A Projection Access Scheme for Iterative Reconstruction Based on the Golden Section

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1 A Projection Access Scheme for Iterative Reconstruction Based on the Golden Section Thomas Köhler Philips Research Laboratories Roentgenstrasse - Hamburg Germany Abstract A new access scheme for projections in iterative image reconstruction is suggested. It is based on a constant angular increment. The constant is chosen such that the range of 10 is divided by the. It is shown in a simulation study that the method behaves superior to the access scheme and the scheme based on. I. INTRODUCTION Iterative reconstruction techniques are widely used in the area of PET and SPECT because they produce superior image quality compared with filtered back-projection type algorithms, if the data are very noisy. Typically, these algorithms operate on subsets of the measured data, e. g. they take one projection to calculate an update for the intermediate image. It is known that the ordering of the projection during the iteration has a large influence on the convergence speed of iterative image reconstruction algorithms like the algebraic reconstruction technique (ART) [1]. Several methods have been proposed and evaluated so far [] []. Mueller et al. also evaluated a method with constant angular increment as it will be used here. However, their choices of.0, 9.,. are not motivated and differ from the choice presented below. A. Reconstruction Algorithm II. METHOD The measurement in PET, SPECT, and CT provides (after proper pre-processing) a set of N line integrals p l through an continuous object function µ( x), which is the image to be recovered. In iterative methods, the image has to be decomposed into a number of basis functions, and it is then represented by a finite number of M coefficients. Here, Kaiser-Bessel functions (blobs) [] are used as basis functions. The discretization results in a finite linear system p = Aµ, (1) where p is an N-dimensional vector of measured line integrals, µ is an M-dimensional vector of image quantities (e. g. absorption coefficients for CT), and A is the discrete version of the integral operator that represents the measurement process. In standard ART [], the linear system is solved iteratively by he following procedure: µ k+1 i p l A lj µ j = µ k j i + λ A li. () j A lj In the original ART, an update step uses just one line integral. In simultaneous ART (SART) [], a full projection is calculated and used for each update step. This improves the stability of the algorithm significantly. The analysis of the new projection ordering scheme was performed using SART. B. Projection Ordering Traditionally, the design rule for the projection ordering is the following: At any time, the next considered projection angle should contain information about the object, which is as independent of the previously taken projections as possible. This is achieved by maximizing the angular distance of subsequent projections. The new method presented here is inspired by the way many plants position their new leafs []. It appears that a lot of plants use an angle between succeeding leafs of approximately 0 times the golden ratio 1 g = 0.1. () The resulting angle of about. might be a result of an optimization problem: A new leaf should be placed such, that there is as little overlap with previously grown leafs as possible in order to avoid unwanted shadowing of previous leafs. Geometrically, this optimization problem is very similar to the problem of selecting the best order for projection angles in iterative reconstruction methods. In contrast to the botanic counterpart of the solution, the proper angle between succeeding projection angles can be chosen as g 10, since parallel projections, which are 10 apart from each other, are already equivalent. Note that there is another symmetry in this particular problem: Going g 0 counter-clockwise is the same as going (1 g) 0 clockwise. Thus, (1 g) 10 is also valid choice. By choosing this angular increment, several key features are achieved: /0/$0.00 (C) 00 IEEE

2 TABLE I FIRST 1 FIBONACCI NUMBERS. k 1 f k k f k Fig. 1. First 1 projections according to the proposed method. 1) Considering at any time the angular distribution of the projection angles taken so far, it appears that only three different values occur for the angular gap between these angles. They have a relative size of 1, g, and g. ) Each new angle divides one of the largest angular gaps according to the golden ratio, see Fig. 1, and therefore, guarantees a maximum of new information to be taken into account. ) The sequence is based upon a constant angular increment. Thus, the method is very simple to implement. ) The sampling pattern is non-periodic because the golden ratio is irrational. This implies that all measured projections are processed during the iteration. This is a basic feature required for all sequences to be independent of the total number of acquired projections. If this method is applied to a data set with discrete angular sampling, the proposed angles have to be mapped onto the discrete samples. This is done by choosing at any time within a full iteration the nearest angle, which has not been taken so far. If the profiles are sampled uniformly in angular direction, the angular increment can be adjusted by adding a small angle to take this into account. The required adjustment is smallest if N is a Fibonacci number f j. For example, in the case N = f = 1, the adjustment is approximately 0.1. Because of the very small fraction of this adjustment with respect to the angular increment of approximately 1., the benefits described so far are still fulfilled, see Fig.. The first 1 Fibonacci numbers are given in Tab. I. Another minor point is the following: If f i denotes the ith Fibonacci number, then after f i processed projections, the angles between processed or measured directions are equal or they differ only by a factor of the g. This implies an almost uniform distribution of angles over the full circle. This is illustrated in Fig. 1 for the case of 1 projections, where the two values g and g for the gaps appear. The next angle of the sequence would partition one of the g 10 segments into a g 10 and a g 10 segment. A 1 B 1 C 1 D E F 1 Fig.. Access sequence for 1 projections over 10. Note that after the first two angles, there is a big and a small gap in the angular range (A). The next angle divides the larger one, resulting in two large gaps and one small gap (B). The next two angles divide the larger gaps, resulting in large and small gaps (C). In general, after f k angles, there are f k 1 large and f k small gaps and the following f k 1 angles will divide the large gaps /0/$0.00 (C) 00 IEEE

