Inpainting of Binary Images Using the Cahn–Hilliard Equation

Bertozzi, Andrea L.; Esedoḡlu, Selim; Gillette, Alan

doi:10.1109/tip.2006.887728

Cited by 358 publications

(341 citation statements)

References 20 publications

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“…Convex splitting scheme. Our choice of splitting is motived by prior work on GL-type functionals for image processing with fidelity [18,7,6]. First we review the algorithm as it applies to differential operators in the classical GinzburgLandau regularization.…”

Section: Proper Normalization Of the Graph Laplacian With Scalementioning

confidence: 99%

“…The method of successive reductions in was used for image inpainting via the Cahn-Hilliard equation [6,7]. In [6] the authors carefully study the space of steady states for a stripe inpainting example in which the problem exhibits an incomplete supercritical pitchfork birfurcation as the scale parameter is varied.…”

Section: Two Moonsmentioning

confidence: 99%

“…One also can exploit variation in the scale parameter during this minimization procedure. This idea was used to great advantage in earlier work on image inpainting using the Cahn-Hilliard equation [7,6]. While the theory of gamma convergence is wellknown for the classical Ginzberg-Landau operator and for its wavelet-based cousin [17], no such theory exists for the graph-based problem and this is a very interesting and important problem.…”

Section: Calculate U (I)mentioning

confidence: 99%

“…This method was popularized by a well-known but unpublished manuscript by David Eyre [24]. It has been used to solve the Cahn Hilliard equation on large domains and on long time intervals [53] and also in imaging applications involving Cahn-Hilliard [7] and a wavelet version of the method proposed here for graphs [18]. This same idea has also been directly discussed in the context of general minimization procedures for nonconvex functionals [59].…”

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Diffuse Interface Models on Graphs for Classification of High Dimensional Data

Bertozzi

Flenner²

2016

SIAM Rev.

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162

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Abstract. There are currently several communities working on algorithms for classification of high dimensional data. This work develops a class of variational algorithms that combine recent ideas from spectral methods on graphs with nonlinear edge/region detection methods traditionally used in in the PDE-based imaging community. The algorithms are based on the Ginzburg-Landau functional which has classical PDE connections to total variation minimization. Convex-splitting algorithms allow us to quickly find minimizers of the proposed model and take advantage of fast spectral solvers of linear graph-theoretic problems. We present diverse computational examples involving both basic clustering and semi-supervised learning for different applications. Case studies include feature identification in images, segmentation in social networks, and segmentation of shapes in high dimensional datasets.Key words. Nyström extension, diffuse interfaces, image processing, high dimensional data AMS subject classifications. Insert AMS subject classifications.This work brings together ideas from different communities and for this reason we review various components of the algorithms in order to make the paper accessible to readers familiar with either the PDE-based or graph-theoretic approaches. In Section 1 we review diffuse interface methods in Euclidean space and convex splitting methods for minimization. These well-known constructions make heavy use of the classical Laplace operator and our new algorithms involve extensions of this idea to a more general graph Laplacian. Section 2 reviews some of the notation and definitions of the graph Laplacian and this discussion contains a level of detail appropriate for readers less familiar with this machinery. Included in this section is a review of segmentation using spatial clustering and a discussion of various normalization conventions for these linear operators on graphs, in connection to real world problems such as machine learning in image analysis. The rest of the paper explains the main computational algorithm and presents different examples involving both sparse connectivity and non-sparse connectivity of the graph. The algorithms have a multi-scale flavor due to (a) the different scales inherent in diffuse interface methods and (b) the role of scale in the eigenfunctions and eigenvalues of the graph Laplacian.1. Background on diffuse interfaces, image processing, and convex splitting methods. Diffuse interface models in Euclidean space are often built around the Ginzburg-Landau functionalwhere W is a double well potential. For example W (u) = 1 4 (u 2 − 1) 2 has minimizers at plus and minus one. The operator ∇ denotes the spatial gradient operator and the first term in GL is /2 times the H 1 semi-norm of u. The small parameter *

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Section: Proper Normalization Of the Graph Laplacian With Scalementioning

confidence: 99%

Section: Two Moonsmentioning

confidence: 99%

Section: Calculate U (I)mentioning

confidence: 99%

mentioning

confidence: 99%

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Diffuse Interface Models on Graphs for Classification of High Dimensional Data

Bertozzi

Flenner²

2016

SIAM Rev.

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162

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“…By contrast, the TV semi-norm contains a nonlinear curvature term. The diffuse interface description has been used successfully in image impainting [5,6] and image segmentation [7]. The standard practice is to introduce an additional fidelity term F to allow for the specification of any known informationû:…”

Section: Ginzburg Landau Functional and Diffuse Interface Modelmentioning

confidence: 99%

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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Bibliography

2014

Mathematical Foundations of Image Processing and Analysis 1

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Inpainting of Binary Images Using the Cahn–Hilliard Equation

Cited by 358 publications

References 20 publications

Diffuse Interface Models on Graphs for Classification of High Dimensional Data

Diffuse Interface Models on Graphs for Classification of High Dimensional Data

Diffuse interface methods for multiclass segmentation of high-dimensional data

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