Table 1 The algorithmic procedure for our new method

Initialise each translation parameter t _k using the mean of landmarks in each corresponding shape ( k=1,2,...,K).

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).

Initialise scale parameters s _k as unity, i.e. original scales.

Initialise measurement covariance matrix as identity matrix.

Compute initial transformed shapes z _k .

Compute the initial mean shape m (and adjust transformation parameters so that the mean orientation is roughly aligned with the reference baseline/plane).

Compute current transformed shapes z _k .

Compute the current mean shape m (Eq. 1).

Compute the whitening matrix W.

Compute current ghost points g _k .

Construct current models ${\mathbf{\text{z}}}_{\mathit{k}}^{\mathit{\prime }}$ based on PCA and the number of eigenvectors e _j chosen J (Eq. 2).

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).

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 ${\mathbf{\text{z}}}_{\mathit{k}}^{\mathit{\prime }}$.

Compute current estimate of the sample covariance matrix C^′ (Eq. 4).

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).

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).

Compute current estimate of the measurement covariance matrix C (Eq. 9).

Repeat steps 7 to 17 until convergence (typically ≈10 iterations).