Robust Subspace Learning: Robust PCA, Robust Subspace Tracking, and Robust Subspace Recovery

PCA is one of the most widely used dimension reduction techniques. A related easier problem is "subspace learning" or "subspace estimation". Given relatively clean data, both are easily solved via singular value decomposition (SVD). The problem of subspace learning or PCA in the presence of outliers is called robust subspace learning or robust PCA (RPCA). For long data sequences, if one tries to use a single lower dimensional subspace to represent the data, the required subspace dimension may end up being quite large. For such data, a better model is to assume that it lies in a low-dimensional subspace that can change over time, albeit gradually. The problem of tracking such data (and the subspaces) while being robust to outliers is called robust subspace tracking (RST). This article provides a magazine-style overview of the entire field of robust subspace learning and tracking. In particular solutions for three problems are discussed in detail: RPCA via sparse+low-rank matrix decomposition (S+LR), RST via S+LR, and "robust subspace recovery (RSR)". RSR assumes that an entire data vector is either an outlier or an inlier. The S+LR formulation instead assumes that outliers occur on only a few data vector indices and hence are well modeled as sparse corruptions.Comment: To appear, IEEE Signal Processing Magazine, July 201

Defect Detection for Patterned Fabric Images Based on GHOG and Low-Rank Decomposition

In contrast to defect-free fabric images with macro-homogeneous textures and regular patterns, the fabric images with the defect are characterized by the defect regions that are salient and sparse among the redundant background. Therefore, as an effective tool for separating an image into a redundant part (the background) and sparse part (the defect), the low-rank decomposition model provides an ideal solution for patterned fabric defect detection. In this paper, a novel patterned method for fabric defect detection is proposed based on a novel texture descriptor and the low-rank decomposition model. First, an efficient second-order orientation-aware descriptor, denoted as GHOG, is designed by combining Gabor and histogram of oriented gradient (HOG). In addition, a spatial pooling strategy based on human vision mechanism is utilized to further improve the discrimination ability of the proposed descriptor. The proposed texture descriptor can make the defect-free image blocks lay in a low-rank subspace, while the defective image blocks have deviated from this subspace. Then, a constructed low-rank decomposition model divides the feature matrix generated from all the image blocks into a low-rank part, which represents the defect-free background, and a sparse part, which represents sparse defects. In addition, a non-convex log det as a smooth surrogate function is utilized to improve the efficiency of the constructed low-rank model. Finally, the defects are localized by segmenting the saliency map generated by the sparse matrix. The qualitative results and quantitative evaluation results demonstrate that the proposed method improves the detection accuracy and self-adaptivity comparing with the state-of-the-art methods