Commit bd5f874d authored by Ying-Qiu Zheng's avatar Ying-Qiu Zheng
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Update 2021JUL21.md

parent 5e771513
......@@ -17,6 +17,19 @@ The marginal distribution of $`\mathbf{x}_{n}^{L}, \mathbf{x}_{n}^{H}`$ is
In summary, in addition to finding the the hyper-parameters $`\pi, \mu, \Sigma_{k}^{H}, \Sigma^{L}_{k}`$, we want to estimate a transformation matrix $`\mathbf{U}`$ such that $`\mathbf{UX}^{H}`$ is as close to $`\mathbf{X}^{L}`$ as possible (or vice versa).
### Pseudo code
Algorithm 1. EM for the Fusion of GMMs
---
---
1. Run K-means clustering on the high-quality data to generate the assignment of the voxels $`R^{(0)}`$.
2. Initialise the means $`\mu_{k}`$, covariances $`\Sigma_{k}`$, and mixing coefficients $`\pi_k`$ using the K-means assignment $`R^{(0)}`$, and evaluate the initial likelihood.
3. Initialise the transformation matrix $`\mathbf{U}`$ using Algorithm 3.
4. For iteration = $`1, 2, ...`$, do
- **E-step.** Evaluate the responsibilities using the current parameter values
```math
\gamma(y_{nk}) = \frac{\pi_{k}\mathcal{N}(\mathbf{x}^{L}_{n} | \mu_{k}, \Sigma_{k}^{L})\mathcal{N}(\mathbf{Ux}^{H}_{n} | \mu_{k}, \Sigma_{k}^{H})}{\sum_{j=1}^{K}\pi_{j}\mathcal{N}(\mathbf{x}^{L}_{n} | \mu_{k}, \Sigma_{k}^{L})\mathcal{N}(\mathbf{Ux}^{H}_{n} | \mu_{k}, \Sigma_{k}^{H})}
```
### Simulation results
#### We considered three scenarios
##### I. Low-quality data noisier than the high-quality data
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