A Weighted Least Squares PET Image Reconstruction Method Using Iterative Coordinate Descent Algorithms

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1 A Weighted Least Squares PET Image Reconstruction Method Using Iterative Coordinate Descent Algorithms Hongqing Zhu, Huazhong Shu, Jian Zhou and Limin Luo Department of Biological Science and Medical Engineering, Southeast University, Nanging, 00 China Abstract In this paper, we present an image reconstruction method for positron emission tomography (PET) based on weighted least squares () obective function. Unlike a usual algorithm, the proposed method, which we call it ICD-, minimizes the obective function using iterative coordinate descent (ICD) method in a sequence of small hidden data spaces. Although ICD- uses a strategy to update parameter sequentially ust like common ICD method, however, the use of these small hidden data spaces makes the ICD- method converge more faster and produce reconstructed images with greater contrast and detail than the usual methods. The OS algorithm is also used to accelerate image reconstruction for ICD-. The experimental results show that the proposed method is more effective even if the proection data include statistic noise. I. INTRODUCTION Positron Emission Tomography (PET) is one of the most important imaging tools in modern diagnosis. Reconstruction of PET scan images is a complex problem, various efforts have been made in the past to speed up both types of PET image reconstruction algorithms, namely, analytic and iterative algorithms. Among iterative algorithms, the statistical reconstruction has become increasingly popular in PET due to its ability to model the noise and the imaging physics and to impose positivity constraints on the reconstruction. So far, all statistical reconstruction algorithms are based on either the maximum likelihood (ML) or least squares (LS) cost function, and their generalizations. A commonly used method for finding ML estimates in PET had been the expectation-maximization (EM) algorithm []. The slow convergence rate is the main disadvantage of EM. Fessler et al. thought this is due essentially to the large Fisher Information of the complete data space used in ML-EM algorithm []. They proposed a space-alternating generalized EM (SAGE) algorithm in which the parameters are updated sequentially using a sequence of small hidden data spaces rather than one large complete-data space to accelerate ML- EM algorithm [,]. Moreover, ML estimates are usually unstable due to the typical limits in fidelity of PET proection data []. To overcome this drawback, some methods such as regularization maximum a posteriori probability (MAP) estimation [,] have been proposed. The other recent algorithm known as iterative coordinate descent (ICD) for image reconstruction has been developed [, ]. It was based on the direct optimization of the MAP criterion, and used the sequential optimization of pixel values in the reconstruction. Both the SAGE algorithm and the ICD algorithm take a similar strategy to update parameters sequentially, which enable them to speed up significantly the convergence rate. The difference is the SAGE algorithm updates each pixel in the limit size complete spaces ( hidden data spaces). The first use of the weighted least squares () method in image reconstruction was Huesman et al. []. The traditional methods are based on proection data, which require the estimation of the error covariance matrix from proection data [0,, ]. Recently, Anderson et al. developed a algorithm, which uses the exact mean instead of the databased variance to describe the weights. The cost function is no longer quadratic, but it is convex [, ]. Anderson et al. demonstrated that the algorithm converges faster than the maximum likelihood expectation-maximization and produced images that had significantly better resolution and contrast []. In this paper, a novel algorithm, namely ICD- that combines the ICD algorithm with the algorithm is proposed for PET image reconstruction. In the new approach, we adopt the same obective function as that proposed by Anderson et al. []. However, unlike the common algorithm, the ICD- algorithm is to minimize each new obective function using iterative coordinate descent in a sequence of small hidden data spaces. This is important since one can cautiously choose hidden data spaces with considerably less Fisher information. According to [,], less informative hidden data spaces can yield fast convergence compared with one large complete-data space with more Fisher information. Although the ICD- algorithm is implemented by sequential updating each pixel of the image, with each updating the current pixel is chosen to minimize the new cost function, however, the use of hidden data spaces makes ICD- algorithm converge faster than the common algorithm. To improve convergence rate, the OS algorithm is used to accelerate image reconstruction for ICD-. We call it OS-ICD- method. In OS-ICD- method, the proection data are grouped into subsets: a pixel in a reconstruction image is updated by using proections in each subset /0/$0.00 (C) 00 IEEE

