Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 491239, 7 pages http://dx.doi.org/10.1155/2014/491239 Research Article Penalized Maximum Likelihood Algorithm for Positron Emission Tomography by Using Anisotropic Median-Diffusion Qian He1,2 and Lihong Huang1,3 1 College of Mathematics and Econometrics, Hunan University, Changsha 410082, China 2 College of Information Science and Engineering, Hunan City University, Yiyang 413000, China 3 Department of Information and Technology, Hunan Women’s University, Changsha 410004, China Correspondence should be addressed to Qian He; [email protected] Received 1 October 2013; Revised 9 December 2013; Accepted 24 December 2013; Published 20 January 2014 Academic Editor: Gelan Yang Copyright © 2014 Q. He and L. Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Nowadays, positron emission tomography (PET) is widely used in engineering. In this paper, a novel penalized maximum likelihood (PML) algorithm is presented for improving the quality of PET images. The proposed algorithm fuses an anisotropic median- diffusion (AMD) filter to the maximum-likelihood expectation-maximization (MLEM) algorithm. The fusing algorithm shows its positive effect on image reconstruction and denoising. Experimental results present that the proposed method denoises and reconstructs images with high quality. Furthermore, by comparing with other classical reconstructing algorithms, this novel algorithm shows better performance in the edge preservation. 1. Introduction algorithm is to find an appropriate energy function, which is defined by Gibbs probability distribution. Unfortunately, PET technology, which has been widely used in neurology, theselectionoftheenergyfunctionisdifficult.Themedian oncology, and new medicine exploitation, is one of the root prior (MRP) algorithm [9], firstly proposed by Alenius, advanced and noninvasive diagnostic techniques in modern is an application of OSL algorithm. This algorithm is good nuclear medical. In order to obtain a high quality recon- at coping with those images that have locally monotonic structed image from clinical projection data with strong structures. However, the images reconstructed by MRP are noise, an excellent image reconstruction algorithm is indis- still noisy because median filter cannot remove Gaussian and pensable. Poisson noise effectively, which dominate in PET images7 [ ]. The MLEM algorithm is a classic method in PET image reconstruction when the measured data follows Poisson The anisotropic diffusion (AD) filter [11] is a nonlinear distribution [1]. One problem of this algorithm is the ill-posed partial differential equation (PDE) based on diffusion pro- problem, which represents that the reconstructed images cess. Overcoming the undesirable effects of linear smoothing cannot remove the noise of projection data [2]. Today, an filter, such as blurring or dislocating the useful edge infor- ill-posed image reconstruction problem, such as MLEM, mation of the images, AD has been widely used in image can be transformed into a well-posed one through the use smoothing, image reconstruction and image segmentation of regularization term. The reconstructed results should be [12–15].ThebasicideaofADisadaptivelychoosingdiffusive not only content with measured data to some extent but coefficients in diffusion process so that diffusion is maximal also be consistent with additional regularization term that within smooth regions and minimal near the edges. is independent of those data at the same time. That is In order to remove noise and preserve edge information usually called PML or Bayesian algorithm. Numerous PML at the same time, image reconstruction based on AD has algorithms have been proposed in the past decades [3–10]. become the research focus [7, 8, 15, 16]. Yan proposed a PML Thereinto, Green proposed a Bayesian algorithm, known algorithm [16] that combined MLEM with AD filter (called as the one-step-late (OSL) algorithm [6]. The key of this MELM-PDE) and could obtain acceptable reconstructed 2 Mathematical Problems in Engineering 8 results. However, MLEM-PDE cannot remove the isolated noise and preserve edge information accurately due to the 7 defects of P-M diffusion model15 [ ]. In this paper, we proposed a new PML algorithm for 6 PET image reconstruction based on AMD. The proposed algorithm can effectively remove noise while preserving 5 edge information accurately. In Section 2,PETimagerecon- struction algorithms such as MLEM, OSL, and MRP are 4 introduced. The AMD filter is presented in Section 3.In g(x) Section 4, our proposed algorithm is described. Simulation 3 experiments are given in Section 5.Finally,Section 6 is the conclusion. 