A Feldkamp-type approximate algorithm for helical multislice CT using extended scanning helix
Introduction
Helical multislice Computed Tomography (CT) has been investigated and developed for rapid, volumetric and high-resolution scanning in medical diagnosis. Normally, the objective of CT reconstruction algorithm is to produce a sequence of planar transversal image slices for representing the objective volume.
Helical multislice CT reconstruction algorithms can be classified into exact and approximate reconstruction algorithms. For exact reconstruction, Katsevich’s algorithm [1] represents a significant breakthrough in solving the reconstruction with filtered backprojection (FBP) based on efficient one-dimensional (1D) shift-invariant filtering. Zou and Pan developed a more efficient exact algorithm which implements the FBP on PI-lines [2], [3]. Compared with the exact reconstruction algorithms, approximate algorithms can provide more flexible tradeoff between the image quality and computational efficiency, therefore they are more widely adopted than the exact ones [10], [11], [12], [13], [14], [15], [16], [17]. Especially, the 2D rebinning algorithms and the well known 3D Feldkamp-type (FDK) algorithms play important roles in modern CT diagnosis [13], [14], [15], [16], [17], [18]. Efforts have been made to improve the FDK image reconstruction in the past decade. In 1982, Yan and Leahy proposed a helix-directed tilted filtering technique which was improved by Sourbelle and Kalender in 2003 [16], [17]. In 2003, Hein et al. proposed an FDK reconstruction in gantry-tilted helical scanning system [14]. In 2005, Yan and Zhang proposed another new tilted reconstruction to improve the large volumetric FDK reconstruction [18]. Recently, Hu et al. proposed an FDK-type approximate algorithm which considered a reconstruction of nutating curved surfaces using short scan data [15]. It was shown that all points of the nutating curved surface satisfy Tuy’s exact reconstruction condition [9] and, therefore, have potential to be exactly reconstructed. The conventional planar transversal image slice is then obtained by reconstructing its points on the corresponding nutating curved surfaces or, alternatively, by interpolating the nutating curved surfaces. The proposed algorithm in this paper is motivated by Hu’s et al. consideration for approximate reconstruction of the image slice which satisfies Tuy’s exact reconstruction condition. Different from Hu’s et al. work, we consider an FDK-type reconstruction of transversal plane which is more direct and practical to implement. An optimized helical scanning trajectory is derived for satisfying Tuy’s condition. As a result, the proposed algorithm can achieve improvement in computational efficiency compared with the existing algorithms.
The rest of this paper is organized as follows. Section 2 presents the coordinates system for helical scanning and Tuy’s condition for the transversal reconstruction plane. Section 3 presents the proposed algorithm, followed by Section 4 on computer simulations and discussions of the results. Conclusion is given in Section 5.
Section snippets
Helical scanning geometry
The geometry of the helical multislice CT scanning is shown in Fig. 1 in a Cartesian coordinate system x–y–z, in which the X-ray source moves along a helical trajectory. Denote R as the radius of the helix, r as the radius of objective cylindrical support, and as the source rotation angle. Let the helix pitch value be h and the detector collimation width be w, then the table translation per angle can be described as . The position of the source is represented as
Reconstruction using extended scanning helix
We now reconstruct the transversal planar image slice using data from the extended half scan helix by FDK-type approximate algorithm. The reconstruction procedure is presented as follows.
- 1.
Collect the data set along the helix segment. Denote it as , where is the index of detector cells with being the detector center and ranges within which is the angular interval of extended half scan helix. Denote as the source–detector distance and define . Before
Phantom reconstruction
In this section, the proposed FDK-type algorithm using the extended half scan helix (abbreviated as ExFDK) is tested by Head phantom and Clock phantom which are defined following the exiting works [19], [16], [17], [10], [11]. The performance of ExFDK is compared with that of Hu’s et al. nutating curved half scan FDK reconstruction [15] (abbreviated as NcFDK), that of the conventional full scan FDK reconstruction (abbreviated as FsFDK) and that of conventional half scan FDK reconstruction
Discussion
We now demonstrate advantages of the proposed ExFDK algorithm in terms of the following three aspects:
- •
Scanning helix length: The objective of CT image reconstruction is to produce transversal planar image slices of an object. Considering the NcFDK algorithm, to obtain a transversal planar slice from the nutating curved slices it needs a large group of data sets from different PI-helices corresponding to different objective points. Differently, ExFDK algorithm directly reconstructs the planar
Conclusion
This paper presents an FDK-type reconstruction algorithm using an extended half scan helix, which enables all points of the reconstructed transversal slice to satisfy Tuy’s condition and have potential of being exactly reconstructed. A formula is derived for obtaining the minimum length of such extended half scan helix, and an approximate FBP algorithm is proposed for constructing the transversal planar slice using the extended half scan data. This algorithm can be adopted to improve the image
HongZhu Liang received BEng degree from Xi’an Jiaotong University, China, in 2000, and MEng degree from Huazhong University of Science and Technology, China, in 2003. Since 2003 he has been with School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, as a research student pursuing PhD degree. His research is mainly on the algorithm for the helical computed tomographic imaging.
References (24)
An improved exact filtered backprojection algorithm for spiral computed tomography
Adv Appl Math
(2004)- et al.
Exact image reconstruction on PI-lines from minimum data in helical cone beam CT
Phys Med Biol
(2004) - et al.
Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT
Phys Med Biol
(2004) - et al.
Exact cone beam CT with a spiral scan
Phys Med Biol
(1998) - et al.
Cone-beam filtered-backprojection algorithm for truncated helical data
Phys Med Biol
(1998) - et al.
A solution to the long object problem in helical cone-beam tomography
Phys Med Biol
(2000) - et al.
Exact Radon rebinning algorithm for the long object problem in helical cone-beam CT
IEEE Trans Med Imaging
(2000) An inversion formula for cone-beam reconstruction
SIAM J Appl Math
(1983)- et al.
Single-slice rebinning method for helical cone-beam CT
Phys Med Biol
(1999) - et al.
Advanced single-slice rebinning in cone-beam spiral CT
Med Phys
(2000)
Multi-slice helical CT: Scan and reconstruction
Med Phys
Practical cone-beam algorithm
J Opt Soc Am
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HongZhu Liang received BEng degree from Xi’an Jiaotong University, China, in 2000, and MEng degree from Huazhong University of Science and Technology, China, in 2003. Since 2003 he has been with School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, as a research student pursuing PhD degree. His research is mainly on the algorithm for the helical computed tomographic imaging.
CiShen Zhang received the BEng degree from Tsinghua University, China, in 1982 and PhD degree in electrical engineering from Newcastle University, Australia, in 1990. Between 1971 and 1978, he was an electrician with Changxindian (February Seven) Locomotive Manufactory, Beijing, China. He carried out research work on control systems at Delft University of Technology, The Netherlands, from 1983 to 1985. After his PhD study from 1986 to 1989 at Newcastle University, he was with the Department of Electrical and Electronic Engineering at the University of Melbourne, Australia as a lecturer, senior lecturer and associate professor and reader till October 2002. He is currently with the School Electrical and Electronic Engineering and School of Chemical and Biomedical Engineering at Nanyang Technological University, Singapore. His research interests include signal processing, medical imaging and control.
Ming Yan received the BEng degree and MSc degree in automation, both from Tsinghua University, China, in 2000 and 2003, respectively. Since 2003, he has been pursuing the PhD degree at the Nanyang Technological University, Singapore. His research interests include computed tomography reconstruction algorithms and medical imaging.