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Synopsis

The work of Katherine Smith investigated warps on a simple bump-like curve (see Figure 9 below).

Figure 9: Reparameterisation along the curve

Description: The first step taken by the application was the generation of some random bumps simpler than the one above. These bumps varied in height and width although the property of height was intended to be ignored during registration. The bumps were all symmetric and the height is one of {hi, low} where $lo=0$ and $0.7<hi<1$. 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 can conjointly define that bump so two real numbers at the minimum would suffice to derive each bump.

As images are being warped, the form of the bump quickly changes to give a smoother curve with more continuous derivatives. This of course will depends on the type of warp which is 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 is obtained using warps and measured using the methods outlined in Subsection 3.6.

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 always 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 at some capacity beforehand in Subsection 3.4, 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 $(n,n)$ 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 evaluated using the creation of an appearance model from the group of images at present state, making this a group-wise optimisation methodology. This was not the first time that such an approach was investigated as Subsection 4.4 shows.

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 matrix⁴¹ 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.

Summary: As the above descriptions imply, this work was able to show how statistical models go hand-by-hand with non-rigid registration. In this case, they simple evaluated the (non-rigid) registration process and distinguished between the many alternatives offered by different families of warps, similarity measures and so forth. Needless to say, the run-time became a real difficulty when ill-chosen strategies were attempted. Smith took this into consideration in the final evaluation and comparison of all different experiments.

Advancements: The work of Marsland, Twining and Taylor [43] went a step ahead and investigated a full 2-D model. However, it concentrated on just a simple contour (defined by 12 control points) of the skull shape as pictured from an overhead perspective. The figure below shows that warps can have an effect on the whole shape, but still lack some control over local structures such as the ventricles. Varying scale can solve problems like this and make the global non-rigid registration approach very robust. While this work produced elegant results, it did not explore many varying options as Smith did.

Figure 10: Brain image warping