Adaptive algebraic reconstruction technique

Adaptive algebraic reconstruction technique Wenkai Lua) Department of Automation, Key State Lab of Intelligent Technology and System, Tsinghua University, Beijing 10084, People s Republic of China Fang-Fang Yinb) Department of Radiation Oncology, Henry Ford Hospital, Detroit, Michigan 48202 (Received 13 February 2004; revised 4 September 2004; accepted for publication 15 September 2004; published 12 ovember 2004) Algebraic reconstruction techniques (ART) are iterative procedures for reconstructing objects from their projections. It is proven that ART can be computationally efficient by carefully arranging the order in which the collected data are accessed during the reconstruction procedure and adaptively adjusting the relaxation parameters. In this paper, an adaptive algebraic reconstruction technique (AART), which adopts the same projection access scheme in multilevel scheme algebraic reconstruction technique (MLS-ART), is proposed. By introducing adaptive adjustment of the relaxation parameters during the reconstruction procedure, one-iteration AART can produce reconstructions with better quality, in comparison with one-iteration MLS-ART. Furthermore, AART outperforms MLS-ART with improved computational efficiency. 2004 American Association of Physicists in Medicine. [DOI: 10.1118/1.1812606] Key words: algebraic reconstruction technique, radon transforms, computerized tomography, image reconstruction I. ITRODUCTIO In computerized tomography, Fourier methods such as Radon transform (RT) and iterative reconstruction methods such as algebraic reconstruction technique (ART) and multilevel scheme algebraic reconstruction technique (MLS-ART) are the two major categories of image reconstruction techniques. While Fourier methods have been widely used in applications where many projections can be taken, algebraic methods have been shown to provide better reconstructions from either very few or limited views.1 3 The practical disadvantage of algebraic methods is their heavy computational burden since all of them are iterative. It is claimed that by a careful adjustment of the order in which the collected data are accessed during the reconstruction procedure and of the so-called relaxation parameters that are to be chosen in an algebraic reconstruction, ART can produce high-quality reconstructions with excellent computational efficiency.4 By arranging and accessing parallel projections in a multilevel scheme, MLS-ART5 can generate high-quality reconstructions with improved computational efficiency. A detailed comparison between MLS and other two conventional projection access orderings in ART (the random permutation scheme and the sequential access scheme) shows that MLS yields the most efficient reconstruction.6 MLS-ART has shown promising results in different situations.7,8 It has been pointed out that ART can be improved by: (1) carefully choosing the data-access ordering; (2) adaptive adjusting the relaxation parameters; and (3) incorporating available a priori knowledge.3 5 In this study, an adaptive image reconstruction technique, namely, AART, is proposed. At first, the same projection access scheme in MLS-ART is adopted by AART. Furthermore, a data-driven adjustment of 3222 Med. Phys. 31 (12), December 2004 relaxation parameters and amplitude constraints are applied during the reconstruction procedure. In other words, the relaxation parameters are adaptively adjusted according to the data. This is the reason why AART is used to name the proposed technique. In Sec. II, a theory describing the proposed method is presented. Section III furnishes the results of a comparative experimental study and Sec. IV yields some conclusions based on the experimental results. II. THEORY CT system can be modeled by 共1兲 Ax = b, where the observed data is b = 关b1, b2, b M 兴, the original image is x = 关x1, x2, x兴t, and x j 艌 0, j = 1,. The number of pixels in the original image is, and the number of rays is M. A = 关aij兴 ; i = 1, M ; j = 1, is a weight matrix. aij represents the contribution of the jth pixel to the ith ray integral and 0 艋 aij 艋 1. The problem is to reconstruct the original image x from the observed data b. A solution by a direct method with complicated computations involving the matrix A might be infeasible, because of the illposedness of the problem (say, limited views case), noisy data b, and huge data dimensions in practice.9,10 Instead, iterative methods, such as ART, are developed for efficient image reconstruction. For each ray, Eq. (1) can be rewritten as T aijx j = bi 兺 j=1 i = 1,...,M. 共2兲 The ART can be mathematically described in the following iterative scheme: 0094-2405/2004/31(12)/3222/9/$22.00 2004 Am. Assoc. Phys. Med. 3222

