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Assessing and Comparing Registration Algorithms

Having shown the validity of the model-based measures of registration quality, we now proceed to the reason for defining these measures, that is, to enable comparison of the performance of various non-rigid registration algorithms in cases where ground truth data is not available.

NRR algorithms can be divided into two general classes: pairwise and groupwise. Pairwise algorithms can be defined as those which register a pair of images. Registration across a group is then defined by successive applications of the basic pairwise algorithm. For example, all images in the training set can each be pairwise-registered to some chosen reference example (e.g., [#!Rueckert!#]). However, this suffers from the problem that, in general, the result obtained depends on the choice of reference. Refinements of this basic approach are possible, where the reference is artificially generated and updated so as to be representative of the group of images as a whole. But the important point to note is that the correspondence for a single training image is defined w.r.t. this reference (which enables consistency of correspondence to be maintained across the group), and that the information used in determining the correct correspondence is limited to that contained in the single training set image and the single reference image.

It can be seen that this approach explicitly does not take advantage of the full information in the group of images when defining correspondence [#!Cootes_Groupwise_ECCV!#]. Making better use of all the available information is the aim of groupwise registration algorithms, where correspondence is determined across the whole set in a principled manner.

One such groupwise method is the Minimum Description Length (MDL) formulation as developed by the authors [#!IPMI_2005_ISBE!#]. The main idea is that the appearance model generated from the current correspondence is made an integral part of the process of further registration, the model being continually updated as the process of registration proceeds. The objective function for this groupwise registration is a minimum description length [#!rissanen-book!#] one, which envisages encrypting the entire training set as a coded message, the length of the message in bits being the objective function. But rather than encoding the raw images, the encoding proceeds by describing each training set image as a series of shape and texture deformations applied to some reference. That is, the encoding explicitly uses the model representation of each image from the appearance model built using the found correspondence. The full encoding hence also has to contain the details of the model itself, and the discrepancy between the actual image and the appearance model representation of that image.

For the purposes of comparing NRR algorithms, we consider the following:

• Pairwise registration of each training set image to a fixed reference image, using an image from the training set as a reference
• Groupwise registration based on the MDL algorithm described above, but admitting two slight variants of the algorithm.

In effect, the differences between the two groupwise variants considered are the way they constrain or do not constrain the allowed spatial deformations. The exact details are not relevant here; what is relevant is that both are groupwise, but with a slight difference.

We hence would expect that the groupwise variants should be close together in performance, but that both should give significantly better registration results than the simple pairwise approach. These three algorithms present a suitable test of the discrimination ability of our proposed evaluation framework.

For these evaluation experiments, we limited ourselves to 2D images, which allows larger-scale experiments to be performed.

The raw dataset consisted of 90#90 3D MR images of normal brains. . These were then affinely aligned, and a single slice extracted from each, at equivalent locations. This hence formed our training set of 104 2D slices.

This training set was then registered using the 3 registration algorithms detailed above. For each algorithm, an appearance model was then built from the found correspondence, with varying numbers of modes included in the model. An example of such a model is shown in Figure .

Figure: Appearance model which was built automatically by group-wise registration. First mode is shown, 12#12 standard deviations.

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The Specificity and Generalisation of each such model was then computed. The results as a function of the number of modes are shown in Figure .

Figure: Generalisation and Specificity of the three registration methods as a function of the number of modes included in the appearance model.

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