Projection Space Maximum A Posterior Method for Low Photon Counts PET Image Reconstruction

Transcription

1 Proection Space Maximum A Posterior Method for Low Photon Counts PET Image Reconstruction Liu Zhen Computer Department / Zhe Jiang Wanli University / Ningbo ABSTRACT In this paper, we proposed a new MAP method more suitable for low signal to noise (SNR) measurements. Different from conventional MAP method, we assume the proection space as a Gibbs field and the penalty term we used was defined in proection space. The spatial resolution of our method was studied and we furthermore modified our method to obtain nearly spatial invariant resolution. Both simulated data and real clinical data were used to testify our method, and future wor was discussed at the end of the paper. Keywords: PET, image reconstruction, MAP, Gibbs, local impulse response I. INTRODUCTION PET image reconstruction problem is an ill posed inverse transform (ill-posed), tiny disturbance observation data reconstruction results may lead to a serious distortion, so often need to adopt regularization method to improve the reconstruction quality. The most common one is the maximum posterior probability density (MAP) algorithm based on Bayes estimation, [1-7]. In MAP algorithm, the distribution rule of image space usually taes the form of prior function as regular item, and the algorithm is constrained. Different prior functions have different effects on reconstruction results. However, for the convenience of computation, people often limit their influence range to a smaller neighborhood when constructing a priori function. This definition is based on prior distribution function, often local features in the image is more sensitive, but the overall characteristics of the image is not well described, so the reconstruction of high noise observation data, the results are often not ideal. Although the weight of the prior function can be increased by modifying the super parameter (hyper parameter), the binding of the prior function to the algorithm can be enhanced, but it may lead to instability of the algorithm. In this paper, we propose a new reconstruction method for high noise pollution observation data, which is called proection space maximum posterior probability density algorithm. Unlie the traditional MAP algorithm, we consider the proection space (the Radon transform space of the image) as a Gibbs random field, and construct a prior distribution function in this space. This transcendental function definition fully considers the overall features of the image, the neighborhood system is equivalent to the definition in the image space of non local and global, in the case of high noise convergence for the algorithm provides effective binding. For the traditional MAP algorithm, even if the spatial invariant (space-invariant) proection method and the spatial invariant penalty term are used, the reconstructed results may still have non-uniform spatial resolution related to obects. Fessler Made a detailed study of this phenomenon and proposed several methods to correct the spatial resolution characteristics of the reconstructed results[8-11]. In this paper, a similar method is used to correct the proection space prior function in order to obtain approximately consistent image spatial resolution. In the MAP algorithm, it is necessary to control the weight relation between the prior function and the lielihood function by the super parameter. The super parameter leads to the lac of the prior term, while the super parameter produces the smoothing effect. Therefore, it is very important to choose the super parameters reasonably. Some literatures put forward the adoption of generalized cross validation criteria (GCV:, Generalized, Cross, Validation), [12], L curve, [13] and other methods to select hyper parameters, but the above method requires a large amount of calculation. Fessler studied the relationship between the super parameters and the reconstructed image resolution, 101

2 and proposed that different sub parameters were selected according to the area of the reconstructed obect, so that different resolutions of [9, 11, 14] were obtained in different regions of the image. Qi proposes another method of selecting hyper parameters, [15, 16]. He used the contrast recovery coefficient (CRC: Contrast Recovery Coefficient) to describe the spatial resolution characteristics of the image, and the noise contrast (CNR: Contrast to Noise Ration) as a function of the observed variables, by modifying the parameters to maximize the CNR, this method realizes automatic selection of hyper parameters, and obtain approximate consistent spatial resolution. In this paper, we use the maximum lielihood estimation (ML: Maximum Lielihood) method to select the global hyper parameter. Zhou has applied this method to the selection of [17] parameters in MAP estimation, but its drawbac lies in the large amount of computation. However, the proection space priors proposed in this paper can effectively reduce the computational complexity of ML estimation. This paper is organized as follows: in section second, first reviewed the MAP estimation model, and then puts forward the proection space prior function method is proposed to modify the spatial resolution, after that, we discuss the selection of hyper parameters. In the third section, the reconstruction effect of the algorithm is analyzed and compared by using simulated data and real proection data. Finally, the algorithm is summarized and discussed. A. MAP reconstruction model As = { y } i II. RECONSTRUCTION ALGORITHM y as the observation data, the parameters q = { q } to be estimated are defined, y and q defined in the observation data space Y and parameter space Q respectively. According to the Bayes formula, the MAP estimate can be expressed as: qˆ ( y) = a r g { L( y q ) + P ( q ) } (2-1) q In the PET image reconstruction problem, the proection data y = { y } obtained by the system is the image to be reconstructed q = { q }.The optimization must satisfy the non negative constraint of Q.It consists of two parts, L ( y q ) is a logarithmic lielihood function.pet observations are generally considered to be a set of independent Poisson random variables, based on the assumption that L ( y q ) can be represented as: y q å q q (2-2) i i i L ( ) = y lo g Y ( ) - Y ( ) + c o n s t a n t Y = { Y ( q ) } ( i = 1, L, I ) i Y ( q) = A q + r (2-3) A is I i represents the ideal proection value to satisfy: J of the large sparse matrix, which represents the system matrix, r is the bacground noise. In the PET reconstruction problem, the most commonly used prior model is the Gibbs prior model. The model regards the image space as neighborhood Gibbs field, and represents the relationship between pixels in the form of potential energy. The distribution function of Gibbs random fields has the following general form: 1 P ( ) = e x p (- bu ( ) ) Z q q (2-4) The energy function U ( q) in the formula is the parameter b that controls the prior weight, that is, the super parameter. The corresponding prior function, that is, the prior function: i P ( q) = lo g P ( q) = - bu ( q) B. proective space priors (2-5) We first review the classical image space two times a priori, 1 U ( ) = ' R 2 q q q (2-6) Define as follows: 102

