Cootes et. al. (1995) used the term ASM as the name of the model fitting algorithm in their paper, where the model includes a point distribution model (PDM). The PDM is a kind of deformable shape model that consists of a linear model to explains the shape variation of the training data. This section explains how the PDM can be constrcted from example data and how to find a set of optimal parameters of the PDM for a new input image.
3.1.1 Point Distribution Model
In the PDMs, a 2D shape is represented by a set of 
vertices, which correspond to the salient points of a non-rigid object. The shape vector s consists of the coordinates of the vertices as
(3.1.1)The shape variation is expressed as a linear combination of a mean shape 
and 
shape bases 
as
(3.1.2) where
are the shape parameters.
The mean and shape bases are learned from a set of training images using a statistical analysis technique, principal component analsys (PCA). The standard approach is to apply the PCA to a set of shape vectors that are gathered from the manually landmarked training images. The mean shape 
is the mean of gathered shape vectors and the basis vectors 
are the 
eigenvectors corresponding to the 
largest eigenvalues.
Usually, the gathered training shape vectors are aligned using the Procrustes analysis (Cootes et. al., 2001d) before the PCA is applied to remove variations due to similarity transformation that scales, rotates, and translates the shape. Thus, the PCA is only concerned with the local shape deformation. Figure 1 illustrates an example of shape bases, where the five shape bases 
to 
are displayed over the mean shape 
.
Figure 1. An example of shape bases, 
to 
Because the PDM does not explain the similarity transformation (scaling, rotation, and translation) of the shape, we need 4 more parameters to explain the similarity transformation of the shape when we synthesize a shape using the PDM that coincident with the object in an image. The similarity transformation can be explained by 4 parameters 
as
.
(3.1.3)The PDM is an accurate, specific, and compact deformable shape model. The PDM is an accurate model, which means that it can synthesize all valid shapes. The PDM is a specific model, which means that it excludes all invalid shapes; The invalid shapes can be excluded by simply limiting the range of the shape parameters into a certain range (usually, +/- 3 standard deviations) because the shape parameters are uncorrelated. The PDM is a compact model, which means that it uses the smallest number of parameters that are enough to describe all the shape variations.