Watersheds, waterfalls, on edge or node weighted graphs

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.

💡 Research Summary

The paper presents a unified algebraic framework for watershed segmentation that works equally well on edge‑weighted and node‑weighted graphs. The authors introduce the concept of a “flooding graph”, a special graph in which node weights and edge weights are mutually derivable through a flooding adjunction. By showing that any node‑weighted graph can be transformed into a flooding graph and likewise any edge‑weighted graph, they prove that there is no intrinsic advantage to choosing one representation over the other; both are mathematically equivalent.

From this foundation they define a family of pruning operators that progressively increase the “steepness” of the graph. Steepness controls the set of never‑ascending paths (NAPs) that can connect a node to regional minima. When steepness is low, many NAPs exist, leading to overlapping catchment basins and a wide watershed zone. As steepness grows, the number of admissible NAPs shrinks, reducing overlap. The most aggressive operator, called the “scissor”, forces each non‑minimum node to retain exactly one incident edge, thereby turning the graph into a forest where every node belongs to a unique basin. The resulting partition is a true watershed: basins are disjoint and cover the whole graph.

The authors further relate NAPs of a given steepness to geodesics of lexicographic distance functions of a prescribed depth. This insight allows the extraction of watershed partitions as skeletons‑by‑zone‑of‑influence (SKIZ) of the minima under those lexicographic distances. Using this connection, they construct a Minimum Spanning Forest (MSF) that spans an initial watershed partition. By contracting each tree of the MSF into a single super‑node, a reduced graph is obtained, to which the same pruning and scissor operations can be reapplied. Repeating this process yields a hierarchical “waterfall” structure: each level provides a coarser segmentation, and the union of all MSFs generated across levels forms a Minimum Spanning Tree (MST) of the original graph.

A key theoretical result is that for natural images, an infinite steepness (i.e., requiring strictly decreasing paths) almost surely yields a unique watershed solution, because the probability of two distinct NAPs of infinite steepness connecting a node to two different minima is negligible. Consequently, the framework guarantees a deterministic, non‑ambiguous segmentation when the steepness parameter is taken to its limit.

Experimentally, the authors demonstrate that their method eliminates the common over‑segmentation and basin‑overlap problems of traditional watershed algorithms, while preserving fine details when the steepness is kept moderate. The approach is computationally comparable to classic graph‑based watershed implementations, yet it offers a clear mathematical interpretation, a systematic way to control segmentation granularity, and a natural link to classical graph structures such as MSTs and MSFs.

In summary, the paper delivers a comprehensive algebraic treatment of watershed segmentation, unifying node‑ and edge‑weighted formulations, introducing steepness‑controlled pruning, connecting to lexicographic distances, and building a hierarchical waterfall scheme that culminates in a globally optimal minimum spanning tree. This work advances both the theoretical understanding and practical applicability of watershed methods in image analysis and related fields.