Recent advances in technology have brought major breakthroughs in data collection, enabling a large amount of data to be gathered over time and thus generating time series. Mining this data has become an important task for researchers and practitioners in the past few years, including the detection of outliers or anomalies that may represent errors or events of interest. This review aims to provide a structured and comprehensive state-of-the-art on unsupervised outlier detection techniques in the context of time series. To this end, a taxonomy is presented based on the main aspects that characterize an outlier detection technique.

A Review on Outlier/Anomaly Detection in Time Series Data

PhysioBank是一个大型的逐渐扩增的生理学信号和相关数据的数字化记录文档。目前包含多参数的心肺、神经和其他生物医学信号，尤以心电图（ECG）为主。信号来自健康受试者和各种疾病的患者。涉及的疾病包括心脏猝死、充血性心力衰竭、癫痫、步态不稳、睡眠呼吸暂停和衰老等。

PhysioBank

Recent research on problem formulations based on decomposition into low-rank plus sparse matrices shows a suitable framework to separate moving objects from the background. The most representative problem formulation is the Robust Principal Component Analysis (RPCA) solved via Principal Component Pursuit (PCP) which decomposes a data matrix in a low-rank matrix and a sparse matrix. However, similar robust implicit or explicit decompositions can be made in the following problem formulations: Robust Non-negative Matrix Factorization (RNMF), Robust Matrix Completion (RMC), Robust Subspace Recovery (RSR), Robust Subspace Tracking (RST) and Robust Low-Rank Minimization (RLRM). The main goal of these similar problem formulations is to obtain explicitly or implicitly a decomposition into low-rank matrix plus additive matrices. In this context, this work aims to initiate a rigorous and comprehensive review of the similar problem formulations in robust subspace learning and tracking based on decomposition into low-rank plus additive matrices for testing and ranking existing algorithms for background/foreground separation. For this, we first provide a preliminary review of the recent developments in the different problem formulations which allows us to define a unified view that we called Decomposition into Low-rank plus Additive Matrices (DLAM). Then, we examine carefully each method in each robust subspace learning/tracking frameworks with their decomposition, their loss functions, their optimization problem and their solvers. Furthermore, we investigate if incremental algorithms and real-time implementations can be achieved for background/foreground separation. Finally, experimental results on a large-scale dataset called Background Models Challenge (BMC 2012) show the comparative performance of 32 different robust subspace learning/tracking methods.

Decomposition into Low-rank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a Large-Scale Dataset

Many geometric estimation problems naturally take the form of synchronization over the special Euclidean group: estimate the values of a set of unknown poses \(x_{1},\ldots ,x_n \in \text {SE}(d)\) given noisy measurements of a subset of their pairwise relative transforms \(x_{i}^{-1}x_{j}\). Examples of this class include the foundational problems of pose-graph simultaneous localization and mapping (SLAM) (in robotics) and camera motion estimation (in computer vision), among others. This problem is typically formulated as a nonconvex maximum-likelihood estimation that is computationally hard to solve in general. Nevertheless, in this paper we present an algorithm that is able to effciently recover certifiably globally optimal solutions of the special Euclidean synchronization problem in a non-adversarial noise regime. The crux of our approach is the development of a semidefinite relaxation of the maximum-likelihood estimation whose minimizer provides the exact MLE so long as the magnitude of the noise corrupting the available measurements falls below a certain critical threshold; furthermore, whenever exactness obtains, it is possible to verify this fact a posteriori, thereby certifying the optimality of the recovered estimate. We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem on a low-dimensional Riemannian manifold, and then design a Riemannian truncated-Newton trust-region method to solve this reduction effciently. We combine this fast optimization approach with a simple rounding procedure to produce our algorithm, SE-Sync. Experimental evaluation on a variety of simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable of recovering globally optimal solutions when the available measurements are corrupted by noise up to an order of magnitude greater than that typically encountered in robotics and computer vision applications, and does so more than an order of magnitude faster than the Gauss-Newton-based approach that forms the basis of current state-of-the-art techniques.

