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Non-rigid registration (NRR) is ubiquitously used as a basis for medical image analysis. Its applications include atlas matching, analysis of change over time or subjects [7], and structural analysis. A wide variety of approaches exist, which solve the NRR problem. They differ in terms of the objective function that defines mis-registration, the representation of spatial deformation fields, and the approach used to minimize mis-registration by selecting good deformations. Ideally, a composition of aggregated deformations brings a set of images into full alignment, which means that corresponding structures across those images overlap.
Most commonly, pairs of images are being registered [27] at any one time, though groups can be considered too [5]. In the former case, which is referred to as pairwise registration, NRR is applied to just two images in isolation. In the latter case, all the different images are handled simultaneously. This approach has become more popular in recent years and is referred to as groupwise registration in the literature. It has real merits since, given a couple of very dissimilar images, the set as a whole can compensate for that dissimilarity, making its contribution in the form of additional information. It has become a recurrent contention that groupwise registration is the more valid approach.
NRR is an under-constrained problem that suffers from subjectivity in its solution. The solution comprises the set of spatial deformations, which deform one image to match another. For any set of images to be registered, different approaches are likely to produce different results. The different objective functions have different minima, which is the direct effect of the way they define similarity between images.
One obvious way to assessing a given solution is by making use of the ground-truth solution. This idea is based on the principle that any solution can be - in one way or another - numerically evaluated in terms of its divergence from the correct solution. Several methods have been demonstrated, which work along these lines [10,12,21,19]. These methods, however, require access to some form of ground truth, which is usually difficult to obtain.
One approach involves the construction of artificial test data, which limits application to 'off-line' evaluation. Furthermore, that approach relies on conditions which are unrealistic, so should be taken with a grain of salt. Other methods can be applied directly to real data, but require that anatomical ground truth be provided, typically involving annotation by an expert. This makes validation expensive and prone to subjective error. In 3D, matters become even more complex. As the correct solution - that which is often based on anatomy - is indeed hard to obtain, NRR assessment without ground truth appears highly valuable.
We consider appearance model, which have been extensively used as the basis for interpretation by synthesis. Such models are derived from sets of training images and, in effect, the models capture statistics about variability within these images. Any model acquires knowledge from its training set and is able to use that knowledge in a variety of ways. Any set of images, which is used to construct an appearance model, is directly related to that model's quality. When the images are not correspondent, the model is fuzzy and often not valuable. When the images are properly correspondent, the model is improved.
As NRR aims to bring sets of images to a state of full pixel-to-pixel correspondence, the output of a good NRR algorithm builds a good model. We embrace this key observation and exploit the relationship between models and NRR. We use existing methods from both ends of the problem and unify the two as to benefit from both.
The paper presents a framework for building appearance models automatically and then evaluating them. In turn, this method facilitates the assessment of NRR, which requires only the image data, and can therefore be applied routinely, while oblivious to any form of ground truth. The method relies on the fact that, for a given set of registered images, a statistical model of appearance can be constructed. When the registration is correct, the model provides the most concise description of the set of images. As the solution to NRR degrades, so does the performance of model synthesis. Thus, the quality of registration affects the quality of the resulting model and the model itself reflects on the quality of NRR, which makes evaluation of the two somewhat mutual.
The remainder of this paper is structured as follows: it begins by covering background on registration (assessment in particular) and statistical models. It outlines some existing NRR assessment methods, explains about the proposed methods, and presents results which support ideas and theory behind our new method. Validation experiments are then performed where brain models are advertently degraded, by mis-registering their training set. Our validation results confirm our method to in tight correlation with ground truth. We show this to be the case by using a generalised measure of label overlap, which uses hand-annotated brain anatomy. Lastly, several registration algorithms are compared to demonstrate one main application of our approach, as applied to brain data. We also show that groupwise registration algorithms produce better results than these of pairwise equivalents. All the same, an algorithm which is based on the minimum description length (MDL) principle, produces results that are comparable, if not better, than the standard groupwise NRR algorithm.