Introduction to the Issue on Filtering and Segmentation With Mathematical Morphology

This issue presents novel contributions and introduces the state-of-art in filtering and segmentation methods using mathematical morphology. Historically, mathematical morphology developed a theoretical framework for non-linear image analysis, which from its inception led to important theoretical and practical results. Due to its algebraic foundation and geometrical intuition, the theory of mathematical morphology is very flexible, capable of handling many diverse image types, including (but not limited to) classical multi-dimensional signals as well as general graphs and surfaces. We received 24 submissions from all over the world, from which we selected 13 articles for publication in this issue spanning a large segment of the research in morphological filtering and segmentation. The first article is an invited paper authored by Jean Serra, who is one of the two founders of Mathematical Morphology and a recent special conference in his honor has been the inspiration for this issue. In a “Tutorial on Connective Morphology,” Jean Serra explores morphological operators from the perspective of connections: set connections, connective segmentations, connected operators, hierarchies of partitions and optimal cuts are the five key concepts under scrutiny. The article “Random Projection Depth forMultivariateMathematical Morphology,” by S. Velasco-Forero and J. Angulo, considers a projection depth map for vector ordering, whereby the authors study a statistical depth function that approximates the Mahalanobis distance from a vector cloud, and use the function in applying mathematical morphological tools to multivalued (vector) images. The article “Non-Local Morphological PDEs and p-Laplacian Equation on Graphs with Applications in Image Processing andMachine Learning,” byA. Elmoataz et al. , investigates connections between non-local morphological PDEs, p-Laplacian equation and non-local average filtering on graphs. It introduces a new class of non-local p-Laplacian operators, involving discrete morphological gradient on graphs, as solutions to several inverse problems in image processing. The article “Active Contours on Graphs: Multiscale Morphology and Graphcuts,” by K. Drakopoulos and P. Maragos, proposes two methods for computing geodesic active contours on arbitrary graphs, based on several ideas stemming from differential morphology: (i) approximations to the calculation of the gradient and the divergence of vector functions defined on graphs and uses these approximations to apply the technique of geodesic active contours for object detection on graphs; (ii) appropriate weights are calculated for each edge for which the Riemannian length of a contour can be approximated by the weighted sum of intersections of the contour with the edges of the graph and applied in a graphcut-based solution to the geodesic active contour problem on graphs. The article “First Departure Algorithms and Image Decompositions into Peaks and Wells,” by F. Meyer, presents view-