Multi-class Graph Mumford-Shah Model for Plume Detection using the MBO Scheme

We focus on the multi-class segmentation problem using the piecewise constant Mumford-Shah model in a graph setting. After formulating a graph version of the Mumford-Shah energy, we propose an efficient algorithm called the MBO scheme using threshold dynamics. Theoretical analysis is developed and a Lyapunov functional is proven to decrease as the algorithm proceeds. Furthermore, to reduce the computational cost for large datasets, we incorporate the Nystr¨om extension method which efficiently approximates eigenvectors of the graph Laplacian based on a small portion of the weight matrix. Finally, we implement the proposed method on the problem of chemical plume detection in hyper-spectral video data