Multi-class Graph Mumford-Shah Model for Plume Detection Using the MBO scheme

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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“…Multiclass diffuse interface algorithm on graphs can be found in [15,19,23]. We state some basic notations.…”

Section: Results For Multiclass Classificationmentioning

confidence: 99%

Convergence of the Graph Allen–Cahn Scheme

Luo

Bertozzi

2017

J Stat Phys

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“…• These methods have been introduced in earlier conference papers [61,42] as serial implementations and as a parallel implementation [56] on a supercomputer, without online code. Here we develop new parallel versions for real time implementations on the IPOL server for both hyperspectral imagery and the non local means functional for segmentation of RGB images.…”

Section: Main Contributionsmentioning

confidence: 99%

“…The graph version of the multi-class piecewise constant Mumford-Shah energy was introduced in [42] for hyperspectral images…”

Section: Unsupervised Algorithmmentioning

confidence: 99%

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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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“…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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Multi-class Graph Mumford-Shah Model for Plume Detection Using the MBO scheme

Cited by 33 publications

References 22 publications

Convergence of the Graph Allen–Cahn Scheme

Convergence of the Graph Allen–Cahn Scheme

Hyperspectral Image Classification Using Graph Clustering Methods

Two-Dimensional Compact Variational Mode Decomposition

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