Improvement of Efficiency and Flexibility in Multi-slice Helical CT

J. Shanghai Jiaotong Univ. (Sci.), 2008, 13(4): 408 412 DOI: 10.1007/s12204-008-0408-x Improvement of Efficiency and Flexibility in Multi-slice Helical CT SUN Wen-wu 1 ( ), CHEN Si-ping 2 ( ), ZHUANG Tian-ge 3 ( ) (1. Nanchang Institute of Aeronautical Technology, Nanchang 330063, China; 2. Shenhen University, Shenhen 518057, China; 3. Department of Biomedical Engineering, Shanghai Jiaotong University, Shanghai 200240, China) Abstract: One of the main aspects in computed tomography (CT) development is to make CT rapidly scan a large longitudinal volume with high -axis resolution. The combination of helical scanning with multi-slice CT is a promising approach. Image reconstruction in multi-slice CT becomes, therefore, the major challenge. Known algorithms need to derive the complementary data or work only for certain range of pitches. A reconstruction algorithm was presented that works with the direct data as well as arbitrary pitches. Filter interpolation based on the proposed method was implemented easy. The results of computer simulations under kinds of conditions for four-slice CT were presented. The proposed method can obtain higher efficiency than the conventional method. Key words: multi-slice computed tomography; helical interpolation; rebinning; filter interpolation CLC number: R 197.39 Document code: A Introduction Step-and-shoot CT and helical CT are two modes used generally for a CT scan. Different from stepand-shoot CT where the table remains stationary while acquiring projection data, helical CT keeps the table transporting at a constant speed as well as continuous gantry rotation. Compared with step-and-shoot CT, helical CT shortens greatly the time of examination and improves the volume coverage speed performance [1]. Single-slice helical CT, however, is not able to obtain high -axis resolution while scanning large longitudinal volume due to a few reasons, e.g., the limitations of the tube loading and pitch, where the pitch is defined as the ratio of table feed per rotation versus entire collimated beam width. In order to meet the needs for many medical CT applications where both high volume coverage speed performance and high image quality, i.e., high - axis resolution and low image artifacts, are necessary, multi-row helical CT, equipping with a multi-row detector array, is a promising approach. Multi-slice CT allows for simultaneous scan of multiple slices at different locations, and enhances the -axis resolution. Due to the distinct differences in scanning geometry, multi-slice CT exhibits complex imaging characteristics and calls for new reconstruction algorithms. In multislice CT, the X-ray beam fan is extended in the -axis, as well as the gantry plane, the reconstruction algorithms for single-slice CT may no longer be adequate to maintain image quality. Therefore, 3D reconstruction algorithms may be required. In last few years, the cone- Received date: 2007-07-13 E-mail: sun anke@yahoo.com.cn beam reconstruction algorithms [2-9] and the nutating surface reconstruction algorithms [10,11] have been addressed in order to eliminate the artifacts induced by beam divergence. Cone-beam back-projection, however, requires expensive calculations, and cannot be implemented efficiently. Fortunately, for multi-slice CT with less than four slices, the effect of beam divergence on image quality is small so that it may be neglected, and the cone-beam geometry is, therefore, approximated as multiple, parallel fan-beams. Several reconstruction algorithms have been addressed in multi-slice CT [1,12-14] that all do not take into account of beam divergent effect. However, the algorithms need to derive the complementary data or work only for certain range of pitches. The purpose of this paper is to propose a reconstruction algorithm in multi-slice helical CT. The proposed method works with the direct data as well as arbitrary pitches. Furthermore, it is also easy to implement filter interpolation based on the proposed method to reduce the artifacts and noise. 1 Multi-slice 180 Linear Interpolation The 360 and 180 linear interpolations in multi-slice CT (see Fig. 1(a)) are, respectively, extended from the 360LI and 180LI for single-slice CT (see Fig. 1(b)) [15,16]. The rays used by the 180 linear interpolation in multislice CT are not restricted from the same slice, both direct and complementary rays of different slices are used. Figure 2 shows the schematic sketch of helical scan in multi-slice CT in the case of pitch P = 0.875. Let β and γ denote, respectively, the view and channel angles, the direct and complementary data for the nth

