Skip to main content

Table 1 The algorithmic procedure for our new method

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 z k 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 z k 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 z k .
14 Compute current estimate of the sample covariance matrix C (Eq. 4).
15 Compute covariance correction term Δ C e 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 Δ C Θ 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).