3 Ã( ) å r, if = ìï Î Ã( ) R = ï r, if ( ) í - Î Ã ï 0, e ls e ïî represents the neighborhood of pixels. ì 1, = ï = í ï 0, ¹ ïî (2-7) Image Space is C, e (2-8) Image space can be regarded as Gibbs field, and the proection space can also be regarded as Gibbs. We construct a proection space prior function by using two times a priori image space: U ( h ) = h ' W h (2-9) In this paper, a first-order neighborhood definition is adopted. Such neighborhood definitions correspond to neighborhood systems that define global in image space. That is, the pixel values of the whole image domain are used to constrain the estimation of the current pixel. However, considering the weighted matrix definition of image space, the neighborhood relation between pixels should be taen into account when the neighborhood of order two or more is considered. The interaction between pixels should be inversely proportional to the distance between pixels. C. Filter design In the filter bac proection reconstruction algorithm, the filter selection has a significant impact on the reconstruction effect. Similarly, in the proection space priors, the selection of the filter is also a ey part. According to the Radon transform and the central slice theorem, the ramp filter can be used in the filter bac proection algorithm: H ( w) = w (2-10) The filter focuses on high frequency space. Because the high frequency signals are often sensitive to noise, the band limited Ram-La filter is often used: H ( w ) w U ( w / w ) = (2-11) s In the reconstruction process, due to the existence of noise, the sharp cut-off frequency is not easy to determine: too large to cause high frequency component suppression, and to reduce the frequency domain resolution. Therefore, the following expansion methods can be used to define: H ( w ) w W ( w ) = (2-12) W is the band limited window function. By designing W, we can choose the appropriate high frequency response according to the requirements, for example: Shepp-Logan Filter: Blacman Filter: H ( w ) s in ( w ) U ( w ) w p w = (2-13) w w w H ( w ) w ( c o s ( p ) c o s (2 p ) ) U ( ) w w w Hamming Filter: = + + (2-14) w w H ( w ) w ( a (1 a ) c o s ( p ) ) U ( ) 0 a 1 w w Generalized Ramp Filter: = + - (2-15) 103

4 H ( w ) w a U ( w ), a 0 w = > (2-16) According to the observed data, the appropriate filter can be reconstructed by noise pollution. The effect of the above filter will be compared in the experiment. D. Local shoc response and resolution adustment MAP estimation is nonlinear biased estimation. For spatial invariant proection schemes and spatially invariant penalty terms, the reconstructed results may still have non-uniform spatial resolution of obect correlation [18]. Fessler describes the resolution characteristics of MAP estimation by using local shoc response [8]. ) q ( y ) represents the estimate of the pair, and m is the expected operator. Then the local shoc response of the pixel position is defined as: l ( ) = lim m( q + de ) - m( q) d 0 d q (2-17) Under the Poisson observation model and the MAP estimation operator, the local shoc response written as: l sy m 1 ( q - ) = ( A D A + b Q ) A D A e (2-18) q q In PET reconstruction, the system matrix A can be represented as: A = C G B (2-19) l ( q) can be Among them, the proection correlation matrix of the system is used to simulate some data correction lins, such as the attenuation factor of LOR in different positions of the system and the compensation matrix of the sensitivity difference between detectors. Used to describe image related factors in the system, including spatial resolution, no uniformity, and spatial correlation attenuation factors. The ideal probability matrix of the photon detected in each location of the imaging region is detected by the system. Unrelated to the imaged obect, it is only related to the geometry of the system. Therefore, it can also be called geometric proection matrix. For simplicity's sae, this article assumes. According to the derivation of the literature [9], the local shoc response can be approximated as: sy m q b (2-20) l ( )» D ( G G + D Q D ) G G e We modify the proection space priors as follows: - 1 Q % = D A F LF W A D (2-21) A. Comparison of reconstruction results III. SIMULATION TEST AND ALGORITHM ANALYSIS In this section, we compare the reconstructed effect of proection space MAP algorithm and traditional MAP algorithm. We test the algorithm using the Shepp-Logan template diagram (Figure 1). The observation data consists of 192 angles, the sampling number of each angle is 192, and the reconstructed image size is pixel Figure 1: Shepp-Logan 104

