Mean Curvature Skeletons

Abstract

Figure 1: Given a watertight surface (a), the well-known medial axis transform (b) often produces too complex of a structure to be of practical use. Our skeletonization algorithm can produce intermediate meso-skeletons (c), which contain medial sheets where needed and curves where appropriate, while converging to a medially centered curve skeleton output (d). AbstractInspired by recent developments in contraction-based curve skeleton extraction, we formulate the skeletonization problem via mean curvature flow (MCF). While the classical application of MCF is surface fairing, we take advantage of its area-minimizing characteristic to drive the curvature flow towards the extreme so as to collapse the input mesh geometry and obtain a skeletal structure. By analyzing the differential characteristics of the flow, we reveal that MCF locally increases shape anisotropy. This justifies the use of curvature motion for skeleton computation, and leads to the generation of what we call "mean curvature skeletons". To obtain a stable and efficient discretization, we regularize the surface mesh by performing local remeshing via edge splits and collapses. Simplifying mesh connectivity throughout the motion leads to more efficient computation and avoids numerical instability arising from degeneracies in the triangulation. In addition, the detection of collapsed geometry is facilitated by working with simplified mesh connectivity and monitoring potential non-manifold edge collapses. With topology simplified throughout the flow, minimal post-processing is required to convert the collapsed geometry to a curve. Formulating skeletonization via MCF allows us to incorporate external energy terms easily, resulting in a constrained flow. We define one such energy term using the Voronoi medial skeleton and obtain a medially centred curve skeleton. We call the intermediate results of our skeletonization motion meso-skeletons; these consist of a mixture of curves and surface sheets as appropriate to the local 3D geometry they capture.

Fast Fully Automatic Segmentation of the Human Placenta from Motion Corrupted MRI

“…A native plane (a) cannot represent all structures of the placenta at once. Therefore, we use our segmentation method (b), correct the motion in this area using [3], project the placenta mask into the the resulting isotropically resolved volume (c), extract the mean curvature flow skeleton [10] (black lines in (d)), use the resulting points to support a curved surface plane (e) and visualize this plane with curved planar reformation [5] (f). The plane in (f) covers only relevant areas, hence gray value mapping can be adjusted automatically to emphasis placental structures.…”

“…It is supported by a mean-curvature flow skeleton [10] generated from the triangulated polygonal mesh of the placenta segmentation and textured similar to curved planar reformation [5], see Figure 3. Although this part is not evaluated thoroughly, clinicians revealed that such a representation is potentially desirable since it compares well to a flattened placenta after birth.…”

An interactive editor for curve-skeletons: SkeletonLab

“…This approach, although less robust than [17,23,11] presents heuristics for collapsing spurious branches and closing loops whenever two terminal nodes have intersecting maximal balls, which are beneficial for the resulting structure. A particularly notable example of point cloud based methods is the work of Huang et al [14], which allows to obtain high quality skeletons from raw and potentially incomplete scans by using the L1-median operator.…”

“…We refer to [15,19] for a qualitative comparison of the most representative methods based on this idea. A recent contraction-based method is [11] which is a robust skeletonization algorithm whose resulting skeletons are topology-preserving, usually well centered and smoother than previous similar methods.…”

“…These are commonly categorised based upon their input, which can be a surface mesh [11,12], a point cloud [13,14] or a voxel grid [15,16]. Volumetric representations, while important, are not handled by our tool, hence they will be not discussed.…”