small samples and/or fine PBI fringes.²⁹ PBI microtomography can nevertheless be achieved with spectral photon-counting detectors if one uses high geometric magnification. The inherently reduced effective propagation distance in a magnifying geometry¹² is manageable considering the limitations of moderate phase contrast effects for a linearized TIE approach. Other possible implementations for spectral x-ray imaging are largely unexplored at synchrotron sources or microfocus x-ray tubes. Examples include rapid energy switching and dual-source concepts, which seem unlikely to be compatible with synchrotron sources, and would suffer from the need for registration accuracy within a few $\mu \mathrm{m}$ at laboratory setups.

Considering the ongoing advances in energy-resolving detector technology,²⁹ we anticipate spectral PBI CT to be used at laboratory micro-CT setups equipped with a spectral photon-counting detector and microfocus x-ray tube. Vazquez et al.³⁰ showed that spectral PBI is feasible with such equipment for 2D imaging, and the extension to 3D CT should be straightforward, given the results shown here. Potential applications include material testing, as well as biomedical imaging, potentially with staining based on heavy elements.³¹

Biographies

Florian Schaff received his PhD in 2018. Currently, he is a postdoctoral researcher at the Technical University of Munich. Prior to this position, he was a postdoctoral researcher at Monash University. His current research interests include x-ray imaging and image processing.

James A. Pollock is a PhD student at Monash University under the supervision of Associate Professor Marcus Kitchen and Dr. Kaye Morgan.

Kaye S. Morgan is an Australian Research Council future fellow at Monash University. Her current research interests include single-grid phase-contrast x-ray imaging and its application to respiratory research.

Marcus J. Kitchen is an associate professor at Monash University. His current research interests include x-ray phase-contrast imaging, lung and brain imaging, and the development of x-ray imaging techniques.

Disclosures

The authors have no relevant conflicts of interest to disclose.