4D Space-Time Mereotopogeometry-Part Connectivity Calculus for Visual Object Representation

Region Connectivity Calculus (RCC) can be used to define the formal grammar describing the relationship between image regions. While RCC provides a possible framework for representation of object part constellations leading to object recognition, little has been done in the direction of RCC for 3D images. Almost all prior RCC representations, such as RCC5/ RCC8/ RCC23/RCC62 are oriented towards 2D projections. While the recently introduced RCC-3D does address this limitation to a certain extent, it heavily depends on other 2D RCC frameworks and as such is limited -- it provides no representation for orthogonally aligned or staggered object parts. It also does not provide convenient representations for shape related pose information (such as "horizontally aligned along principal axis" or "vertically aligned" etc.). While it is possible to use oriented matroids for projected 3D representations using cocircuits and chirotopes, these are again sub-optimal given that there is considerable information loss (through dimensionality reduction) and multiple object models map to the same structures. In this paper, we introduce a new hierarchical graph based 3D Region/ Surface/ Object/ Part Connectivity Calculus (OCC/PCC), given the domain of affordance based equivalence recognition. We call our PCC -- 4D Space Time Mereo-topo-geometry (4D-STMTG) and the equivalent OCC as 4D Space Time Joint Topo-geometry (4D-STJTG). The modeling of simple objects using the calculus is demonstrated in practical scenarios. Comparisons of the proposed calculus with respect to the state-of-art RCC-3D is also presented, demonstrating the flexibility, suitability and superiority of 4D-STMTG.