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Connecting Array Processing and Sparse Optimization in Oceanography
Compressive Sensing Array Framework May Provide an Efficient Way To Study Large Areas With Sparsely Distributed Information of Interest

By Ananya Sen Gupta
Postdoctoral Researcher
Department of Applied Ocean Physics and Engineering
Woods Hole Oceanographic Institution
Woods Hole, Massachusetts

Many oceanographic applications necessitate data acquisition over a large support (spatial, temporal or frequency) where the information of interest is sparsely distributed. Sparsity refers to scenarios where the significant components of the data are few and occur over a much wider realm of possible values; for example, stars in a night sky represent a sparse distribution of light across the three-dimensional spatial support of the sky visible to the naked eye. Sparse distributions abound in oceanography—for example, the delay-Doppler spread in shallow-water acoustics, organisms of interest in the deep sea and sparse spikes of interest in marine seismic signals, among many others.

Array technology and associated signal processing have been employed for decades for efficient measurement and monitoring of oceanographic processes, including high-resolution synthetic aperture sonar, hydrophone arrays used for underwater acoustic communications and light-emitting diode arrays for underwater imaging.

This article connects concepts in adaptive array processing and sparse sensing to meet data acquisition challenges in oceanography based on a previously proposed compressive sensor array framework. The goal is to motivate the marine technology audience to seek innovations in the combined framework of sparse optimization and array processing to potentially attain significant gains in energy-efficiency, array size and the ability to efficiently capture information on sparsely distributed processes in the ocean.

Compressive Sensing: A Brief Overview
In recent years, signal-processing research has witnessed tremendous growth in data acquisition and reconstruction techniques for sparsely distributed data. Compressive sensing and sparse processing approaches essentially utilize the sparse distribution of the data to acquire it at sub-Nyquist rates, i.e., acquisition rates far below what is theoretically needed to capture the full bandwidth. Data acquisition is typically performed using random projections of the data space, i.e., the vector space where the data of interest lies, along a measurement matrix. The measurement matrix is chosen such that it satisfies sufficient incoherence with the basis of the data space to guarantee precise reconstruction of the data from a much smaller set of samples.

Sparse reconstruction is achieved using a variety of L1-regularized least squares (L1-LS) techniques that pursue the highest components of the data that best match the measured samples. Figure 1 shows an example of sparse reconstruction for a simulated dataset generated using 400 coefficients, of which only 10 are significant. The sparsely distributed coefficients are estimated using an estimation window length of only 100 samples with a signal-to-noise ratio of 15 decibels. The graph shows true values of the simulated coefficients (red stars), estimates made using sparse (L1-LS) optimization (green stars) and the estimates derived from a traditional least squares algorithm with the same averaging window (blue circles). The sparse optimization algorithm was found to significantly outperform the least squares technique that seeks a compromise between the significant components (outliers) and the insignificant components (inliers).

Figure 1: Comparison of performance between a least squares technique and a sparse optimization technique over a simulated sparsely distributed dataset.

Compressive Sensor Array
Sparse-array processing is well known in the broader signal processing community and has been successfully applied to sparse magnetic resonance imaging, video communications and many other applications. A classic example of successful implementation of sparse sensing and array processing is the recent development of cameras based on compressive imaging techniques. These highly efficient data acquisition devices use micromirror imaging arrays that project the incident light across random directions to enable sparse reconstruction from these image projections.

However, despite these advancements in combining array processing and sparse sensing techniques in other fields, much remains to be done in the sphere of marine technology. To close this gap, research has been done at Woods Hole Oceanographic Institution (WHOI) proposing a compressive sensor array (CSA) structure that has great potential for a variety of oceanographic applications.

Specifically, this is a geometric framework that exploits the spatial redundancy of the sensor array for efficient beamforming and employs sparse reconstruction techniques to detect and recover data in the volume of interest. Figure 2 illustrates the basic structure of the CSA framework, though individual implementations may vary depending on the application, the problem at hand and the specific array geometry desired.

The compressive sensor array consists of a three-dimensional array of transmitters and receivers. As a typical application, we may use this compressive sensor array to image sparsely distributed objects over a large volume of interest. Overlapping transmitter beams illuminate the total volume, and each receiver sees a potentially incoherent mixture of the transmitted signals reflected along random directions from sparsely distributed objects of interest. The degree of incoherence may be controlled using array geometry and adaptive beamforming techniques that exploit the spatial diversity of the array.

If the compressive sensor framework is specifically used for optical imaging systems, then the spatial diversity of the array may also be utilized to eliminate the backward optical backscatter on the receiving end. From the principles of compressive sensing, we expect a marked reduction in required array size, i.e., we may reconstruct the data (or detect the image in this case) from much fewer array elements than what would be required for covering the entire volume using nonsparse reconstruction methods. We also expect sparse array processing to significantly reduce the illumination intensity required per unit volume for high-resolution imaging by eliminating backscatter more effectively. This is because sparse sensing techniques glean out the significant components of sparsely distributed data much more effectively than traditional least square minimization techniques that compromise between the high and low values of the data, and thus may provide higher immunity toward backscatter noise.