3223 3223 FIG. 1. Illustration of the computational procedures for locating the solution from an initial guess by Eq. (4) (circle) and Eq. (5) (diamond) for the cases of two unknown. x共m兲 = x共m 1兲 共i,m兲共a共i兲关x共m 1兲兴T bi兲, 共3兲 where i is the index of current ray from 1 to M, m is the iteration index, a共i兲关x共m 1兲兴T is the product of the 1 vector a共i兲 with the 1 vector 关x共m 1兲兴T, a共i兲 = 关ai1, ai2,..., ai兴,, x共m 1兲,..., x共m 1兲兴, and 共i,m兲 are the relaxation x共m 1兲 = 关x共m 1兲 1 2 parameters. The reconstruction at the current iteration is an update of that at the last iteration. To improve the performance of ART, different strategies for reconstruction update are proposed.1 5 A. Multilevel scheme for data access To improve the computation efficiency of ART, Guan and Gordon5 proposed a projection access order for ART in which the projections are arranged and accessed in a nominal multilevel scheme. Using this scheme, maximum orthogonality among projections can be maintained during the reconstruction. A detailed comparison between MLS and the other two conventional projection access orderings in ART (the random permutation scheme and the sequential access scheme) shows that MLS yields the most efficient reconstruction.6 Applications in different situations have shown that MLS-ART obtains promising results.7,8 In this proposed study, AART is modified based on MLS-ART with the same projection access scheme. Suppose the number of views P is a power of two, say P = 2L. And all the views are indexed as 0, 1,..., P 1 sequentially. In the MLS scheme,5 views are arranged in a total of L levels for accessing. In level l = 1, 0 and P / 2; level l = 2, P / 4 and 3P / 4; level l = 3, P / 8, 5P / 8, 3P / 8, 7P / 8; and so on. Views in one level halve the views in all previous levels and hence double the total number. FIG. 2. Image reconstruction results when angle interval = 1. (a) The original phantom and reconstructions obtained by (b) RT, (c) one-iteration MLS-ART, (d) one-iteration AART, (e) five-iteration MLS-ART, and (f) five-iteration AART.

3224 3224 of the relaxation parameters can lead to better-quality reconstruction and usually at the expense of convergence. In basic ART, the relaxation parameters 共i,m兲 are chosen to be 共i,m兲 = a共i兲/共a共i兲关a共i兲兴t兲 共m兲, FIG. 3. Choice of ROI (white) and BR (gray) for CR calculation. (a) The ROI is the boundary with high contrast while the BR is the area inside it, and (b) the ROI is the three small tumors while the BR is the area surrounding. B. Adaptive adjustment of relaxation parameters The practical importance of the relaxation parameter 共i,m兲 has also been demonstrated in the literatures. Careful choice 共4兲 where 共m兲 can be fixed or changes with the number of iterations. In Eq. (4), it is seen that the relaxation parameters 共i,m兲 are only dependent on the choice of 共m兲 because vector a共i兲 is not changed during the reconstruction procedure. It is noticed that small relaxation parameters can usually reduce the artifacts noise in ART reconstruction but need more computation time. And in some cases, by making the relaxation parameters as a function of the iteration number, that is, the relaxation parameters become progressively smaller with an increase in the number of iterations, speedy convergence can be achieved. The problem is that there is no single choice for the best relaxation parameters. The choice depends on the ultimate medical purpose of the reconstruction, the method of data collection, and on the number of iterations.1,4 FIG. 4. Plots of the measures (a) 1, (b) 2, (c) cnr1, and (d) cnr2 versus angle interval for three image reconstruction techniques (RT, one-iteration MLS-ART and one-iteration AART).

