A Shape Median Based on Symmetric Area Differences
Benjamin Berkels, Gina Linkmann, Martin Rumpf · 2008
Median averaging is a powerful averaging concept on sets of vector data in finite dimensions. A generalization of the median for shapes in the plane is introduced. The underlying distance measure for shapes is based on the area of the symmetric difference of shapes and takes into account different invariance classes. These classes are generated by classical transformation groups such as translation, rotation, anisotropic scaling, and shear. As in the finite dimensional case, non-uniqueness of the median is observed. The numerical approximation of shape medians is based on a level set approach for the description of the shape contour. The level set function and the parameter sets of the group action on every given shape are incorporated in a joint variational functional, which is minimized based on step size controlled, regularized gradient descent. Various applications show in detail the qualitative behavior of the method. and the minimum is attained for each convex combination of x1 and x2. Generalizing this averaging approach to shapes requires first a suitable definition of distances between shapes and then a transfer of the optimality property into the context of shapes. Our intention is to derive a rigorous definition of shape medians and to highlight some of the resulting properties by a set of characteristic examples. Chen and Parent [6] investigated averages of 2D contours already in 1989. Jiang et al. [9] defined median shapes of polygonal curves based on weights for edit operations, which transfer one curve into the other. A generalization of this approach has been presented by Jiang et al. [8]. Furthermore, the computation of shape distances naturally appears when matching shapes in images. Yezzi and Soatto [16] have investigated shape averages in image structure reconstruction and joint registration. Beg et al. [3] introduced a geodesic 1