### 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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