Analogous to the 2-D case, a subdivision scheme is employed to generate a hierarchy of triangles, which is subject to the governing equations of the physical model. The modified Butterfly subdivision scheme for triangulated surfaces has been suggested for this purpose [54]. It was originally introduced by the same authors [55] as the DLG subdivision scheme in two dimensions and exhibits similar properties: it has interpolating behavior and has a tangent-continuous limit surface, i.e., Cx. As for most subdivision methods for triangulated surfaces, quaternary subdivision is used. To correct for degeneracies resulting from topo-logically irregular neighborhoods, i.e. for vertices with valence other then six, the extensions proposed by Zorin [56] have to be incorporated, hence the term "modified" in the name of the scheme.

Considering the extensions, the weights for the new vertices Vf are computed as a function of the valence of the vertices v*. To solve Eq. (14.22) on triangular meshes, the Laplace operator has to be replaced by a discrete operator L. One example of such an operator is the so-called umbrella-operator U introduced by Kobbelt [57]:

where n is the valence of the vertex v*. This operator clearly does not consider the geometric constellation of the neighborhood of v*, but results in a simple computation of the Laplace operator with reasonable accuracy for regular meshes.

The approximation of differential operators on arbitrary, discrete 2-D manifolds poses a complex problem. In contrast to the 1-D situation, where the adjacent vertices are always the left and right neighbors of the current vertex, there exists no such fixed relationship for 2-D manifolds. Many different methods for the computation of discrete operators have been proposed in the past few years [58,59].

At this point it has to be noted that practical implementations of 3-D snakes pose additional challenges that have to be considered. In general, the 3-D

situation requires a stronger shape regularization in order to preserve a valid mesh structure. The projection of the image forces into normal directions, as suggested in Eq. (14.24) can only be applied on finer levels, as the normals of the coarse mesh may point in rather odd directions. In the context of tamed models, fixing all even vertices of the coarser mesh V[ can have an adverse effect on the optimization leading to strong parametric distortions. Hug [54], therefore, recommends to "freeze" as few vertices as possible, depending on the quality of the underlying data.

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