Classification and image processing with a semi‐discrete scheme for fidelity forced Allen–Cahn on graphs

Abstract

This paper introduces a semi‐discrete implicit Euler (SDIE) scheme for the Allen‐Cahn equation (ACE) with fidelity forcing on graphs. The continuous‐in‐time version of this differential equation was pioneered by Bertozzi and Flenner in 2012 as a method for graph classification problems, such as semi‐supervised learning and image segmentation. In 2013, Merkurjev et. al. used a Merriman‐Bence‐Osher (MBO) scheme with fidelity forcing instead, as heuristically it was expected to give similar results to the ACE. The current paper rigorously establishes the graph MBO scheme with fidelity forcing as a special case of an SDIE scheme for the graph ACE with fidelity forcing. This connection requires the use of the double‐obstacle potential in the ACE, as was already demonstrated by Budd and Van Gennip in 2020 in the context of ACE without a fidelity forcing term. We also prove that solutions of the SDIE scheme converge to solutions of the graph ACE with fidelity forcing as the discrete time step converges to zero. In the second part of the paper we develop the SDIE scheme as a classification algorithm. We also introduce some innovations into the algorithms for the SDIE and MBO schemes. For large graphs, we use a QR decomposition method to compute an eigendecomposition from a Nyström extension, which outperforms the method used by, for example, Bertozzi and Flenner in 2012, in accuracy, stability, and speed. Moreover, we replace the Euler discretization for the scheme's diffusion step by a computation based on the Strang formula for matrix exponentials. We apply this algorithm to a number of image segmentation problems, and compare the performance with that of the graph MBO scheme with fidelity forcing. We find that while the general SDIE scheme does not perform better than the MBO special case at this task, our other innovations lead to a significantly better segmentation than that from previous literature. We also empirically quantify the uncertainty that this segmentation inherits from the randomness in the Nyström extension.

An Empirical Study of Graph-Based Approaches for Semi-supervised Time Series Classification

“…Details of the definition of the potential and the fidelity term incorporating the training data are found in [50]. Further extensions of this approach have been suggested in [20,22,[51][52][53][54][55].…”

Gradient flows and randomised thresholding: sparse inversion and classification

“…There, this idea has been used in the context of mean curvature flows. Its use in segmentation and classification has been considered by, e.g., [9,10,15,18]. The connection that we make here between MBO and Allen-Cahn has been thoroughly discussed by [9] in the setting of graph diffusions.…”

“…The meaning of 'neighbouring' is determined by the choice of the gradient, see also Remark 3.1. A similar double-well potential has been employed by [10] -they actually used a related double-obstacle formulation. ε > 0 controls whether the process is closer to a diffusion (ε ≫ 0) or to a strict binary classification…”

“…There are again multiple ways to discretise the differential inclusion (3.1) via explicit, semi-implicit, and fully implicit methods, see, e.g. [10,24]. Again, we make the observation that we can split the right-hand side of (3.1) into two parts that are relatively simple to solve separately.…”

Effective time step analysis for the Allen–Cahn equation with a high‐order polynomial free energy

“…The classical AC equation and modified forms have been widely employed for phase transition, 1,2 image processing, 3,4 topology optimization, 5,6 multiphase flows, [7][8][9] mechanical behavior of thin-walled superalloys, 10 dendrite growth, 11,12 motion by mean curvature, 13,14 microscopic fluctuating strain, 15 damage field of sensitivity-uncertainty quantification framework 16 and so forth. As such, many scientific problems have been solved through the numerical application of the AC equation.…”