Statistical Shape Model (SSM)

What is shape?

The shape is usually defined as the geometric information invariant to a particular class of transformation (similarity transform: translation, rotation and scaling)

The statistical models represent shape of an object by a set of points. The points can be parameterized and controlled so that the shape of the object can be changed.
The purpose of Statistical Shape Model (SSM) is to derive models which allow us to 1) analyse new shapes, and 2) synthesise shapes similar to those in a training set.

Note: The shapes don't need to be represented in space, but also in time or intensity. For examples:
3D shapes: composed of points in 3D space/ points in 3D space + time (image sequence)
2D shapes: composed of points in 2 D space/ space + time
1D shapes: points along a line/ intensity values sample in an image

Suitable Landmarks

Good landmarks are points which can be consistently located from one image to another. A training set can be generated by a human expert who annotates each of a series images with a set of corresponding points. Since it can be time consuming and tedious, some automatic and semi-automatic method are being developed.

In two dimensions suitable landmarks can be placed at clear corners of object boundaries, 'T' junctions between boundaries or easily located biological landmarks.

Representation: If a shape is described n points in d dimensions we represent the shape by a nd element vector formed by concatenating the elements of the individual point position vectors. For example, in a 2D image the n landmark points, {(xi,yi)}, can be represented as

(this is an example in the training set)

Aligning the Training Set

One of the most popular approach to align two shape is Procrustes Analysis which aligns each shape so that the sum of distances of each shape to the mean is minimised.