Euler Number and Percolation Threshold on a Square Lattice with Diagonal Connection Probability and Revisiting the Island-Mainland Transition

We report some novel properties of a square lattice filled with white sites, randomly occupied by black sites (with probability $p$). We consider connections up to second nearest neighbours, according to the following rule. Edge-sharing sites, i.e. nearest neighbours of similar type are always considered to belong to the same cluster. A pair of black corner-sharing sites, i.e. second nearest neighbours may form a 'cross-connection' with a pair of white corner-sharing sites. In this case assigning connected status to both pairs simultaneously, makes the system quasi-three dimensional, with intertwined black and white clusters. The two-dimensional character of the system is preserved by considering the black diagonal pair to be connected with a probability $q$, in which case the crossing white pair of sites are deemed disjoint. If the black pair is disjoint, the white pair is considered connected. In this scenario we investigate (i) the variation of the Euler number $\chi(p) \ [=N_B(p)-N_W(p)]$ versus $p$ graph for varying $q$, (ii) variation of the site percolation threshold with $q$ and (iii) size distribution of the black clusters for varying $p$, when $q=0.5$. Here $N_B$ is the number of black clusters and $N_W$ is the number of white clusters, at a certain probability $p$. We also discuss the earlier proposed 'Island-Mainland' transition (Khatun, T., Dutta, T. & Tarafdar, S. Eur. Phys. J. B (2017) 90: 213) and show mathematically that the proposed transition is not, in fact, a critical phase transition and does not survive finite size scaling. It is also explained mathematically why clusters of size 1 are always the most numerous

IOS Press A Co-processor for Computing the Euler Number of a Binary Image using Divide-and-Conquer Strategy

Abstract. Euler number is a fundamental topological feature of an image. The efficiency of computation of topological features of an image is critical for many digital imaging applications such as image matching, database retrieval, and computer vision that require real time response. In this paper, a novel algorithm for computing the Euler number of a binary image based on divide-andconquer paradigm, is proposed, which outperforms significantly the conventional techniques used in image processing tools. The algorithm can be easily parallelized for computing the Euler number of an ¤ ¥ ¤ image in ¦§¤ ¨ time, with ¦§¤ ¨ processors. Using a simple architecture, the proposed method can be implemented as a special purpose VLSI chip to be used as a co-processor. Keywords: Binary image, digital imaging, Euler number, VLSI. 1