Mathematical morphology

Mathematical morphology (MM) is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions. MM is most commonly applied to digital images, but it can be employed as well on graphs, surface meshes, solids, and many other spatial structures. Mathematical morphology (MM) is a theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions. MM is most commonly applied to digital images, but it can be employed as well on graphs, surface meshes, solids, and many other spatial structures. Topological and geometrical continuous-space concepts such as size, shape, convexity, connectivity, and geodesic distance, were introduced by MM on both continuous and discrete spaces. MM is also the foundation of morphological image processing, which consists of a set of operators that transform images according to the above characterizations. The basic morphological operators are erosion, dilation, opening and closing. MM was originally developed for binary images, and was later extended to grayscale functions and images. The subsequent generalization to complete lattices is widely accepted today as MM's theoretical foundation. Ever since the 1960s, all sorts of schemes for nonlinear processing of images have been discussed and used in particular communities. An example originally popular in the earth and environmental sciences is mathematical morphology, based on 'dilation' of data consisting of 0's and 1's with a 'structuring element' σ according to Sign] (as well as the dual operation of 'erosion'). Mathematical Morphology was developed 1964 by the collaborative work of Georges Matheron and Jean Serra, at the École des Mines de Paris, France. Matheron supervised the PhD thesis of Serra, devoted to the quantification of mineral characteristics from thin cross sections, and this work resulted in a novel practical approach, as well as theoretical advancements in integral geometry and topology. In 1968, the Centre de Morphologie Mathématique was founded by the École des Mines de Paris in Fontainebleau, France, led by Matheron and Serra. During the rest of the 1960s and most of the 1970s, MM dealt essentially with binary images, treated as sets, and generated a large number of binary operators and techniques: Hit-or-miss transform, dilation, erosion, opening, closing, granulometry, thinning, skeletonization, ultimate erosion, conditional bisector, and others. A random approach was also developed, based on novel image models. Most of the work in that period was developed in Fontainebleau. From the mid-1970s to mid-1980s, MM was generalized to grayscale functions and images as well. Besides extending the main concepts (such as dilation, erosion, etc.) to functions, this generalization yielded new operators, such as morphological gradients, top-hat transform and the Watershed (MM's main segmentation approach).