3225 a11x1 + a12x2 = b1, 3225 a21x1 + a22x2 = b2. 共6兲 The computational procedures for locating the solution from an initial guess by Eqs. (4) and (5) are displayed in Fig. 1. Like the solutions obtained by Eq. (4), the solutions obtained by Eq. (5) in all steps are from one line to another when 共m兲 is 1. It is seen that the solution obtained by Eq. (5) are closer to the true solution than that obtained by Eq. (4) after 20 iterations. The only difference between AART and MLS-ART is that the relaxation parameters are adjusted by Eq. (5) in AART, and by Eq. (4) in MLS-ART. So, for one iteration to update the data along a ray using Eq. (3), the computation burden of AART and MLS-ART are the same. These two methods all need 4 + 1 multiplications and 3 1 additions. Extra memory is needed to save the vector y共m 1兲 in AART. It should be noticed that we only need to calculate a共i兲 / 共a共i兲 关a共i兲兴T兲 in Eq. (4) one time if we can have enough memory to save them. But it might be infeasible, because of the huge memory space required. For an example, in the case to reconstruct an object of 256* 256 pixels with 179 views, and each view has 362 rays, we need 256* 256* 179* 362* 4 bytes, which is about 15.8 gigabytes, to save a共i兲 / 共a共i兲关a共i兲兴T兲. In the following section, we suppose we need to calculate the relaxation parameters for MLS-ART during each iteration. FIG. 5. The marked profiles through the original and reconstructed images shown in Fig. 2. For the purpose of comparison, only the segments including the three small tumors are shown, and the true profile shown in Fig. 5(a) is also plotted in Figs. 5(b) 5(f) as a solid line. In AART, a data-driven adjustment of the relaxation parameters 共i,m兲 is given as follows: 共i,m兲 = y共m 1兲/共a共i兲 y共m 1兲兲 共m兲, 共5兲 where y共m 1兲 = 关y 共m 1兲, y 共m 1兲,..., y 共m 1兲兴T, and y 共m 1兲 1 2 j 共m 1兲 = xj aij. ote that in case 共m兲 is 1, substituting x共m兲 for x can satisfy Eq. (2). With this approach, the relaxation parameter for each pixel is adjusted according to its value obtained in the last iteration. It is noticed that aij only represents the geometry contribution of the jth pixel to the ith ray integral. The true contribution of the jth pixel to the ith ray integral is y 共m 1兲. In Eq. (4), we can see that the pixels with larger j geometry contribution aij to the ith ray integral will have a larger adjustment step during the reconstruction procedure. It is more reasonable to adjust the pixels that have a larger to the ith ray integral with a larger adjustcontribution y 共m 1兲 j ment step. This adaptive adjustment of relaxation parameters leads to not only to speedy convergence but also high-quality reconstruction, as shown in the next section. To illustrate how the speedy convergence can be achieved by Eq. (5), we have considered the case of only two variables x1 and x2 satisfying the following equations: III. EXPERIMETAL RESULTS Computer simulations are conducted to evaluate the performance of the proposed approach. Here we only evaluate the performance of the proposed AART using parallel geometry because the projection data obtained by fan-beam collimators are usually rebined to yield parallel projections.11 And hence the same reconstruction algorithm can be used for both sets of projections. The primary purpose of this study is to present an ART reconstruction technique to further improve the reconstruction efficiency of the previous published MLS-ART technique. Therefore, the evaluation of the presented technique is focused on the efficient improvements between AART and MLS-ART. Although results from RT technique were also presented in the limited comparison, it dose not necessarily mean that the AART technique is superior to the RT technique since the nature of reconstruction methods are quite different. Comprehensive comparison between ART and RT techniques has been discussed by several investigators and will not be discussed in this paper. The Shepp-Logan phantom shown in Fig. 2(a) is used in the following experiments. The size of the phantom is 256 256 pixels. In all experiments, the views are arranged from 0 to 179 at an equal angle interval. The number of views in an experiment depends on the chosen angle interval. In each view, there are 362 equally spaced parallel rays. The width of each ray is equal to the image pixel size. In all experiments, 共m兲 for both AART and MLS-ART is set to be 1. The inverse RT uses the filtered back-projection algorithm to reconstruct the image. In our experiments, we adopt the

3226 3226 FIG. 6. Plots of the measures (a) 1, (b) 2, (c) cnr1, and (d) cnr2 versus iteration number for MLS-ART and AART when = 1. MATLAB RT tools, in which spline interpolation is chosen and Hamming filter, which is obtained by multiplying the RamLak filter by a Hamming window, is applied. Here we denote the width of the hamming window as w. In our experiments, w is changed according to the angle interval. Supposing the whole frequency range is 1, w = 1 when = 1, and w = 0.5 when = 18. w for other is obtained by a linear interpolation. For convenience of description, in the following section, one iteration means that all rays are used to update the reconstruction once. For a quick visual comparison, reconstruction images obtained using RT, one-iteration MLS-ART, one-iteration AART, fiveiteration MLS-ART, and five-iteration AART with = 1 are shown in Figs. 2(b) 2(f), respectively. These reconstructed images indicate that AART gives promising results. For objective and quantitative evaluation of the reconstruction image, three measures, namely, error, correlation, and contrast-to-noise ratio (CR) are used. The error and correlation measures are defined as 1 = 2 = 兺i=1 共xi xi 兲2 2 兺i=1 xi 共7兲, 关兺i=1 共xi x 兲共xi x 兲兴 关兺i=1 共xi x 兲2兺i=1 共xi x 兲2兴1/2, 共8兲 where xi共x 兲 and x i 共x 兲 each represent the pixel value (image average value) in the original and the reconstructed images, respectively. By choosing region-of-interest (ROI) and background region (BR), we can calculate CR as follows: CR = 兩mr mb兩 冑 r2 + 2b, 共9兲 where mr is the mean value of ROI, mb is the mean value of BR, r is the standard deviation of ROI, and b is the standard deviation of BR. In general, lower error 1, higher correlation 2 and higher CR yield a better result.

