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Cone-Beam Reconstruction for the 2-Circles Plus Line Trajectory

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Cone-Beam Reconstruction for the 2-Circles Plus Line Trajectory Cone-beam reconstruction for the 2-circles plus line trajectory Yanbin Lua,*, Jiansheng Yanga, John W. Emersonb, Heng Maoa, Yuanzheng Sia and Ming Jianga a LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China b Department of Statistics, Yale University, New Haven, CT, 06511, USA Abstract. The 2-circles plus lines trajectory is designed to solve the problem of limited physical space in a multimodality imaging system for the introduction of X-ray cone-beam CT. We apply the backprojection filtration algorithm proposed by Pack et al for X-ray cone-beam data to the standard 2-circles plus lines trajectory. We find that for every point inside the region between the two circles there exists at least one redundantly measured line (a line segment that connects two sources points, R-line), and so the region can be fully covered by R-lines. Furthermore, for about 2/3 of the points inside the region there exist two or more such R-lines. The length of the line curve segment required by the algorithm can be as short as the distance between the two circles when the projections have no truncation. Numerical simulations validate the feasibility of the algorithm and the spiral resolution of cone-beam CT in the imaging system. Key words: Cone-beam CT, backprojection filtration algorithm, 2-circles plus line trajectory. 1. Introduction [Molecular imaging and multimodal imaging] Imaging modalities can be roughly classified into two categories: those which primarily provide structural information (such as Computed Tomography (CT) and Magnetic Resonance Imaging (MRI)) and those which primarily provide functional information (such as Positron Emission Tomography (PET) and Near Infrared Optical Tomography (OT)). Optical tomography is an emerging functional imaging technique. Its objective is to reconstruct the spatial distribution of optical parameters inside the medium based on measurements of transmitted light intensities on the surface [1]. Depending on the physical problem, OT can fall into many sub-categories, such as diffuse optical tomograph (DOT), florescence molecular tomography (FMT), and bioluminescence tomography (BLT). It has been recognized as a promising molecular imaging modality because of its sensitivity, specificity, low-cost and non-radiation characteristics. However, it often suffers from poor spatial resolution because of the high scattering nature of photons inside the tissue [2]. Thus hybrid systems that utilize multimodality data to combine the advantages of different imaging techniques have been explored. The use of structural information from CT or MRI to penalize the objective function of OT has been studied by many research groups [1, 3, 4]. [The earlier cone-beam CT] The main aim of X-ray CT reconstruction is to recover the object’s absorption coefficient using projection data acquired along a scanning source trajectory. Since Hounsfield reported the first x-ray computed tomography (CT) scanner in 1973 [5], reconstruction methods have made remarkable progress. Tuy proposed an exact reconstruction formulae with Tuy’s data sufficiency condition for cone-beam CT [6]. Related to the Radon transform, Smith’s formula [7] and Grangeat’s formula [8] provided two other classic formulae for cone-beam CT. To solve the long object problem for helical * Corresponding author: E-mail: [email protected]. 1 cone-beam CT, Wang et al proposed a Feldkamp-type algorithm [9, 10], and Kudo, Noo and Defrise proposed exact and approximate algorithms in [11-13]. [Recent development of cone-beam CT, helical CT] A series of new cone-beam formulae were developed in the past decade. They can be roughly classified into two categories: filtered-backprojection (FBP) approaches and backprojection-filtration (BPF) approaches. Katsevich reported the first exact shift-invariant FBP reconstruction formula for helical CT using more data than the Tam-Dannielson window [14], and then reduced the requirements in [15]. In his formula, one-dimensional filtrating of a derivative of the projection data was first done along particular lines on the detector, and the filtration results were backprojected to form the image. Its computation is more efficient than previous exact formulae. Zou and Pan then proposed a BPF format formula using data only in the Tam-Dannielson window [16]. They claim that the Hilbert transform of the absorption coefficient along a PI-line (which is also 1D filtrating) can be obtained by backprojection of a derivative of the projection data on this PI-line. Helical CT motivated theoretical research of cone-beam CT. Katsevich developed a general cone-beam FBP formula based on the Grangeat’s formula using a general weight function [17]. Chen et al proposed an alternative derivation of the Katsevich algorithm