5 We simulate actual observations in the following manner: * = (2-22) g( s, q) P o isso n ( g ( s, q) a ( s, q) V) P o isso n ( a ( s, q) ) In order to evaluate the reconstruction effect of the algorithm, the mean square error(mse) and correlation coefficient of the reconstructed results (CORR)are calculated respectively. MSE is defined as: P f - f P 2 M S E = % * P f P * 2 (2-23) Theoretically, the smaller the MSE, the smaller the difference between the reconstruction results and the template. We first set the total number of photons to 500K, and Figure 2 and figure 3 are the reconstruction results and error curves respectively. As can be seen from figures 2 and 3, the proection space MAP algorithm is close to the traditional MAP method in the case of moderate photon number. (a) (b) (c) (d) (e) Figure 2.Reconstruction results of 500K photon number observation data (a) (b) Figure3: 500K photon number observation data reconstruction error curve: (a) MSE (b) CORR We reduce the photon number to 100 and repeat the above experiments. The reconstruction results and error curves are shown in Figure 4 and figure 5 respectively. Since the photon number condition, the MLEM algorithm is very poor, and is not compared here. 105

6 (a) (b) (c) (d) Figure 4.Reconstruction results of 100 photon number observation data Figure5: 100K photon number observation data reconstruction error curve: (a) MSE (b) CORR From Figure 4 and figure 5 can be seen in the photon number less, serious noise pollution situation, the FBP method due to the lac of effective restraint mechanism of the noise, the reconstruction results become difficult to identify; and the image space of traditional MAP method because of constraints using only neighborhood information, the reconstruction results of noise was still obvious. In contrast, proection space MAP algorithm can provide better global constraints, and use the information of the entire image domain to correct the current pixel values, so as to obtain more smooth reconstruction results. B. Filter Effects Figure 6.Filter curves in different frequency domain 106

7 In the reconstruction process, the filter L is replaced, which has different effects on the reconstruction results. In this experiment, Shepp-Logan template is still used for simulation. The different filters used are shown in figure 6. We reconstructed the 500K photon number data and the 100 photon number data respectively, and the reconstruction results are shown in Figure 7, shown in figure 8. Figure7.Reconstruction effect of different filters under 500K photon number(a):ram-la, MSE=4.42%; (b): Shepp-Logan, MSE=4.63%; (c):blacman, MSE=5.93%; (d):hamming, MSE=5.46%; (e):generalized ramp with α = 0:5, MSE=5.02%; (f):generalized ramp filter with α= 1:25,MSE=4.38% Figure8: Reconstruction effect of different filters under 100K photon number(a):ram-la, MSE=4.42%; (b):shepp-logan, MSE=4.63%; (c):blacman, MSE=5.93%; (d):hamming, MSE=5.46%; (e):generalized ramp with α = 0:5, MSE=5.02%; (f):generalized ramp filter with α= 1:25,MSE=4.38% As can be seen from Figure 7, the Ram-La filter and the generalized slope filter (500K) have better results for medium photon number proection data. Fig. 8 shows that the Shepp-Logan filter and the generalized ramp filter are more suitable for reconstruction of low photon number proection data (100). The reason is that the Shepp-Logan filter and the generalized ramp filter are more effective in suppressing the high-frequency components, and thus have better noise reduction effects. In addition, although the Blacman filter and the generalized Hamming filter 107

