Threshold dynamics for the piecewise constant Mumford–Shah functional

Esedoḡlu, Selim; Tsai, Yen‐Hsi Richard

doi:10.1016/j.jcp.2005.05.027

Cited by 176 publications

(178 citation statements)

References 27 publications

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“…Our results suggest that the two should be connected but no rigorous results exist to date. Finally we mention that the GL functional leads to a simplified algorithm for piecewise-constant image segmentation using a carefully designed alternation between evolution of the heat equation and thresholding [23]. The same kind of procedure could be extended to our method although again it would be important to develop a theoretical framework for this idea and its best practice.…”

Section: Calculate U (I)mentioning

confidence: 99%

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Diffuse Interface Models on Graphs for Classification of High Dimensional Data

Bertozzi

Flenner²

2016

SIAM Rev.

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: Calculate U (I)mentioning

confidence: 99%

“…Some examples of GL in image processing include the motivation for the EsedogluTsai [23] threshold method for Chan-Vese segmentation [12]. Although the GL is not ultimately used in their method the construction of their method is directly built on the GL functional, rather than the TV method of the original Chan-Vese paper [12].…”

mentioning

confidence: 99%

Diffuse Interface Models on Graphs for Classification of High Dimensional Data

Bertozzi

Flenner²

2016

SIAM Rev.

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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“…The original MBO algorithm that we described above has been rigorously shown to converge to motion by mean curvature of ∂Σ in [14,2]. Several generalizations of the basic algorithm are presented in [19,25,16,27,28,29] and some applications to image segmentation are given in [17,13]. In [15], Grzibovskis and Heintz propose a generalization of the MBO algorithm to Willmore flow for two-dimensional surfaces in three-space, which is a fourth order evolution of the interface obtained as gradient descent for the Willmore functional.…”

Section: Previous Workmentioning

confidence: 99%

Threshold dynamics for high order geometric motions

Esedoḡlu¹,

Ruuth²,

Tsai³

2008

Interfaces Free Bound.

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A class of algorithms for the high order geometric motion of planar curves is developed. The algorithms alternate two simple steps-a convolution and a thresholding step-to evolve planar curves according to combinations of Willmore flow, surface diffusion flow and curvature motion. A distinguishing feature of the methods is that they possess much better stability than typical explicit algorithms. Formal expansions and numerical examples are provided for a variety of high order flows to validate the methods and illustrate their behaviors.

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Threshold dynamics for the piecewise constant Mumford–Shah functional

Cited by 176 publications

References 27 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

Threshold dynamics for high order geometric motions

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