2D PET Image Reconstruction Using Robust Estimation of the Gaussian Mixture Model
Volume 33, Issue 3 (2022), pp. 653–669
Pub. online: 2 May 2022
Type: Research Article
Open Access
Open Access
Received
1 February 2021
1 February 2021
Accepted
1 April 2022
1 April 2022
Published
2 May 2022
2 May 2022
Abstract
An image or volume of interest in positron emission tomography (PET) is reconstructed from gamma rays emitted from a radioactive tracer, which are then captured and used to estimate the tracer’s location. The image or volume of interest is reconstructed by estimating the pixel or voxel values on a grid determined by the scanner. Such an approach is usually associated with limited resolution of the reconstruction, high computational complexity due to slow convergence and noisy results.
This paper presents a novel method of PET image reconstruction using the underlying assumption that the originals of interest can be modelled using Gaussian mixture models. Parameters are estimated from one-dimensional projections using an iterative algorithm resembling the expectation-maximization algorithm. This presents a complex computational problem which is resolved by a novel approach that utilizes ${L_{1}}$ minimization.
References
Alessio, A., Kinahan, P. (2006). PET image reconstruction. Nuclear Medicine, 1, 1–22. https://doi.org/10.1118/1.2358198.
Chen, J. (1995). Optimal rate of convergence for finite mixture models. The Annals of Statistics, 23(1), 221–233. https://doi.org/10.1214/aos/1176324464.
Cheng, Y.-m. (2005). Maximum weighted likelihood via rival penalized EM for density mixture clustering with automatic model selection. IEEE Transactions on Knowledge and Data Engineering, 17(6), 750–761. https://doi.org/10.1109/TKDE.2005.97.
Dempster, A.P., Laird, N.M., Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. Series B, 39, 1–38. https://doi.org/10.2307/2984875.
Friedman, N., Russell, S. (1997). Image segmentation in video sequences: a probabilistic approach. In: Proceedings of the Thirteenth Conference on Uncertainty in Artificial Intelligence, UAI’97, pp. 175–181. http://dl.acm.org/citation.cfm?id=2074226.2074247.
Golub, G.H., Van Loan, C.F. (2012). Matrix Computations, Vol. 3. JHU Press. https://doi.org/10.1137/1028073.
Hudson, H.M., Larkin, R.S. (1994). Accelerated image reconstruction using ordered subsets of projection data. IEEE Transactions on Medical Imaging, 13(4), 601–609. https://doi.org/10.1109/42.363108.
Layer, T., Blaickner, M., Knäusl, B., Georg, D., Neuwirth, J., Baum, R.P., Schuchardt, C., Wiessalla, S., Matz, G. (2015). PET image segmentation using a Gaussian mixture model and Markov random fields. EJNMMI Physics, 2(1), 9. https://doi.org/10.1186/s40658-015-0110-7.
Leroux, B.G. (1992). Consistent estimation of a mixing distribution. The Annals of Statistics, 1350–1360. https://doi.org/10.1214/aos/1176348772.
Lloyd, S. (1982). Least squares quantization in PCM. IEEE Transactions on Information Theory, 28(2), 129–137. https://doi.org/10.1109/TIT.1982.1056489.
Natterer, F. (1986). The Mathematics of Computerized Tomography, Vol. 32. Siam. https://doi.org/10.1137/1.9780898719284.
Nguyen, T.M., Wu, Q.J. (2013). Fast and robust spatially constrained Gaussian mixture model for image segmentation. IEEE Transactions on Circuits and Systems for Video Technology, 23(4), 621–635. https://doi.org/10.1109/TCSVT.2012.2211176.
O’Sullivan, F., Pawitan, Y. (1993). Multidimensional density estimation by tomography. Journal of the Royal Statistical Society: Series B (Methodological), 55(2), 509–521. https://doi.org/10.1111/j.2517-6161.1993.tb01919.x.
Ralašić, I., Seršić, D., Petrinović, D. (2019). Off-the-shelf measurement setup for compressive imaging. IEEE Transactions on Instrumentation and Measurement, 68(2), 502–511. https://doi.org/10.1109/TIM.2018.2847018.
Ralašić, I., Tafro, A., Seršić, D. (2018). Statistical compressive sensing for efficient signal reconstruction and classification. In: 2018 4th International Conference on Frontiers of Signal Processing (ICFSP), pp. 44–49. https://doi.org/10.1109/ICFSP.2018.8552059.
Rani, M., Dhok, S., Deshmukh, R. (2018). A systematic review of compressive sensing: concepts, implementations and applications. IEEE Access, 6, 4875–4894. https://doi.org/10.1109/ACCESS.2018.2793851.
Reader, A.J., Zaidi, H. (2007). Advances in PET image reconstruction. PET Clinics, 2(2), 173–190. PET Instrumentation and Quantification. https://doi.org/10.1016/j.cpet.2007.08.001.
Reynolds, D. (2015). Gaussian mixture models. In: Li, S.Z., Jain, A.K. (Eds.), Encyclopedia of Biometrics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-7488-4_196.
Reynolds, D.A., Quatieri, T.F., Dunn, R.B. (2000). Speaker verification using adapted Gaussian mixture models. Digital Signal Processing, 10(1–3), 19–41. https://doi.org/10.1006/dspr.1999.0361.
Rousseeuw, P.J., Croux, C. (1993). Alternatives to the median absolute deviation. Journal of the American Statistical association, 88(424), 1273–1283. https://doi.org/10.1080/01621459.1993.10476408.
Shepp, L.A., Vardi, Y. (1982). Maximum likelihood reconstruction for emission tomography. IEEE Transactions on Medical Imaging, 1(2), 113–122. https://doi.org/10.1109/TMI.1982.4307558.
Sović Kržić, A., Seršić, D. (2018). L1 minimization using recursive reduction of dimensionality. Signal Processing, 151, 119–129. https://doi.org/10.1016/j.sigpro.2018.05.002.
Tafro, A., Seršić, D. (2019). Iterative algorithms for Gaussian mixture model estimation in 2D PET Imaging. In: 2019 11th International Symposium on Image and Signal Processing and Analysis (ISPA), pp. 93–98. https://doi.org/10.1109/ISPA.2019.8868570.
Tong, S., Alessio, A.M., Kinahan, P.E. (2010). Image reconstruction for PET/CT scanners: past achievements and future challenges. Imaging in Medicine, 2.5, 529–545. https://doi.org/10.2217/iim.10.49.
Yu, G., Sapiro, G. (2011). Statistical compressed sensing of Gaussian mixture models. IEEE Transactions on Signal Processing, 59(12), 5842–5858. https://doi.org/10.1109/TSP.2011.2168521.
Zhang, J., Modestino, J.W., Langan, D.A. (1994). Maximum-likelihood parameter estimation for unsupervised stochastic model-based image segmentation. IEEE Transactions on Image Processing, 3(4), 404–420. https://doi.org/10.1109/83.298395.
Zhang, Y., Brady, M., Smith, S. (2001). Segmentation of brain MR images through a hidden Markov random field model and the expectation-maximization algorithm. IEEE Transactions on Medical Imaging, 20(1), 45–57. https://doi.org/10.1109/42.906424.
