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 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 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 . |
14 | Compute current estimate of the sample covariance matrix C′ (Eq. 4). |
15 | Compute covariance correction term 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 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). |