3227 3227 FIG. 7. Plots of the measures (a) 1, (b) 2, (c) cnr1, and (d) cnr2 versus iteration number for MLS-ART and AART when = 10. In our study, we choose two pairs of ROI and BR shown in Figs. 3(a) and 3(b) respectively. The white regions are ROI while the gray regions are BR. In Fig. 3(a) the ROI is the boundary with high contrast, and the BR is the area inside the boundary. We denote the CR obtained using the ROI and BR shown in Fig. 3(a) as cnr1. In Fig. 3(b), the ROI is the three small tumors while the BR is the area surrounding. We denote the CR obtained using the ROI and BR shown in Fig. 3(b) as cnr2. To compare the image quality of reconstructed images obtained by RT, one-iteration MLS-ART and one-iteration AART methods, the measures 1, 2, cnr1, and cnr2 versus angle interval for the above three methods are plotted in Figs. 4(a) 4(d), respectively. The marked profiles through the original and reconstructed images shown in Fig. 2 are plotted in Figs. 5(a) 5(f), respectively. For the purpose of comparison, only the segments including the three small tumors are shown, and the true profile shown in Fig. 5(a) is also plotted in Figs. 5(b) 5(f) as a solid line. It is shown that AART gives the best results according to the measures 1, 2, cnr1 in this specific example. The measure cnr2 and the profiles shown in Fig. 5 indicate that both AART and MLSART give comparable results for the reconstruction of the three small tumors. The furthermore comparisons between AART and MLSART are made based on multi-iteration reconstructions. For = 1, the measures 1, 2, cnr1, and cnr2 versus iteration number for AART and MLS-ART are plotted in Figs. 6(a) 6(d), respectively. For = 10, the measures 1, 2, cnr1, and cnr2 versus iteration number for AART and MLSART are plotted in Figs. 7(a) 7(d), respectively. It is seen that AART has faster convergence speed. To evaluate the improvement of convergence speed resulted from AART for different angle interval, the iteration number IMLS for MLS-ART to obtain reconstruction with almost the same image quality of that obtained by oneiteration AART is used as a measure since the computation time of one iteration for AART and MLS-ART is the same. Let us denote a as the error measure 1 of the reconstruction obtained by one-iteration AART, MLS共i兲 the error measure

3228 3228 FIG. 8. The iteration number IMLS versus angle interval for MLS-ART to obtain reconstruction with almost the same image quality of that obtained by one-iteration AART according to the error measure 1. FIG. 10. The marked profiles through the reconstructed images shown in Fig. 9. For the purpose of comparison, the true profile is also plotted in Figs. 10(a) 10(e) as a solid line. 1 of the reconstruction obtained by ith-iteration MLS-ART. Since the image quality can not be equal exactly, IMLS is defined to satisfy the following condition: MLS共IMLS兲 艌 a MLS共IMLS + 1兲. FIG. 9. Image reconstruction results when angle interval = 6. The reconstructions obtained by (a) RT, (b) one-iteration MLS-ART, (c) one-iteration AART, (d) five-iteration MLS-ART, and (e) five-iteration AART. 共10兲 The iteration number IMLS versus angle interval is plotted in Fig. 8. The average value of IMLS for = 1, 2,..., 18 is about 13. From this result, it is seen that the calculation speed could be improved significantly using the AART method in comparison with MLS-ART. In Figs. 9(a) 9(e), we show the reconstructions by RT, one-iteration MLS-ART, one-iteration AART, five-iteration MLS-ART, and five-iteration AART with = 6, respectively. In the reconstructions obtained by our method, we can see that the artifact noises are almost disappeared in the area outside the boundary. The marked profiles through the reconstructed images shown in Fig. 9 are plotted in Figs. 10(a) 10(e), respectively. For the purpose of comparison, the true profile is also plotted in Figs. 10(a) 10(e) as a solid line. The measures 1 and 2 of these reconstructed profiles are shown in Table I. These data show that AART gives the best result in the case of limited views. To evaluate the performance of AART under noise situation, we repeat the above experiments by adding the Poisson noise to the projection data with different signal-noise-ratio (SR). In all the following experiments, we do not apply

