Determining the correct number of clusters is essential for efficient clustering and cluster validity indices are widely used for the same. Generally, the effectiveness of a cluster validity index relies on two factors: first, separation, defined by the distance between a pair of cluster centroids or a pair of data points belonging to different clusters and second, compactness, which is determined in terms of the distance between a data point and a centroid or between a pair of data points belonging to the same cluster. However, the existing cluster validity indices for centroid-based clustering are unreliable when the clusters are too close, but corresponding centroids are distant. To mitigate this, a new cluster validity index, Saraswat-and-Mittal index, has been proposed in this article for hyperellipsoid or hyperspherical shape close clusters with distant centroids, generated by fuzzy c-means. The proposed index computes compactness in terms of the distance between data points and corresponding centroids, whereas the distance between data points of disjoint clusters defines separation. These parameters benefit the proposed index in the analysis of close clusters with distinct centroids efficiently. The performance of the proposed index is validated against ten state-of-the-art cluster validity indices on artificial, UCI, and image datasets, clustered by the fuzzy c-means.

Digital watermarking embeds a watermark to minimise the problem of illegal copying and disseminating multimedia contents. However, the existing techniques do not maintain the imperceptibility and robustness simultaneously. To achieve the same, this study proposes an optimised robust watermarking technique using chaotic kbest gravitational search algorithm. The chaotic kbest gravitational search algorithm is used to obtain the optimal values of embedding factors. The efficacy of the proposed technique has been experimented on a standard images and compared with the six recent state‐of‐the‐art techniques in terms of imperceptibility and robustness. The experimental results validate that the proposed technique outperforms the other considered techniques.