Convex Optimization Approaches for Blind Sensor Calibration using Sparsity

We investigate a compressive sensing framework in which the sensors introduce a distortion to the measurements in the form of unknown gains. We focus on blind calibration, using measures performed on multiple unknown (but sparse) signals and formulate the joint recovery of the gains and the sparse signals as a convex optimization problem. We divide this problem in 3 subproblems with different conditions on the gains, specifially (i) gains with different amplitude and the same phase, (ii) gains with the same amplitude and different phase and (iii) gains with different amplitude and phase. In order to solve the first case, we propose an extension to the basis pursuit optimization which can estimate the unknown gains along with the unknown sparse signals. For the second case, we formulate a quadratic approach that eliminates the unknown phase shifts and retrieves the unknown sparse signals. An alternative form of this approach is also formulated to reduce complexity and memory requirements and provide scalability with respect to the number of input signals. Finally for the third case, we propose a formulation that combines the earlier two approaches to solve the problem. The performance of the proposed algorithms is investigated extensively through numerical simulations, which demonstrates that simultaneous signal recovery and calibration is possible with convex methods when sufficiently many (unknown, but sparse) calibrating signals are provided

Bilinear inverse problems with sparsity: Optimal identifiability conditions and efficient recovery

Bilinear inverse problems (BIPs), the resolution of two vectors given their image under a bilinear mapping, arise in many applications. Without further constraints, BIPs are usually ill-posed. In practice, parsimonious structures of natural signals (e.g., subspace or sparsity) are exploited. However, there are few theoretical justifications for using such structures for BIPs. We consider two types of BIPs, blind deconvolution (BD) and blind gain and phase calibration (BGPC), with subspace or sparsity structures. Our contributions are twofold: we derive optimal identifiability conditions, and propose efficient algorithms that solve these problems. In previous work, we provided the first algebraic sample complexities for BD that hold for Lebesgue almost all bases or frames. We showed that for BD of a pair of vectors in \bbC^n, with subspace constraints of dimensions $m_1$ and $m_2$, respectively, a sample complexity of $n\geq m_1m_2$ is sufficient. This result is suboptimal, since the number of degrees of freedom is merely $m_1+m_2-1$. We provided analogous results, with similar suboptimality, for BD with sparsity or mixed subspace and sparsity constraints. In Chapter 2, taking advantage of the recent progress on the information-theoretic limits of unique low-rank matrix recovery, we finally bridge this gap, and derive an optimal sample complexity result for BD with generic bases or frames. We show that for BD of an arbitrary pair (resp. all pairs) of vectors in \bbC^n, with sparsity constraints of sparsity levels $s_1$ and $s_2$, a sample complexity of $n > s_1+s_2$ (resp. $n > 2(s_1+s_2)$) is sufficient. We also present analogous results for BD with subspace constraints or mixed constraints, with the subspace dimension replacing the sparsity level. Last but not least, in all the above scenarios, if the bases or frames follow a probabilistic distribution specified in Chapter 2, the recovery is not only unique, but also stable against small perturbations in the measurements, under the same sample complexities. In previous work, we proposed studying the identifiability in bilinear inverse problems up to transformation groups. In particular, we studied several special cases of blind gain and phase calibration, including the cases of subspace and joint sparsity models on the signals, and gave sufficient and necessary conditions for identifiability up to certain transformation groups. However, there were gaps between the sample complexities in the sufficient conditions and the necessary conditions. In Chapter 3, under a mild assumption that the signals and models are generic, we bridge the gaps by deriving tight sufficient conditions with optimal or near optimal sample complexities. Recently there has been renewed interest in solutions to BGPC with careful analysis of error bounds. In Chapter 4, we formulate BGPC as an eigenvalue/eigenvector problem, and propose to solve it via power iteration, or in the sparsity or joint sparsity case, via truncated power iteration (which we show is equivalent to a sparsity-projected gradient descent). Under certain assumptions, the unknown gains, phases, and the unknown signal can be recovered simultaneously. Numerical experiments show that power iteration algorithms work not only in the regime predicted by our main results, but also in regimes where theoretical analysis is limited. We also show that our power iteration algorithms for BGPC compare favorably with competing algorithms in adversarial conditions, e.g., with noisy measurement or with a bad initial estimate. A problem related to BGPC is multichannel blind deconvolution (MBD) with a circular convolution model, i.e., the recovery of an unknown signal $f$ and multiple unknown filters $x_i$ from circular convolutions $y_i=x_i \circledast f$ ($i=1,2,\dots,N$). In Chapter 5, we consider the case where the $x_i$'s are sparse, and convolution with $f$ is invertible. Our nonconvex optimization formulation solves for a filter $h$ on the unit sphere that produces sparse outputs $y_i\circledast h$. Under some technical assumptions, we show that all local minima of the objective function correspond to the inverse filter of $f$ up to an inherent sign and shift ambiguity, and all saddle points have strictly negative curvatures. This geometric structure allows successful recovery of $f$ and $x_i$ using a simple manifold gradient descent algorithm with random initialization. Our theoretical findings are complemented by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods

