Exploring generalized shape analysis by topological representations

One of the most common properties of various data in pattern recognition is the shape, and the shape matters. However, the shape can appear with uncertain appearances, e.g., the shapes of a person in different poses. We realize that the most fundamental feature of any shape is the number of connected components, the number of holes and its higher dimensional counterparts. These are what we call topological invariants. This is the place where topology comes into play for pattern recognition. Persistent homology, one of the most powerful tools in algebraic topology, is proposed to compute these topological invariants at different resolutions. The proposed method, by firstly transferring the given data into a topological graph representation, i.e., the simplicial complex, can assemble discrete points into a global structure. Then by integrating with multiple filtrations and metric learning, both the global structure and different local parts can be taken into account at the same time. We test the proposed method in 2D shape classification, 2.5D gait identification and 3D facial expression recognition. Experimental results demonstrate the effectiveness of this generalized shape analysis method and show its potentials in different applications. Moreover, we provide a new insight for the generalized shape analysis.