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Description of the Approach

The first step taken by the application was the generation of some random bumps simpler than the ones described in Subsection . These bumps varied in their height and width; the step size of the bump (the steep ends of the flat pinnacle) was fixed, i.e. the bump was initially flat at the top.

Although the property of height was not intended to be ignored during registration, it was expected that it would remain unchanged due to CPS being perfectly diffeomorphic^B.1. The bumps were all symmetric and the height was one of {hi, low} where and . The data was therefore far simpler than any 1-D data which is not constrained in any way. The height of the bump and the position at which the bump goes high could conjointly define that bump so two real numbers (a tuple) at the minimum would suffice to reconstruct each bump.

As images were being warped, the form of the bump quickly changed to give a smoother curve with more continuous derivatives. This of course depended on the type of warp which had been applied to the bump. At each iteration, new similarity with respect to some reference image or similarity with reference to the whole set of images was obtained using warps and measured using the methods outlined in Section sub:Minimum-Description-Length in modelling.

The similarity measures used in these experiments to evaluate similarity were mean-squared-difference (MSD) and mutual information (MI). The latter was more computationally expensive so although it gave better results, it needed to be used with caution. Likewise, the type of warp applied was often, but not alway,s a simple one which is controlled by a single allocated control point. In some cases, many control points were assembled to form an expensive warp of increased complexity. The choice of these points was often decided to be random as a successful rational choice would have required much more speed, consequently slowing down the whole process.

As explained to some extent beforehand in sub:Reparameterisation, reparameterisation was used to perform points placements in the image of the bump. These points did not directly express the form of the bump, but rather controlled the warps that affected the bump point coordinates. Initially, the curve to be reparameterised was an ordinary linear function stretching from the origin to a point where $n$ is the number that is chosen to be the image width (the only dimension of the single-dimensional data). Points were later chosen according to the change imposed on the curve due to warping.

The experimentation Smith carried out allowed for many combinations of different options to be set, applied and appraised comparatively. The estimates of the ``goodness'' of warps were calculated using the creation of an appearance model from the group of images at present state, making this a group-wise optimisation methodology.

The images after warping had been applied were treated as training data for the creation of an appearance model. PCA reduced the complexity of that model as required. The compactness of the model which could be derived from the the sum of variances or the determinant of the covariance matrix^B.2 was then scoring the choice of warps after they had been applied. In this way, a better choice of warps could be made so that bad ones quickly get discarded and the state of all affected images reverted.