Point Distribution Model - Details

Details

First, a set of training images are manually landmarked with enough corresponding landmarks to sufficiently approximate the geometry of the original shapes. These landmarks are aligned using the generalized procrustes analysis, which minimizes the least squared error between the points.

aligned landmarks in two dimensions are given as

.

It's important to note that each landmark should represent the same anatomical location. For example, landmark #3, might represent the tip of the ring finger across all training images.

Now the shape outlines are reduced to sequences of landmarks, so that a given training shape is defined as the vector . Assuming the scattering is gaussian in this space, PCA is used to compute normalized eigenvectors and eigenvalues of the covariance matrix across all training shapes. The matrix of the top eigenvectors is given as, and each eigenvector describes a principal mode of variation along the set.

Finally, a linear combination of the eigenvectors is used to define a new shape, mathematically defined as:

where is defined as the mean shape across all training images, and is a vector of scaling values for each principal component. Therefore, by modifying the variable an infinite number of shapes can be defined. To ensure that the new shapes are all within the variation seen in the training set, it is common to only allow each element of to be within 3 standard deviations, where the standard deviation of a given principal component is defined as the square root of its corresponding eigenvalue.

PDM's can be extended to any arbitrary number of dimensions, but are typically used in 2D image and 3D volume applications (where each landmark point is or ).