8 suppress the high-frequency component more, the reconstruction result produces an over smoothing effect. From the MSE numerical value, the reconstruction result is different from the template map. C. Spatial Resolution Figure. 9: Simplified heart lung template In order to test the spatial resolution property of proective space MAP method, the local shoc response of some pixels in simplified heart lung template is selected (Figure 9). In Figure 9, the U region represents the heart, and the left bright spot represents the tumor. The U region and the bright spot are surrounded by the thoracic region, and the heart, tumor, and chest pixel values are 4:5:1. For the sae of simplification, the attenuation coefficient is set to 0. The traditional MAP method uses the space invariant two times a priori. In order to ensure convergence, all iterative algorithms are iterated 100 times (also using OS, SAGE and other methods to accelerate convergence). The correlation results are shown in figure 10. Figure10.Simplified local impact response of vertical points of heart lung template CONCLUSION In this paper, we extend the traditional MAP method and propose an image reconstruction algorithm suitable for low photon number and high noise pollution observation of PET data. The algorithm regards the forward proection space (Radon transform space) of the image as Gibbs random field, and constructs a prior function in this space. In 108

9 the process of constructing transcendental functions, we through the filter, simulation system of generalized inverse matrix, distance weighted on the prior model, and studied the influence of different filters on the reconstruction results of different levels of noise data. In order to obtain approximately consistent image spatial resolution, we study the local shoc response of the algorithm, and modify a priori term definition accordingly. Finally, the ML estimation is used to estimate the super parameters, and an alternating iterative algorithm is designed to optimize the reconstructed image and hyper parameters simultaneously. Experimental results show that the proposed algorithm can effectively improve the quality of reconstruction in low photon number, and provide smooth reconstruction results. REFERENCES [1]. L. A. Shepp, Y. Vardi. Maximum lielihood reconstruction for emission tomography. IEEE Trans. Med. Imaging [J], (2): [2]. J. A. Fessler, A. O. Hero Iii. Penalized maximum-lielihood image reconstruction using space-alternating generalized EM algorithms. Image Processing, IEEE Transactions on [J], (10): [3]. J. A. Fessler. Penalized weighted least-squares image reconstruction for positron emission tomography. Medical Imaging, IEEE Transactions on [J], (2): [4]. C. V. Alvino, A. J. Yezzi Jr. Tomographic reconstruction of piecewise smooth images. Computer Vision and Pattern Recognition, CVPR Proceedings of the 2004 IEEE Computer Society Conference on [J]. 1. [5]. M. C. Hong, M. G. Kang, A. K. Katsaggelos. Regularized multichannel restoration approach for globally optimal high-resolution video sequence. Proceedings of SPIE [J], : [6]. J. Kalifa, A. Laine, P. D. Esser. Regularization in tomographic reconstruction using thresholding estimators. Medical Imaging, IEEE Transactions on [J], (3): [7]. E. Levitan, G. T. Herman. A Maximum a Posteriori Probability Expectation Maximization Algorithm for Image Reconstruction in Emission Tomography. Medical Imaging, IEEE Transactions on [J], (3): [8]. J. A. Fessler. Resolution properties of regularized image reconstruction methods. Ann Arbor, MI [J], 1995: [9]. J. A. Fessler, W. L. Rogers. Spatial resolution properties of penalized-lielihood image reconstruction: spaceinvariant tomographs. Image Processing, IEEE Transactions on [J], (9): [10]. J. W. Stayman, J. A. Fessler. Compensation for no uniform resolution using penalized-lielihood reconstruction in space-variant imaging systems. Medical Imaging, IEEE Transactions on [J], (3): [11]. J. W. Stayman, J. A. Fessler. Regularization for uniform spatial resolution properties in penalized-lielihood image reconstruction. Medical Imaging, IEEE Transactions on [J], (6): [12]. V. E. Johnson, W. H. Wong, X. Hu, et al. Image restoration using Gibbs priors: boundary modeling, treatment of blurring, and selection of hyper parameter. IEEE Transactions on Pattern Analysis and Machine Intelligence [J], (5): [13]. P. C. Hansen. Analysis of Discrete Ill-Posed Problems by Means of the L-Curve. SIAM Review [J], : 561. [14]. J. A. Fessler, S. D. Booth. Conugate-gradient preconditioning methods for shift-variant PET image reconstruction. Image Processing, IEEE Transactions on [J], (5): [15]. J. Qi, R. M. Leahy. Resolution and noise properties of MAP reconstruction for fully 3-DPET. Medical Imaging, IEEE Transactions on [J], (5): [16]. J. Qi, R. H. Hues man. Theoretical study of lesion detectability of MAP reconstruction using computer observers. Medical Imaging, IEEE Transactions on [J], (8): [17]. Z. Zhou, R. Leahy, E. Mumguoclu. A comparative study of using anatomical boundary in PET reconstruction [C] [18]. J. A. Fessler, W. L. Rogers. Uniform quadratic penalties cause no uniform image resolution (and sometimes vice versa) [C]