Gaussian Mixture Estimation from Lower-Dimensional Data with Application to PET Imaging
Abstract
1. Introduction
2. Gaussian Estimation from PET Data
2.1. Estimation of
2.2. Estimation of
3. Gaussian Mixtures in PET Imaging
3.1. Gaussian Mixture Model
3.2. EM-like Algorithm
- E:
- Given a set of parameters , calculate the probability that each observation originated from the k-th component:These probabilities are sometimes also called responsibilities, or posterior probabilities.
- M:
- Given the set of probabilities , estimate the parameters of each component. This step is named after the maximum likelihood method that is used to estimate the parameters.
3.2.1. Initialization
- Each line is initially randomly assigned to exactly one component, while ensuring that each component has approximately the same number of lines.
- Mean vectors of each component are estimated as described in Section 2.
- Lines are reassigned to the component whose estimated mean vectors are nearest to them in terms of Euclidean distance. In this step, we still adhere to the so-called hard classification; i.e., each line is assigned to only one component.Steps 2 and 3 are repeated until the changes in mean vectors are sufficiently small.
3.2.2. E Step
- A Gaussian distribution retains properties when rotated.
- Marginal distributions of a Gaussian are again Gaussian.
3.2.3. M-like Step
- Find the nearest points to the (previous iterations’) center for each line.
- Calculate the weighted mean and covariance of with probabilities h as weights:
- Calculate from as in Section 2.
- Mixture weights are calculated, as in the original EM algorithm, as the proportion of all lines (events) assigned to each component: .
4. Experiments and Results
4.1. Single-Component Estimation
4.2. Gaussian Mixture Estimation
4.3. Noise Resistance
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
PET | positron emission tomography |
LOR | line of response |
GMM | Gaussian mixture model |
EM | expectation maximization |
VOR | volume of response |
Appendix A
Appendix A.1. Different Covariances
Appendix A.2. Equal Component Sizes
Appendix A.3. Four Components
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Tafro, A.; Seršić, D. Gaussian Mixture Estimation from Lower-Dimensional Data with Application to PET Imaging. Mathematics 2024, 12, 764. https://doi.org/10.3390/math12050764
Tafro A, Seršić D. Gaussian Mixture Estimation from Lower-Dimensional Data with Application to PET Imaging. Mathematics. 2024; 12(5):764. https://doi.org/10.3390/math12050764
Chicago/Turabian StyleTafro, Azra, and Damir Seršić. 2024. "Gaussian Mixture Estimation from Lower-Dimensional Data with Application to PET Imaging" Mathematics 12, no. 5: 764. https://doi.org/10.3390/math12050764
APA StyleTafro, A., & Seršić, D. (2024). Gaussian Mixture Estimation from Lower-Dimensional Data with Application to PET Imaging. Mathematics, 12(5), 764. https://doi.org/10.3390/math12050764