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

Recent Progress in Image Deblurring

This paper comprehensively reviews the recent development of image deblurring, including non-blind/blind, spatially invariant/variant deblurring techniques. Indeed, these techniques share the same objective of inferring a latent sharp image from one or several corresponding blurry images, while the blind deblurring techniques are also required to derive an accurate blur kernel. Considering the critical role of image restoration in modern imaging systems to provide high-quality images under complex environments such as motion, undesirable lighting conditions, and imperfect system components, image deblurring has attracted growing attention in recent years. From the viewpoint of how to handle the ill-posedness which is a crucial issue in deblurring tasks, existing methods can be grouped into five categories: Bayesian inference framework, variational methods, sparse representation-based methods, homography-based modeling, and region-based methods. In spite of achieving a certain level of development, image deblurring, especially the blind case, is limited in its success by complex application conditions which make the blur kernel hard to obtain and be spatially variant. We provide a holistic understanding and deep insight into image deblurring in this review. An analysis of the empirical evidence for representative methods, practical issues, as well as a discussion of promising future directions are also presented.Comment: 53 pages, 17 figure

Optimization for Probabilistic Machine Learning

We have access to great variety of datasets more than any time in the history. Everyday, more data is collected from various natural resources and digital platforms. Great advances in the area of machine learning research in the past few decades have relied strongly on availability of these datasets. However, analyzing them imposes significant challenges that are mainly due to two factors. First, the datasets have complex structures with hidden interdependencies. Second, most of the valuable datasets are high dimensional and are largely scaled. The main goal of a machine learning framework is to design a model that is a valid representative of the observations and develop a learning algorithm to make inference about unobserved or latent data based on the observations. Discovering hidden patterns and inferring latent characteristics in such datasets is one of the greatest challenges in the area of machine learning research. In this dissertation, I will investigate some of the challenges in modeling and algorithm design, and present my research results on how to overcome these obstacles. Analyzing data generally involves two main stages. The first stage is designing a model that is flexible enough to capture complex variation and latent structures in data and is robust enough to generalize well to the unseen data. Designing an expressive and interpretable model is one of crucial objectives in this stage. The second stage involves training learning algorithm on the observed data and measuring the accuracy of model and learning algorithm. This stage usually involves an optimization problem whose objective is to tune the model to the training data and learn the model parameters. Finding global optimal or sufficiently good local optimal solution is one of the main challenges in this step. Probabilistic models are one of the best known models for capturing data generating process and quantifying uncertainties in data using random variables and probability distributions. They are powerful models that are shown to be adaptive and robust and can scale well to large datasets. However, most probabilistic models have a complex structure. Training them could become challenging commonly due to the presence of intractable integrals in the calculation. To remedy this, they require approximate inference strategies that often results in non-convex optimization problems. The optimization part ensures that the model is the best representative of data or data generating process. The non-convexity of an optimization problem take away the general guarantee on finding a global optimal solution. It will be shown later in this dissertation that inference for a significant number of probabilistic models require solving a non-convex optimization problem. One of the well-known methods for approximate inference in probabilistic modeling is variational inference. In the Bayesian setting, the target is to learn the true posterior distribution for model parameters given the observations and prior distributions. The main challenge involves marginalization of all the other variables in the model except for the variable of interest. This high-dimensional integral is generally computationally hard, and for many models there is no known polynomial time algorithm for calculating them exactly. Variational inference deals with finding an approximate posterior distribution for Bayesian models where finding the true posterior distribution is analytically or numerically impossible. It assumes a family of distribution for the estimation, and finds the closest member of that family to the true posterior distribution using a distance measure. For many models though, this technique requires solving a non-convex optimization problem that has no general guarantee on reaching a global optimal solution. This dissertation presents a convex relaxation technique for dealing with hardness of the optimization involved in the inference. The proposed convex relaxation technique is based on semidefinite optimization that has a general applicability to polynomial optimization problem. I will present theoretical foundations and in-depth details of this relaxation in this work. Linear dynamical systems represent the functionality of many real-world physical systems. They can describe the dynamics of a linear time-varying observation which is controlled by a controller unit with quadratic cost function objectives. Designing distributed and decentralized controllers is the goal of many of these systems, which computationally, results in a non-convex optimization problem. In this dissertation, I will further investigate the issues arising in this area and develop a convex relaxation framework to deal with the optimization challenges. Setting the correct number of model parameters is an important aspect for a good probabilistic model. If there are only a few parameters, model may lack capturing all the essential relations and components in the observations while too many parameters may cause significant complications in learning or overfit to the observations. Non-parametric models are suitable techniques to deal with this issue. They allow the model to learn the appropriate number of parameters to describe the data and make predictions. In this dissertation, I will present my work on designing Bayesian non-parametric models as powerful tools for learning representations of data. Moreover, I will describe the algorithm that we derived to efficiently train the model on the observations and learn the number of model parameters. Later in this dissertation, I will present my works on designing probabilistic models in combination with deep learning methods for representing sequential data. Sequential datasets comprise a significant portion of resources in the area of machine learning research. Designing models to capture dependencies in sequential datasets are of great interest and have a wide variety of applications in engineering, medicine and statistics. Recent advances in deep learning research has shown exceptional promises in this area. However, they lack interpretability in their general form. To remedy this, I will present my work on mixing probabilistic models with neural network models that results in better performance and expressiveness of the results

Proximal Methods for Hierarchical Sparse Coding

Sparse coding consists in representing signals as sparse linear combinations of atoms selected from a dictionary. We consider an extension of this framework where the atoms are further assumed to be embedded in a tree. This is achieved using a recently introduced tree-structured sparse regularization norm, which has proven useful in several applications. This norm leads to regularized problems that are difficult to optimize, and we propose in this paper efficient algorithms for solving them. More precisely, we show that the proximal operator associated with this norm is computable exactly via a dual approach that can be viewed as the composition of elementary proximal operators. Our procedure has a complexity linear, or close to linear, in the number of atoms, and allows the use of accelerated gradient techniques to solve the tree-structured sparse approximation problem at the same computational cost as traditional ones using the L1-norm. Our method is efficient and scales gracefully to millions of variables, which we illustrate in two types of applications: first, we consider fixed hierarchical dictionaries of wavelets to denoise natural images. Then, we apply our optimization tools in the context of dictionary learning, where learned dictionary elements naturally organize in a prespecified arborescent structure, leading to a better performance in reconstruction of natural image patches. When applied to text documents, our method learns hierarchies of topics, thus providing a competitive alternative to probabilistic topic models

Deep Learning for Single Image Super-Resolution: A Brief Review

Single image super-resolution (SISR) is a notoriously challenging ill-posed problem, which aims to obtain a high-resolution (HR) output from one of its low-resolution (LR) versions. To solve the SISR problem, recently powerful deep learning algorithms have been employed and achieved the state-of-the-art performance. In this survey, we review representative deep learning-based SISR methods, and group them into two categories according to their major contributions to two essential aspects of SISR: the exploration of efficient neural network architectures for SISR, and the development of effective optimization objectives for deep SISR learning. For each category, a baseline is firstly established and several critical limitations of the baseline are summarized. Then representative works on overcoming these limitations are presented based on their original contents as well as our critical understandings and analyses, and relevant comparisons are conducted from a variety of perspectives. Finally we conclude this review with some vital current challenges and future trends in SISR leveraging deep learning algorithms.Comment: Accepted by IEEE Transactions on Multimedia (TMM

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