Diffuse interface methods for multiclass segmentation of high-dimensional data

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

doi:10.1016/j.aml.2014.02.008

Cited by 38 publications

(43 citation statements)

References 17 publications

(26 reference statements)

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“…Similar approaches have been used in [4,5], where the problem is formulated as a minimization of the Ginzburg-Laudau (GL) functional (in graph form) with a fidelity term. In [60], the authors propose an MBO scheme to solve the binary classification problem; a multi-class extension of that algorithm is described in [34,59].…”

Section: Semi-supervised Algorithmmentioning

confidence: 99%

“…encountered also in [34,60,59]. The first two terms of (5) comprise the graph form of the GinzburgLandau functional, where L s is the symmetric Laplacian, is a small positive constant, and W (u i ) is the multi-well potential inn dimensions, wheren is the number of classes…”

Section: Semi-supervised Algorithmmentioning

confidence: 99%

“…Moreover, in the case of image processing, the framework allows one to easily capture texture and patterns throughout the image [16,60,37,36] using the non-local means feature vector. The graphical approach has appeared in many recent works for different kinds of applications: semi-supervised learning [21,57,2,34,60,59,61,3], image processing [26,80,25], machine learning [81], graph cuts [10,39,67,69,11,13,12,58].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Hyperspectral Image Classification Using Graph Clustering Methods

Zhang¹,

Merkurjev²,

Koniges³

et al. 2017

Image Processing on Line

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Hyperspectral imagery is a challenging modality due to the dimension of the pixels which can range from hundreds to over a thousand frequencies depending on the sensor. Most methods in the literature reduce the dimension of the data using a method such as principal component analysis, however this procedure can lose information. More recently methods have been developed to address classification of large datasets in high dimensions. This paper presents two classes of graph-based classification methods for hyperspectral imagery. Using the full dimensionality of the data, we consider a similarity graph based on pairwise comparisons of pixels. The graph is segmented using a pseudospectral algorithm for graph clustering that requires information about the eigenfunctions of the graph Laplacian but does not require computation of the full graph. We develop a parallel version of the Nyström extension method to randomly sample the graph to construct a low rank approximation of the graph Laplacian. With at most a few hundred eigenfunctions, we can implement the clustering method designed to solve a variational problem for a graph-cut-based semi-supervised or unsupervised classification problem. We implement OpenMP directive-based parallelism in our algorithms and show performance improvement and strong, almost ideal, scaling behavior. The method can handle very large datasets including a video sequence with over a million pixels, and the problem of segmenting a data set into a pre-determined number of classes. Source CodeThe reviewed source code and documentation for this algorithm are available from the web page of this article 1 . Compilation and usage instructions are included in the README.txt file of the archive.

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Section: Semi-supervised Algorithmmentioning

confidence: 99%

Section: Semi-supervised Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hyperspectral Image Classification Using Graph Clustering Methods

Zhang¹,

Merkurjev²,

Koniges³

et al. 2017

Image Processing on Line

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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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“…This graph Ginzburg-Landau method has found many applications, for example in data clustering and classification and image segmentation [4,16,20] and has also been extended to deal with clustering and classification into more than two classes [21][22][23][24][25]. Recent papers prove convergence of the graph Allen-Cahn algorithm (both the spectrally untruncated and truncated versions) and extend the method to non-smooth potentials and hypergraphs [26,27].…”

mentioning

confidence: 99%

Introduction: Big data and partial differential equations

Gennip¹,

Schönlieb²

2017

Eur. J. Appl. Math

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Partial differential equations (PDEs) are expressions involving an unknown function in many independent variables and their partial derivatives up to a certain order. Since PDEs express continuous change, they have long been used to formulate a myriad of dynamical physical and biological phenomena: heat flow, optics, electrostatics and -dynamics, elasticity, fluid flow and many more. Many of these PDEs can be derived in a variational way, i.e. via minimization of an ‘energy’ functional. In this globalised and technologically advanced age, PDEs are also extensively used for modelling social situations (e.g. models for opinion formation, mathematical finance, crowd motion) and tasks in engineering (such as models for semiconductors, networks, and signal and image processing tasks). In particular, in recent years, there has been increasing interest from applied analysts in applying the models and techniques from variational methods and PDEs to tackle problems in data science. This issue of the European Journal of Applied Mathematics highlights some recent developments in this young and growing area. It gives a taste of endeavours in this realm in two exemplary contributions on PDEs on graphs [1, 2] and one on probabilistic domain decomposition for numerically solving large-scale PDEs [3].

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Diffuse interface methods for multiclass segmentation of high-dimensional data

Cited by 38 publications

References 17 publications

Hyperspectral Image Classification Using Graph Clustering Methods

Hyperspectral Image Classification Using Graph Clustering Methods

Two-Dimensional Compact Variational Mode Decomposition

Introduction: Big data and partial differential equations

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