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Variational and PDE-based methods for big data analysis, classification and image processing using graphs.

机译：基于图的变体和基于PDE的大数据分析，分类和图像处理方法。

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摘要

We present several graph-based algorithms for image processing and classification of high- dimensional data. The first (semi-supervised) method uses a graph adaptation of the classical numerical Merriman-Bence-Osher (MBO) scheme, and can be extended to the multiclass case via the Gibbs simplex. We show examples of the application of the algorithm in the areas of image inpainting (both binary and grayscale), image segmentation and classification on benchmark data sets. We have also applied this algorithm to the problem of object detection using hyperspectral video sequences as a data set. In addition, a second related model is introduced. It uses a diffuse interface model based on the Ginzburg-Landau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs simplex, with the functional's double-well potential modified to handle the multiclass case. The version minimizes the functional using a convex splitting numerical scheme. In our computations, we 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 on benchmark data sets show that our models produce results that are comparable with or outperform the state-of-the-art algorithms.;The second (semi-supervised) method develops a global minimization framework for binary classification of high-dimensional data. It combines recent convex optimization methods for image processing 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. We perform a thorough comparison to recent MBO (Merriman-Bence-Osher) and phase field methods, and show the advantage of the proposed algorithms.;Lastly, we present the current work (unsupervised method) related to normalized cuts. The method uses a clever alternative to the normalized cut to solve the binary classification problem. In particular, we work with the Ginzburg-Landau functional. In addition, we use a generalized graphical framework, so several different graph Laplacians are tested and their results are compared.
机译：我们提出了几种基于图的算法，用于图像处理和高维数据分类。第一种（半监督）方法使用经典数值Merriman-Bence-Osher（MBO）方案的图形自适应，并且可以通过Gibbs单形方法扩展到多类情况。我们展示了该算法在图像修复（二进制和灰度），图像分割和基准数据集分类方面的应用示例。我们还将这种算法应用于使用高光谱视频序列作为数据集的物体检测问题。另外，引入了第二种相关模型。它使用基于Ginzburg-Landau功能的扩散接口模型，与总变化压缩感测和图像处理有关。使用Gibbs单形函数引入了多类扩展，并修改了该函数的双阱势能以处理多类情况。该版本使用凸分裂数值方案将功能最小化。在我们的计算中，我们使用快速数值求解器来查找图拉普拉斯算子的特征向量和特征值，并利用矩阵的稀疏性。在基准数据集上进行的实验表明，我们的模型所产生的结果可与先进的算法相媲美或优于现有的算法。第二种（半监督）方法为高维数据的二进制分类开发了全局最小化框架。它结合了用于图像处理的最新凸优化方法和用于数据分割的基于最近图的变异模型。提出了两种凸分裂算法，其中基于图的PDE技术用于解决一些子问题。结果表明，对于两个区域的半监督分割，可以保证全局最小化器。如果纳入对区域数量的限制，则不能保证全局最小化器，但通常可以在实践中获得，否则可以近似地近似。我们对最新的MBO（Merriman-Bence-Osher）和相场方法进行了彻底的比较，并展示了所提出算法的优势。最后，我们介绍了与归一化割相关的当前工作（无监督方法）。该方法使用归一化割的巧妙替代方法来解决二进制分类问题。特别是，我们使用Ginzburg-Landau功能。此外，我们使用广义的图形框架，因此测试了几种不同的图拉普拉斯算子并比较了它们的结果。

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  • 作者

    Merkurjev, Ekaterina.;

  • 作者单位

    University of California, Los Angeles.;

  • 授予单位 University of California, Los Angeles.;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2015
  • 页码 132 p.
  • 总页数 132
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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