Figure 2: Basic geometrical framework of the compressive sensor array depicted for an imaging application using active lighting: Each arrow from a transmitter (Tx) represents the light incident on the object of interest from a non-collimated optical beam from that transmitter. Each arrow reaching an optical detector (Rx) represents the light received from the object of interest by that detector.

By applying adaptive array processing to control backscatter across the common volumes and taking advantage of sparse sensing techniques to reconstruct the data from significantly smaller array sizes, the total volume illumination may be drastically reduced. Thus the compressive sensor framework combines adaptive array processing and compressive sensing principles to potentially enable surveillance of large volumes with sparsely distributed objects of interest with relatively low environmental impact.

Passive Compressive Sensor Array
The sparse array framework depicted in Figure 2 assumes active transmission of energy to detect and collect data. However, a passive variant of the compressive sensor array, which only consists of receivers, is equally feasible, and depending on the application, may be easier to build. Such an array can efficiently detect sparsely distributed targets that emit energy in bursts that can be easily detected using a small array employing sparse detection over a large volume. A unique advantage of the passive form of the compressive sensor array is that it is unobtrusive and noninvasive, in terms of both energy emission and array size. It is therefore ideal for passive biological observation or stealth surveillance by coast guard security.

Compressive Sensor Array Framework Applications
The compressive sensor array framework that has been presented is widely applicable to any oceanographic problem that involves recovering data that is sparsely distributed over a much bigger support. To aid the visualization of this approach, this article used an imaging system as a typical application. However, the transmitter-receiver structure may be applied more generally to a wider variety of detection problems. Both the active and passive versions of the framework may be used over optical, acoustic and infrared wavelengths, though performance may vary depending on the application and the degree of backscatter and penetrability of the wavelength used across the range to potential targets.

The CSA structure may be used to implement hydrophones using wide-beam, low-energy acoustics to detect torpedoes and mines in the ocean without being visible to enemy submarines. It may be used in source localization and detection of anthropogenic objects embedded sparsely in the seabed by setting up a sensor array of acoustic transmitters and receivers. The directionality of the beams may be tuned adaptively using the spatial diversity of the array to cover a large area with relatively low impact in terms of acoustic energy, thus making acoustic surveillance less invasive to marine mammals. Moreover, the ability to acquire data using smaller array sizes enables deployment of a network of small arrays, passive or active, over large areas of the ocean for naval surveillance, bathymetric applications, remote sensing of oil wells and many other applications of oceanographic relevance.

Continuing Challenges
The sparse array processing approach has been successfully applied in non-oceanographic paradigms such as medical imaging, wireless communications and multimedia data acquisition, among many others. However, transporting sparse array processing as a signal processing technique to oceanography needs to account for the additional challenge that most oceanic processes are fundamentally time-varying and often nonlinear in nature.

The uncertain nature of the oceanographic environment renders direct implementation of sparse sensing techniques difficult and sometimes infeasible. The mathematical precision of sparse reconstruction relies on incoherence assumptions being met between the measurement space and the basis of the vector space spanned by the full support of the sparse distribution. The often ill-posed nature of inverse problems in oceanography makes it hard to guarantee sufficient incoherence for precise reconstruction to be feasible. Moreover, the nonstationary nature of physical and biological processes in the ocean also poses difficulties to correct modeling of the degree of sparsity of the data distribution. In some scenarios, such as zooplankton model validation in uncertain environments, the data collection itself is sparse, i.e., key elements of the observation samples are missing (or poorly known). Therefore, estimating the underlying sparse distribution based on poorly known observations is not possible using standard sparse optimization techniques.

The takeaway from these issues is that the ocean environment is challenging to implement using traditional sparse array processing. However, the benefits of marked reduction in array size and energy requirements, as well as performance improvement for detecting and tracking sparse data distributions, provide a strong incentive to make the CSA framework work for the oceanographic paradigm. We have observed in current and ongoing research that most of these issues can be resolved within the application by incorporating known parameters and constraints of the physical environment, which can often be separately measured or modeled.

For a full list of references, contact Ananya Sen Gupta at asengupta@whoi.edu.

Ananya Sen Gupta, a postdoctoral researcher at Woods Hole Oceanographic Institution, holds a master's degree and a Ph.D. in electrical engineering from the University of Illinois at Urbana-Champaign. Her research interests involve signal processing challenges commonly encountered in real environments, with particular focus on interference mitigation and channel estimation in shallow water acoustics and underwater optics, petroleum fingerprinting and sparse optimization across a variety of oceanographic applications.

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