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Intensity Model

The next stage involves the sampling of texture. In principle, having got the description of some shapes from a set with their given spatial correspondences, it is possible to estimate homologous points in between these correspondences. This essentially allows the prediction of the denser correspondence - that which involves larger sections of the image, rather than points only. Below lies a description of one special case; the descriptions are aimed to illustrate one possible way of sampling intensities. Construction of an intensity model is the more significant step which is carried out in the exact same way as was done for shapes (Equation ).

At this stage, each of the images should be aligned to fit a common volume in space^2.14. In practice, the properties of that space are implicitly defined by the mean shape^2.15. Rigid (or Euclidean similarity) transformations, namely translation, scale and rotation, are not always sufficient to warp all images into that common space, e.g. in the ubiquitous case of human faces, different head sizes and facial expressions introduce difficulties. Nonetheless, it is crucial that good fitting is obtained before the sampling of grey-level commences. Following these basic transformations which align all images, the displaced control points of each image overlap and contain in between them shape-normalised patches. These patches are available for construction of texture vectors. Barycentric arithmetics, known for their frequent utility in computer graphics and stereo vision, are used to describe the location of all corresponding points within a patch^2.16. This location of point is directly affected by the warps applied to shift a given shape onto the space of the mean shape.

Triangle meshes are subsequently created by stretching lines between neighbouring control points and intensity values are captured one by one (along a chosen grid of points to be sampled) and stored in a vector representative of texture. Each component in such a vector captures the intensity (or colour) of one single pixel as was learned from the examples. Statistical analysis, which is not different from the one above, results in the following formulation for texture:

$$\mathbf{g}=\overline{\mathbf{g}}+\mathbf{P}_{g}\mathbf{b}_{g}.$$ (2.3)

It is again worth the while to emphasise that the process if no different to dimensionality reduction in the case of shape. The use of the algorithm above implies that for short vectors and a low number of pixels sampled, noncontinuous appearances will be easy to spot^2.17. In fact, objects will often appear to be nothing more than a collection of polygons that do not quite resemble realistic appearances^2.18. To compensate for this, algorithms from the related field of computer graphics can be used, e.g. Phong and Gouraud shading. In practical use, geodesic interpolation [WWW-12] is used and the results can be quite astounding considering the low dimensionality of the available data. Compression here is dependent upon reconstruction strategy and assumptions about natural phenomena.

The models above (Equations , ) are expressed linearly and quite compactly - a highly desirable and manageable form. This is due to PCA which reduces the length of the vectors describing shape and texture. As earlier mentioned, although Eigen-analysis is involved in the process, its derivation, proofs, or characteristics are less than essential for the understanding of PCA which works as follows.