Statistical Models and Optimization Algorithms for High-Dimensional Computer Vision Problems

Data-driven and computational approaches are showing significant promise in solving several challenging problems in various fields such as bioinformatics, finance and many branches of engineering. In this dissertation, we explore the potential of these approaches, specifically statistical data models and optimization algorithms, for solving several challenging problems in computer vision. In doing so, we contribute to the literatures of both statistical data models and computer vision. In the context of statistical data models, we propose principled approaches for solving robust regression problems, both linear and kernel, and missing data matrix factorization problem. In computer vision, we propose statistically optimal and efficient algorithms for solving the remote face recognition and structure from motion (SfM) problems. The goal of robust regression is to estimate the functional relation between two variables from a given data set which might be contaminated with outliers. Under the reasonable assumption that there are fewer outliers than inliers in a data set, we formulate the robust linear regression problem as a sparse learning problem, which can be solved using efficient polynomial-time algorithms. We also provide sufficient conditions under which the proposed algorithms correctly solve the robust regression problem. We then extend our robust formulation to the case of kernel regression, specifically to propose a robust version for relevance vector machine (RVM) regression. Matrix factorization is used for finding a low-dimensional representation for data embedded in a high-dimensional space. Singular value decomposition is the standard algorithm for solving this problem. However, when the matrix has many missing elements this is a hard problem to solve. We formulate the missing data matrix factorization problem as a low-rank semidefinite programming problem (essentially a rank constrained SDP), which allows us to find accurate and efficient solutions for large-scale factorization problems. Face recognition from remotely acquired images is a challenging problem because of variations due to blur and illumination. Using the convolution model for blur, we show that the set of all images obtained by blurring a given image forms a convex set. We then use convex optimization techniques to find the distances between a given blurred (probe) image and the gallery images to find the best match. Further, using a low-dimensional linear subspace model for illumination variations, we extend our theory in a similar fashion to recognize blurred and poorly illuminated faces. Bundle adjustment is the final optimization step of the SfM problem where the goal is to obtain the 3-D structure of the observed scene and the camera parameters from multiple images of the scene. The traditional bundle adjustment algorithm, based on minimizing the l_2 norm of the image re-projection error, has cubic complexity in the number of unknowns. We propose an algorithm, based on minimizing the l_infinity norm of the re-projection error, that has quadratic complexity in the number of unknowns. This is achieved by reducing the large-scale optimization problem into many small scale sub-problems each of which can be solved using second-order cone programming

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

Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches

Imaging spectrometers measure electromagnetic energy scattered in their instantaneous field view in hundreds or thousands of spectral channels with higher spectral resolution than multispectral cameras. Imaging spectrometers are therefore often referred to as hyperspectral cameras (HSCs). Higher spectral resolution enables material identification via spectroscopic analysis, which facilitates countless applications that require identifying materials in scenarios unsuitable for classical spectroscopic analysis. Due to low spatial resolution of HSCs, microscopic material mixing, and multiple scattering, spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus, accurate estimation requires unmixing. Pixels are assumed to be mixtures of a few materials, called endmembers. Unmixing involves estimating all or some of: the number of endmembers, their spectral signatures, and their abundances at each pixel. Unmixing is a challenging, ill-posed inverse problem because of model inaccuracies, observation noise, environmental conditions, endmember variability, and data set size. Researchers have devised and investigated many models searching for robust, stable, tractable, and accurate unmixing algorithms. This paper presents an overview of unmixing methods from the time of Keshava and Mustard's unmixing tutorial [1] to the present. Mixing models are first discussed. Signal-subspace, geometrical, statistical, sparsity-based, and spatial-contextual unmixing algorithms are described. Mathematical problems and potential solutions are described. Algorithm characteristics are illustrated experimentally.Comment: This work has been accepted for publication in IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensin

Sparse Modeling for Image and Vision Processing

In recent years, a large amount of multi-disciplinary research has been conducted on sparse models and their applications. In statistics and machine learning, the sparsity principle is used to perform model selection---that is, automatically selecting a simple model among a large collection of them. In signal processing, sparse coding consists of representing data with linear combinations of a few dictionary elements. Subsequently, the corresponding tools have been widely adopted by several scientific communities such as neuroscience, bioinformatics, or computer vision. The goal of this monograph is to offer a self-contained view of sparse modeling for visual recognition and image processing. More specifically, we focus on applications where the dictionary is learned and adapted to data, yielding a compact representation that has been successful in various contexts.Comment: 205 pages, to appear in Foundations and Trends in Computer Graphics and Visio

Optimization with Sparsity-Inducing Penalties

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel selection. It turns out that many of the related estimation problems can be cast as convex optimization problems by regularizing the empirical risk with appropriate non-smooth norms. The goal of this paper is to present from a general perspective optimization tools and techniques dedicated to such sparsity-inducing penalties. We cover proximal methods, block-coordinate descent, reweighted $\ell_2$-penalized techniques, working-set and homotopy methods, as well as non-convex formulations and extensions, and provide an extensive set of experiments to compare various algorithms from a computational point of view