based on the Tuy’s formula with relaxed Tuy’s data sufficiency condition [18]. Zhao et al gave their proofs for Katsevich’s formula and extended Zou and Pan’s formula to any interior point on an R-line on a continuous scanning curve in the Tuy framework [19]. In this case, a redundantly measured line (R-line) is a line segment connecting two separate source positions. Ye et al proposed a generalized BPF formula for reconstruction on R-lines for a single smoothing scanning curve [20]. Ye and Wang proposed a FBP formula and gave suggestions of filtering directions for a smooth curve [21]. Pack et al proposed BPF reconstruction formulae for both a single smoothing curve and multiple smooth curves [22]. Reconstructions by their formulae could be more flexible by choosing R-lines and M-lines in the image space. A measured line (M-line) is a radial line starting from a source position such that for every point in the intersection of this M-line and the object there is at least one R-line passing through it. Pack and Noo also developed FBP reconstruction formula for multiple smooth curves [23]. All these formulae demanded that for every interior point there existed an R-line passing through it. [Our work] In this paper we study a special scanning trajectory for X-ray cone-beam CT in a multimodality imaging system called the Kodak In-Vivo Imaging System FX (Carestream Health, Woodbridge, CT). The system can provide 2D X-ray images, florescence images, bioluminescence images and radioisotopic images, but the precise 3D reconstruction images can’t be achieved from these images directly. In order to introduce the X-ray cone-beam CT, an innovation of the system is proposed here by implementing object movement mechanisms (such as rotation and translation mechanisms) into the system. Because the physical space in the imaging cabinet is very limited, we apply the general BPF algorithm for multiple smooth curves proposed by Pack et al [22] to a special scanning trajectory: 2-circles plus line trajectory. This 2-circles plus line trajectory consists of two parallel circles with radius R and a line orthogonal to the plane of the circles. It can be described as R cos, Rsin, z1 if [0,2 ) 2 (1) a R,0, z1 z2 z1 if [2 ,4 ) 2 R cos, Rsin, z2 if [4 ,6 ) We find that the two-circles plus line trajectory has nice geometry properties for reconstruction. It is proved that for every point inside the region between the two circles, there exists at least one R-line. This means that the region between two circles can be fully covered by R-lines. Assuming 2 data sufficiency for reconstruction, that region can be fully reconstructed by using the general algorithm on many-curve source trajectory. Finally, the number of R-lines for each point can be determined. [Organization of paper] The paper is organized as follows. In Section 2, we review the BPF algorithm in [22] and derive it for the 2-circles plus line trajectory. The geometry property of the 2-circles plus line trajectory is described, including the existence and calculation of R-lines. Section 3 shows numerical simulations and Section 4 discusses characteristics of this trajectory and the data requirement of the algorithm. Section 5 concludes the paper. 2. Method 2.1 Notations Let be a bounded convex neighbourhood of the object in R3 . The 3D source trajectory a() consists of a union of smooth curves. One example is the 2-circles plus line trajectory defined by (1). Two assumptions of the trajectory are needed here: (1) a() locates outside ; (2) For interior source points of every single smooth curve, the tangent vector a'() da()/ d is bounded, continuous and nonzero. Let r be a 3D point and f (r) : R3 R be the 3D function of the objects’ absorption coefficient of X-ray. f (r) equals to zero for r . The cone-beam projection at a() is defined as g(,) f (a() t)dt (2) 0 where S 2 is the unit vector in R3 . The local coordinate can be introduced into the detector plane as show in Figure 1. Let the orthogonal projection of a() onto the detector plane be the origin of the detector plane and ew perpendicular to the detector plane. eu and ev are orthogonal unit vectors on the detector plane. Figure 1. Local orientation on detector plane. The origin of the detector plane locates at the orthogonal projection of a() onto the detector plane. ew perpendicular to the detector plane. eu and ev are orthogonal unit vectors on the detector plane. 3 2.2 General formula of Pack et.al The idea behind BPF algorithms is that the backprojection of derivatives of the projection data can be linearly transferred to the Hilbert transform of f (r) on a line segment. Here we introduce the formula for multiple source curves based on [22]. First, two important concepts must be introduced. As shown in Figure 2, redundantly measured lines (R-lines) are line segments that connect two different source positions, and measured lines (M-lines) are radial lines from the source trajectory.
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