Deformable pedal curves and surfaces: Hybrid geometric active models for shape recovery

In this paper, we propose significant extensions to the "snake pedal" model , a powerful geometric shape modeling scheme introduced in (Vemuri and Guo, 1998). The extension allows the model to automatically cope with topologic al changes and for the first time, introduces the concept of a compact glob al shape into geometric active models. The ability to characterize global s hape of an object using very few parameters facilitates shape learning and recognition. In this new modeling scheme, object shapes are represented usi ng a parameterized function-called the generator-which accounts for the glo bal shape of an object and the pedal curve (surface) of this global shape w ith respect to a geometric snake to represent any local detail. Traditional ly, pedal curves (surfaces) are defined as the loci of the feet of perpendi culars to the tangents of the generator from a fixed point called the pedal point. Local shape control is achieved by introducing a set of pedal point s-lying on a snake-for each point on the generator. The model dubbed as a " snake pedal" allows for interactive manipulation via forces applied to the snake. In this work, we replace the snake by a geometric snake and derive a ll the necessary mathematics for evolving the geometric snake when the snak e pedal is assumed to evolve as a function of its curvature. Automatic topo logical changes of the model may be achieved by implementing the geometric snake in a level-set framework. We demonstrate the applicability of this mo deling scheme via examples of shape recovery from a variety of 2D and 3D im age data.