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.
Exact medial axis computation is possible only for simple or special shapes, such as polyhedra @cite_25 @cite_17 . For free-form shapes, medial axis approximations are widely used in practice. There are several main approaches to computing the medial axis approximation: pixel or voxel-based methods that compute the medial axis using a thinning operation @cite_26 ; methods based on distance transform @cite_21 @cite_13 @cite_8 @cite_28 , often performed on a regular or adaptive grid; the divide-and-conquer methods @cite_5 , performed on spline curve boundaries; the tracing approaches @cite_3 , by tracing along the shape boundary or the seam curves; and the Voronoi diagram (VD) based methods @cite_0 @cite_23 @cite_9 @cite_29 @cite_15 .
@cite_26 @cite_8 @cite_28 @cite_9 @cite_21 @cite_29 @cite_3 @cite_0 @cite_23 @cite_5 @cite_15 @cite_13 @cite_25 @cite_17
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Abstract Medial axis as a compact representation of shapes has evolved as an essential geometric structure in a number of applications involving 3D geometric shapes. Since exact computation of the medial axis is difficult in general, efforts continue to approximate them. One line of research considers the point cloud representation of the boundary surface of a solid and then attempts to compute an approximate medial axis from this point sample. It is known that the Voronoi vertices converge to the medial axis for a curve in 2D as the sample density approaches infinity. Unfortunately, the same is not true in 3D. Recently, it is discovered that a subset of Voronoi vertices called poles converge to the medial axis in 3D. However, in practice, a continuous approximation as opposed to a discrete one is sought. Recently few algorithms have been proposed which use the Voronoi diagram and its derivatives to compute this continuous approximation. These algorithms are scale or density dependent. Most of them do not have convergence guarantees, and one of them computes it indirectly from the power diagram of the poles. Recently, we proposed a new algorithm that approximates the medial axis straight from the Voronoi diagram in a scale and density independent manner with convergence guarantees. In this paper, we present several experimental results with this algorithm that support our theoretical claims and also show its effectiveness on practical data sets. We present an accurate algorithm to compute the internal Voronoi diagram and medial axis of a 3-D polyhedron. It uses exact arithmetic and exact representations for accurate computation of the medial axis. The algorithm works by recursively finding neighboring junctions along the seam curves. To speed up the computation, we have designed specialized algorithms for fast computation with algebraic curves and surfaces. These algorithms include lazy evaluation based on multivariate Sturm sequences, fast resultant computation, culling operations, and floating-point filters. The algorithm has been implemented and we highlight its performance on a number of examples. We present a simple, efficient, and stable method for computing-with any desired precision-the medial axis of simply connected planar domains. The domain boundaries are assumed to be given as polynomial spline curves. Our approach combines known results from the field of geometric approximation theory with a new algorithm from the field of computational geometry. Challenging steps are (1) the approximation of the boundary spline such that the medial axis is geometrically stable, and (2) the efficient decomposition of the domain into base cases where the medial axis can be computed directly and exactly. We solve these problems via spiral biarc approximation and a randomized divide & conquer algorithm. 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.