3 image resolution / mm Fig.. Resolution of the reconstructed images as a function of the number of iterations for the case of 0 projections. III. RESULTS One-dimensional parallel projections of the forbild head phantom were simulated. Apart from the proposed method, the access and the [1] were evaluated. Three different numbers of projections were taken in order to evaluate the performance of the different methods with respect to this number: 0, 10, and 0. 0 projections were used because this is the number a modern CT scanner takes within is considered because it is a Fibonacci number, which is expected to work in particular well for the proposed method. Finally, 0 contains only small prime factors, which is expected to work well for the prime number decomposition. The images were evaluated after each full iteration. The correlation method introduced by Liow and Strother [9] was used for quantitative evaluation. In this method, the correlation between the reconstructed image and a smoothed version of the true image is calculated. Gaussian smoothing was used here, and the size of the Gauss-kernel was alway chosen such that the correlation with the reconstructed image is maximized. The smoothing width represents an estimate for the resolution of the reconstructed image. The correlation serves as a measure for the image quality. The resolution obtained by the different access schemes differs only very little. Exemplarily, the resolution as a function of the is shown in Fig. for the case of 0 projections. For the obtained correlation, the results differ significantly between the different number of projections. Fig. shows 1 the correlation as a function of the number of iterations. Note that the correlation increases until iteration and decreases again. This is a side effect of the additional smoothing before the correlation is calculated. In all cases, the resolution improves during the iteration. Once the resolution reaches a certain limit, the result is influenced by the discrete representation of the image, which results e. g. in overshoots and undershoots at sharp edges. This is hidden at lower resolution by the smoothing of the true image. Clearly, the approach suffers from the large prime factors in the cases with 0 and 10 projections. Only for 0 projections, provides the same good values for the correlation than the. For all cases, the new method provides better results than the access and better or equal results than the. IV. SUMMARY The proposed method appears to be very insensitive to the number of projections. It performs better than the access scheme and better or equal to the. In particular for the case of practical importance (0), it performs better. Furthermore, it is easier to implement than the or other known methods like the weighted distance approach []. Results have been obtained for ART only, but it is anticipated that other iterative reconstruction algorithms like OS- ML [10] will behave similar. ACKNOWLEDGMENT I would like to thank Samuel Matej for providing the code for the. REFERENCES [1] G. T. Herman and L. B. Meyer, Algebraic reconstruction technique can be made computationally efficient, IEEE Trans. Med. Imag., vol. 1, no., pp , 199. [] H. Guan and R. Gordon, A projection access order for speedy convergence of ART (algebraic reconstruction technique): A multilevel scheme for computed tomography, Phys. Med. Biol., vol. 9, pp. 00 0, 199. [] K. Mueller, R. Yagel, and F. Cornhill, The weighted distance scheme: A globally optimizing projection ordering method for ART, IEEE Trans. Med. Imag., vol. 1, no., pp. 0, 199. [] 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., vol. 1, pp. 1 1, 199. [] R. M. Lewitt, Alternatives to voxels for image representation in iterative reconstruction algorithms, Phys. Med. Biol., vol., pp. 0 1, 199. [] R. Gordon, R. Bender, and G. T. Herman, Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography, J. Theor. Biol., vol. 9, pp. 1 1, 190. [] A. Andersen and A. Kak, Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm, Ultrason. Imag., vol., pp. 1 9, 19. [] E. Strassburger, Lehrbuch der Botanik, rd ed. Stuttgart: Gustav Fischer Verlag, [9] J.-S. Liow and S. C. Strother, The convergence of object dependent resolution in maximum likelihood based tomographic image reconstruction, Phys. Med. Biol., vol., pp. 0, 199. [10] C. Kamphuis and F. Beekman, Accelerated Iterative Transmission CT Reconstruction Using an Ordered Subsets Convex Algorithm, IEEE Trans. Med. Imag., vol. 1, no., pp. 01 0, /0/$0.00 (C) 00 IEEE

e- 0 projections e- e- 1e- e- e- 0 1 9 e- 10 projections e- e- 1e- e- e- 0 1 9 e- 0 projections e- e- 1e- e- e- 0 1 9 Fig.

4 e- 0 projections e- e- 1e- e- e e- 10 projections e- e- 1e- e- e e- 0 projections e- e- 1e- e- e Fig.. Image quality of different projection access schemes for 0 (top), 10 (middle), and 0 (bottom) projections. Fig.. Example reconstruction for different access schemes after one iteration for 0 projections. From top to bottom: Golden section,, /0/$0.00 (C) 00 IEEE

5 Fig.. Example reconstruction for different access schemes after one iteration for 10 projections. From top to bottom: Golden section,,. Fig.. Example reconstruction for different access schemes after one iteration for 0 projections. From top to bottom: Golden section,, /0/$0.00 (C) 00 IEEE

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