Compositional Hierarchical Representation of Shape Manifolds for Classification of Non-manifold Shapes

We address the problem of statistical learning of shape models which are invariant to translation, rotation and scale in compositional hierarchies when data spaces of measurements and shape spaces are not topological manifolds. In practice, this problem is observed while modeling shapes having multiple disconnected components, e.g. partially occluded shapes in cluttered scenes. We resolve the aforementioned problem by first reformulating the relationship between data and shape spaces considering the interaction between Receptive Fields (RFs) and Shape Manifolds (SMs) in a compositional hierarchical shape vocabulary. Then, we suggest a method to model the topological structure of the SMs for statistical learning of the geometric transformations of the shapes that are defined by group actions on the SMs. For this purpose, we design a disjoint union topology using an indexing mechanism for the formation of shape models on SMs in the vocabulary, recursively. We represent the topological relationship between shape components using graphs, which are aggregated to construct a hierarchical graph structure for the shape vocabulary. To this end, we introduce a framework to implement the indexing mechanisms for the employment of the vocabulary for structural shape classification. The proposed approach is used to construct invariant shape representations. Results on benchmark shape classification outperform state-of-the-art methods.