Invariant representative cocycles of cohomology generators using irregular graph pyramids

Structural pattern recognition describes and classifies data based on the relationships of features and parts. Topological invariants, like the Euler number, characterize the structure of objects of any dimension. Cohomology can provide more refined algebraic invariants to a topological space than does homology. It assigns ‘quantities’ to the chains used in homology to characterize holes of any dimension. Graph pyramids can be used to describe subdivisions of the same object at multiple levels of detail. This paper presents cohomology in the context of structural pattern recognition and introduces an algorithm to efficiently compute representative cocycles (the basic elements of cohomology) in 2D using a graph pyramid. An extension to obtain scanning and rotation invariant cocycles is given.

► We compute representative cocycles invariant to scanning and rotation of the object. ► A graph pyramid provides a reduced object-representation (ROR), preserving topology. ► Cocycles in the ROR are down-projected to the original object, in the pyramid. ► Rotation invariance is achieved by a rotation invariant construction of the pyramid.