Skeletonization via Distance Maps and Level Sets

A level set approach for computing solutions to incompressible two-phase ow. A viscosity solutions approach to shape-from-shading. SIAM. 19 Proof: Every simply connected shape has a simply connected skeleton 20]. Only circles have skeletons that are single points. Therefore, the skeletons of all other simply connected shapes have at least two end points. The proof follows immediately from Lemma 4. 2 18 Let us prove the rst statement: All points propagating from the arc segments quench each other at the center of the original maximal disk. Assume by contradiction that a propagating arc segment point p 0 meets the another propagating boundary point before it meets any other propagating arc segment point. This implies that the maximal disk touching point p 0 in the new shape has a smaller radius then the maximal disk. This cannot happen, since any such disk is contained in the maximal disk, which is contained in both new shapes. We now show that pairs of generating points on the boundary of the full shape cannot be separated into diierent new shapes. Assume by contradiction that two points that generated a skeleton point of the full shape are located each on the boundary of a diierent new shape. In that case the radius connecting one of those points with the skeleton point corresponding to it must intersect one of the radii connecting the center of the maximal disk to either p or q, in contradiction to Corollary 2. p and q are the points generating the original maximal disk, see Figure 8. To prove the second statement we must show that in the new shapes, each boundary point is quenched by its original twin from the full shape. This is so since otherwise it must be quenched by another point from the boundary of the new shape. But this is impossible: it will not be quenched by another point that belongs to the full shape since it did not happen in the full shape at the rst place; it will not be quenched by a point from the newly introduced circular segments, since we have shown that all the arc segment points are quenched together. 2 The following corollary is a direct consequence of Lemma 3. Corollary 3 The center of a maximal disk that touches the boundary in two separate points, connects two diierent parts of the skeleton. The two parts are the skeletons of …