Multiclass Data Segmentation Using Diffuse Interface Methods on Graphs

Garcia-Cardona, Cristina; Merkurjev, Ekaterina; Bertozzi, Andrea L.; Flenner, Arjuna; Percus, Allon G.

doi:10.1109/tpami.2014.2300478

Cited by 107 publications

(133 citation statements)

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“…In [14], a different well function is defined using the L 1 norm instead of L 2 . However, the algorithm in [14] uses a subgradient descent followed by a projection onto the Gibbs simplex.…”

Section: Results For Multiclass Classificationmentioning

confidence: 99%

Convergence of the Graph Allen–Cahn Scheme

Luo

Bertozzi

2017

J Stat Phys

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The graph Laplacian and the graph cut problem are closely related to Markov random fields, and have many applications in clustering and image segmentation. The diffuse interface model is widely used for modeling in material science, and can also be used as a proxy to total variation minimization. In Bertozzi and Flenner (Multiscale Model Simul 10(3):1090-1118, 2012), an algorithm was developed to generalize the diffuse interface model to graphs to solve the graph cut problem. This work analyzes the conditions for the graph diffuse interface algorithm to converge. Using techniques from numerical PDE and convex optimization, monotonicity in function value and convergence under an a posteriori condition are shown for a class of schemes under a graph-independent stepsize condition. We also generalize our results to incorporate spectral truncation, a common technique used to save computation cost, and also to the case of multiclass classification. Various numerical experiments are done to compare theoretical results with practical performance.

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Section: Results For Multiclass Classificationmentioning

confidence: 99%

Convergence of the Graph Allen–Cahn Scheme

Luo

Bertozzi

2017

J Stat Phys

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“…We first summarize the results in [1], which contains a more detailed description of our algorithms. The main data sets used in the paper were the WebKB [17], MNIST [18] and COIL [19] benchmark data sets.…”

Section: Previous Results For Mnist Webkb Coil and Three Moons Datamentioning

confidence: 99%

“…Ours however has the advantages: (i) it is direct and thus does not require one to solve further minimization problems, (ii) one can adjust the accuracy by changing the number of eigenfunctions, and (iii) its complexity is close to linear in |V|. Details about (iii) can be found in [1].…”

Section: Energy Minimizationmentioning

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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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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“…This idea has been discussed more rigorously in [20]. For an application of a similar strategy to graph-based image processing, see [26,42,49]. The projection-based partitioning update for A k becomes:…”

Section: Multiphase Mbo and Rearrangementmentioning

confidence: 99%

Two-Dimensional Compact Variational Mode Decomposition

et al. 2017

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Decomposing multidimensional signals, such as images, into spatially compact, potentially overlapping modes of essentially wavelike nature makes these components accessible for further downstream analysis. This decomposition enables space-frequency analysis, demodulation, estimation of local orientation, edge and corner detection, texture analysis, denoising, inpainting, or curvature estimation. Our model decomposes the input signal into modes with narrow Fourier bandwidth; to cope with sharp region boundaries, incompatible with narrow bandwidth, we introduce binary support functions that act as masks on the narrow-band mode for image recomposition. L 1 and TV terms promote sparsity and spatial compactness. Constraining the support functions to partitions of the signal domain, we effectively get an image segmentation model based on spectral homogeneity. By coupling several submodes together with a single support function, we are able to decompose an image into several crystal grains. Our efficient algorithm is based on variable splitting and alternate direction optimization; we employ Merriman-Bence-Osher-like threshold dynamics to handle efficiently the motion by mean curvature of the support function boundaries under the sparsity promoting terms. The versatility and effectiveness of our proposed model is demonstrated on a broad variety of example images from different modalities. These demonstrations include the decomposition of images into overlapping modes with smooth or sharp boundaries, segmentation of images of crystal grains, and inpainting of damaged image regions through artifact detection.

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Multiclass Data Segmentation Using Diffuse Interface Methods on Graphs

Cited by 107 publications

References 50 publications

Convergence of the Graph Allen–Cahn Scheme

Convergence of the Graph Allen–Cahn Scheme

Diffuse interface methods for multiclass segmentation of high-dimensional data

Two-Dimensional Compact Variational Mode Decomposition

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