One approach to shape analysis (after Kendall) uses a fixed number of points to define a shape. The points may describe an object boundary or an interior morphology such as the veins on a leaf. They may be selected randomly; alternatively they may be landmarks--i.e. points of significance. A set of points, so selected, constitutes a shape summary and the original shape data, that had been extracted from the raw image, is discarded.
One problem with representing a shape in this way is that it introduces a source of variability. Different shapes can be reconstructed with the same set of points. This is especially true if the number of points selected for that particular shape are few.
There are, of course, other approaches to shape analysis that do not involve selecting points (at least not at this stage of shape representation). Deformable templates is one such methodology and its use is quite common in medical image analysis. The shape could also be represented by a continuous function (see for example Younes et al. ). But these methods are also computationally expensive.
1) Younes, Laurent; Michor, Peter W.; Shah, Jayant; Mumford, David. A metric on shape space with explicit geodesics. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (2008), no. 1, 25--57.