Volkan Cevher

Model-Based Compressive Sensing

Compressive sensing (CS) is an alternative to Shannon/Nyquist sampling for the acquisition of sparse or compressible signals that can be well approximated by just K ¿ N elements from an N -dimensional basis. Instead of taking periodic samples, CS measures inner products with M < N random vectors and then recovers the signal via a sparsity-seeking optimization or greedy algorithm. Standard CS dictates that robust signal recovery is possible from M = O(K log(N/K)) measurements. It is possible to substantially decrease M without sacrificing robustness by leveraging more realistic signal models that go beyond simple sparsity and compressibility by including structural dependencies between the values and locations of the signal coefficients. This paper introduces a model-based CS theory that parallels the conventional theory and provides concrete guidelines on how to create model-based recovery algorithms with provable performance guarantees. A highlight is the introduction of a new class of structured compressible signals along with a new sufficient condition for robust structured compressible signal recovery that we dub the restricted amplification property, which is the natural counterpart to the restricted isometry property of conventional CS. Two examples integrate two relevant signal models-wavelet trees and block sparsity-into two state-of-the-art CS recovery algorithms and prove that they offer robust recovery from just M = O(K) measurements. Extensive numerical simulations demonstrate the validity and applicability of our new theory and algorithms.

Compressive Sensing for Background Subtraction

Compressive sensing (CS) is an emerging field that provides a framework for image recovery using sub-Nyquist sampling rates. The CS theory shows that a signal can be reconstructed from a small set of random projections, provided that the signal is sparse in some basis, e.g., wavelets. In this paper, we describe a method to directly recover background subtracted images using CS and discuss its applications in some communication constrained multi-camera computer vision problems. We show how to apply the CS theory to recover object silhouettes (binary background subtracted images) when the objects of interest occupy a small portion of the camera view, i.e., when they are sparse in the spatial domain. We cast the background subtraction as a sparse approximation problem and provide different solutions based on convex optimization and total variation. In our method, as opposed to learning the background, we learn and adapt a low dimensional compressed representation of it, which is sufficient to determine spatial innovations; object silhouettes are then estimated directly using the compressive samples without any auxiliary image reconstruction. We also discuss simultaneous appearance recovery of the objects using compressive measurements. In this case, we show that it may be necessary to reconstruct one auxiliary image. To demonstrate the performance of the proposed algorithm, we provide results on data captured using a compressive single-pixel camera. We also illustrate that our approach is suitable for image coding in communication constrained problems by using data captured by multiple conventional cameras to provide 2D tracking and 3D shape reconstruction results with compressive measurements.

Convex Optimization for Big Data

This article reviews recent advances in convex optimization algorithms for big data, which aim to reduce the computational, storage, and communications bottlenecks. We provide an overview of this emerging field, describe contemporary approximation techniques such as first-order methods and randomization for scalability, and survey the important role of parallel and distributed computation. The new big data algorithms are based on surprisingly simple principles and attain staggering accelerations even on classical problems.

Low-Dimensional Models for Dimensionality Reduction and Signal Recovery: A Geometric Perspective

We compare and contrast from a geometric perspective a number of low-dimensional signal models that support stable information-preserving dimensionality reduction. We consider sparse and compressible signal models for deterministic and random signals, structured sparse and compressible signal models, point clouds, and manifold signal models. Each model has a particular geometrical structure that enables signal information to be stably preserved via a simple linear and nonadaptive projection to a much lower dimensional space; in each case the projection dimension is independent of the signals ambient dimension at best or grows logarithmically with it at worst. As a bonus, we point out a common misconception related to probabilistic compressible signal models, namely, by showing that the oft-used generalized Gaussian and Laplacian models do not support stable linear dimensionality reduction.

A compressive beamforming method

Compressive sensing (CS) is an emerging area which uses a relatively small number of non-traditional samples in the form of randomized projections to reconstruct sparse or compressible signals. This paper considers the direction-of-arrival (DOA) estimation problem with an array of sensors using CS. We show that by using random projections of the sensor data, along with a full waveform recording on one reference sensor, a sparse angle space scenario can be reconstructed, giving the number of sources and their DOAs. The number of projections can be very small, proportional to the number sources. We provide simulations to demonstrate the performance and the advantages of our compressive beamformer algorithm.

