An MBO Scheme on Graphs for Classification and Image Processing

Merkurjev, Ekaterina; Kostic, Tijana; Bertozzi, Andrea L.

doi:10.1137/120886935

Cited by 128 publications

(267 citation statements)

References 58 publications

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“…The motion by mean curvature of the MBO scheme was generalized to graphs by Merkurjev et al in [3], where a modified MBO scheme on graphs has been used to formulate an algorithm for binary classification and image processing. The authors apply a two-step time splitting scheme to (4) so that the second step is the same as the one in the original MBO scheme, and then replace the ∆u term with a more general graph term −L s u.…”

Section: Graph Formulationmentioning

confidence: 99%

“…The standard Gibbs-simplex Σ K defined in equation (8) allows the multiclass extension of the algorithm in [3] to be straightforward.…”

Section: Extension To Multiclass Segmentationmentioning

confidence: 99%

“…The first method extends this procedure to a multiclass case using the idea of the Gibbs simplex. The second method applies the simplex idea to the graph-based MerrimanBence-Osher (MBO) scheme developed in [3]. Both algorithms result in efficient and accurate ways to perform multiclass segmentation.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Diffuse interface methods for multiclass segmentation of high-dimensional data

Merkurjev

Garcia-Cardona

Bertozzi

et al. 2014

Applied Mathematics Letters

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We present two graph-based algorithms for multiclass segmentation of high-dimensional data, motivated by the binary diffuse interface model. One algorithm generalizes Ginzburg-Landau (GL) functional minimization on graphs to the Gibbs simplex. The other algorithm uses a reduction of GL minimization, based on the Merriman-Bence-Osher scheme for motion by mean curvature. These yield accurate and efficient algorithms for semi-supervised learning. Our algorithms outperform existing methods, including supervised learning approaches, on the benchmark data sets that we used. We refer to [1] for a more detailed illustration of the methods, as well as different experimental examples.

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Section: Graph Formulationmentioning

confidence: 99%

“…The standard Gibbs-simplex Σ K defined in equation (8) allows the multiclass extension of the algorithm in [3] to be straightforward.…”

Section: Extension To Multiclass Segmentationmentioning

confidence: 99%

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Diffuse interface methods for multiclass segmentation of high-dimensional data

Merkurjev

Garcia-Cardona

Bertozzi

et al. 2014

Applied Mathematics Letters

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“…In addition, we apply this Gibbs simplex idea to the graph-based Merriman-Bence-Osher (MBO) scheme developed in [38]. The MBO scheme [39] is a well-established PDE method for evolving an interface by mean curvature.…”

Section: Introductionmentioning

confidence: 99%

“…The MBO scheme [39] is a well-established PDE method for evolving an interface by mean curvature. As with the diffuse interface model, tools for nonlocal calculus [27] are used in [38] to generalize the PDE formulation to the graph setting. By introducing the phase-field representation to the graph-based MBO scheme, we develop another new and highly efficient algorithm for multiclass segmentation in a SSL framework.…”

Section: Introductionmentioning

confidence: 99%

Fast Multiclass Segmentation using Diffuse Interface Methods on Graphs

Garcia-Cardona¹,

Merkurjev²,

Bertozzi³

et al. 2013

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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.

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

2021

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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.

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An MBO Scheme on Graphs for Classification and Image Processing

Cited by 128 publications

References 58 publications

Diffuse interface methods for multiclass segmentation of high-dimensional data

Diffuse interface methods for multiclass segmentation of high-dimensional data

Fast Multiclass Segmentation using Diffuse Interface Methods on Graphs

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

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