Congratulations! Dual Graphs are Now Orientated!

A digital image can be perceived as a 2.5D surface consisting of pixel coordinates and the intensity of pixel as height of the point in the surface. Such surfaces can be e ciently represented by the pair of dual plane graphs: neighborhood (primal) graph and its dual. By de ning ori- entation of edges in the primal graph and use of Local Binary Patters (LBPs), we can categorize the vertices corresponding to the pixel into critical (maximum, minimum, saddle) or slope points. Basic operation of contraction and removal of edges in primal graph result in con guration of graphs with di erent combinations of critical and non-critical points. The faces of graph resemble a slope region after restoration of the contin- uous surface by successive monotone cubic interpolation. In this paper, we de ne orientation of edges in the dual graph such that it remains consistent with the primal graph. Further we deliver the necessary and su cient conditions for merging of two adjacent slope regions