Research
Summary
The majority of the research I have undertaken is focused on the broad area of Compressive Sensing (CS). In the past decade years I have developed five separate lines of research. The first one continues the focus from my Ph.D. thesis on the use of structured sparsity models, evolving into the study of structured measurement settings for CS. The second line broadens CS into com- pressive signal processing by focusing on parameter estimation problems from compressive sensing measurements, with a focus on line spectral estimation and time delay of arrival estimation; these problems underlie a host of applications in communications and radar. The third line considers the application of statistical signal models in hyperspectral imaging to extract numerical features that convey the semantic information commonly used by practitioners and expert systems, following the premise of big data. The fourth line focused on the application of manifold models for parameter estimation in images captured by cyber-physical systems; we developed feature selection schemes for manifold learning, which I plan to expand in the near future to additional unsupervised learning tasks that haven’t been considered in the literature. We also developed noise-robust learning methods that leverage temporal correlations between images in a temporal sequence, an effort that can be expanded to other applications of manifold models that exhibit correlation between data points (such as the identification of reflectance spectra for adjacent pixels in a hyperspectral image). The fifth line focuses on the design of image-based localization and mapping algorithms designed to alleviate shortcomings observed with sight-impaired operators.
Structured Sparsity and Structured Compressive Sensing
During my Ph.D. and postdoctoral research studies, I was involved in the formulation of structured sparsity models that capture signal structure beyond that available from standard sparsity models. Extending previous work I performed at Duke University on the use of statistical models for group-sparse signal approximation, and as part of a collaboration with Waheed Bajwa at Rutgers, I formulated theoretical guarantees on the use of the lasso and group lasso algorithms for the problems of CS recovery and linear regression when group sparsity is present in the signals of interest. On a related direction, we investigated practical aspects of CS-based analog-to-digital converters, focusing on the nonideal (nonlinear) behavior of the devices involved due to the presence of strong interferers in the channel. We determined that the presence of periodic interferers yields a predetermined structure on the sets of signal samples that can be leveraged in CS acquisition and reconstruction. This realization led to the formulation of a theoretical framework that accounted for the presence of structure in the set of measurements leveraged in CS, providing a multiplicative penalty on the size of the set of samples needed for accurate signal recovery. Additional work has considered the idea of designing the modulation sequences in compressive ADCs to perform filtering of the acquired signal, with the goal of mitigating the strength of the interferer to improve the linearity of the acquisition process. Finally, we have studied the effect of enforcing non-negativity constraints on the performance of the lasso algorithm for linear regression and sparse recovery. Our main interest is in the application of non-negatively constrained lasso for hyperspectral unmixing. Our results allow us to theoretically characterize the success of sparse support identification (e.g., identification of mixed materials) in terms of the corruption with respect to the ideal (linearly mixed) observation. The most promising application of our results considers the characterization of the performance of linear model-based unmixing in real-world settings where mixing is decidedly non-linear; the nonlinear component an be characterized as the corruption of the observation, allowing us to consistently predict the performance of nonnegative lasso for nonlinear mixture unmixing.
Compressive Parameter Estimation
The core of my research work at UMass has considered the extension of the CS framework for signal acquisition beyond full signal recovery, focusing in particular on parameter estimation. This is a problem that I started to consider late during my Ph.D. studies and into my postdoctoral research studies, where the fact that a signal can be represented by a few parameters was used to formulate a parametric dictionary that provides an appropriate sparse representation; when combined with the aforementioned structured sparsity models, parametric dictionaries allowed for increased accuracy in signal recovery. This work, however, still focused on signal recovery, rather than parameter estimation. During my first year at UMass I developed compressive parameter estimation (CPE) algorithms that can be applied to a variety of problems based on these models, but found their performance to be limited by their discrete nature. Thus, I pursued alternative approaches with my graduate students, as detailed below.
Manifold-Based Interpolation methods solve the issues in my prior work due to parameter discretization by providing a tractable model for parametric signals corresponding to a continuum of parameter values. The key to the use of these manifold models, which are usually intractable, is to apply them only to small portions of the parameter space, focusing on bridging the gap between any two parameter values present in the discretized representation provided by parametric dictionaries. Leveraging prior knowledge on the problem of interest allows for the formulation of tractable manifold models (such as hyperspheres and polynomials) that provide significant improvements in the performance of CPE. Our work showed significant performance improvements in time delay of arrival estimation with respect to existing approaches that rely on signal recovery or discretized parameter models, and has proven to be a key component in the formulation of later improved CPE algorithms described in the sequel.
