We present an information-theoretic approach for structural similarity for assessing gray scale image quality. The structural similarity measure SSIM, proposed in 2004, has been successflly used and verfied. SSIM is based on statistical similarity between the two images. However, SSIM can produce confusing results in some cases where it may give a non-trivial amount of similarity for two different images. Also, SSIM cannot perform well (in detecting similarity or dissimilarity) at low peak signal to noise ratio (PSNR). In this study, we present a novel image similarity measure, HSSIM, by using information -theoretic technique based on joint histogram. The proposed method has been tested under Gaussian noise. Simulation results show that the proposed measure HSSIM outperforms statistical similarity SSIM by ability to detect similarity under very low PSNR. The average difference is about 20dB.

In this work, we associate a new topology to undirected graph G = (V, E) which may contain one isolated vertex or more and we named it Independent (non-adjacent vertices) Topology. A new sub-basis family to generate the Independent Topology is introduced on the set of n vertices V and for every vertex v of V the number of adjacent vertices is not greater than n − 2 (In simple graph we can say : for every vertex v of V, Δ(G) = n − 2, where Δ(G) is the maximum degree of vertices in a graph G). Then we give a fundamental step toward investigation of some properties of undirected graphs by their corresponding Independent Topology which we introduce in this work. Furthermore, an application to our new model (Independent Topology) are presented, that to carry out a framework in practical life like biomathematics (applied examples in biomathematics).

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