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Reparameterisation

Taking again an example from work on shapes^3.12, a shape can be described by a collection of landmarks as shown in Figure earlier in this report. The landmarks are usually located at corners, T-junctions and edges that are easy to locate. Also, other additional points in between these landmarks can chosen to expand the representation of that shape and make it richer, though ideally curves should be continuous and the number of points that make them up arbitrary. To register multiple images, all corresponding landmarks and points must overlap in as accurate a way as possible. They must correspond to one another in one common spatial reference^3.13 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.

As shapes continue to be discussed, it is worth stressing that it is never obvious what choice of landmarks and intermediate points will result in an optimal overlap or even a good one. The quality varies depending on the pre-defined objective function. To automatically shift points and evaluate the subsequent global (or pair-wise) effects, reparameterisation of these points must take place in a way that preserves their order along the 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 points^3.14 from an arbitrary point on the curve in such a way that will not violate their sequential order.

Figure: Monotonically-increasing function.

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

$$\forall(u\in S\wedge v\in S\wedge u<v)\rightarrow f_{mon}(u)<f_{mon}(v)$$ (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 Figure , 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 evaluating the selection of landmark points. For each such reparameterisation, the specificity, generalisability and compactness were evaluated at some stage although minimum description length was ultimately chosen (to be discussed in Chapter cha:MDL Models). The first and second of these terms were coined in the thesis published by Davies.

Figure: Reparameterisation example. A point moves along the curve a distance $S'$ from the origin. All other points will do so as well to make this a continuous reparameterisation, often defined by a Cauchy.

Prior to the invention of this technique, points were often chosen to be allocated a position 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.