In this paper, an efficient and polynomial edge detection algorithm based on a hierarchical graph-partition approach is presented. After transforming a digital image into a graph network, the proposed algorithm proceeds by iteratively dividing the image into regions, and then transforming this hierarchical region map into a sequence of boundary maps. This allows the proposed algorithm to operate naturally with colour or hyperspectral images, as well as to detect edges at different levels of detail in a simultaneous and consistent manner. Such a sequence of edge maps can be seen as jointly approximating the different levels of detail that humans may use when recognizing objects in an image. This idea is taken to base the evaluation methodology of the proposed algorithm, that extends the usual boundary-based evaluation methodology based on the matching of the automatic maps with a set of human ground truth, reference maps. The computational experiences carried out to benchmark the performance of the proposed algorithm over the BSDS500 dataset suggest that the proposed method attains a statistically significant better performance than some well-known detectors as Canny or Sobel.

In this article, we derive a closed-form expression for computing the probabilities of p-dimensional rectangles by means of a multivariate skew-normal distribution. We use a stochastic representation of the multivariate skew-normal/independent distributions to derive expressions that relate their probability density functions to the expected values of positive random variables. We also obtain an analogous expression for probabilities of p-dimensional rectangles for these distributions. Based on this, we propose a procedure based on Monte Carlo integration to evaluate the probabilities of p-dimensional rectangles through multivariate skew-normal/independent distributions. We use these findings to evaluate the probability density functions of a truncated version of this class of distributions, for which we also suggest a scheme to generate random vectors by using a stochastic representation involving a truncated multivariate skew-normal random vector. Finally, we derive distributional properties involving affine transformations and marginalization. We illustrate graphically several of our methodologies and results derived in this article.