Ana B. Ramirez

Reconstruction of Sparse Signals From

Dimension reduction methods via linear random projections are used in numerous applications including data mining, information retrieval and compressive sensing (CS). While CS has traditionally relied on normal random projections, corresponding to ℓ₂ distance preservation, a large body of work has emerged for applications where ℓ₁ approximate distances may be preferred. Dimensionality reduction in ℓ₁ often use Cauchy random projections that multiply the original data matrix B ∈ R^n×D with a Cauchy random matrix R ∈^k×n (k≪n), resulting in a projected matrix C ∈^k×D. In this paper, an analogous of the Restricted Isometry Property for dimensionality reduction in is ℓ₁ proposed using explicit tail bounds for the geometric mean of the random projections. A set of signal reconstruction algorithms from the Cauchy random projections are then developed given that the large suite of reconstruction algorithms developed in compressive sensing perform poorly due to the lack of finite second-order statistics in the projections. These algorithms are based on regularized coordinate-descent Myriad estimates using both ℓ₀ and Lorentzian norms as sparsity inducing terms.

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Traditional hyperspectral imaging sensors acquire high-dimensional data that are used for the discrimination of objects and features in a scene. Recently, a novel architecture known as the coded-aperture snapshot spectral imaging (CASSI) system has been developed for the acquisition of compressive spectral image data with just a few coded focal plane array measurements. This paper focuses on developing a classification approach with hyperspectral images directly from CASSI compressive measurements, without first reconstructing the full data cube. The proposed classification method uses the compressive measurements to find the sparse vector representation of the test pixel in a given training dictionary. The estimated sparse vector is obtained by solving a sparsity-constrained optimization problem and is then used to directly determine the class of the unknown pixel. The performance of the proposed classifier is improved by taking optimal CASSI compressive measurements obtained when optimal coded apertures are used in the optical system. The set of optimal coded apertures is designed such that the CASSI sensing matrix satisfies a restricted isometry property with high probability. Several simulations illustrate the performance of the proposed classifier using optimal coded apertures and the gain in the classification accuracy obtained over using traditional aperture codes in CASSI.

Dimensionality-Reduced Cauchy Random Projections

Compressive sensing generally relies on the ℓ2 norm for data fidelity, whereas in many applications, robust estimators are needed. Among the scenarios in which robust performance is required, applications where the sampling process is performed in the presence of impulsive noise, i.e., measurements are corrupted by outliers, are of particular importance. This article overviews robust nonlinear reconstruction strategies for sparse signals based on replacing the commonly used ℓ2 norm by M-estimators as data fidelity functions. The derived methods outperform existing compressed sensing techniques in impulsive environments, while achieving good performance in light-tailed environments, thus offering a robust framework for CS.

Spectral Image Classification From Optimal Coded-Aperture Compressive Measurements

While compressive sensing (CS) has traditionally relied on l2 as an error norm, a broad spectrum of applications has emerged where robust estimators are required. Among those, applications where the sampling process is performed in the presence of impulsive noise, or where the sampling of the high-dimensional sparse signals requires the preservation of a distance different than l2. This article overviews robust sampling and nonlinear reconstruction strategies for sparse signals based on the Cauchy distribution and the Lorentzian norm for the data fidelity. The derived methods outperform existing compressed sensing techniques in impulsive environments, thus offering a robust framework for CS.

Robust compressive sensing of sparse signals: a review

Dimensionality reduction via linear random projections are used in numerous applications including data streaming, information retrieval, data mining, and compressive sensing (CS). While CS has traditionally relied on normal random projections, corresponding to l 2 distance preservation, a large body of work has emerged for applications where l 1 approximate distances may be preferred. Dimensionality reduction in l 1 use Cauchy random projections that multiply the original data matrix B ∈ 葷D×n with a Cauchy random matrix R ∈ 葷n×k (k « min(n,D)), resulting in a projected matrix C ∈ 葷D×k. This paper focuses on developing signal reconstruction algorithms from Cauchy random projections, where the large suite of reconstruction algorithms developed in compressive sensing perform poorly due to the lack of finite second-order statistics in the projections. In particular, a set of regularized coordinate-descent Myriad regression based reconstruction algorithms are developed using, both l 0 and Lorentzian norms as sparsity inducing terms. The l 0 -regularized algorithm shows superior performance to other standard approaches. Simulations illustrate and compare accuracy of reconstruction.