Earth Mover’s Distance (EMD) is a commonly used metric in computer vision and was recently leveraged in CS when applied to images. Our work has borrowed this metric, as applied to parametric dictionary coefficient vectors, due to the provable relationship between the value of the EMD and the parameter estimation error derived from these coefficient vectors; this is in sharp contrast with existing metrics such as
distances, which only capture any information on the performance of CPE when there is no
error. Furthermore, the EMD can also be leveraged to formulate new CPE algorithms that do
not require the use of structured sparsity models. In contrast to the existing literature,
our formulation includes theoretical performance guarantees that consider the parameter
estimation error (instead of the signal recovery error) and that are generic to a variety
of parameter estimation problems 21. Furthermore, our algorithm is agnostic to the
signal acquisition method, and thus can be applied both to CS and to super-resolution. We
have successfully applied our framework to line spectral estimation and time delay of
arrival estimation from subsampled and compressively sensed data.Approximate Message Passing (AMP) has become one of the most popular algorithms for CS recovery due to its simple implementation and tractable performance guarantees. The algorithm was recently shown to be amenable to signal models beyond standard sparsity, requiring only the formulation of a statistical denoising algorithm that must be optimal for Gaussian noise; the denoiser can then be integrated inside the AMP iterations, and the resulting algorithm shares the tractable performance guarantees of the original. Our work leverages well-known algorithms for statistical parameters estimation (such as MUSIC) to design parametric (or analog) denoisers that are integrated within AMP and return parameter estimates along with the recovered signal at each iteration. Our proposed approach is immediately applicable to multiple settings (including line spectral estimation and bearing estimation) thanks to decades of work in statistical signal processing. Our latest results show that these new algorithms inherit the tractable performance analysis from their standard counterpart.
Modeling Semantics in Hyperspectral Imaging
I have been interested in hyperspectral imaging (HSI) problems since my Ph.D. studies. HSI routinely deals with very high-dimensional datasets (both in physical dimensions and in the size of the acquired data) and exhibits high degrees of structure and redundancy, yet the work in signal processing aspects of this area has been somewhat limited. While HSI has been pitched as one of the areas where CS can provide significant savings in the complexity of sensing systems, the recovery of HSI data is seldom the final goal of acquisition; rather, practitioners are interested in the extraction of relevant information from the hyperspectral datacube, such as the presence or absence of a suite of materials (minerals, clays, etc.) from the scene under consideration.
We have identified the potential for automation of HSI signal processing tasks that currently rely on the formulation of ad-hoc rules posed by expert practitioners with prior knowledge of the problem at hand that represent the semantics relevant to a problem; common examples include the presence or absence of a fluctuation (i.e., a discontinuity) in a material’s reflectivity spectrum, as well as the location and shape of such fluctuations. We are studying the use of statistical models, applied to wavelet decompositions of spectra, to identify and model structural (or semantic) information that is relevant to signal processing tasks in HSI. In particular, we have studied the use of non-homogeneous Markov chains, which have been popular in image processing, to generate numerical features to be used in hyperspectral signal and image processing problems. We have shown the applicability of such semantic features in multiple problems in hyperspectral image processing, including mineral identification, hyperspectral unmixing, and band selection.
Manifold Models for Image Processing in Cyber-Physical Systems
We have developed schemes that allow for the design of masks for low-power imaging devices that (i) are matched to the use of manifold models for parameter estimation and (ii) provide significant savings on the energy consumption of the sensing platform. This is the opening of a new effort in the design of feature selection schemes for unsupervised learning problems in machine learning (beyond clustering, the one example discussed in prior literature). To mitigate the nontrivial amount of noise present in the images captured by niche devices such as the low-power architecture above, we have also developed a scheme for manifold learning and extension that leverages correlation between points in the manifold (such as temporal correlation between frames in a video) in order to increase the robustness of manifold learning to additive observation noise. The scheme has been shown experimentally to be robust to a variety of image noise types, including Gaussian, salt-pepper, blur, and motion noise. The scheme is also flexible enough that we expect its impact to expand to a variety of fields; for example, we can exploit the spatial correlation between hyperspectral signatures for adjacent pixels in an image.
Image-Based Localization for Indoor Navigation
We focus on indoor localization as part of the design of PERCEPT-V: an organic vision-driven, smartphone-based indoor navigation system, in which the user can navigate in open spaces without requiring retrofit of the environment. When the user seeks a smartphone app to obtain navigation instructions to a chosen destination, the smartphone will record observations from multiple onboard sensors in order to perform user localization. Once the location and orientation of the user are estimated, they are used to calculate the coordinates of the navigation landmarks surrounding the user. The system can then provide directions to the chosen destination, as well as an optional description of the landmarks around the user.
Our efforts have focused on improvements to the baseline approaches to indoor localization from images such as COLMAP. First, we perform foreground removal for the images taken from a video recorded by the user via algorithms using robust PCA, showing that it improves the performance of image-based localization. Second, we show that exploitation of the spatiotemporal relationships in training datasets obtained from multiple simultaneous video recordings provide improvements in the performance of 3D point cloud generation for the indoor environment that is necessary for image-based localization