Sequential Compressed Sensing

Compressed sensing allows perfect recovery of sparse signals (or signals sparse in some basis) using only a small number of random measurements. Existing results in compressed sensing literature have focused on characterizing the achievable performance by bounding the number of samples required for a given level of signal sparsity. However, using these bounds to minimize the number of samples requires a-priori knowledge of the sparsity of the unknown signal, or the decay structure for near-sparse signals. Furthermore, there are some popular recovery methods for which no such bounds are known. In this paper, we investigate an alternative scenario where observations are available in sequence. For any recovery method, this means that there is now a sequence of candidate reconstructions. We propose a method to estimate the reconstruction error directly from the samples themselves, for every candidate in this sequence. This estimate is universal in the sense that it is based only on the measurement ensemble, and not on the recovery method or any assumed level of sparsity of the unknown signal. With these estimates, one can now stop observations as soon as there is reasonable certainty of either exact or sufficiently accurate reconstruction. They also provide a way to obtain "run-time" guarantees for recovery methods that otherwise lack a-priori performance bounds. We investigate both continuous (e.g. Gaussian) and discrete (e.g. Bernoulli) random measurement ensembles, both for exactly sparse and general near-sparse signals, and with both noisy and noiseless measurements.Comment: to appear in IEEE transactions on Special Topics in Signal Processin

One-bit compressive sensing with norm estimation

Consider the recovery of an unknown signal ${x}$ from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that ${x}$ is sparse, and that the measurements are of the form $\operatorname{sign}(\langle {a}_i, {x} \rangle) \in \{\pm1\}$. Since such measurements give no information on the norm of ${x}$, recovery methods from such measurements typically assume that $\| {x} \|_2=1$. We show that if one allows more generally for quantized affine measurements of the form $\operatorname{sign}(\langle {a}_i, {x} \rangle + b_i)$, and if the vectors ${a}_i$ are random, an appropriate choice of the affine shifts $b_i$ allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. Additionally, we show that for arbitrary fixed ${x}$ in the annulus $r \leq \| {x} \|_2 \leq R$, one may estimate the norm $\| {x} \|_2$ up to additive error $\delta$ from $m \gtrsim R^4 r^{-2} \delta^{-2}$ such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors, in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors ${x}$ within a Euclidean ball of known radius.Comment: 20 pages, 2 figure

System Level Synthesis

This article surveys the System Level Synthesis framework, which presents a novel perspective on constrained robust and optimal controller synthesis for linear systems. We show how SLS shifts the controller synthesis task from the design of a controller to the design of the entire closed loop system, and highlight the benefits of this approach in terms of scalability and transparency. We emphasize two particular applications of SLS, namely large-scale distributed optimal control and robust control. In the case of distributed control, we show how SLS allows for localized controllers to be computed, extending robust and optimal control methods to large-scale systems under practical and realistic assumptions. In the case of robust control, we show how SLS allows for novel design methodologies that, for the first time, quantify the degradation in performance of a robust controller due to model uncertainty -- such transparency is key in allowing robust control methods to interact, in a principled way, with modern techniques from machine learning and statistical inference. Throughout, we emphasize practical and efficient computational solutions, and demonstrate our methods on easy to understand case studies.Comment: To appear in Annual Reviews in Contro

A Distributed Frank-Wolfe Algorithm for Communication-Efficient Sparse Learning

Learning sparse combinations is a frequent theme in machine learning. In this paper, we study its associated optimization problem in the distributed setting where the elements to be combined are not centrally located but spread over a network. We address the key challenges of balancing communication costs and optimization errors. To this end, we propose a distributed Frank-Wolfe (dFW) algorithm. We obtain theoretical guarantees on the optimization error $\epsilon$ and communication cost that do not depend on the total number of combining elements. We further show that the communication cost of dFW is optimal by deriving a lower-bound on the communication cost required to construct an $\epsilon$-approximate solution. We validate our theoretical analysis with empirical studies on synthetic and real-world data, which demonstrate that dFW outperforms both baselines and competing methods. We also study the performance of dFW when the conditions of our analysis are relaxed, and show that dFW is fairly robust.Comment: Extended version of the SIAM Data Mining 2015 pape

Mixed marker-based/marker-less visual odometry system for mobile robots

When moving in generic indoor environments, robotic platforms generally rely solely on information provided by onboard sensors to determine their position and orientation. However, the lack of absolute references often leads to the introduction of severe drifts in estimates computed, making autonomous operations really hard to accomplish. This paper proposes a solution to alleviate the impact of the above issues by combining two vision‐based pose estimation techniques working on relative and absolute coordinate systems, respectively. In particular, the unknown ground features in the images that are captured by the vertical camera of a mobile platform are processed by a vision‐based odometry algorithm, which is capable of estimating the relative frame‐to‐frame movements. Then, errors accumulated in the above step are corrected using artificial markers displaced at known positions in the environment. The markers are framed from time to time, which allows the robot to maintain the drifts bounded by additionally providing it with the navigation commands needed for autonomous flight. Accuracy and robustness of the designed technique are demonstrated using an off‐the‐shelf quadrotor via extensive experimental test

An Automated Approach for Sub-Pixel Registration of Landsat-8 Operational Land Imager (OLI) and Sentinel-2 Multi Spectral Instrument (MSI) Imagery

Moderate spatial resolution satellite data from the Landsat-8 OLI and Sentinel-2A MSI sensors together offer 10 m to 30 m multi-spectral reflective wavelength global coverage, providing the opportunity for improved combined sensor mapping and monitoring of the Earth’s surface. However, the standard geolocated Landsat-8 OLI L1T and Sentinel-2A MSI L1C data products are currently found to be misaligned. An approach for automated registration of Landsat-8 OLI L1T and Sentinel-2A MSI L1C data is presented and demonstrated using contemporaneous sensor data. The approach is computationally efficient because it implements feature point detection across four image pyramid levels to identify a sparse set of tie-points. Area-based least squares matching around the feature points with mismatch detection across the image pyramid levels is undertaken to provide reliable tie-points. The approach was assessed by examination of extracted tie-point spatial distributions and tie-point mapping transformations (translation, affine and second order polynomial), dense-matching prediction-error assessment, and by visual registration assessment. Two test sites over Cape Town and Limpopo province in South Africa that contained cloud and shadows were selected. A Landsat-8 L1T image and two Sentinel-2A L1C images sensed 16 and 26 days later were registered (Cape Town) to examine the robustness of the algorithm to surface, atmosphere and cloud changes, in addition to the registration of a Landsat-8 L1T and Sentinel-2A L1C image pair sensed 4 days apart (Limpopo province). The automatically extracted tie-points revealed sensor misregistration greater than one 30 m Landsat-8 pixel dimension for the two Cape Town image pairs, and greater than one 10 m Sentinel-2A pixel dimension for the Limpopo image pair. Transformation fitting assessments showed that the misregistration can be effectively characterized by an affine transformation. Hundreds of automatically located tie-points were extracted and had affine-transformation root-mean-square error fits of approximately 0.3 pixels at 10 m resolution and dense-matching prediction errors of similar magnitude. These results and visual assessment of the affine transformed data indicate that the methodology provides sub-pixel registration performance required for meaningful Landsat-8 OLI and Sentinel-2A MSI data comparison and combined data applications