2 II. METHODOLOGY A. ICD- Algorithm Assume an obective image be discretized into n pixels with emission rates x =[x,...x,...x n ] T. Assume that the emission source is viewed by m detectors (i =,...,m). According to the assumption that the observed photon counts are independent Poisson random variables over the region of interest [], we have: Y i = N i + R i P oisson{ p i x + r i } () where Y i denotes the ith detector recording emissions which include the photon counts emitted by all pixels and the numbers of emissions brought by background events. {Ri} are also independent Poisson variables: R i P oisson{r i }. Background rates {r i } are assumed to be known. N i is the actual numbers of photons emitted from the th pixel and detected by the ith detector bin. The system matrix element p i represents the probability that an emission from pixel is recorded at detector tube i. x is the expected value of pixel. Under the assumption of {r i } being zero, according to [, ], we can propose the following estimator: { ˆx = arg min Φ(x) x () s.t. x 0 where Φ(x) = (Px y) T W (Px y) = m ((Px) i y i ) (Px) i = m ( = p ix y i ) = p () ix where W is m m weight matrix. Anderson et al. defined it as: w i = diag((px), (Px),...,(Px) m ) () To apply iterative coordinate descent directly to this, we might try to update x by equating the partial derivative of Φ(x) the to zero. x (Φ(x)) = (Px) i y i (Px) i =0 () Unfortunately equation () has no analytical solution. A line-search method would require multiple evaluations of (), which would be expensive. The complete-data space for the classical algorithm for this problem is the set of unobservable random variables. X = {{N i } n =, {R i }} m () According to the SAGE algorithm [,], in ICD-, we employ a series of small complete-data spaces, namely, hidden data spaces. Here, we choose a most obvious and simple hidden data space which consists of those photons emitted from a given pixel, assuming the th pixel, that is, X S = {N i,r i } m () where S = {} is an individual pixel index set, Obviously, we have S S = {,...,n}, =,...,n. Thus, the new obective function in each hidden data spaces Φ S for the th parameter is Φ S (x ; x (k) (p i x N i ) )= () p i x N i = E{N i Y = y; x (k) } = x(k) p i y i = p () ix (k) where N i is the expected photon counts emitted from pixel and detected by the ith detector. x (k) is the estimate of x after k iterations. To minimize the new obective function Φ S in each hidden data space, we calculate the partial derivate for each given pixel x in () and yield: (Φ S ) = x (p i x N i )p i x (p i x N i ) p i (p i x ) (p i x ) N i = p i x (0) Let Eq.(0) be zero; we then have the formula for the th pixel s update: or N i p i = p i x = x x = m p i N i p i () N i p i () Substituting () into () yields x = x p i yi m p i ( = p () ix ) We obtain the fixed point iterative formula for the th pixel s update as follows: x (k+) = x (k) p i yi m p i ( = p () ix (k) ) According to the ICD algorithm, the detail ICD- algorithm for Poisson data can be outlined as follows. Initialization: y i = p i x () + r i () where x () be a starting positive image vector. We can set it via the FBP reconstructed image for k =,,... until convergence of x (k) { /0/$0.00 (C) 00 IEEE

3 for =,...,n{ updating each pixel: x (k+) = x (k) m p i p i y i y i () x (k+) q = x (k) q, q () updating proection: y i = y i +(x (k+) x (k) )p i i : p i 0 () endfor endfor The above ICD- algorithm updates the parameters sequentially and the predicted measurement y i is updated within the inner loop. B. OS Version ICD- Algorithm The OS algorithm is a useful method to accelerate PET image reconstruction. OS can be applied into any algorithm, which involved a sum over proection. The ML-EM method combined with the OS algorithm is called the OSEM method []. Here, we attempt to apply the OS algorithm and develop the OS version ICD- (OS-ICD-) method. x (k+) = x (k) p iy i i S p t i i S ( t l p () ilx (k) l ) here S t denotes the chosen order subset, l S t, t is the order subset level. OS-ICD- use a strategy to update parameters grouped on a sequence of order subsets (or blocks) of proection that enable it to obtain a very good reconstructed image in few iterations. Fig.. A simulated emission thorax phantom. III. EXPERIMENTAL RESULTS In order to test the effectiveness of the proposed methods, we compare the reconstructed images of, ICD- and OS-ICD- algorithms. In our experiment, a thorax phantom is used to test the presented method. The relative activities of the elements are shown in Fig.. The sinogram has radial bins and 0 angles. The proection data including % uniform Poisson background noise is calculated. Fig. shows the reconstructed results. The iteration Fig.. (From up to down), reconstructed images using the, ICD- and OS-ICD-, the first column with noise-free and the second column with Poisson noise. numbers are 0 for, ICD- and the iteration number are for OS-ICD- method. We can observe from Fig. that the whether the proections with noise or not, the ICD- methods can yield some high-quality images in comparison with the algorithm [,]. We can also observe that OS-ICD- is more efficient than ICD- method. The former obtains a very good reconstructed image in few iterations. Fig. displays the obective function Φ versus iteration numbers for the and ICD- algorithms. Fig. shows that the obective function of ICD- decreases more rapidly than the obective function of. It is because that ICD method update parameters sequentially in hidden data spaces allows us to more accelerate the convergence. We also use the mean absolute error (MAE) and chi-square error (CSE) to evaluate the quality of reconstructed images. The MAE is defined as: where x org MAE = rec x x org n = (0) and x rec denote the value of pixel of the original activity image and the reconstructed image, respectively. MAE measures the average discrepancy between the /0/$0.00 (C) 00 IEEE

