14#14

The key requirement in building an appearance model from a set of images, is the existence of a dense correspondence across the set. This is often defined by interpolating between the correspondences of a limited number of userdefined landmarks. Shape variation is then represented in terms of the motions of these sets of landmark points. Using the notation of Cootes et al [9], the shape (configuration of landmark points) of a single example can be represented as a vector 15#15 formed by concatenating the coordinates of the positions of all the landmark points for that example. The texture is represented by a vector 16#16, formed by concatenating image values (texture) sampled over a regular grid on the registered image. This means that the a given element in 16#16 is sampled from an equivalent point in each image, assuming the registration is correct.
In the simplest case, we model the variation of shape and texture
in terms of multivariate gaussian distributions, using Principal
Component Analysis (PCA) [15] to obtain linear
statistical models of the form:
17#17  18#18  19#19  
20#20  (2)  
21#21  18#18  22#22  (3) 
In generative mode, the input shape ( 23#23) and texture ( 24#24) parameters can be varied continuously, allowing the generation of sets of images whose statistical distribution matches that of the training set.
In many cases, the variations of shape and texture are correlated.
If this correlation is taken into account, we obtain a
combined statistical model of the more general form:
17#17  18#18  29#29  
20#20  (4)  
21#21  18#18  30#30  (5) 
Generally, we wish to distinguish between the meaningful shape variation of the objects under consideration, and the apparent variation in shape 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. Point positions 35#35 in the original image frame are then obtained by applying the relevant pose transformation 36#36:
37#37  (6) 
In an analogous manner, we can also normalise the image set with respect to the mean image intensities and image variance,
41#41  (7) 
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 [11,12]. It is this link that forms the basis of our new approach to NRR evaluation.
The link between registration and modelling is further exploited in the Minimum Description Length (MDL) [16] approach to groupwise NRR, where modelling becomes an integral part of the registration process. This is one of the registration strategies evaluated in this paper.