On the representation of shapes using implicit functions

In this chapter, we explore shape representation, registration, and modeling through implicit functions. To this end, we propose novel techniques for global and local registration of shapes through the alignment of the corresponding distance transforms by defining objective functions that minimize metrics between the implicit representations of shapes. Registration methods in the space of implicit functions like the sum of squares differences (SSD), which can account for primitive transformations (similarity), and more advanced methods like mutual information, which are able to handle more generic parametric transformations, are considered. To address local correspondences we also propose an objective function on the space of implicit representations where the displacement field is represented with a free form deformation that can guarantee one-to-one mapping. In order to address outliers as well as introduce confidence in the registration process, we extend our registration paradigm to estimate uncertainties through the formulation of local registration as a statistical inference problem in the space of implicit functions. Validation of the method through various applications is proposed: (i) parametric shape modeling and segmentation through active shapes for medical image analysis, (ii) variable bandwidth non-parametric shape modeling for recognition, and (iii) object extraction through a level set method. Promising results demonstrate the potentials of implicit shape representations.