2 1 2. Image Reconstruction Algorithms for PET 0 In PET, the maximum likelihood (ML) algorithm seeks a −0.5 −0.4 −0.3 −0.2 −0.10 0.1 0.2 0.3 0.4 0.5 solution that makes the measured data most likely to occur x and maximizes the conditional probability ( ,where| ) is the measured data and is the emission image. It is = 0.2 Figure 1: The plot of diffusion function with . described in the following: ̂ =arg max (), ≥0 1 ()=log ( ( | )) (1) 0.8 0.6 = ∑ (− ∑ , + log (∑ ,)) . =1 =1 =1 0.4 0.2 In (2), is the probability of photons emitted by pixel , which can be detected by the detector , is the number 0 of photons emitted by the pixel ,and is the number of photons captured by the detector . g(x)∗x −0.2 In order to solve (1), Shepp and Vardi [1]haveproposed −0.4 theEMalgorithm.Theiterativeformulacanbedescribedin the following: −0.6 ∑ ( / ∑ ) −0.8 +1 =1 =1 = , (2) ∑ −1 =1 −0.5 −0.4 −0.3 −0.2 −0.10 0.1 0.2 0.3 0.4 0.5 where is the iteration. x Although MLEM algorithm is better than filtered back- Figure 2: The plot of flux function with = 0.2. projection (FBP) algorithm [17], its convergence rate is extremely slow, and as the iteration number increases, the reconstructed results suffer from noise artifacts. The usual method to solve this problem is to introduce a regularization term, and the objective function is ̂ =arg max [ () + ()], ≥0 (3) where () has been explained above and () is regulariza- tion term or penalty term. The OSL algorithm uses the current image when calculating the value of the derivative of the energy function, and the iterative formula can be defined as [6] ∑ ( /∑ ) +1 =1 =1 = , (4) ∑ +(/)() Figure 3: Modified Shepp-Logan phantom. =1 Mathematical Problems in Engineering 3 (a) (b) (c) (d) Figure 4: The modified Shepp-Logan phantom reconstructed by different algorithms after 50 iterations: (a) MLEM-AMD with ℎ=40and = 1.5; (b) MLEM-PDE with ℎ=40and =40; (c) MRP with = 0.1; (d) MLEM. (a) (b) (c) (d) Figure 5: The zoomed images of Figure 4: (a) MLEM-AMD; (b) MLEM-PDE; (c) MRP; (d) MLEM. 70 15 60 10 50 5 40 0 30 SNR −5 NRMSE (%) NRMSE 20 −10 10 −15 0 −20 010203040 50 0 10 20 30 40 50 Iterations Iterations MLEM-AMD MRP MLEM-AMD MRP MLEM-PDE MLEM MLEM-PDE MLEM Figure 6: The plots of NRMSE along with iterations for different Figure 7: The plots of SNR along with iterations for different algorithms. algorithms. local median. Edge preservation is an intrinsic characteristic of median filter, and its update equation is [9] where (⋅) is the energy function and is the Bayes weight of the prior. ∑ ( / ∑ ) +1 =1 =1 = , (5) The MRP algorithm can be coped with monotonic struc- ∑ + (( −()) / ()) tures in a neighborhood by comparing the pixel against the =1 4 Mathematical Problems in Engineering incorporates a median filter into the diffusion step, and the discrete form is defined as +1 = + ∑ ( (∇ )∇ ), , , ∈ (9) +1 = Median ( ,), where ∈[0,1]controls the rate of diffusion, is iteration number, is the gray value of pixel , || is the number of neighbor at pixel (usuallyfourdirections,north,south,east, Figure 8: Real thorax phantom. and west, resp.), ∇, = −, is the window for the median operator (such as a 3×3square), and the diffusion coefficient (⋅) is where ( ) is the median of pixel within its neighbor- 2 { 25 2 hood. { [1 − ( ) ] , || ≤ √5, () = {16 √5 (10) {0, otherwise. 3. Anisotropic Median-Diffusion Filter The flux function () is defined as () = () ⋅. The AD filter (usually called P-M diffusion model), firstly The diffusion coefficient and the flux function arepre- proposed by Perona and Malik, is a nonlinear filter that sented, respectively, in Figures 1 and 2. These figures together purports to remove noise without blurring edges, and the with (9) suggest that diffusion is maximal within smooth basicequationis[11] regions and stopped near the edges. In AMD, diffusion process can smooth the regions with (,,) small gradient between current pixel and its neighborhood, = [ ( ∇(,,))∇(,,)], (6) div while regions with large gradient (edge or noise spike) are left unchanged. Noise spike that generated the large gradients where is the time parameter, div is the divergence operator, will be removed effectively by the median filter subsequently. (, , 0) is the original image, ∇(,,)is the gradient However, those large gradients generated by the edges will of the image at time ,and(⋅) is the diffusion coefficient, not be affected by the median filter. Thus, low-level noise which is a function of local gradient. This function should be is smoothed by diffusion, and high-level noise is removed satisfied: by median filter. In brief, the AD and median filter are always mutually complementary to gradually eliminate noise () =1, without blurring the edges [22]. →0lim (7) () =0, →∞lim 4. Proposed Algorithm Although MLEM-PDE algorithm is better than classical so that the diffusion is more in smooth regions and less reconstructing algorithms, the images reconstructed by near the edges. They put forward the following two diffusion MLEM-PDE are still noisy because P-M diffusion model coefficients: isnotgoodenoughfortheremovalofisolatednoiseand accurate preservation of edges. Therefore, it is natural that a 2 novel diffusion model should be introduced into the MLEM 1 () = exp [−( ) ], algorithm.