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Generalisation

The Generalisation ability of a generative appearance model measures the extent to which it is able to represent images of the modelled class both seen (in the training set) and unseen (not in the training set). A model that comprehensively captures the variation in the modelled class should generate a distribution of images that overlaps the training distribution as completely as possible. This means that, if we generate a large set of synthetic images, $\{I_{\alpha}:\alpha=1,\ldots m\}$, from the model, each image in the training set should be close to a synthetic image. Given a measure, $\vert\cdot\vert$, of the distance between images, we define the Generalisation $G$ of a model and its standard error, $\sigma_{G}$, as follows:

$$G=\frac{1}{n} \sum\limits_{i=1}^{n}{\bf min}_{\alpha}\vert I_{i}-I_{\alpha}\vert,$$ (3)

$$\mathbf{\sigma_{G}}=\frac{SD({\bf min}_{\alpha}\vert I_{i}-I_{\alpha}\vert)}{\sqrt{n-1}},$$ (4)

where $I_{i}$ is the $i^{th}$ training image, $\min_{\alpha}$ is the minimum over $\alpha$ (the set of synthetic images), and SD is standard deviation. That is, Generalisation is the average distance from each training image to its nearest neighbour in the synthetic image set. A good model exhibits a low value of Generalisation, indicating that the modelled class is well-represented by the model.