4 0 x 00 0 x 00 x 0 x Obection Function ICD Obection Function ICD ICD ICD 0 0 Fig.. Comparison of obective function versus iteration numbers, proections with noise free (left), proections with Poisson background noise (right). Fig.. Chi-square error of reconstructed images, proections without noise (left), proections with Poisson background noise (right). reconstructed image and the original activity image. The CSE is defined as CSE = [y i log(y i /y (k) i ) (y i y (k) i )] () y (k) i = n = p i x (k) () CSE measures the discrepancy between the calculated proections and the original proections. Figs. and display the MAE and CSE values of the reconstructed images using and ICD-, respectively. The results show that the MAE and CSE values using ICD- decrease more rapidly than those obtained with even if the statistic noise is presented. Fig. indicate the CSE of OS version ICD- and. The CSE quantitative valuation results express the OS-ICD- is superior to the and OS- method. The OS technique can improve convergence rate and image quality for a certainly. Mean Absolute Error (MAE) ICD Mean Absolute Error (MAE) ICD x 0 0 OS OS ICD 0 x 0 OS OS ICD Fig.. Chi-square error of reconstructed images, proections without noise (left), proections with Poisson background noise (right). algorithm. In addition, the OS technique is used to accelerate the proposed ICD- method. We attempted a fair comparison between, ICD- and their OS version. The effectiveness of the proposed ICD- and OS-ICD- reconstructed approach were confirmed by simulation experiment. ACKNOWLEDGMENT The authors gratefully acknowledge The University of Michigan for their amiably providing with thorax phantom and some Matlab code in their website. This work was supported by National Basic Research Program of China under grant, No.00CB REFERENCES Fig.. Mean absolute error of reconstructed images, proections without noise (left), proections with Poisson background noise (right). IV. CONCLUSION We have proposed a new approach named ICD- that combines the ICD algorithm and algorithm for PET image reconstruction. In the new approach, the sequential updating of parameters was completed in some small hidden data spaces, and during each update, the current pixel was chosen to minimize each new obective function, so, the convergence rate and the quality of reconstructed images have been greatly improved in comparison with the common [] L. A. Shepp and Y. Vardi, Maximum likelihood reconstruction for emission tomography, IEEE Trans. Med. Imag., vol. MI-, pp.-,. [] J. A. Fessler and A. O. Hero, Space-alternating generalized expectationmaximization algorithm, IEEE Tran. Signal Proc., vol., no.0, pp. -, Oct.. [] J. A. Fessler and A. O. Hero, New complete-data spaces and faster algorithms for penalized-likelihood emission tomography, IEEE Conference on Nuclear Science Symposium and Medical Imaging, vol., pp.-0,. [] J. A. Fessler and A. O. Hero, Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms, IEEE Trans. IP., vol., no., pp.-, Oct.. [] C. A. Bouman and K. Sauer, A unified approach to statistical tomography using coordinate descent optimization, IEEE Trans. Imag. Proc., vol., no., pp. 0-, March. [] E. Levitan and G. T. Herman, A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography, IEEE Trans. Med. Imag., vol. MI-, /0/$0.00 (C) 00 IEEE

5 [] P. J. Green, Bayesian reconstructions from emission tomography data using a modified EM algorithm, IEEE Trans. Med. Imag., vol., pp.-, 0. [] C. A. Bouman and K. Sauer, A local update strategy for iterative reconstruction from proections, IEEE Trans. Signal Proc., vol., no., pp.-, Feb.. [] R. H. Huesman, G. T. Gullberg, W. L. Greeberg, and T, F, Budinger, User manual, Donner algorithms for reconstruction tomography, Lawrence Berkeley Lab., Univ, California,. [0] B. M. W. Tsui, Z. Zhao, E. C. Frey, and G. T. Gullberg, Comparison between ML-EM and -CG algorithms for SPECT image reconstruction, IEEE Trans. Nucl. Sci., vol., pp.-,. [] J. A. Fessler, Penalized weighted least-squares image reconstruction for positron emission tomography, IEEE Trans. Med. Imag., vol., no., pp.0-00, Jun.. [] N. H. Clinthorne, Constrained least-squares vs. maximum likelihood reconstructions for Poisson data, In Proc. IEEE Med. Imag. Conf., vol., pp. -,. [] J. M. M. Anderson, B. A. Mair, M. Rao and C. -H. Wu, Weighted leastsquares reconstruction algorithms for positron emission tomography, IEEE Trans. Med. Imag., vol., no., pp.-, April. [] J. M. M. Anderson, B. A. Mair, M. Rao and C. -H. Wu, A weighted least squares methods for PET, IEEE Nuclear Science Symposium and Medical Imaging Conference Record, vol., pp.-, San Francisco,. [] H. M. Hudson and R. S. Larkin, Accelerated image reconstruction using ordered subsets of proection data, IEEE Trans. Med Imag., vol., pp.0-0, /0/$0.00 (C) 00 IEEE

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