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Finding the Correct Solution

One essential step, which can be used to understand the algorithm's behaviour and the direction it takes, is that which infers a state which is optimal - that is -- a state that the objective function must reach when it behaves correctly. Analysis was performed to discover which warps, when applied to the data available, give a sensible (or perfect) result, i.e. align the data. Figure shows a set of correct warps with the corresponding set of results. These help in verifying that registration of data is finally obtained.

Figure: Data alignment to discover correct solution. On the left: piece-wise linear warps to be applied to original data; on the right: data after alignment.

The choice of a reference of course effects the 'correct' set of warps, but the notion of correctness is unchanged. In the case shown, reference is chosen to be the first of the data instances, meaning that it is assumed all data needs to be transformed based on that one image. It is warped to fit the first image. In practice, only the actual value is used to evaluate the performance of the model-based objective function. This means that this alignment is one amongst several possibilities, but its value provides an excellent estimate. Also worthy of mentioning is the fact that an immutable reference is maintained for the model-based case. This is why results are expected to be ultimately similar the the ones above, where that very same reference is chosen.

In AART, advanced caution is dispensed to avoid reference choice that damages data integrity. It is safer to choose as reference data that is representative of the whole set (essentially one which lies in the centre of the multi-dimensional cloud in vector scape). Reference is chosen which lies closest to the mean of the set. Definitions of distance are alterable. Currently, sum of squared distances^7.17 or fixed geometrical distances can be used in AART to locate a good reference.