Image reconstruction in regions-of-interest from truncated projections in a reduced fan-beam scan.pdf
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1、INSTITUTE OFPHYSICSPUBLISHINGPHYSICS INMEDICINE ANDBIOLOGY Phys. Med. Biol. 50 (2005) 1327doi:10.1088/0031-9155/50/1/002 Image reconstruction in regions-of-interest from truncated projections in a reduced fan-beam scan Yu Zou, Xiaochuan Pan and Emil Y Sidky Department of Radiology, The University of
2、 Chicago, 5841 S Maryland Avenue, Chicago, IL 60637, USA Received 19 July 2004, in fi nal form 27 October 2004 Published 16 December 2004 Online at stacks.iop.org/PMB/50/13 Abstract In a reduced fan-beam scan, the scanned angular range is smaller than that in a short scan (i.e., a half-scan). In thi
3、s work, we have developed a new algorithm,whichisreferredtoasthebackprojection-fi ltration(BPF)algorithm, for exact image reconstruction within ROIs from reduced-scan data containing truncations. Explicit conditions on data acquisition have also been derived for exact image reconstruction within an
4、ROI. We have performed a preliminary quantitative study whose results demonstrated and verifi ed the proposed fan- beam BPF algorithm and the derived conditions on data acquisition.The proposed BPF algorithm can have signifi cant implications for clinical and animalCTimaging,therapyimaging,electronp
5、aramagneticresonanceimaging and other tomographic imaging because it allows for reconstruction from truncated data and for a potentially drastic reduction of radiation dose and/or of imaging time. 1. Introduction A fan-beam scanning confi guration is widely used in current clinical computed tomograp
6、hy (CT) systems for data acquisition. Even though future generation of CT is expected to adopt a helical cone-beam confi guration for rapid volumetric coverage and for effi cient usage of the x-ray source, it can also be used in a fan-beam mode, which may still fi nd useful applications in many imag
7、ing protocols. When the fan-beam scan covers an angular range of 2 or plus the fan-beam angle, it is referred to as the full- or short-scan, respectively. It is well known that the image over its entire, compact support can be reconstructed exactly from full- or short-scan data, and algorithms have
8、been developed for obtaining images in the cases of full-scan (Kak and Slaney 1988, Besson 1999, Pan and Yu 2003) and short-scan (Parker 1982, Yu and Pan 2003). We refer to the fan-beam scan over an angular range smaller than that in a short scan as a reduced scan.Algorithms have been proposed for r
9、econstructing exactly the image 0031-9155/05/010013+15$30.00 2005 IOP Publishing LtdPrinted in the UK13 14Y Zou et al within a region-of-interest (ROI) from reduced-scan data (Noo et al 2002). Recently, we have also developed an algorithm for image reconstruction within an ROI from reduced-scan data
10、 (Pan et al 2004). The existing algorithms are mathematically identical, and, when they are in the discrete forms, however, they may respond differently to data inconsistencies such as sample aliasing (Pan et al 2004). From the operational point of view, both algorithms reconstruct images by perform
11、ing a shift-invariant fi ltering of the modifi ed data followed by a weighted backprojection of the fi ltered data. Therefore, we refer to them as the fi ltered- backprojection(FBP)-based algorithms.The FBP-based algorithms with a shift-invariant fi ltration can generally reconstruct images within a
12、n ROI only from reduced-scan data containing no truncations (Noo et al 2002, Pan et al 2004).Recently, an algorithm has been developed for ROI-image reconstruction from parallel-beam data containing truncations (Clackdoyle and Noo 2004). In this work, following the strategy that was used to develop
13、our algorithms for three- dimensional (3D) image reconstruction in a helical cone-beam scan (Zou and Pan 2004a), we derive a new algorithm for image reconstruction from reduced-scan fan-beam data. Unlike the existing FBP-based algorithms, the new algorithm reconstructs the image within an ROI by fi
14、rst backprojecting the weighted data and then performing a shift-invariant fi ltering of the backprojection and is referred to as the backprojection-fi ltration (BPF) algorithm. Furthermore, we derive explicit conditions on data acquisition under which the image within a given ROI can exactly be rec
15、onstructed by using the proposed BPF algorithm. An important property of this BPF algorithm is that it can exactly reconstruct the image within an ROI directly from reduced-scan data that do contain truncations. It should be pointed out that the problem discussed here is not a truly interior problem
16、 (or, the local tomography) in which, truncations occur on both ends of the detector array for all possible projection views. It is well known that the image within any ROI cannot be obtained exactly in an interior problem (Natterer 1986). In this work, instead, we are addressing an ROI reconstructi
17、on problem in which, although data can contain truncations at all the views in a reduced scan, there must be some views in the reduced scan at which data truncations do not occur or occur only on one end of the detector array. In this situation, we show that ROI images can exactly be reconstructed b
18、y using the proposed BPF algorithm. We have performed a preliminary numerical study to demonstrate and validate the proposed fan-beam BPF algorithm. The paper is organized as follows. In section 2, we introduce the fan-beam BPF algorithm and illustrate why it can exactly reconstruct the image within
19、 an ROI from reduced-scan data that contain truncations. In section 3, we present the results of our preliminary numerical study that validate and demonstrate the fan-beam BPF algorithm. In section 4, we discuss the implications and generalization of the fan-beam BPF algorithm. 2. Theory In this sec
20、tion, we introduce the fan-beam BPF algorithm for exact image reconstruction within an ROI from reduced-scan data containing truncations. 2.1. Data function in a fan-beam scan As shown in fi gure 1(a), we use two coordinate systems x,y and u,w to characterize the geometry in a fan-beam scan, which a
21、re fi xed on the centre of rotation of the source and on the rotating source, respectively, and thus are referred to as the fi xed-coordinate and rotation-coordinate systems. In the fi xed-coordinate system, for a given view , the unit Image reconstruction in regions-of-interest from truncated proje
22、ctions in a reduced fan-beam scan15 x y w 1 2 x (a)(b) Figure1. (a)Illustrationofthefi xed-coordinatesystemx,y, whichisfi xedontheobject, andthe rotation-coordinate system u,v, which centres on the source point. (b) Illustration of a PI-line segment, which is a straight line segment intersecting wit
23、h the trajectory at two points labelled by 1and 2. In both (a) and (b), the trajectories are displayed as the thick curves. vectors of the rotation-coordinate system can be written as eu() = (sin,cos)Tand ew() = (cos,sin)T, and the circular trajectory of the source point (i.e., the origin of the rot
24、ation-coordinate system) can be written as ? r0() = (R cos,R sin)T,(1) where R denotes the distance from the source point to the centre of rotation. We assume that the two-dimensional (2D) image f(? r) has a compact support within ?Renclosed by the source trajectory. The fi xed- and rotation-coordin
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