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We propose an algorithm for surface reconstruction from unorganized points based on a view of the sampling process as a deformation from the original surface. In the course of this deformation the Medial Scaffold (MS) - a graph representation of the 3D Medial Axis (MA) - of the original surface undergoes abrupt changes (transitions) such that the MS of the unorganized point set is significantly different from that of the original surface. The algorithm seeks a sequence of transformations of the MS to invert this process. Specifically, some MS curves (junctions of 3 MA sheets) correspond to triplets of points on the surface and represent candidates for generating a (Delaunay) triangle to mesh that portion of the surface. We devise a greedy algorithm that iteratively transforms the MS by "removing" suitable candidate MS curves (gap transform) from a rank-ordered list sorted by a combination of properties of the MS curve and its neighborhood context. This approach is general and applicable to surfaces which are: non-closed, non-orientable, non-uniformly sampled. In addition, the method is comparable in speed and complexity to current popular Voronoi/Delaunay-based algorithms, and is applicable to very large datasets.