Ridge–Valley graphs: Combinatorial ridge detection using Jacobi sets

Abstract

Ridges are one of the key feature of interest in areas such as computer vision and image processing. Even though a significant amount of research has been directed to defining and extracting ridges some fundamental challenges remain. For example, the most popular ridge definition (height ridge) is not invariant under monotonic transformations and its global structure is typically ignored during numerical computations. Furthermore, many existing algorithm are based on numerical heuristics and are rarely guaranteed to produce consistent results. In this paper a small change to the height ridge definition gives a definition that is consistent with the desired invariants. Nevertheless, we show that this definition results in similar structures compared to the most common traditional approach and that both formulations are equivalent for quadratic functions. Furthermore, this definition can be cast in the form of a degenerate Jacobi set, which allows insights into the global structure of ridges. In particular, we introduce the Ridge-Valley graph as the complete description of all ridges in an image. Analyzing this graph reveals similar invariants to those found for traditional ridge formulations, such as the fact that ridges do not merge. Finally, using the connection to Jacobi sets we describe a new combinatorial algorithm to extract the Ridge-Valley graph from sampled images that is guaranteed to produce a valid structure.

Local bilinear computation of Jacobi sets

“…As part of topological data analysis tools, the Jacobi set can be used to compare multiple scalar fields [10], track critical points [4,8], define silhouettes of objects [12], and extract ridges in image analysis [26]. It has also been employed in various scientific applications [1,2,8,20,22].…”

Interactive Visualization for Singular Fibers of Functions f : R3 → R2

“…The Jacobi set has since been simplified [4,33] and applied to ridgevalley extraction [25]. However, in these cases, the Jacobi set is illustrated in the R R R 2 → R R R 2 case, where the Jacobi set divides the domain into regions.…”

Do Finite-Size Lyapunov Exponents detect coherent structures?

“…The advantage of Definition 2 is that it guarantees robustness for the ridge based on powerful persistence results for normally hyperbolic invariant manifolds (see Proposition 2). Other available ridge definitions do not provide a well-defined set of conditions for ridge-persistence under changes in the underlying scalar field; only partial results exist for specific cases [6,25].…”