Vehicle Speed Estimation Using Acoustic Wave Patterns

We estimate a vehicles speed, its wheelbase length, and tire track length by jointly estimating its acoustic wave pattern with a single passive acoustic sensor that records the vehicles drive-by noise. The acoustic wave pattern is determined using the vehicles speed, the Doppler shift factor, the sensors distance to the vehicles closest-point-of-approach, and three envelope shape (ES) components, which approximate the shape variations of the received signals power envelope. We incorporate the parameters of the ES components along with estimates of the vehicle engine RPM, the number of cylinders, and the vehicles initial bearing, loudness and speed to form a vehicle profile vector. This vector provides a fingerprint that can be used for vehicle identification and classification. We also provide possible reasons why some of the existing methods are unable to provide unbiased vehicle speed estimates using the same framework. The approach is illustrated using vehicle speed estimation and classification results obtained with field data.

Bilinear Generalized Approximate Message Passing—Part I: Derivation

In this paper, we extend the generalized approximate message passing (G-AMP) approach, originally proposed for high-dimensional generalized-linear regression in the context of compressive sensing, to the generalized-bilinear case, which enables its application to matrix completion, robust PCA, dictionary learning, and related matrix-factorization problems. Here, in Part I of a two-part paper, we derive our Bilinear G-AMP (BiG-AMP) algorithm as an approximation of the sum-product belief propagation algorithm in the high-dimensional limit, where central-limit theorem arguments and Taylor-series approximations apply, and under the assumption of statistically independent matrix entries with known priors. In addition, we propose an adaptive damping mechanism that aids convergence under finite problem sizes, an expectation-maximization (EM)-based method to automatically tune the parameters of the assumed priors, and two rank-selection strategies. In Part II of the paper, we will discuss the specializations of EM-BiG-AMP to the problems of matrix completion, robust PCA, and dictionary learning, and we will present the results of an extensive empirical study comparing EM-BiG-AMP to state-of-the-art algorithms on each problem.

Compressive wireless arrays for bearing estimation

Joint processing of sensor array outputs improves the performance of parameter estimation and hypothesis testing problems beyond the sum of the individual sensor processing results. When the sensors have high data sampling rates, arrays are tethered, creating a disadvantage for their deployment and also limiting their aperture size. In this paper, we develop the signal processing algorithms for randomly deployable wireless sensor arrays that are severely constrained in communication bandwidth. We focus on the acoustic bearing estimation problem and show that when the target bearings are modeled as a sparse vector in the angle space, low dimensional random projections of the microphone signals can be used to determine multiple source bearings by solving an l 1-norm minimization problem. Field data results are shown where only 10 bits of information is passed from each microphone to estimate multiple target bearings.

Sparse Signal Recovery and Acquisition with Graphical Models

A great deal of theoretic and algorithmic research has revolved around sparsity view of signals over the last decade to characterize new, sub-Nyquist sampling limits as well as tractable algorithms for signal recovery from dimensionality reduced measurements. Despite the promising advances made, real-life applications require more realistic signal models that can capture the underlying, application-dependent order of sparse coefficients, better sampling matrices with information preserving properties that can be implemented in practical systems, and ever faster algorithms with provable recovery guarantees for real-time operation.

Target Tracking Using a Joint Acoustic Video System

In this paper, a multitarget tracking system for collocated video and acoustic sensors is presented. We formulate the tracking problem using a particle filter based on a state-space approach. We first discuss the acoustic state-space formulation whose observations use a sliding window of direction-of-arrival estimates. We then present the video state space that tracks a targets position on the image plane based on online adaptive appearance models. For the joint operation of the filter, we combine the state vectors of the individual modalities and also introduce a time-delay variable to handle the acoustic-video data synchronization issue, caused by acoustic propagation delays. A novel particle filter proposal strategy for joint state-space tracking is introduced, which places the random support of the joint filter where the final posterior is likely to lie. By using the Kullback-Leibler divergence measure, it is shown that the joint operation of the filter decreases the worst case divergence of the individual modalities. The resulting joint tracking filter is quite robust against video and acoustic occlusions due to our proposal strategy. Computer simulations are presented with synthetic and field data to demonstrate the filters performance