and thus is very useful in image characterization to match shapes, recognize objects, image database retrieval and other image processing, and computer vision applications. These types of topological properties remain invariant under any arbitrary rubber-sheet transformation, i.e., stretching, shrinking, rotation etc. The Euler number (or genus) is defined as the difference between the number of “connected components” and the number of “holes” in an image. The concept of Euler number is an important topological property inspired from ideas useful to the field of image processing. Another quantity which survives finite size scaling is the Euler number which has therefore many practical applications. The identification of the percolation transition as a critical phase transition has been a significant finding with deep theoretical as well as practical implications. Yet, there are certain lattice properties which have not been as well-studied as the others. It is also explained mathematically why clusters of size 1 are always the most numerous.ĭifferent aspects of the properties of two-dimensional square lattices has been an ongoing challenge for over half a century. We also discuss the earlier proposed “Island-Mainland” transition and show mathematically that the proposed transition is not in fact a critical phase transition and does not survive finite size scaling. Here N B is the number of black clusters and N W is the number of white clusters, at a certain probability p. 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. In this scenario we investigate (i) the variation of the Euler number χ( p) vs. If the black pair is disjoint, the white pair is considered connected. 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. In this case assigning connected status to both pairs simultaneously, makes the system quasi-three dimensional, with intertwined black and white clusters. A pair of black corner-sharing sites, i.e., second nearest neighbors may form a “cross-connection” with a pair of white corner-sharing sites. Edge-sharing sites, i.e., nearest neighbors of similar type are always considered to belong to the same cluster. We consider connections up to the second nearest neighbors, according to the following rule. In our study we report on some of the novel properties of a square lattice filled with white sites, randomly occupied by black sites (with probability p).
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |