We present a full pipeline for computing the medial axis transform of an arbitrary 2D shape. The instability of the medial axis transform is overcome by a pruning algorithm guided by a user-defined Hausdorff distance threshold. The stable medial axis transform is then approximated by spline curves in 3D to produce a smooth and compact representation. These spline curves are computed by minimizing the approximation error between the input shape and the shape represented by the medial axis transform. Our results on various 2D shapes suggest that our method is practical and effective, and yields faithful and compact representations of medial axis transforms of 2D shapes.
We describe a novel continuous medial repre- sentation for describing object geometry and a deformable templates method for tting the representation to images. Our representation simultaneously describes the boundary and medial loci of geometrical objects, always maintaining Blum's symmetric axis transform (SAT) relationship. Cu- bic b-splines dene the continuous medial locus and the as- sociated thickness eld, which in turn generate the object boundary. We present geometrical properties of the rep- resentation and derive a set of constraints on the b-spline parameters. The 2D representation encompasses branching medial loci; the 3D version can model objects with a sin- gle medial surface, and the extension to branching medial surfaces is a subject of ongoing research. We present prelim- inary results of segmenting 2D and 3D medical images. The representation is ultimately intended for use in statistical shape analysis. A model of object shape by nets of medial and boundary primitives is justified as richly capturing multiple aspects of shape and yet requiring representation space and image analysis work proportional to the number of primitives. Metrics are described that compute an object representation's prior probability of local geometry by reflecting variabilities in the net's node and link parameter values, and that compute a likelihood function measuring the degree of match of an image to that object representation. A paradigm for image analysis of deforming such a model to optimize a posteriori probability is described, and this paradigm is shown to be usable as a uniform approach for object definition, object-based registration between images of the same or different imaging modalities, and measurement of shape variation of an abnormal anatomical object, compared with a normal anatomical object. Examples of applications of these methods in radiotherapy, surgery, and psychiatry are given.