J. Shanghai Jiaotong Univ. (Sci.), 2008, 13(4): 408 412 409 Helical orbit Helical orbit 2D detector 1D detector Axis of rotation Axis of rotation Fig. 1 X-ray source X-ray source (a) Multi-slice CT (b) Single-slice CT The helical scanning geometry for multi-slice and single-slice CT 0 180 360 Fig. 2 Filter width Schematic sketch of helical scan in multi-slice CT with pitch P = 0.875. The slice position of direct data (solid line) and complementary data at γ = 0 (dashed line) are shown in whole helical scan. Filters may be used to average data along the -axis and provide more trade-off options between effective slice thickness and image noise slice are defined as p d (β, γ, n) and p c (β, γ, n), respectively, and d (β, n) and c (β, γ, n) are the corresponding slice positions. For the slice = 0, the 180 linear interpolation in multi-slice CT is described as where p(β, γ) = w(β, γ) p h (β, γ, q + 1) w(β, γ) = + (1 w(β, γ) p h (β, γ, q), (1) 0 h (β, γ, q) h (β, γ, q + 1) h (β, γ, q) ; p h (β, γ, q) and p h (β, γ, q + 1) are the lower and upper data sets closest to the slice, respectively; and h (β, γ, q) and h (β, γ, q+1) are the corresponding slice positions. As Fig. 2 indicates, the data p h (β, γ, q) and p h (β, γ, q + 1) can be the direct data p d (β, γ, n) or the complementary data p c (β, γ, n). 2 Proposed Method 2.1 Helical Interpolation According to the 180 linear interpolation in multislice CT given by Eq. (1), the complementary data have to be derived from the direct data, which makes the interpolation expensive. Actually, not only does there exist the relationship between the direct and complementary data as p c (β, γ, n) = p d (β + 2γ ± π, γ, n), (2) but it is easy to prove that the weights w(β, γ) have the following relationship: w(β, γ) = w(β + 2γ ± π, γ). (3) The contribution w(β, γ) p c (β, γ, n) of the complementary data p c (β, γ, n) to projection measurement p(β, γ) is, therefore, same as that of the direct data p d (β + 2γ ± π, γ, n) to p(β + 2γ ± π, γ), i.e., w(β + 2γ ± π, γ) p d (β + 2γ ± π, γ, n). The contributions of the projections from the nth slice to projection measurement p(β, γ) are then known by combining the contributions of both direct data p d (β, γ, n) to p(β, γ) and p d (β + 2γ ± π, γ, n) to p(β + 2γ ± π, γ). As a consequence, the derivations of the complementary data p c (β, γ, n) can be omitted so as to reduce the computation load while the weights of the direct data p d (β + 2γ ± π, γ, n) are doubled over the interval β [0, 2π].

410 J. Shanghai Jiaotong Univ. (Sci.), 2008, 13(4): 408 412 2.2 Parallel Rebinning Different from the fan-beam geometry where β and γ are used, two parameters θ and t are usually denoted as the aimuthal angle and radial position of a ray, respectively, for the parallel geometry. There are two simple relationships between the two sets of parameters (θ, t) and (β, γ) as θ = β + γ, (4) t = R sin γ, (5) where R is the source-to-isocenter distance. Using Eqs. (4) and (5), we can perform rebinning from sampled fan-beam projections p(β i, γ j ) to parallel projections. In general, the rebinning process can be separated into the following two steps: first, data are interpolated in the aimuth to obtain samples on radial lines that are arranged at equiangular intervals [17] ; second, a radial interpolation is performed to obtain equidistant samples on the radial lines. Let β 1,=0 (n) and β 2,=0 (γ j, n) be, respectively, the view angles when d (β, n) and c (β, γ j, n) equal to 0, and g(θ l, t j ) be the equiangular sampled parallel projections, where t j corresponds to γ j by Eq. (5). In order to maintain image quality, the rebinning in the proposed method is different from that given by Ref. [17]. Not only the data p(β i, γ j ) but the interpolated data p d (β 1,=0 (n), γ j, n) are used for the linear interpolation for g(θ l, t j ) as given by g(θ l, t j ) = h(β, γ j ) f(β, γ j )+(1 h(β, γ j )) f( ˆβ, γ j ), (6) where h(β, γ j ) is the weight, and β and ˆβ are the two view angles adjacent to the view angle θ l γ j β and ˆβ may be β i, β 1,=0 (n) and β 2,=0 (γ j, n). The interpolated data f(β, γ j ) for the cases are p(β i, γ j ), β =β i f(β, γ j )= 2p d (β 1,=0 (n), γ j, n), β =β 1,=0 (n). 0, β = β 2,=0 (δ j, n) However, if the channel angle increment γ equals to half the view angle increment β and there is the following relationship for all slice index n : β 1,=0 (n) = k n β, where k n is an integer, then, the view angles β 1,=0 (n) and β 2,=0 (γ j, n) coincide with β i, the proposed rebinning is simplified into that given by Ref. [17]. 2.3 Full-scan Parallel Reconstruction The normal full-scan parallel reconstruction technique, which uses filtered back-projection, is applied to obtain slice image. 2.4 Filter Interpolation The problem with helical interpolation in multi-slice CT is the existence of several discontinuous changeovers in pairs of data samples in the interpolated data sets obtained for one rotation. The most effective method of eliminating these changeover effects is to use filter interpolation that is a filtering process performed in the -axis with several data sets [13]. The concept is shown in Fig. 1. A width defined as the filter width (FW) is assumed in the direction, and several data points lie within FW. Filter interpolation can be divided into the following two steps: 1 resampling by linear interpolation using adjacent data points; 2 filtering the resampled data set. In the proposed method, it seems that the weights of the direct data for helical interpolation must be calculated at each resampling position when using filter interpolation based on the proposed method. Let v be the projection index, the weights w(β v, γ j, n) for the direct data of the nth slice are found to have the following relationship between the positions at = 0 and = 0 + as w =0+ (β v, γ j, n) = w =0 (β v v, γ j, n), where v is the projection spacing. Furthermore, the weights h =0 (β, γ j ) and h =0+ (β, γ j ) for rebinning the projections from fan-beam to equiangular parallel corresponding to the positions = 0 and = 0 +, respectively, also show the following characteristic similar to Eq. (7): h =0+ (β, γ j ) = h =0 (β v β, γ j ). As a consequence of the observations, it is sufficient to know the weights w =0 (β v, γ j, n) for helical interpolation and the weights h = 0 (β, γ j ) for rebinning at the position = 0. The convolution for filtering can be operated on the equiangular parallel data g(θ l, t j ). 3 Experiments and Discussions The results are simulated by the conventional and proposed methods. In the simulations, four ball phantoms ( = 20 mm) locate uniformly at the plane perpendicular to the -axis along a circle ( = 150 mm). The contrasts of the ball and background are 0 and 1 000 HU, respectively. The beam width is fixed at 2 mm. The ratios of the proposed versus conventional method in terms of the time of reconstruction are given in Table 1 under kinds of conditions. The proposed method can obtain higher efficiency than the conventional method, and the advantages of the proposed over conventional method become more obvious with the increase of the number of resampling points.

