Computational and Mathematical Methods in Medicine

Volume 2015, Article ID 623236, 7 pages

http://dx.doi.org/10.1155/2015/623236

## Explicit Filtering Based Low-Dose Differential Phase Reconstruction Algorithm with the Grating Interferometry

^{1}Key Laboratory of Particle & Radiation Imaging (Tsinghua University), Ministry of Education, Beijing 100084, China^{2}Department of Engineering Physics, Tsinghua University, Beijing 100084, China^{3}Department of Radiology, Beijing Tongren Hospital, Capital Medical University, Beijing 100730, China^{4}Department of Radiology, Beijing Friendship Hospital, Capital Medical University, Beijing 100050, China

Received 15 September 2014; Revised 8 December 2014; Accepted 20 December 2014

Academic Editor: Yi Gao

Copyright © 2015 Xiaolei Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

X-ray grating interferometry offers a novel framework for the study of weakly absorbing samples. Three kinds of information, that is, the attenuation, differential phase contrast (DPC), and dark-field images, can be obtained after a single scanning, providing additional and complementary information to the conventional attenuation image. Phase shifts of X-rays are measured by the DPC method; hence, DPC-CT reconstructs refraction indexes rather than attenuation coefficients. In this work, we propose an explicit filtering based low-dose differential phase reconstruction algorithm, which enables reconstruction from reduced scanning without artifacts. The algorithm adopts a differential algebraic reconstruction technique (DART) with the explicit filtering based sparse regularization rather than the commonly used total variation (TV) method. Both the numerical simulation and the biological sample experiment demonstrate the feasibility of the proposed algorithm.

#### 1. Introduction

X-ray grating interferometry [1–4] with conventional X-ray tubes develops rapidly in recent years and is becoming the most promising technology among various phase contrast imaging methods for clinical applications. Three kinds of information, that is, the attenuation, differential phase contrast (DPC), and dark-field images, can be obtained through one single scanning, and the latter two images provide additional and complementary information to the conventional attenuation image. The DPC method measures phase shifts of X-rays by obtaining the line integral of the directional derivatives of refractive index decrements (*δ*) [5], that is, the refraction angle of the beam. The refraction index reconstructed afterwards is 1000 times larger than the absorption index.

The phase-stepping approach of the grating interferometry, which requires a number of images to retrieve information, significantly increases the examine time and the dose delivered to the patient [6]. The problem becomes even more severe for DPC-CT because of the requirement of multiangle scanning. Therefore, reducing the number of projections, the exposure time, and the delivered dose is of great value. And that is why the low dose DPC reconstruction algorithm is proposed.

As mentioned above, the reconstruction problem for DPC-CT is to obtain the refraction index from the refraction angle data. The analytical method, the filtered backprojection (FBP) algorithm with the Hilbert transform, was first applied [7, 8]. Afterwards, several iterative algorithms, such as the maximum likelihood (ML) algorithm [9] and the differential algebraic reconstruction technique (DART) [10], were proposed. However, these algorithms rely on the completeness of data and the large number of projections.

The recently proposed compressed sensing (CS) theory [11] makes image reconstruction from incomplete data possible. Essentially, it illustrates that if the image is sparse in a domain which has small coherence with the sampling domain, according to the Shannon/Nyquist sampling theorem, fewer projections can almost accurately recover the images. A typical image reconstruction method exploits TV as the sparse regularization [12] (from CS measurements). Applications in both the absorption imaging [12] and the DPC imaging [10] have been implemented. Instead of the implicit regularization coming from the penalty, another sparse regularization method based on explicit filtering is proposed, which exploits spatially adaptive filters sensitive to image features and details [13]. However, no similar algorithms for DPC imaging have been suggested so far, which is just the problem to be solved in this paper.

In this work, we propose an explicit filtering based low-dose differential phase reconstruction algorithm. The algorithm combines the DART iterative algorithm and the explicit filtering based CS method. It has the potential to exactly reconstruct the refractive index distribution using few-view projections, thus reducing the exposure time and the delivered dose, making DPC-CT closer to clinical applications. The feasibility of the low dose reconstruction algorithm is verified by both the numerical simulation and the biological sample experiments.

#### 2. Methods

##### 2.1. Grating-Based Imaging

Figure 1 illustrates the schematic diagram of a typical grating interferometry. Two kinds of apparatuses are shown, the Talbot effect based interferometry with coherent source, that is, Figure 1(a), and the Talbot-Lau effect based interferometry with incoherent source, that is, Figure 1(b). The first grating G1 creates its self-image through Talbot-Lau effect or classical optics in the position of G2 where Moire fringes occur. The source grating G0 splits the source into an array of line sources, enabling the use of the large-focal-spot X-ray tube, that is, the incoherent source. The phase-stepping approach is adopted for image acquisition, capturing a series of raw images at every step of one of the gratings along the transverse direction, obtaining the intensity oscillation curve, Figure 1(c). The changes of the background oscillation curves determine three kinds of information, namely, the attenuation image, the DPC image, and the dark-field image.