Ekaterina Merkurjev

Multiclass Data Segmentation Using Diffuse Interface Methods on Graphs

We present two graph-based algorithms for multiclass segmentation of high-dimensional data on graphs. The algorithms use a diffuse interface model based on the Ginzburg-Landau functional, related to total variation and graph cuts. A multiclass extension is introduced using the Gibbs simplex, with the functionals double-well potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm uses a graph adaptation of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates between diffusion and thresholding. We demonstrate the performance of both algorithms experimentally on synthetic data, image labeling, and several benchmark data sets such as MNIST, COIL and WebKB. We also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments indicate that the results are competitive with or better than the current state-of-the-art in multiclass graph-based segmentation algorithms for high-dimensional data.

An MBO Scheme on Graphs for Classification and Image Processing

In this paper we present a computationally efficient algorithm utilizing a fully or seminonlocal graph Laplacian for solving a wide range of learning problems in binary data classification and image processing. In their recent work [Multiscale Model. Simul., 10 (2012), pp. 1090--1118], Bertozzi and Flenner introduced a graph-based diffuse interface model utilizing the Ginzburg--Landau functional for solving problems in data classification. Here, we propose an adaptation of the classic numerical Merriman--Bence--Osher (MBO) scheme for minimizing graph-based diffuse interface functionals, like those originally proposed by Bertozzi and Flenner. We also make use of fast numerical solvers for finding eigenvalues and eigenvectors of the graph Laplacian. Various computational examples are presented to demonstrate the performance of our algorithm, which is successful on images with texture and repetitive structure due to its nonlocal nature. The results show that our method is multiple times more efficient than o...

Detection and tracking of gas plumes in LWIR hyperspectral video sequence data

Automated detection of chemical plumes presents a segmentation challenge. The segmentation problem for gas plumes is difficult due to the diffusive nature of the cloud. The advantage of considering hyperspectral images in the gas plume detection problem over the conventional RGB imagery is the presence of non-visual data, allowing for a richer representation of information. In this paper we present an effective method of visualizing hyperspectral video sequences containing chemical plumes and investigate the effectiveness of segmentation techniques on these post-processed videos. Our approach uses a combination of dimension reduction and histogram equalization to prepare the hyperspectral videos for segmentation. First, Principal Components Analysis (PCA) is used to reduce the dimension of the entire video sequence. This is done by projecting each pixel onto the first few Principal Components resulting in a type of spectral filter. Next, a Midway method for histogram equalization is used. These methods redistribute the intensity values in order to reduce icker between frames. This properly prepares these high-dimensional video sequences for more traditional segmentation techniques. We compare the ability of various clustering techniques to properly segment the chemical plume. These include K-means, spectral clustering, and the Ginzburg-Landau functional.

Graph MBO method for multiclass segmentation of hyperspectral stand-off detection video

We consider the challenge of detection of chemical plumes in hyperspectral image data. Segmentation of gas is difficult due to the diffusive nature of the cloud. The use of hyperspectral imagery provides non-visual data for this problem, allowing for the utilization of a richer array of sensing information. In this paper, we present a method to track and classify objects in hyperspectral videos. The method involves the application of a new algorithm recently developed for high dimensional data. It is made efficient by the application of spectral methods and the Nyström extension to calculate the eigenvalues/eigenvectors of the graph Laplacian. Results are shown on plume detection in LWIR standoff detection.

Diffuse Interface Methods for Multiclass Segmentation of High-Dimensional Data

Abstract 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 datasets that we used. We refer to Garcia-Cardona (2014) for a more detailed illustration of the methods, as well as different experimental examples.

Global Binary Optimization on Graphs for Classification of High-Dimensional Data

This work develops a global minimization framework for segmentation of high-dimensional data into two classes. It combines recent convex optimization methods from imaging with recent graph- based variational models for data segmentation. Two convex splitting algorithms are proposed, where graph-based PDE techniques are used to solve some of the subproblems. It is shown that global minimizers can be guaranteed for semi-supervised segmentation with two regions. If constraints on the volume of the regions are incorporated, global minimizers cannot be guaranteed, but can often be obtained in practice and otherwise be closely approximated. Experiments on benchmark data sets show that our models produce segmentation results that are comparable with or outperform the state-of-the-art algorithms. In particular, we perform a thorough comparison to recent MBO (Merriman–Bence–Osher, AMS-Selected Lectures in Mathematics Series: Computational Crystal Growers Workshop, 1992) and phase field methods, and show the advantage of the algorithms proposed in this paper.

Convex Variational Methods on Graphs for Multiclass Segmentation of High-Dimensional Data and Point Clouds

Graph-based variational methods have recently shown to be highly competitive for various classification problems of high-dimensional data, but are inherently difficult to handle from an optimization perspective. This paper proposes a convex relaxation for a certain set of graph-based multiclass data segmentation models involving a graph total variation term, region homogeneity terms, supervised information and certain constraints or penalty terms acting on the class sizes. Particular applications include semi-supervised classification of high-dimensional data and unsupervised segmentation of unstructured 3D point clouds. Theoretical analysis shows that the convex relaxation closely approximates the original NP-hard problems, and these observations are also confirmed experimentally. An efficient duality-based algorithm is developed that handles all constraints on the labeling function implicitly. Experiments on semi-supervised classification indicate consistently higher accuracies than related non-convex approaches and considerably so when the training data are not uniformly distributed among the data set. The accuracies are also highly competitive against a wide range of other established methods on three benchmark data sets. Experiments on 3D point clouds acquire by a LaDAR in outdoor scenes and demonstrate that the scenes can accurately be segmented into object classes such as vegetation, the ground plane and human-made structures.

Modified Cheeger and ratio cut methods using the Ginzburg-Landau functional for classification of high-dimensional data

Abstract : Recent advances in clustering have included continuous relaxations of the Cheeger cut problem and those which address its linear approximation using the graph Laplacian. In this paper, we show how to use the graph Laplacian to solve the fully nonlinear Cheeger cut problem, as well as the ratio cut optimization task. Both problems are connected to total variation minimization, and the related Ginzburg-Landau functional is used in the derivation of the methods. The graph framework discussed in this paper is undirected. The resulting algorithms are efficient ways to cluster thedata into two classes, and they can be easily extended to case of multiple classes, or used on a multiclass data set via recursive bipartitioning. In addition to showing results on benchmark data sets, we also show an application of the algorithm to hyperspectral video data.

Hyperspectral Image Classification Using Graph Clustering Methods

Author(s): Meng, Z; Merkurjev, E; Koniges, A; Bertozzi, AL | Abstract:

Asymptotics for Optimal Design Problems for the Schrödinger Equation with a Potential

We study the problem of optimal observability and prove time asymptotic observability estimates for the Schrodinger equation with a potential in , with , using spectral theory. An elegant way to model the problem using a time asymptotic observability constant is presented. For certain small potentials, we demonstrate the existence of a nonzero asymptotic observability constant under given conditions and describe its explicit properties and optimal values. Moreover, we give a precise description of numerical models to analyze the properties of important examples of potentials wells, including that of the modified harmonic oscillator.