J. Shanghai Jiaotong Univ. (Sci.), 2008, 13(4): 408 412 411 Fig. 3 Reconstructed images of ball phantoms with the diameter of 20 mm for various scan and filter parameters Furthermore, as reconstructed images that are 8 mm away from the centers of the balls show (see Fig. 3, the window level and width are 1 000 and 150 HU, respectively), image quality is improved by filter interpolation, and enhanced significantly with the increase of FW as indicated in the bottom two rows. Table 1 The ratios of the proposed versus conventional method in terms of the reconstruction time while the number of resampling points is fixed at 11 and 21 Ratios of time P FW=2 mm FW=4 mm num 11 num 21 num 11 num 21 0.625 0.747 0.711 0.748 0.717 0.875 0.735 0.717 0.741 0.714 1.125 0.742 0.715 0.746 0.713 4 Conclusion The reconstruction algorithm for multi-slice CT works well with the direct data as well as arbitrary pitches. Filter interpolation based on the proposed method can also be implemented easily. The results of computer simulations under kinds of conditions for four slice CT are presented. The quantitative gains show that the proposed method can obtain higher efficiency than the conventional method. References [1] Hu H. Multi-slice helical CT: scan and reconstruction [J]. Med Phys, 1999, 26(1): 5 18. [2] Feldkamp L A, Davis L C, Kress J W. Practical conebeam algorithm [J]. J Opt Soc Am, 1984, 1(6): 612 619. [3] Flohr T, Stierstorfer K, Bruder H, et al. Image reconstruction and image quality evaluation for a 16-slice CT scanner [J]. Med Phys, 2003, 30(5): 832 845. [4] Silver M D, Taguchi K, Han K S. Field-of-view dependent helical pitch in multi-slice CT [C]//Proc SPIE. San Diego CA: [s.n.], 2001: 839 850. [5] Katsevich A, Lauritsch G, Bruder H, et al. Evaluation and empirical analysis of an exact FBP algorithm for spiral cone-beam CT. [C]//Proc SPIE. San Diego, California: [s.n.], 2003: 663 674. [6] Zhao S, Wang G. Feldkamp-type cone-beam tomography in the wavelet framework [J]. IEEE Transactions on Medical Imaging, 2000, 19(9): 922 929. [7] Tam K, Samarasekera S, Sauer F. Exact cone beam CT with a spiral scan [J]. Phys Med Biol, 1998, 43(4): 1015 1024. [8] Proksa R, Kohler T, Grass M, et al. The n-pi-method for helical cone-beam CT [J]. IEEE Transactions on Medical Imaging, 2000, 19(9): 848 863. [9] Wang G, Lin T H, Cheng P C, et al. A general conebeam reconstruction algorithm [J]. IEEE Transactions on Medical Imaging, 1993, 12(9): 486 496. [10] Kachelriess M, Schaller S, Kalender W A. Advanced single-slice rebinning in cone-beam helical CT [J]. Med Phys, 2000, 27(4): 754 772.

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