3229 3229 TABLE I. The measures 1 and 2 of these reconstructed profiles shown in Fig. 10. 1 2 RT One-iteration MLS-ART One-iteration AART Five-iteration MLS-ART Five-iteration AART 0.2975 0.7753 0.2404 0.8313 0.0551 0.9589 0.0914 0.9381 0.0446 0.9674 filter on the noisy projection data for AART and MLS-ART. Here some results obtained when SR= 15 db are given. In Figs. 11(a) 11(e), we show the reconstructions by RT, oneiteration MLS-ART, one-iteration AART, five-iteration MLSART, and five-iteration AART with = 3, respectively. Same as the other ARTs, AART has to be stopped before the reconstructed image noise is significantly amplified due to the nature of the ill-posed and ill-condition tomography problem, especially in the case that limited views with lower FIG. 12. Image reconstruction results using noisy projection data when angle interval = 3 and SR= 5 db. The reconstructions obtained by (a) RT, (b) one-iteration MLS-ART, and (c) one-iteration AART. TABLE II. The measures 1, 2, cnr1, and cnr2 of these reconstructions shown in Fig. 12. FIG. 11. Image reconstruction results using noisy projection data when angle interval = 3 and SR= 15 db. The reconstructions obtained by (a) RT, (b) one-iteration MLS-ART, (c) one-iteration AART, (d) five-iteration MLS-ART, and (e) five-iteration AART. 1 2 cnr1 cnr2 RT One-iteration MLS-ART One-iteration AART 0.2137 0.8613 1.8834 0.6712 0.1632 0.9077 2.3197 1.0736 0.0931 0.9517 3.4232 1.0405

3230 SR can be obtained. Since AART has speedy convergence, we may be able to obtain reasonable good quality reconstruction after one iteration even if the noise is strong as indicated in our experiments. In Figs. 12(a) 12(c), we show the reconstructions by RT, one-iteration MLS-ART, and one-iteration AART with = 3 when SR= 5 db, respectively. The measures 1, 2, cnr1, and cnr2 of these reconstructions are shown in Table II. IV. COCLUSIOS We have developed an image reconstruction technique, an adaptive ART, for computerized tomography in this paper. Besides the MLS data-access scheme, our method exploits data-driven adjustment of relaxation parameter. Comparisons are made between our method, RT, and MLS-ART. The results from our test cases are shown that AART gives promising results in comparison with the other two methods. In comparison with MLS-ART, AART converge fast significantly in our experiments. In summary, simulation data show the effectiveness of the proposed image reconstruction method. a) Author to whom correspondence should be addressed; electronic mail: lwkmf@mail.tsinghua.edu.cn b) Electronic mail: fyinl@hfhs.org 1 3230 G. T. Herman, Image Reconstructions from Projections: The Fundamentals of Computerized Tomography (Academic, ew York, 1989). 2 A. H. Andersen, Algebraic reconstruction in CT from limited views, IEEE Trans. Med. Imaging 8, 50 55 (1989). 3 P. Oskoui and Henty Stark, A comparative study of three reconstruction methods for a limited-view computer tomography problem, IEEE Trans. Med. Imaging 8, 43 49 (1989). 4 G. T. Herman and L. B. Meyer, Algebraic reconstruction techniques can be made computationally efficient, IEEE Trans. Med. Imaging 12, 600 609 (1993). 5 H. Guan and R. Gordon, A projection access order for speedy convergence of algebraic reconstruction techniques (ART): A multilevel scheme (MLS) for computed tomography, Phys. Med. Biol. 39, 2005 2022 (1994). 6 H. Guan and R. Gordon, Computed tomography using algebraic reconstruction techniques (ARTs) with different projection access schemes: A comparison study under practical situations, Phys. Med. Biol. 41, 1727 1743 (1996). 7 H. Guan, R. Gordon, and Y. Zhu, Combining various projection access schemes with the algebraic reconstruction technique for low-contrast detection in computed tomography, Phys. Med. Biol. 43, 2413 2421 (1998). 8 H. Guan, F. Yin, Y. Zhu, and J. J. Kim, Adaptive portal CT reconstruction: A simulation study, Med. Phys. 27, 2209 2214 (2000). 9 Y. Censor and G. T. Herman, On some optimization techniques in image reconstruction from projections, Appl. umer. Math. 3, 365 391 (1987). 10 Y. Censor and S. A. Zenios, Parallel Optimization: Theory, Algorithms and Applications (Oxford University Press, ew York, 1997). 11 A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, ew York, 1987).