An overview of robust compressive sensing of sparse signals in impulsive noise

Hyperspectral remote sensing often captures imagery where the spectral profiles of the spatial pixels are the result of the reflectance contribution of numerous materials. Spectral unmixing is then used to extract the collection of materials, or endmembers, contained in the measured spectra and a set of corresponding fractions that indicate the abundance of each material present at each pixel. This paper aims at developing a spectral unmixing algorithm directly from compressive measurements acquired using the coded-aperture snapshot spectral imaging (CASSI) system. The proposed method first uses the compressive measurements to find a sparse vector representation of each pixel in a 3-D dictionary formed by a 2-D wavelet basis and a known spectral library of endmembers. The sparse vector representation is estimated by solving a sparsity-constrained optimization problem using an algorithm based on the variable splitting augmented Lagrangian multipliers method. The performance of the proposed spectral unmixing method is improved by taking optimal CASSI compressive measurements obtained when optimal coded apertures are used in the optical system. The optimal coded apertures are designed such that the CASSI sensing matrix satisfies a restricted isometry property with high probability. Simulations with synthetic and real hyperspectral cubes illustrate the accuracy of the proposed unmixing method.

Reconstruction of sparse signals from ℓ 1 dimensionality-reduced Cauchy random-projections

This article presents an overview of the fundamental optical phenomena behind compressive spectral imaging systems based on coded apertures. The key mathematical concepts embodying the sensing and reconstruction methods, the framework developed to design optimal coded apertures together with the computational spectral imaging strategies for classification, unmixing, and spectral selectivity purposes are also presented. Special attention is given for describing and discussing many practical aspects of compressive spectral imaging, including discretization models of the optical system, optimal parameters design, and physical limitations. The performance of the sensing strategies and computational methods is illustrated in this article with real data and imagery for different applications. Keywords: coded aperture; compressive sensing; computational imaging; spectral imaging

Spectral Image Unmixing From Optimal Coded-Aperture Compressive Measurements

This paper describes a new approach and its associated theoretical performance guarantees for supervised hyperspectral image classification from compressive measurements obtained by a Coded Aperture Snapshot Spectral Imaging System (CASSI). In one snapshot, the two-dimensional focal plane array (FPA) in the CASSI system captures the coded and spectrally dispersed source field of a three-dimensional data cube. Multiple snapshots are used to construct a set of compressive spectral measurements. The proposed approach is based on the concept that each pixel in the hyper-spectral image lies in a low-dimensional subspace obtained from the training samples, and thus it can be represented as a sparse linear combination of vectors in the given subspace. The sparse vector representing the test pixel is then recovered from the set of compressive spectral measurements and it is used to determine the class label of the test pixel. The theoretical performance bounds of the classifier exploit the distance preservation condition satisfied by the multiple shot CASSI system and depend on the number of measurements collected, code aperture pattern, and similarity between spectral signatures in the dictionary. Simulation experiments illustrate the performance of the proposed classification approach.

Snapshot Compressive Multispectral Cameras

A classification method of compressed hyperspectral images acquired by using a Coded Aperture Snapshot Spectral Imaging (CASSI) system is proposed. The CASSI system captures spectral imaging information of a 3-dimensional cube with just a single 2-dimensional measurement containing the coded and spectrally dispersed source field. The proposed method is based on the concept that each pixel in the hyperspectral image lies in a low-dimensional subspace, and thus it can be represented as a sparse linear combination of vectors in a dictionary which is obtained from training samples. The method incorporates interpixel correlation within the image by assuming a sparse multidimensional representation of the scene. The recovered sparse vector is then used directly to determine the class label of the test pixel. The proposed algorithm is used to classify real hyperspectral data cubes directly from their CASSI measurements.a

Hyperspectral pixel classification from coded-aperture compressive imaging

Full waveform inversion of Ground Penetrating Radar (GPR) data is a promising strategy to estimate quantitative characteristics of the subsurface such as permittivity and conductivity. In this paper, we propose a methodology that uses Full Waveform Inversion (FWI) in time domain of 2D GPR data to obtain highly resolved images of the permittivity and conductivity parameters of the subsurface. FWI is an iterative method that requires a cost function to measure the misfit between observed and modeled data, a wave propagator to compute the modeled data and an initial velocity model that is updated at each iteration until an acceptable decrease of the cost function is reached. The use of FWI with GPR are expensive computationally because it is based on the computation of the electromagnetic full wave propagation. Also, the commercially available acquisition systems use only one transmitter and one receiver antenna at zero offset, requiring a large number of shots to scan a single line.