New parallel thinning algorithms for 2D grayscale images

Several approaches have been proposed for the study of topological properties of binary digital images: the digital topology, the connected ordered topological space approach, and the cell complex approach. The topology which is used in the last two approaches is a discrete topology. In fact, the datum of a discrete topology is equivalent to the datum of a partially ordered set (order). One of the authors has proposed a study of homotopy in orders, and has introduced the notion of (alpha) -simple point. An (alpha) -simple point is an 'inessential point' for the topology in orders. A fundamental characteristic concerning (alpha) -simple points is that they can be deleted in parallel from an object, while preserving its topological properties. This led us to propose, in a recent work, two parallel thinning algorithms based on the parallel deletion of (alpha) -simple points, for 2D and 3D binary images. Very few parallel thinning algorithms for 2D grayscale images have already been proposed. The most recent ones have been developed by extending binary notions with the cross-section topology. In the same way, we extend our previous works in orders to the case of 2D grayscale images, and we propose two parallel thinning algorithms for such images.