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.
@cite_27 propose a continuous medial representation by modeling the MAT with cubic B-splines, as an extension of its discrete counterpart called the @cite_10 . The m-rep is built upon a sparse set of medial atoms, each of which encapsulates the 2D position of a medial point and the corresponding spoke vectors from the 2D medial point to the closest points on the object boundary. The continuous m-rep @cite_27 (cm-rep), on the other hand, uses control points in cubic B-splines to describe the MAT, which must meet specific constraints defined on the implied boundary. In applying cm-reps to object modeling and image segmentation, a template cm-rep model is first built manually which is then deformed to fit a target shape. There is currently no method for automatically computing a smooth curve approximation to the MAT of a 2D shape.
@cite_27 @cite_10
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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. We present an algorithm that automatically establishes dense correspondences between a large number of 3D faces. Starting from automatically detected sparse correspondences on the outer boundary of 3D faces, the algorithm triangulates existing correspondences and expands them iteratively by matching points of distinctive surface curvature along the triangle edges. After exhausting keypoint matches, further correspondences are established by generating evenly distributed points within triangles by evolving level set geodesic curves from the centroids of large triangles. A deformable model (K3DM) is constructed from the dense corresponded faces and an algorithm is proposed for morphing the K3DM to fit unseen faces. This algorithm iterates between rigid alignment of an unseen face followed by regularized morphing of the deformable model. We have extensively evaluated the proposed algorithms on synthetic data and real 3D faces from the FRGCv2, Bosphorus, BU3DFE and UND Ear databases using quantitative and qualitative benchmarks. Our algorithm achieved dense correspondences with a mean localisation error of 1.28 mm on synthetic faces and detected 14 anthropometric landmarks on unseen real faces from the FRGCv2 database with 3 mm precision. Furthermore, our deformable model fitting algorithm achieved 98.5 percent face recognition accuracy on the FRGCv2 and 98.6 percent on Bosphorus database. Our dense model is also able to generalize to unseen datasets. The medial axis transform (MAT) of a shape, better known as its skeleton, is frequently used in shape analysis and related areas. In this paper a new approach for determining the skeleton of an object is presented. The boundary is segmented at points of maximal positive curvature and a distance map from each of the segments is calculated. The skeleton is then located by applying simple rules to the zero sets of distance map differences. A framework is proposed for numerical approximation of distance maps that is consistent with the continuous case and hence does not suffer from digitization bias due to metrication errors of the implementation on the grid. Subpixel accuracy in distance map calculation is obtained by using gray-level information along the boundary of the shape in the numerical scheme. The accuracy of the resulting efficient skeletonization algorithm is demonstrated by several examples. The paper describes an algorithm for generating an approximation of the medial axis transform (MAT) for planar objects with free form boundaries. The algorithm generates the MAT by a tracing technique that marches along the object boundary rather than the bisectors of the boundary entities. The level of approximation is controlled by the choice of the step size in the tracing procedure. Criteria based on distance and local curvature of boundary entities are used to identify the junction or branch points and the search for these branch points is more efficient than while tracing the bisectors. The algorithm works for multiply connected objects as well. Results of implementation are provided.