Effective Sampling: Fast Segmentation Using Robust Geometric Model
  Fitting

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Abstract

Identifying the underlying models in a set of data points contaminated by noise and outliers, leads to a highly complex multi-model fitting problem. This problem can be posed as a clustering problem by the projection of higher order affinities between data points into a graph, which can then be clustered using spectral clustering. Calculating all possible higher order affinities is computationally expensive. Hence in most cases only a subset is used. In this paper, we propose an effective sampling method to obtain a highly accurate approximation of the full graph required to solve multi-structural model fitting problems in computer vision. The proposed method is based on the observation that the usefulness of a graph for segmentation improves as the distribution of hypotheses (used to build the graph) approaches the distribution of actual parameters for the given data. In this paper, we approximate this actual parameter distribution using a k-th order statistics based cost function and the samples are generated using a greedy algorithm coupled with a data sub-sampling strategy. The experimental analysis shows that the proposed method is both accurate and computationally efficient compared to the state-of-the-art robust multi-model fitting techniques. The code is publicly available from https://github.com/RuwanT/model-fitting-cbs.

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Sparse subspace clustering: algorithm, theory, and applications.

Ehsan Elhamifar, Rene Vidal (2013)

Many real-world problems deal with collections of high-dimensional data, such as images, videos, text, and web documents, DNA microarray data, and more. Often, such high-dimensional data lie close to low-dimensional structures corresponding to several classes or categories to which the data belong. In this paper, we propose and study an algorithm, called sparse subspace clustering, to cluster data points that lie in a union of low-dimensional subspaces. The key idea is that, among the infinitely many possible representations of a data point in terms of other points, a sparse representation corresponds to selecting a few points from the same subspace. This motivates solving a sparse optimization program whose solution is used in a spectral clustering framework to infer the clustering of the data into subspaces. Since solving the sparse optimization program is in general NP-hard, we consider a convex relaxation and show that, under appropriate conditions on the arrangement of the subspaces and the distribution of the data, the proposed minimization program succeeds in recovering the desired sparse representations. The proposed algorithm is efficient and can handle data points near the intersections of subspaces. Another key advantage of the proposed algorithm with respect to the state of the art is that it can deal directly with data nuisances, such as noise, sparse outlying entries, and missing entries, by incorporating the model of the data into the sparse optimization program. We demonstrate the effectiveness of the proposed algorithm through experiments on synthetic data as well as the two real-world problems of motion segmentation and face clustering.

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Robust Recovery of Subspace Structures by Low-Rank Representation

, , … (2012)

In this work we address the subspace recovery problem. Given a set of data samples (vectors) approximately drawn from a union of multiple subspaces, our goal is to segment the samples into their respective subspaces and correct the possible errors as well. To this end, we propose a novel method termed Low-Rank Representation (LRR), which seeks the lowest-rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary. It is shown that LRR well solves the subspace recovery problem: when the data is clean, we prove that LRR exactly captures the true subspace structures; for the data contaminated by outliers, we prove that under certain conditions LRR can exactly recover the row space of the original data and detect the outlier as well; for the data corrupted by arbitrary errors, LRR can also approximately recover the row space with theoretical guarantees. Since the subspace membership is provably determined by the row space, these further imply that LRR can perform robust subspace segmentation and error correction, in an efficient way.