分水岭、瀑布、边或节点加权图
Watersheds, waterfalls, on edge or node weighted graphs
摘要：
我们提出了一个代数方法的分水岭适应边或节点加权图。从洪泛附加开始，我们引入了洪泛图，其中节点和边的权重可以从另一个推导出来。每个节点加权图或边加权图都可以在洪泛图中变换，这表明使用一个或另一个没有优势，两者都是等价的。然后，我们引入修剪算子提取子图的陡度增加。对于增加的陡度，从不上升的路径的数量变得越来越少。这减少了集水区，在那里集水盆地重叠。最后一个修剪算子称为边缘关联到区域最小值之外的每个节点，并且只有一个边缘。这个新图形的集水盆地不重叠，并形成一个分水岭分区。同样，随着陡度的增加，图中包含的不同分水岭分区的数量变得越来越小。最终，对于自然图像，无限陡度导致唯一解，因为两个绝对相同的无限陡度的非上升路径不太可能连接具有两个不同极小值的节点。它碰巧是给定陡度的非上升路径是给定深度的字典距离函数的测地线。这允许提取的分水岭分区作为骨架的影响区的最小值，这样的字典距离。瀑布层次结构是通过一系列操作获得的。第一种方法是构造一个跨越初始分水岭分区的最小生成森林。将树收缩成一个节点产生一个简化图，该简化图可以被提交给相同的处理。重复该过程，直到只剩下一个区域。所有生成的森林的边的并集构成初始图的最小生成树。
英文摘要：
We present an algebraic approach to the watershed adapted to edge or node weighted graphs. Starting with the flooding adjunction, we introduce the flooding graphs, for which node and edge weights may be deduced one from the other. Each node weighted or edge weighted graph may be transformed in a flooding graph, showing that there is no superiority in using one or the other, both being equivalent. We then introduce pruning operators extract subgraphs of increasing steepness. For an increasing steepness, the number of never ascending paths becomes smaller and smaller. This reduces the watershed zone, where catchment basins overlap. A last pruning operator called scissor associates to each node outside the regional minima one and only one edge. The catchment basins of this new graph do not overlap and form a watershed partition. Again, with an increasing steepness, the number of distinct watershed partitions contained in a graph becomes smaller and smaller. Ultimately, for natural image, an infinite steepness leads to a unique solution, as it is not likely that two absolutely identical non ascending paths of infinite steepness connect a node with two distinct minima. It happens that non ascending paths of a given steepness are the geodesics of lexicographic distance functions of a given depth. This permits to extract the watershed partitions as skeletons by zone of influence of the minima for such lexicographic distances. The waterfall hierarchy is obtained by a sequence of operations. The first constructs the minimum spanning forest which spans an initial watershed partition. The contraction of the trees into one node produces a reduced graph which may be submitted to the same treatment. The process is iterated until only one region remains. The union of the edges of all forests produced constitutes a minimum spanning tree of the initial graph.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
