The Point Distribution Model (Cootes, 1995) has become a very popular tool for statistical shape analysis and has been widely used in various computer-vision and medical-imaging applications such as image segmentation and shape based diagnosis. The major challenge in constructing a Point Distribution Model (PDM), especially in 3D, is the step of landmark-based shape correspondence. Shape correspondence aims at identifying a set of accurately corresponding landmarks from a population of given shape instances. The non-linearity of the shape description and shape variation for most anatomical structures leads to a problem where it is very difficult to find an optimal solution.
Various 3D shape correspondence methods have been proposed for PDM construction. However, most of these methods are aimed at closed-surface shape correspondence. For example, both the Minimum Description Length (Davies, 2002; Heimann, 2005) and Spherical Harmonics (Brechbuhler, 1995; Gerig, 2001) methods map each shape instance to a sphere and reduce the shape correspondence problem to that of parameterizing the sphere. It is usually difficult to apply such a sphere-mapping step to open-surface shapes. Hence, we require a method that can perform 3D shape correspondence for both closed-surface and open-surface shapes.
The aim of shape correspondence is to obtain a set of corresponding landmarks on a population of shape instances. As shown in Figure 1, we can represent each shape instance S as:
Illustration of each representation of a shape instance: (a) Point cloud SP representing the surface and the surface boundary SB. (b) Landmark-based triangle mesh ST where each vertex is a landmark in SL. (c) Discrete triangle mesh SM to approximate the surface S. Note that SM and ST are not the same.
SP, a dense point cloud defining the entire closed or open surface;
SB, the subset of SP that describe the closed boundary of the surface if SP is an open surface (SB is empty if SP is a closed surface);
SM, a triangle mesh constructed to approximate S;
SL, the set of landmarks identified by the shape correspondence method;
ST, the triangle mesh on SL.
With correspondence across all shape instances, the triangle mesh ST can be used to ensure topology consistency among all instances in the population. Different from SL and ST, SM is constructed independently and may contain different number of vertices and triangles for each shape instance. Further, SB is required to ensure that the landmarks along the boundary of an open-surface shape instance are correctly corresponded with points along the boundary of another instance in the population.
First we consider a simplified form of the shape correspondence where the population consists of only two instances: a template U and a target V. If we identify a set of landmarks UL on the template, the problem then is to identify target landmarks VL. Following this same method for a population of N instances, we can first select an instance as the template and identify template landmarks UL. Then, we can construct corresponding landmarks on each target Vi in a pair-wise manner. In this way, corresponding landmarks can be identified on the entire population of shape instances. Note that these landmarks may not coincide with any anatomically significant locations or features.