Bayesian Robust Tensor Factorization for Incomplete Multiway Data

We propose a generative model for robust tensor factorization in the presence of both missing data and outliers. The objective is to explicitly infer the underlying low-CP-rank tensor capturing the global information and a sparse tensor capturing the local information (also considered as outliers), thus providing the robust predictive distribution over missing entries. The low-CP-rank tensor is modeled by multilinear interactions between multiple latent factors on which the column sparsity is enforced by a hierarchical prior, while the sparse tensor is modeled by a hierarchical view of Student-$t$ distribution that associates an individual hyperparameter with each element independently. For model learning, we develop an efficient closed-form variational inference under a fully Bayesian treatment, which can effectively prevent the overfitting problem and scales linearly with data size. In contrast to existing related works, our method can perform model selection automatically and implicitly without need of tuning parameters. More specifically, it can discover the groundtruth of CP rank and automatically adapt the sparsity inducing priors to various types of outliers. In addition, the tradeoff between the low-rank approximation and the sparse representation can be optimized in the sense of maximum model evidence. The extensive experiments and comparisons with many state-of-the-art algorithms on both synthetic and real-world datasets demonstrate the superiorities of our method from several perspectives.Comment: in IEEE Transactions on Neural Networks and Learning Systems, 201

Robust Subspace Estimation via Low-Rank and Sparse Decomposition and Applications in Computer Vision

PhDRecent advances in robust subspace estimation have made dimensionality reduction and noise and outlier suppression an area of interest for research, along with continuous improvements in computer vision applications. Due to the nature of image and video signals that need a high dimensional representation, often storage, processing, transmission, and analysis of such signals is a difficult task. It is therefore desirable to obtain a low-dimensional representation for such signals, and at the same time correct for corruptions, errors, and outliers, so that the signals could be readily used for later processing. Major recent advances in low-rank modelling in this context were initiated by the work of Cand`es et al. [17] where the authors provided a solution for the long-standing problem of decomposing a matrix into low-rank and sparse components in a Robust Principal Component Analysis (RPCA) framework. However, for computer vision applications RPCA is often too complex, and/or may not yield desirable results. The low-rank component obtained by the RPCA has usually an unnecessarily high rank, while in certain tasks lower dimensional representations are required. The RPCA has the ability to robustly estimate noise and outliers and separate them from the low-rank component, by a sparse part. But, it has no mechanism of providing an insight into the structure of the sparse solution, nor a way to further decompose the sparse part into a random noise and a structured sparse component that would be advantageous in many computer vision tasks. As videos signals are usually captured by a camera that is moving, obtaining a low-rank component by RPCA becomes impossible. In this thesis, novel Approximated RPCA algorithms are presented, targeting different shortcomings of the RPCA. The Approximated RPCA was analysed to identify the most time consuming RPCA solutions, and replace them with simpler yet tractable alternative solutions. The proposed method is able to obtain the exact desired rank for the low-rank component while estimating a global transformation to describe camera-induced motion. Furthermore, it is able to decompose the sparse part into a foreground sparse component, and a random noise part that contains no useful information for computer vision processing. The foreground sparse component is obtained by several novel structured sparsity-inducing norms, that better encapsulate the needed pixel structure in visual signals. Moreover, algorithms for reducing complexity of low-rank estimation have been proposed that achieve significant complexity reduction without sacrificing the visual representation of video and image information. The proposed algorithms are applied to several fundamental computer vision tasks, namely, high efficiency video coding, batch image alignment, inpainting, and recovery, video stabilisation, background modelling and foreground segmentation, robust subspace clustering and motion estimation, face recognition, and ultra high definition image and video super-resolution. The algorithms proposed in this thesis including batch image alignment and recovery, background modelling and foreground segmentation, robust subspace clustering and motion segmentation, and ultra high definition image and video super-resolution achieve either state-of-the-art or comparable results to existing methods

Clutter Suppression in Ultrasound: Performance Evaluation of Low-Rank and Sparse Matrix Decomposition Methods

Vessel diseases are often accompanied by abnormalities related to vascular shape and size. Therefore, a clear visualization of vasculature is of high clinical significance. Ultrasound Color Flow Imaging (CFI) is one of the prominent techniques for flow visualization. However, clutter signals originating from slow-moving tissue is one of the main obstacles to obtain a clear view of the vascular network. Enhancement of the vasculature by suppressing the clutters is an essential step for many applications of ultrasound CFI. In this thesis, we focus on a state-of-art algorithm framework called Decomposition into Low-rank and Sparse Matrices (DLSM) framework for ultrasound clutter suppression. Currently, ultrasound clutter suppression is often performed by Singular Value Decomposition (SVD) of the data matrix, which is a branch of eigen-based filtering. This approach exhibits two well-known limitations. First, the performance of SVD is sensitive to the proper manual selection of the ranks corresponding to clutter and blood subspaces. Second, SVD is prone to failure in the presence of large random noise in the data set. A potential solution to these issues is the use of DLSM framework. SVD, as a means for singular values, is also one of the widely used algorithms for solving the minimization problem under the DLSM framework. Many other algorithms under DLSM avoid full SVD and use approximated SVD or SVD-free ideas which may have better performance with higher robustness and lower computing time due to the expensive computational cost of full SVD. In practice, these models separate blood from clutter based on the assumption that steady clutter represents a low-rank structure and the moving blood component is sparse. In this thesis, we exploit the feasibility of exploiting low-rank and sparse decomposition schemes, originally developed in the field of computer vision, in ultrasound clutter suppression. Since ultrasound images have different texture and statistical properties compared to images in computer vision, it is of high importance to evaluate how these methods translate to ultrasound CFI. We conduct this evaluation study by adapting 106 DLSM algorithms and validating them against simulation, phantom and in vivo rat data sets. The advantage of simulation and phantom experiments is that the ground truth vessel map is known, and the advantage of the in vivo data set is that it enables us to test algorithms in a realistic setting. Two conventional quality metrics, Signal-to-Noise Ratio (SNR) and Contrast-to-Noise Ratio (CNR), are used for performance evaluation. In addition, computation times required by different algorithms for generating the clutter suppressed images are reported. Our extensive analysis shows that the DLSM framework can be successfully applied to ultrasound clutter suppression