From: Quantitative shape analysis with weighted covariance estimates for increased statistical efficiency
Step | Process |
---|---|
1 | Initialise each translation parameter t_{ k } using the mean of landmarks in each corresponding shape ( k=1,2,...,K). |
2 | Initialise each rotation parameter R_{ k } based on the orientation of each shape relative to the 2-point baseline in 2D or 3-point reference plane in 3D (Figure 1). |
3 | Initialise scale parameters s_{ k } as unity, i.e. original scales. |
4 | Initialise measurement covariance matrix as identity matrix. |
5 | Compute initial transformed shapes z_{ k }. |
6 | Compute the initial mean shape m (and adjust transformation parameters so that the mean orientation is roughly aligned with the reference baseline/plane). |
7 | Compute current transformed shapes z_{ k }. |
8 | Compute the current mean shape m (Eq. 1). |
9 | Compute the whitening matrix W. |
10 | Compute current ghost points g_{ k }. |
11 | Construct current models ${\mathbf{\text{z}}}_{\mathit{k}}^{\mathit{\prime}}$ based on PCA and the number of eigenvectors e_{ j } chosen J (Eq. 2). |
12 | Minimise the Mahalanobis distance corresponding to every shape z_{ k } (Eq. 3) using simplex optimisation (where e_{ j } and W are fixed while t_{ k }, R_{ k } and s_{ k }, and so, z_{ k }, m, g_{ k } and ${\mathbf{\text{z}}}_{\mathit{k}}^{\mathit{\prime}}$ are varied). |
13 | Update current estimates of t_{ k }, R_{ k } and s_{ k } based on the outcome of the optimisation, and then update current estimates of z_{ k }, m, g_{ k } and ${\mathbf{\text{z}}}_{\mathit{k}}^{\mathit{\prime}}$. |
14 | Compute current estimate of the sample covariance matrix C^{′} (Eq. 4). |
15 | Compute covariance correction term $\Delta {\mathit{C}}_{{\mathbf{\text{e}}}_{\mathit{j}}}$ due to degrees of freedom in the model (Eqs 5-6) for every eigenvector used e_{ j } ( J=1,2,...,J). |
16 | Skip this step for the first iteration (as it requires an estimate of C); compute covariance correction term $\Delta {\mathit{C}}_{{\Theta}_{\mathit{i}}}$ due to parameter orthogonalisation (Eqs. 7-8) for every direction vector Θ_{ i } corresponding to transformation parameters, i=1,2,...,I (where I=4 in 2D and I=7 in 3D). |
17 | Compute current estimate of the measurement covariance matrix C (Eq. 9). |
18 | Repeat steps 7 to 17 until convergence (typically ≈10 iterations). |