Application of the Ring Theory in the Segmentation of Digital Images

Ring theory is one of the branches of the abstract algebra that has been broadly used in images. However, ring theory has not been very related with image segmentation. In this paper, we propose a new index of similarity among images using Zn rings and the entropy function. This new index was applied as a new stopping criterion to the Mean Shift Iterative Algorithm with the goal to reach a better segmentation. An analysis on the performance of the algorithm with this new stopping criterion is carried out. The obtained results proved that the new index is a suitable tool to compare images.

Edge Detection based on Kernel Density Estimation

Edges of an image are considered a crucial type of information. These can be extracted by applying edge detectors with different methodology. Edge detection is a vital step in computer vision tasks, because it is an essential issue for pattern recognition and visual interpretation. In this paper, we propose a new method for edge detection in images, based on the estimation by kernel of the probability density function. In our algorithm, pixels in the image with minimum value of density function are labeled as edges. The boundary between two homogeneous regions is defined in two domains: the spatial/lattice domain and the range/color domain. Extensive experimental evaluations proved that our edge detection method is significantly a competitive algorithm.

New similarity index based on entropy and group theory

In this work, we propose a new similarity index for images considering the entropy function and group theory. This index considers an algebraic group of images, it is defined by an inner law that provides a novel approach for the subtraction of images. Through an equivalence relationship in the field of images, we prove the existence of the quotient group, on which the new similarity index is defined. We also present the main properties of the new index, and the immediate application thereof as a stopping criterion of the "Mean Shift Iterative Algorithm".

A new stopping criterion for the mean shift iterative algorithm

The mean shift iterative algorithm was proposed in 2006, for using the entropy as a stopping criterion. From then on, a theoretical base has been developed and a group of applications has been carried out using this algorithm. This paper proposes a new stopping criterion for the mean shift iterative algorithm, where stopping threshold via entropy is used now, but in another way. Many segmentation experiments were carried out by utilizing standard images and it was verified that a better segmentation was reached, and that the algorithm had better stability. An analysis on the convergence, through a theorem, with the new stopping criterion was carried out. The goal of this paper is to compare the new stopping criterion with the old criterion. For this reason, the obtained results were not compared with other segmentation approaches, since with the old stopping criterion were previously carried out.