Deformable models have proved effective for interpreting images of objects whose shape can vary [2]. Where a training set of example images is available, a successful approach is to build a statistical model of the shape variation seen in the training set. Such a model can be used for image search to locate objects in new images. A good model will be sufficiently general that it can fit to valid unseen examples, but specific in that it will not allow significantly different shapes.
Here we deal with the case where we can place n landmark points repeatably
on each example object (for instance around the boundary), and thus represent
a shape by this set of landmarks 
[2].
Given a set of such shapes, aligned into a common co-ordinate frame, each shape
corresponds to a vector 
in a 2n dimensional space. The set of shapes then forms a cloud of points
in this space, which can be thought of as drawn from a probability distribution. If we can estimate the probability density function (p.d.f.) 
for the distribution of shapes, we can decide whether any new shape is
plausible, and can use this information when attempting to locate examples of
the object in new images.
A general approach is to use a density estimation technique such as the kernel method [7]. This represents the distribution as a sum of gaussians, one placed at every original data point. However, when there are many points this becomes far too expensive (in both time and memory). It is necessary to further approximate the distribution, for instance using a mixture of a small number of gaussians, which can be fit to the kernel estimate using a modification of the Expectation Maximisation (EM) algorithm.
We have previously described Active Shape Model (ASM) search - an efficient approach to interpreting images containing known objects represented by statistically defined deformable templates. The method is iterative; at each step an initial hypothesis is deformed using the image evidence, then regularized to the nearest plausible shape (as defined by the p.d.f.). Given a `good enough' starting point, this can converge rapidly to locate objects in new images. The method was originally intended for use with PDMs, but can be extended to use mixture models.
In the following we will demonstrate how mixtures of gaussians can be used to approximate the p.d.f. for a shape model, and how the Active Shape Model approach can be used to find objects in new images.