Revisiting the Isoperimetric Graph Partitioning Problem

Abstract

Isoperimetric graph partitioning, which is also known as the Cheeger cut, is NP-hard in its original form. In the literature, multiple modifications to this problem have been proposed to obtain approximation algorithms for clustering applications. In the context of image segmentation, a heuristic continuous relaxation to this problem introduced by Leo Grady and Eric L. Schwartz has yielded good quality results. This algorithm is based on solving a linear system of equations involving the Laplacian of the image graph. Furthermore, the same algorithm applied to a maximum spanning tree (MST) of the image graph was shown to produce similar results at a much lesser computational cost. However, the data reduction step (i.e., considering an MST, a much sparser graph compared to the original graph) leading to a faster yet useful algorithm has not been analyzed. In this paper, we revisit the isoperimetric graph partitioning problem and rectify a few discrepancies in the simplifications of the heuristic continuous relaxation, leading to a better interpretation of what is really done by this algorithm. We then use the power watershed (PW) framework to show that is enough to solve the relaxed isoperimetric graph partitioning problem on the graph induced by the union of MSTs (UMST) with a seed constraint. The UMST has a lesser number of edges compared to the original graph, thus improving the speed of sparse matrix multiplication. Furthermore, given the interest of PW framework in solving the relaxed seeded isoperimetric partitioning problem, we discuss the links between the PW limit of the discrete isoperimetric graph partitioning and watershed cuts. We then illustrate with experiments, a detailed comparison of solutions to the relaxed seeded isoperimetric partitioning problem on the original graph with the ones on the UMST and an MST. Our study opens many research directions, which are discussed in the conclusion section.INDEX TERMS Image segmentation, isoperimetric partitioning, Cheeger cut, spectral clustering, power watersheds.

Power Spectral Clustering

“…In [27], the power watershed framework was extended and a simple generic algorithm was proposed to calculate the limit of minimizers under some conditions. Other applications of the power watershed framework can be found in [16,15,40]. The methods described in [16,15,40] use power watershed framework to reduce the size of the graph.…”

“…Other applications of the power watershed framework can be found in [16,15,40]. The methods described in [16,15,40] use power watershed framework to reduce the size of the graph. In this article, we propose to use power watershed framework for spectral clustering -by computing a Γ −limit of spectral clustering.…”

A tutorial on applications of power watershed optimization to image processing

“…Also, it is worth noting that Algorithm 1 decomposes the cost function on the original graph into costs on smaller subgraphs. It is shown in [17,18,21,11,10,23,22,12] that the computation of said limit is easier than minimizing the original cost Q(x). In Sec 4, Sec 5, Sec 6 and Sec 7, it will be demonstrated that the quality of the segmentation is retained while reducing the computational cost.…”

“…This is because of the cost in Eq 20 is always non-negative (a graph Laplacian is a positive semi-definite [53]). For the relaxed constraints, the cost can be made arbitrarily close to zero (which would be the minimum cost) for every possible partition of the graph (see [22] for details). Hence an additional constraint is added i.e.…”

“…We revisit the power watershed (PW) framework [39] and show that this optimization framework provides a formalism to impose the constraint that the solutions are contrast-agnostic. It is well known that an application of PW to classic graph-based cost minimization methods results in high-quality approximation solutions to the cost minimization problem [1,17,18,21,11,10,23,22,12,56,54,55]. In other words, PW exploits contrast-agnostic nature of solutions and allows one to apply classic graph-based cost minimization methods to images with very large number of pixels which were otherwise prohibited by computational constraints.…”