Eikonal Equation Adaptation on Weighted Graphs: Fast Geometric Diffusion Process for Local and Non-local Image and Data Processing

Abstract

In this paper we propose an adaptation of the Eikonal equation on weighted graphs, using the framework of Partial difference Equations, and with the motivation of extending this equation's applications to any discrete data that can be represented by graphs. This adaptation leads to a finite difference equation defined on weighted graphs and a new efficient algorithm for multiple labels simultaneous propagation on graphs, based on such equation. We will show that such approach enables the resolution of many applications in image and high dimensional data processing using a unique framework. Keywords Eikonal equation • weighted graph • non-local image processing • active contour • PdE • fast marching • high dimensional data This work was supported under a doctoral grant of the Conseil Régional de Basse-Normandie and the Coeur et Cancer association.

Eikonal-based region growing for efficient clustering

“…In comparison to traditional gradient-based approaches [3] where F(p) = ‖∇I ‖, the proposed formulation favors the grouping of similar pixels, even for pixels that are far from the initial seeds (Fig. 2).…”

“…Perform a full pass of ERGC with the seeds placed on a grid, 2. add a new seed to the location of the maximum geodesic distance found at the previous step, 3. let R i be the superpixel in which there is the new seed (red superpixel of Fig.…”

“…Recent works [3] adapt the eikonal equation to graphs in order to perform over-clustering from an initial set of annotated vertices V 0 . Let f : V → ℝ be a real-valued function that assigns a real value f(u) to each vertex u ∈ V. The reformulation of the Eikonal equation in the graph domain leads to the equation:…”

Morphological PDEs on Graphs for Image Processing on Surfaces and Point Clouds

“…Based on the previous definition of discrete dilation and erosion on graphs [9,37], the front propagation can be expressed as a morphological process with the following sum of dilation and erosion:…”

“…Then, the algorithm consists of sorting increasingly the values a i and computing temporary solution x m until the condition x m ≤ a m+1 is satisfied. To compute the solution of Eikonal equation at each vertex, we used Fast Marching's updating scheme [37].…”