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Reparameterisation

A shape can be described by a collection of landmarks as shown in Figure 1 earlier in this report. The landmarks are usually located at corners, T-junctions and edges that are easy to locate and other points in between these landmarks are chosen to expand the representation of that shape and make it richer. To register multiple images, all corresponding landmarks and points must overlap in as accurate a way as possible. They must correspond to one another on one common grid so that image analysis can proceed. One way of doing this is to apply diffeomorphic warps to the space in which images will be embedded. That newly-defined plane is supposed to bring the collection of landmarks across the set of input images closer together. This ends up bringing a number of images to correspondence of some quality.

It is not, however, obvious what choice of landmarks and intermediate points will finally result in an optimal overlap or even a good one. To automatically shift points and evaluate the the subsequent global (or pair-wise) effects, reparameterisation of these points must take place in a way that preserves their order along the imaginary contour they form. A new spread of the points needs to be chosen iteratively and the results recorded. The spread of the points can be defined purely by a function and the reparameterisation alters this function to find preferable results. A monotonically-increasing function describes the distance of all points²⁹ from an arbitrary point on the curve in such a way that will not violate their sequential order.

Figure 6: Monotonically-increasing function

Figure 6 shows what is meant by a monotonically-increasing function. The following must hold although its inverse may hold instead (a monotonically-decreasing function):

$$\forall(u\in S\cap v\in S\cap u<v)\rightarrow f_{mon}(u)<f_{mon}(v)$$ (3.1)

where $f_{mon}$ is the monotonically-increasing function used and $f_{mon}(S)=S'$. More simply, the derivative at any point must be positive, i.e. $0<\theta<90$ so that $0<tan(\theta)<1$. In the figure below, the meaning of reparameterisation as it is applied to points of a shape is made clearer. The distance or offset along the curve is guided by the value which was determined by the function above. In this particular way, all points which lie on the curve can be moved simultaneously without colliding with one another and new autonomous descriptors of shape become available. Instead of describing the movement of each individual point, an arbitrary number of points can be shifted according to one modifiable function. Davies et al. used this technique to optimise a shape model by appraising any selection of points. For each such reparameterisation, the specificity, generalisability and compactness were evaluated. The first and second of these terms were coined in the thesis published by Davies.

Figure 7: Reparameterisation example

Prior to the invention of this technique, points were often chosen to be put on the curve so that they are equally spaced. This approach was often a straight-forward and computationally inexpensive, but its results were unsatisfactory for more complex shape where the curve bends sharply. Some attempts were made at placing more points at regions of high-curvature, but these were still inferior to the aforementioned reparameterisation-based approach.