reverse engineering – recent advances and applications - OpenLibra

124 Reverse Engineering – Recent Advances and Applications
8 Will-be-set-by-IN-TECH
(a) (b)
Fig. 5. (a) example in which a non-manifold mesh structure is resolved by a vertex-split, (b)
resolving non-manifold mesh structure by first applying an edge-split followed by two
vertex-splits.
3.6 Vertex-mapping
The next step concerns moving the mesh toward the “unknown“ surface by projecting the
mesh vertices onto the local tangent planes defined by the set of proximate sample points.
This is accomplished for each vertex v of the mesh M by first identifying the voxel set W
directly adjoining v and collecting the enclosed points Pv. Next, we fit a plane to the points Pv
by computing the centroid �c and the plane normal �n obtained from the covariance matrix C
of Pv. In order to improve the accuracy of the fitting the points of Pv can be filtered according
their distance to v yielding P ′ v = {p ∈ Pv| 2�p − v� < l}, with l representing the edge length of
the voxel complex W. The normal �n is defined by the eigenvector associated with the smallest
eigenvalue of C. Together with the former position of the vertex �v we are able to compute the
new coordinates of �vn by
�vn = �v − ((�v −�c) ·�n)�n . (1)
To be able to perform this projection the number of points of Pv has to be |Pv| ≥3. Otherwise,
points from adjacent voxels need to be added from surrounding voxels. By extending W to
Fig. 6. Vertex projected onto the tangent plane defined by the points Pv of the adjacent voxel
set W.