Characterizing Obstacle- Avoiding Paths using Cohomology Theory

In this paper, we investigate the problem of analyzing theshape of obstacle-avoiding paths in a space. Given a d-dimensional spacewith holes, representing obstacles, we ask if certain paths are equivalent,informally if one path can be continuously deformed into another,within this space. Algebraic topology is used to distinguish between topologicallydifferent paths. A compact yet complete signature of a path isconstructed, based on cohomology theory. Possible applications includeassisted living, residential, security and environmental monitoring. Numericalresults will be presented in the final version of this paper.