Convergence Of The Mean Shift Using The Infinity Norm In Image Segmentation

In this work a comparison between two algorithms for image segmentation via the mean shift is carried out. These algorithms apply recursively the mean shift filtering by using the euclidean and infinity norms in order to define pixel neighborhoods. In the conventional mean shift algorithm for image segmentation euclidean norm is used. Due to matrix representation of images and the rectangular windows used in the implementation of the algorithm, with the aim of evaluating pixels membership to neighborhoods, the use of euclidean norm always implies evaluate membership of pixels wich certainly cannot be inside each defined neighborhood. However, the use of the infinity norm ensures that each evaluated pixel could be inside the neighborhood. This fact avoids unnecessary calculations. In the work the convergence of the algorithm, by using the infinity norm, is proven. Additionaly, the runtimes of both algorithms were compared using different spatial and range bandwidth sizes. Through an extensive experimentation using standard images was evidenced that the use of the infinity norm, instead of the euclidean norm, decreases the runtime of the mean shift when the values of spatial and range bandwidths were increased.