Joint Sensing Matrix and Sparsifying Dictionary Optimization for Tensor Compressive Sensing.

Tensor compressive sensing (TCS) is a multidimensional framework of compressive sensing (CS), and it is advantageous in terms of reducing the amount of storage, easing hardware implementations, and preserving multidimensional structures of signals in comparison to a conventional CS system. In a TCS system, instead of using a random sensing matrix and a predefined dictionary, the average-case performance can be further improved by employing an optimized multidimensional sensing matrix and a learned multilinear sparsifying dictionary. In this paper, we propose an approach that jointly optimizes the sensing matrix and dictionary for a TCS system. For the sensing matrix design in TCS, an extended separable approach with a closed form solution and a novel iterative nonseparable method are proposed when the multilinear dictionary is fixed. In addition, a multidimensional dictionary learning method that takes advantages of the multidimensional structure is derived, and the influence of sensing matrices is taken into account in the learning process. A joint optimization is achieved via alternately iterating the optimization of the sensing matrix and dictionary. Numerical experiments using both synthetic data and real images demonstrate the superiority of the proposed approache

Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)

The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist1

Nonlocal tensor sparse representation and low-rank regularization for hyperspectral image compressive sensing reconstruction

Hyperspectral image compressive sensing reconstruction (HSI-CSR) is an important issue in remote sensing, and has recently been investigated increasingly by the sparsity prior based approaches. However, most of the available HSI-CSR methods consider the sparsity prior in spatial and spectral vector domains via vectorizing hyperspectral cubes along a certain dimension. Besides, in most previous works, little attention has been paid to exploiting the underlying nonlocal structure in spatial domain of the HSI. In this paper, we propose a nonlocal tensor sparse and low-rank regularization (NTSRLR) approach, which can encode essential structured sparsity of an HSI and explore its advantages for HSI-CSR task. Specifically, we study how to utilize reasonably the l1 -based sparsity of core tensor and tensor nuclear norm function as tensor sparse and low-rank regularization, respectively, to describe the nonlocal spatial-spectral correlation hidden in an HSI. To study the minimization problem of the proposed algorithm, we design a fast implementation strategy based on the alternative direction multiplier method (ADMM) technique. Experimental results on various HSI datasets verify that the proposed HSI-CSR algorithm can significantly outperform existing state-of-the-art CSR techniques for HSI recovery

Using the Proximal Gradient and the Accelerated Proximal Gradient as a Canonical Polyadic tensor decomposition Algorithms in difficult situations

Canonical Polyadic (CP) tensor decomposition is useful in many real-world applications due to its uniqueness, and the ease of interpretation of its factor matrices. This work addresses the problem of calculating the CP decomposition of tensors in difficult cases where the factor matrices in one or all modes are almost collinear - i.e. bottleneck or swamp problems arise. This is done by introducing a constraint on the coherences of the factor matrices that ensures the existence of a best low-rank approximation, which makes it possible to estimate these highly correlated factors. Two new algorithms optimizing the CP decomposition based on proximal methods are proposed. Simulation results are provided and demonstrate the good behaviour of these algorithms, as well as a better compromise between accuracy and convergence speed than other algorithms in the literature

Structured Sub-Nyquist Sampling with Applications in Compressive Toeplitz Covariance Estimation, Super-Resolution and Phase Retrieval

Sub-Nyquist sampling has received a huge amount of interest in the past decade. In classical compressed sensing theory, if the measurement procedure satisfies a particular condition known as Restricted Isometry Property (RIP), we can achieve stable recovery of signals of low-dimensional intrinsic structures with an order-wise optimal sample size. Such low-dimensional structures include sparse and low rank for both vector and matrix cases. The main drawback of conventional compressed sensing theory is that random measurements are required to ensure the RIP property. However, in many applications such as imaging and array signal processing, applying independent random measurements may not be practical as the systems are deterministic. Moreover, random measurements based compressed sensing always exploits convex programs for signal recovery even in the noiseless case, and solving those programs is computationally intensive if the ambient dimension is large, especially in the matrix case. The main contribution of this dissertation is that we propose a deterministic sub-Nyquist sampling framework for compressing the structured signal and come up with computationally efficient algorithms. Besides widely studied sparse and low-rank structures, we particularly focus on the cases that the signals of interest are stationary or the measurements are of Fourier type. The key difference between our work from classical compressed sensing theory is that we explicitly exploit the second-order statistics of the signals, and study the equivalent quadratic measurement model in the correlation domain. The essential observation made in this dissertation is that a difference/sum coarray structure will arise from the quadratic model if the measurements are of Fourier type. With these observations, we are able to achieve a better compression rate for covariance estimation, identify more sources in array signal processing or recover the signals of larger sparsity. In this dissertation, we will first study the problem of Toeplitz covariance estimation. In particular, we will show how to achieve an order-wise optimal compression rate using the idea of sparse arrays in both general and low-rank cases. Then, an analysis framework of super-resolution with positivity constraint is established. We will present fundamental robustness guarantees, efficient algorithms and applications in practices. Next, we will study the problem of phase-retrieval for which we successfully apply the sparse array ideas by fully exploiting the quadratic measurement model. We achieve near-optimal sample complexity for both sparse and general cases with practical Fourier measurements and provide efficient and deterministic recovery algorithms. In the end, we will further elaborate on the essential role of non-negative constraint in underdetermined inverse problems. In particular, we will analyze the nonlinear co-array interpolation problem and develop a universal upper bound of the interpolation error. Bilinear problem with non-negative constraint will be considered next and the exact characterization of the ambiguous solutions will be established for the first time in literature. At last, we will show how to apply the nested array idea to solve real problems such as Kriging. Using spatial correlation information, we are able to have a stable estimate of the field of interest with fewer sensors than classic methodologies. Extensive numerical experiments are implemented to demonstrate our theoretical claims

Occlusion reasoning for multiple object visual tracking

Thesis (Ph.D.)--Boston UniversityOcclusion reasoning for visual object tracking in uncontrolled environments is a challenging problem. It becomes significantly more difficult when dense groups of indistinguishable objects are present in the scene that cause frequent inter-object interactions and occlusions. We present several practical solutions that tackle the inter-object occlusions for video surveillance applications. In particular, this thesis proposes three methods. First, we propose "reconstruction-tracking," an online multi-camera spatial-temporal data association method for tracking large groups of objects imaged with low resolution. As a variant of the well-known Multiple-Hypothesis-Tracker, our approach localizes the positions of objects in 3D space with possibly occluded observations from multiple camera views and performs temporal data association in 3D. Second, we develop "track linking," a class of offline batch processing algorithms for long-term occlusions, where the decision has to be made based on the observations from the entire tracking sequence. We construct a graph representation to characterize occlusion events and propose an efficient graph-based/combinatorial algorithm to resolve occlusions. Third, we propose a novel Bayesian framework where detection and data association are combined into a single module and solved jointly. Almost all traditional tracking systems address the detection and data association tasks separately in sequential order. Such a design implies that the output of the detector has to be reliable in order to make the data association work. Our framework takes advantage of the often complementary nature of the two subproblems, which not only avoids the error propagation issue from which traditional "detection-tracking approaches" suffer but also eschews common heuristics such as "nonmaximum suppression" of hypotheses by modeling the likelihood of the entire image. The thesis describes a substantial number of experiments, involving challenging, notably distinct simulated and real data, including infrared and visible-light data sets recorded ourselves or taken from data sets publicly available. In these videos, the number of objects ranges from a dozen to a hundred per frame in both monocular and multiple views. The experiments demonstrate that our approaches achieve results comparable to those of state-of-the-art approaches