Industrial Geometry

The overall topic of this subproject has been variational techniques for shape recovery that take into account topological constraints and shape priors. The considered variational techniques have been based on explicit or implicit (that are level sets) shape representations. Moreover, novel variational filtering and enhancing technique have been developed that are also defined on manifolds.

Medial axes representations (m-reps) have been investigated in combination with variational regularization methods. The particular advantage of m-reps is that they allow sparse representations of shapes, that is, representations by a few parameters only. On the other hand, optimization problems with respect to medial axes representations are significantly more difficult since the domain of the optimization problem is a manifold. In this context, our suggested approach is a variational regularization method with a similarity measure based on the Mahalanobis distance. Moreover, we have done research on taking into account topological shape constraints (in particular homology constraints), which have been used for segmentation by active contour models.

Combined filtering and enhancing techniques can be modelled as variational techniques with Bregman distance regularization term. The asymptotical limits of these variational methods are inverse scale space equations. Based on recent results of Ambrosio, it has been possible to establish an analysis of such flow equations (existence, stability, convergence).

We have found that density estimation in statistics can be implemented with a two level approach combining variational methods and methods from computational geometry. The computational geometry methods are used to determine first an approximation of the density and subsequently a variational filtering method is used to clear the data. It turns out that this approach is equivalent to the taut-string algorithm for filtering one-dimensional data.

• M-Reps Shape Priors in Variational Methods
• Density Estimation in Statistics
• Homology Methods
• Non-Convex Variational Level Set Segmentation Models
• Bregman Distance Regularization and Inverse Scale Space Equations