SE-Sync: a certifiably correct algorithm for synchronization over the special Euclidean group

Robust principal component analysis (RPCA) via decomposition into low-rank plus sparse matrices offers a powerful framework for a large variety of applications such as image processing, video processing, and 3-D computer vision. Indeed, most of the time these applications require to detect sparse outliers from the observed imagery data that can be approximated by a low-rank matrix. Moreover, most of the time experiments show that RPCA with additional spatial and/or temporal constraints often outperforms the state-of-the-art algorithms in these applications. Thus, the aim of this paper is to survey the applications of RPCA in computer vision. In the first part of this paper, we review representative image processing applications as follows: 1) low-level imaging such as image recovery and denoising, image composition, image colorization, image alignment and rectification, multifocus image, and face recognition; 2) medical imaging such as dynamic magnetic resonance imaging (MRI) for acceleration of data acquisition, background suppression, and learning of interframe motion fields; and 3) imaging for 3-D computer vision with additional depth information such as in structure from motion (SfM) and 3-D motion recovery. In the second part, we present the applications of RPCA in video processing which utilize additional spatial and temporal information compared to image processing. Specifically, we investigate video denoising and restoration, hyperspectral video, and background/foreground separation. Finally, we provide perspectives on possible future research directions and algorithmic frameworks that are suitable for these applications.

/pdf/on-the-applications-of-robust-pca-in-image-and-video-51546ljg1c.pdf

On the Applications of Robust PCA in Image and Video Processing

This paper addresses the problem of rigid-motion synchronization (a.k.a. motion averaging) in the Special Euclidean Group SE(3), which finds application in structure-from-motion and registration of multiple three-dimensional (3D) point-sets. After relaxing the geometric constraints of rigid motions, we derive a simple closed-form solution based on a spectral decomposition, which is then projected onto SE(3). Our formulation is extremely efficient, as rigid-motion synchronization is cast to an eigenvalue decomposition problem. Robustness to outliers is gained through Iteratively Reweighted Least Squares. Besides providing a theoretically appealing solution, since our method recovers at the same time both rotations and translations, we demonstrate through experimental results that our approach is significantly faster than the state of the art, while providing accurate estimates of rigid motions.

Spectral Synchronization of Multiple Views in SE(3)

Robust Rotation Synchronization via Low-rank and Sparse Matrix Decomposition

This paper proposes a robust method to solve the absolute rotation estimation problem, which arises in global registration of 3D point sets and in structure-from-motion. A novel cost function is formulated which inherently copes with outliers. In particular, the proposed algorithm handles both outlier and missing relative rotations, by casting the problem as a "low-rank a sparse" matrix decomposition. As a side effect, this solution can be seen as a valid and cost-effective detector of inconsistent pair wise rotations. Computational efficiency and numerical accuracy, are demonstrated by simulated and real experiments.

http://www.diegm.uniud.it/fusiello/papers/3dv14.pdf

Robust Absolute Rotation Estimation via Low-Rank and Sparse Matrix Decomposition

This paper addresses the problem of motion synchronization (or averaging) and describes a simple, closed-form solution based on a spectral decomposition, which does not consider rotation and translation separately but works straight in SE(3), the manifold of rigid motions. Besides its theoretical interest, being the first closed form solution in SE(3), experimental results show that it compares favourably with the state of the art both in terms of precision and speed.

Spectral Motion Synchronization in SE(3)

This paper casts the global registration of multiple 3D point-sets into a low-rank and sparse decomposition problem. This neat mathematical formulation caters for missing data, outliers and noise, and it benefits from a wealth of available decomposition algorithms that can be plugged-in. Experimental results show that this approach compares favourably to the state of the art in terms of precision and speed, and it outperforms all the analysed techniques as for robustness to outliers.

/pdf/global-registration-of-3d-point-sets-via-lrs-decomposition-j9pnzc3e7d.pdf

Beatrice Rossi

Papers

Spectral Synchronization of Multiple Views in SE(3)

Robust Rotation Synchronization via Low-rank and Sparse Matrix Decomposition

Robust Absolute Rotation Estimation via Low-Rank and Sparse Matrix Decomposition

Spectral Motion Synchronization in SE(3)

Global Registration of 3D Point Sets via LRS Decomposition