Wrapping up: Week 2

In this week we have set the mathematical foundations for the rest of the course. Most importantly, we have seen that we can think of any shape in the shape family as a deformed version of a given reference shape.

Thinking of shapes as deformations of a reference shape makes clear that our task is to model the possible deformations that lead to the shapes in the shape family. It is best to think of these deformations as functions (deformation fields), which are defined on the reference surface. The corresponding mathematical concept to define a normal distribution over these deformation fields is called a Gaussian Process.

To define a Gaussian Process, we need to define a mean function and a covariance function. We have seen that one way to define these parameters is to estimate them from example data; i.e. known deformation fields that relate the reference shape with some normal instances of the shape family. This construction is the main idea behind point distribution models, the arguably most widely used type of shape models.

We have also learned that, in accordance with the definition of shape, we have to standardize the pose of the example surfaces before the deformation can be estimated. We have seen how to perform this pose standardization in Scalismo Lab. This enables us to prepare a dataset of shapes for the actual analysis and will be very useful to those of you implementing the project.

There is one big assumption that we have made when we estimated the parameters of the Gaussian Processes, namely that we have correspondence between all the example surfaces. In practice this is not given and we first have to establish correspondence using a registration algorithm. We will learn how to deal with this problem in Week 6.