Tamed Snakes

Despite their ability to approximate objects with little user input, the approaches reported so far lack an intuitive manipulation semantic. The primitive manipulation metaphors presented, spring and volcano, can only be applied directly on the Snake's curve v. It would be desirable though to have a more powerful interaction method at hand, for example, to modify parts of the shape on a coarse scale, while keeping small details on a finer scale intact. Tamed Snakes combine the hierarchical modeling and Snake-like edge delineation. They adhere to the concept of hierarchical shape representations with several scales of resolution to provide the necessary interactive modeling capabilities while being suitable for numerical simulations.

Hierarchical modeling consists of (a) an iterative refinement of the geometry, which defines a hierarchy of representations and (b) a local detail encoding, which represents the details on a finer level with respect to the next coarser one. Subdivision curves are best suited for such hierarchical modeling, as their representation implicitly comprises a hierarchy of refined shapes. These curves are constructed using univariate subdivision schemes defined as the iterative application of an operator that maps a given polygon Pl = [v(l)] to a refined polygon Pl+1 = [v(l+1)], where l denotes the level of the hierarchy. Such an operator is given by two rules for computing the new so-called even vertices x = {v2i+r)} and the new odd vertices Vf+ x = {v2li+11')}.

The Tamed Snakes as introduced by Hug [52] employ the DLG-subdivision scheme [53], given by

- V v2i+1 - I 2 + W J Vvi + Vi+l) - nvi-1 + vi+2j. (14-23)

As the even vertices remain unchanged the subdivision operator has interpolating behavior. The free tension parameter rn has to be chosen inside the interval 0 < rn < 8 to obtain a limit curve that has a continuous tangent vector.

Local details, i.e. transformations of the vertices v(l} from their given position, are encoded by establishing a local coordinate system f® in each vertex v(l) and by representing the details with respect to this frame.

The subdivision scheme suggests to start the segmentation process with a reasonably coarse model and to iteratively adjust and refine the control vertices of the resulting curve. Since the subdivision scheme is interpolating, only the odd vertices of the current level must be adjusted to align with a correct boundary position before proceeding to the next finer level. In doing so, the prediction of the refinement operator improves continuously with respect to the vertex position on the next finer level and converges to the correct boundary position.

The traditional Snake energy has to be modified to combine the described coarse-to-fine approach with the Snake-like edge tracking. Tamed Snakes replace the elastic rod model term (Eq. 14.9) by a spring energy similar to the external energy term introduced earlier for mouse interaction with Snakes. The springs are attached to each odd vertex Vi e Vf, so that the vertices Vi snap to the correct boundary position within the vicinity of their starting positions vi(0) = Vi |i=0. Assuming a good initialization, the imposed restriction on the search space to the local neighborhood is reasonable. The spring constant k® can be increased with each subdivision step to further restrict the search area, as the error of the subdivision operator's prediction tends to decrease.

Besides the spring energy, Tamed Snakes incorporate a kinetic energy EK, an image energy Eimage and a Rayleigh dissipation functional D(vt). The resulting Euler-Lagrange equation of motion for Tamed Snakes describing the motion of all odd mass points Vi at time t is f d 2Vj(t) dVi(t) (1)

Vvi e Vf : —dtT^ + Yi^f1 + kfWt) - Vj(0)) = -Vj_P(I^(t) (14.24)

In order to prevent the control vertices from drifting along the boundary, the gradient of the potential is projected onto the normal direction of the curve, denoted by V± in the previous equation.

The segmentation process using Tamed Snakes is depicted in Fig. 14.11. The initialization has a strong impact on the additional manual editing required in finer levels. For the presented case, user interaction was required on a few points in the first and second subdivision level. Because of the limited number of vertices in these levels and the ability to better predict new positions at finer levels, Tamed Snakes proof themselves to alleviate the interactive segmentation

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Figure 14.11: Segmentation using Tamed Snakes. The adaptive character of the subdivision scheme is clearly visible. User interaction was required in the second and third image.

task. In case of clear boundaries though, the segmentation is not as fast and elegant as with traditional Snakes.

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