Alternative formulations to compute the binary shape Euler number

The authors propose two equations based on the pixel geometry and connectivity properties, which can be used to compute, efficiently, the Euler number of a binary digital image with either thick or thin boundaries. Although computing this feature, the authors' technique extracts the underlying topological information provided by the shape pixels of the given image. The correctness of computing the Euler number using the new equations is also established theoretically. The performance of the proposed method is compared against other available alternatives. Experimental results on a large image database demonstrate that the authors technique for computing the Euler number outperforms the earlier approaches significantly in terms of the number of basic arithmetic operations needed per pixel. Both equations are specialised only for 4-connectivity cases.