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Analysis of Warps

Throughout the entire year, there was some general interest in how clamped-plate splines affect the data and how the model-based objective function affects the choice of warps. Several of the drawbacks of various families of warps and the problems concerned with diffeomorphism were identified, yet these were of greater interest to Marsland and Twining who posses knowledge of the more theoretical grounds. In Figure lies a representation of a warp - that is - a reparameterisation curve that maps one point coordinate to another (and being a strict one-to-one mapping, it is a bijection as well). The idea was explained in some detail in sub:Reparameterisation.

Figure: Warps shown as the MSD objective function runs. Each row shows the reparameterisation which is applied to one of the 5 images in the same row.

It should be noted that one data instance remains unchanged. That is in fact the reference which avoids the data from drifting away interminably. The left-hand side corresponds to the former iterations, the right-hand side - to the latter ones.

It was, at the earlier stages of this investigative study, vital to ensure that no cases of tearing and folding issues could arise. It turned out that one certain type of warp was problematic. A warp which was in essence made of a composition of knot-points (or control points in a more orthodox terminology for functions), also known as the multi-point warps, could produce unwanted effects and usage of that warp immediately ceased. Instead, a simpler single-point^7.8 warp has been used since, while the other was permanently conceded.