PDF version of this entire document

Representing Shapes

If a skilled person annotates to learn the shape of an object within an image, this is typically done by tracing the outline of that object. However, such a continuous outline is not directly suitable for analysis purposes.

The simplest way to represent such an outline in a compact fashion is using some set of points (called landmarks or shape points). The entire shape is then obtained from the positions of these landmarks by interpolation. The simplest case is linear interpolation, where the landmarks are the vertexes of a polygon, but more complicated spline interpolants are possible if a smoother outline is required.

This method can describe a single shape, but what is required is a representation which describes a set of shapes within a single common framework. This is usually achieved by identifying corresponding landmarks across a set of shapes (see Figure cap:Landmark-identification). For example, if we are modelling the shapes of biological objects, landmarks can be placed at anatomically relevant points, which can be reliably located in any example of the object. Further shape points can then be created semi-automatically along edges in the images, to produce the final dense polygonal representation.

Mathematically, the position of each landmark point within each image is encoded in terms of the Cartesian coordinates of each point. The entire set of landmark points within a given shape is then represented by a single shape vector $\mathbf{S}_{i}$, formed by concatenating the coordinates of the individual landmarks

$$(x_{1},y_{1},x_{2},y_{2},...,x_{n},y_{n})\Rightarrow\mathbf{S}_{i}.$$ (3.1)