Computational Imaging and Artificial Intelligence: The Next Revolution of Mobile Vision

Signal capture stands in the forefront to perceive and understand the environment and thus imaging plays the pivotal role in mobile vision. Recent explosive progresses in Artificial Intelligence (AI) have shown great potential to develop advanced mobile platforms with new imaging devices. Traditional imaging systems based on the "capturing images first and processing afterwards" mechanism cannot meet this unprecedented demand. Differently, Computational Imaging (CI) systems are designed to capture high-dimensional data in an encoded manner to provide more information for mobile vision systems.Thanks to AI, CI can now be used in real systems by integrating deep learning algorithms into the mobile vision platform to achieve the closed loop of intelligent acquisition, processing and decision making, thus leading to the next revolution of mobile vision.Starting from the history of mobile vision using digital cameras, this work first introduces the advances of CI in diverse applications and then conducts a comprehensive review of current research topics combining CI and AI. Motivated by the fact that most existing studies only loosely connect CI and AI (usually using AI to improve the performance of CI and only limited works have deeply connected them), in this work, we propose a framework to deeply integrate CI and AI by using the example of self-driving vehicles with high-speed communication, edge computing and traffic planning. Finally, we outlook the future of CI plus AI by investigating new materials, brain science and new computing techniques to shed light on new directions of mobile vision systems

Sparse and Redundant Representations for Inverse Problems and Recognition

Sparse and redundant representation of data enables the description of signals as linear combinations of a few atoms from a dictionary. In this dissertation, we study applications of sparse and redundant representations in inverse problems and object recognition. Furthermore, we propose two novel imaging modalities based on the recently introduced theory of Compressed Sensing (CS). This dissertation consists of four major parts. In the first part of the dissertation, we study a new type of deconvolution algorithm that is based on estimating the image from a shearlet decomposition. Shearlets provide a multi-directional and multi-scale decomposition that has been mathematically shown to represent distributed discontinuities such as edges better than traditional wavelets. We develop a deconvolution algorithm that allows for the approximation inversion operator to be controlled on a multi-scale and multi-directional basis. Furthermore, we develop a method for the automatic determination of the threshold values for the noise shrinkage for each scale and direction without explicit knowledge of the noise variance using a generalized cross validation method. In the second part of the dissertation, we study a reconstruction method that recovers highly undersampled images assumed to have a sparse representation in a gradient domain by using partial measurement samples that are collected in the Fourier domain. Our method makes use of a robust generalized Poisson solver that greatly aids in achieving a significantly improved performance over similar proposed methods. We will demonstrate by experiments that this new technique is more flexible to work with either random or restricted sampling scenarios better than its competitors. In the third part of the dissertation, we introduce a novel Synthetic Aperture Radar (SAR) imaging modality which can provide a high resolution map of the spatial distribution of targets and terrain using a significantly reduced number of needed transmitted and/or received electromagnetic waveforms. We demonstrate that this new imaging scheme, requires no new hardware components and allows the aperture to be compressed. Also, it presents many new applications and advantages which include strong resistance to countermesasures and interception, imaging much wider swaths and reduced on-board storage requirements. The last part of the dissertation deals with object recognition based on learning dictionaries for simultaneous sparse signal approximations and feature extraction. A dictionary is learned for each object class based on given training examples which minimize the representation error with a sparseness constraint. A novel test image is then projected onto the span of the atoms in each learned dictionary. The residual vectors along with the coefficients are then used for recognition. Applications to illumination robust face recognition and automatic target recognition are presented

Widely Distributed Radar Imaging: Unmediated ADMM Based Approach

This paper presents a novel approach to reconstruct a unique image of an observed scene via synthetic apertures (SA) generated by employing widely distributed radar sensors. The problem is posed as a constrained optimization problem in which the global image which represents the aggregate view of the sensors is a decision variable. While the problem is designed to promote a sparse solution for the global image, it is constrained such that a relationship with local images that can be reconstructed using the measurements at each sensor is respected. Two problem formulations are introduced by stipulating two different establishments of that relationship. The proposed formulations are designed according to consensus ADMM (CADMM) and sharing ADMM (SADMM), and their solutions are provided accordingly as iterative algorithms. We drive the explicit variable updates for each algorithm in addition to the recommended scheme for hybrid parallel implementation on the distributed sensors and a central processing unit. Our algorithms are validated and their performance is evaluated by exploiting the Civilian Vehicles Dome dataset to realize different scenarios of practical relevance. Experimental results show the effectiveness of the proposed algorithms, especially in cases with limited measurements