Refraction angle extracting strategy for fan-beam differential phase contrast CT

Abstract

In this paper, the fan-beam differential phase contrast computed tomography (DPC-CT) reconstruction method is studied. We first present a new vision of how to implement the Reverse-Projection (RP) method to extract the refraction-angle data efficiently in fan-beam geometry, and then provide a Katsevich-type formula for fan-beam DPC-CT reconstruction. The proposed method has two key properties. First, it is essentially a filtered back projection (FBP) reconstruction formula. Second, it can deal with incomplete data sets. The main contributions of this paper lie in the following three aspects: First, the physical principle of the bent-grating based fan-beam DPC imaging is discussed and the RP-method is extended to the fan-beam case. Second, an implementation strategy of Katsevich algorithm for fan-beam DPC-CT is proposed. Third, a semi-quantitative research on the influence of the approximation errors introduced by the RP-method is carried out by using several numerical simulations. It should be pointed out that the RP-method will certainly introduce some errors. The effect of these errors on our reconstruction algorithm is discussed by several numerical simulations.

Introduction

Phase-contrast (PC) imaging is a novel X-ray inspection method which uses the refraction rather than the absorption as the imaging signal and offers more internal structure details [4]. Until now, several X-ray PC imaging methods have been proposed and they can be divided into four categories: interferometer based methods [5], [6], [7], propagation based methods [8], [9], [10], analyzer based methods [11], [12], [13] and grating [14], [15], [16], [17], [18], [19], [20] based methods, which are differentiated from one another by measuring either the zero-order, the first-order or the second-order derivatives of the phase distribution. These three methods rely on the high coherence of synchrotron radiation sources or micro-focus tube [9], [11], which is the main obstacle to clinical diagnosis of PC imaging.

In recent years, grating-based methods, which were initially implemented at X-ray synchrotron radiation sources [14], [16], have been transferred to conventional X-ray tube sources [17], [18], [19], [20], [21]. In 2006, Pfeiffer et al., firstly, implemented PC imaging by using a conventional x-ray source [17], [18]. In their methods, a source grating is adopted to create an array of line sources from the X-ray beams generated by a conventional X-ray source. Within each line source, the X-rays are highly coherent. If the coherence length of each line source is much larger than the period of the phase grating, the self-imaging replica of the phase grating appears at the fractional Talbot distance and then the gradient of the phase distribution can be measured by a Talbot–Lau interferometer. In synchrotron radiation experiments, the radiation beams can be well approximated as parallel beams and the plane gratings are adopted. In conventional X-ray tube experiments, the source emits a fan/cone beam [1], [2], [3]. If the plane gratings are still used, only the small central part angle of the fan/cone beam near the optical axis of the imaging system can be effectively employed, which limits the application of PC imaging. To overcome this problem, Revol et al. used the cylindrical gratings instead of the plane gratings to realize the fan-beam differential phase-contrast imaging [14]. One major advantage of the aforementioned methods is to realize the compact X-ray differential phase-contrast CT (DPC-CT) system [18], [22], [23], [24], [25], [26], [27], [28], [29], which shows a great potential in medical diagnosis and industrial nondestructive test.

The aim of DPC-CT is to reconstruct the refractive index decrement from the refraction-angle data by using the X-ray scanner of a given geometry [41]. Generally, the refraction-angle data should be extracted by using the phase-stepping (PS) method [17], [19]. This method assumes the sample to be stationary and requires several sampling attempts at each view angle, which leads to unacceptably long exposure time and huge X-ray doses. In CT scanning, the sample is rotating round a fixed axis. The reverse projection (RP) [25] method has been proposed recently and extracted the refraction-angle data through rotation, which does not require phase stepping and is much simpler than the PS method. After the refraction-angle data was obtained, the DPC-CT algorithms [22], [23], [24], [25], [26], [27], [28], [29] can be performed to reconstruct the refractive index distribution decrement. The reconstruction problem for parallel-beam DPC-CT is solved by the FBP algorithm [24], [25] which has also been extended to fan/cone beam [26], [27] scanning geometries. However, the FBP algorithm is known to be sensitive to incomplete data sets. Several algorithms have been developed to overcome the FBP׳s drawback. Two famous examples are BPF [30], [31], [32], [33] and Katsevich-type [34], [35], [36], [37] algorithms. The key difference between the two algorithms is the filtering-process. This difference leads to an interesting conclusion that in some cases the BPF algorithm is faster, whereas in another case the Katsevich algorithm is faster [33]. In 2008, the concept of BPF algorithm has been transferred to fan beam DPC-CT [28] by Chen and Qi, which shows great advantages over the conventional FBP algorithm. In 2011, Fu et al. provided a formulation of BPF algorithm for flat detector geometry [29]. However, they did not discuss the extraction of the refraction-angle under the fan-beam condition. In this paper, we first extend the RP method to the fan-beam condition and then discuss and analyze the implementation of the Katsevich algorithm for fan-beam DPC-CT. It should be pointed out that the RP-method will introduce some approximation errors. The influence of these errors on our reconstruction algorithm is analyzed by numerical simulations.

The rest of the paper is organized as follows. In Section 2, the physical theory of bent-grating interferometer is introduced and the RP method is extended to the fan-beam case. In Section 3, the 2D Katsevich׳s formula is briefly reviewed and the implementation of this formula for fan-beam DPC-CT reconstruction is proposed and analyzed. In Section 4, some numerical simulation results are given and discussed. The summary of our work is provided in Section 5.