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Modelling Intensity

To model texture, differences in shapes are removed by morphing each training image to the mean shape^3.3. A shape-free texture patch can then be estimated from the image by sampling on a regular grid and forming a vector $\mathbf{g}$. Statistical analysis proceeds as for shapes and it results in the following linear expression for texture

$$\mathbf{g}=\overline{\mathbf{g}}+\mathbf{P}_{g}\mathbf{b}_{g}.$$ (3.3)

$\mathbf{g}$ is the intensity vector. $\mathbf{\mathbf{P}_{g}}$ contains the eigenvectors of the covariance matrix of the training data and $\mathbf{b}_{g}$ controls the intensity. The process is hardly different from dimensionality reduction in the case of shape.

Generally, we wish to distinguish between the meaningful shape/texture variation of the objects under consideration, and the apparent variation in shape/texture that is due to the positioning of the object within the image (the pose of the imaged object). In this case, the appearance model is generated from an (affinely) aligned set of images, just as was the case for shape models considered earlier. Point positions $\mathbf{x}_{im}$ in the original image frame are then obtained by applying the relevant pose transformation $T_{\mathbf{t}}(\cdot)$:

$$\mathbf{x}_{im}=T_{\mathbf{t}}(\mathbf{x}_{model})$$ (3.4)

where $\mathbf{x}_{model}$ are the points in the model frame, and $\mathbf{t}$ are the pose parameters. For example, in 2-D, $T_{\mathbf{t}}$ could be a similarity transform with four parameters describing the translation, rotation, and scale of the object.

In an analogous manner, the image can also be normalised wrt the mean image intensities and image variance,

$$\mathbf{g}_{im}=T_{\mathbf{u}}(\mathbf{g}_{model}),$$ (3.5)

where $T_{\mathbf{u}}$ consists of a shift and scaling of the image intensities. For further implementation details see [,].

As noted above, a meaningful, dense, groupwise correspondence is required before an appearance model can be built. NRR provides a natural method of obtaining such a correspondence, as noted by Frangi and Rueckert [,]. It is this link that forms the basis of the new approach to NRR evaluation.

The link between registration and modelling is further exploited in the Minimum Description Length (MDL) [] approach to groupwise NRR, where modelling becomes an integral part of the registration process. This is one of the registration strategies which is discussed in later chapters.