Multiclass Data Segmentation using Diffuse Interface Methods on Graphs

TL;DR

This paper introduces two graph-based algorithms for multiclass data segmentation using diffuse interface models, demonstrating competitive performance on various datasets and leveraging efficient numerical solvers.

Contribution

It extends diffuse interface methods to multiclass segmentation on graphs with novel algorithms and practical implementations.

Findings

Algorithms perform well on synthetic and real datasets.

Results are competitive with or surpass current state-of-the-art methods.

Efficient eigenvector computations improve scalability.

Abstract

We present two graph-based algorithms for multiclass segmentation of high-dimensional data. The algorithms use a diffuse interface model based on the Ginzburg-Landau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs simplex, with the functional's double-well potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm is a uses a graph adaptation of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates between diffusion and thresholding. We demonstrate the performance of both algorithms experimentally on synthetic data, grayscale and color images, and several benchmark data sets such as MNIST, COIL and WebKB. We also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments indicate that the results are competitive with or better than the current state-of-the-art multiclass segmentation algorithms.