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We now have our mean shape and restricted set of PCA axes that provide a concise description of the training set of shapes.

However, this model can also be used to generate new shapes. Let such a new shape be x, generated from a set of shape parameters $\mathbf{b}_{s}$:

$$\mathbf{x}=\mathbf{\overline{x}}+\mathbf{P}_{s}\mathbf{b}_{s}.$$ (3.2)

The matrix $\mathbf{P}_{s}$ contains the eigenvectors of the covariance matrix of the training data.

The generated shapes can be constrained to be similar to those seen in the training set by constraining the allowed shape parameters, $\mathbf{b}_{s}$, to be similar to those extracted or learned from the training shapes. Typically, the distribution of training set shape parameters is modelled by a multivariate Gaussian pdf, and new shapes are generated by sampling from this pdf.

Figure davies-manual shows an example where model parameters are altered. The effect of varying parameters that correspond to first of second modes of variation is shown using hand data.