  The reconstruction of an object's shape or surface from a set of 3D points
plays an important role in medical image analysis, e.g. in anatomy
reconstruction from tomographic measurements or in the process of aligning
intra-operative navigation and preoperative planning data. In such scenarios,
one usually has to deal with sparse data, which significantly aggravates the
problem of reconstruction. However, medical applications often provide
contextual information about the 3D point data that allow to incorporate prior
knowledge about the shape that is to be reconstructed. To this end, we propose
the use of a statistical shape model (SSM) as a prior for surface
reconstruction. The SSM is represented by a point distribution model (PDM),
which is associated with a surface mesh. Using the shape distribution that is
modelled by the PDM, we formulate the problem of surface reconstruction from a
probabilistic perspective based on a Gaussian Mixture Model (GMM). In order to
do so, the given points are interpreted as samples of the GMM. By using mixture
components with anisotropic covariances that are "oriented" according to the
surface normals at the PDM points, a surface-based fitting is accomplished.
Estimating the parameters of the GMM in a maximum a posteriori manner yields
the reconstruction of the surface from the given data points. We compare our
method to the extensively used Iterative Closest Points method on several
different anatomical datasets/SSMs (brain, femur, tibia, hip, liver) and
demonstrate superior accuracy and robustness on sparse data.
