We propose a novel approach to reconstruct Hyperspectral images from very few number of noisy compressive measurements. Our reconstruction approach is based on a convex minimization which penalizes both the nuclear norm and the 2,1 mixed-norm of the data matrix. Thus, the solution tends to have a simultaneous low-rank and joint-sparse structure. We explain how these two assumptions fit Hyperspectral data, and by severals simulations we show that our proposed reconstruction scheme significantly enhances the state-ofthe-art tradeoffs between the reconstruction error and the required number of CS measurements.

With the development of numbers of high resolution data acquisition systems and the global requirement to lower the energy consumption, the development of efficient sensing techniques becomes critical. Recently, Compressed Sampling (CS) techniques, which exploit the sparsity of signals, have allowed to reconstruct signal and images with less measurements than the traditional Nyquist sensing approach. However, multichannel signals like Hyperspectral images (HSI) have additional structures, like inter-channel correlations, that are not taken into account in the classical CS scheme.In this paper we exploit the linear mixture of sources model, that is the assumption that the multichannel signal is composed of a linear combination of sources, each of them having its own spectral signature, and propose new sampling schemes exploiting this model to considerably decrease the number of measurements needed for the acquisition and source separation. Moreover, we give theoretical lower bounds on the number of measurements required to perform reconstruction of both the multichannel signal and its sources. We also proposed optimization algorithms and extensive experimentation on our target application which is HSI, and show that our approach recovers HSI with far less measurements and computational effort than traditional CS approaches. Index TermsCompressed sensing, source separation, hyperspectral image, linear mixture model, sparsity, proximal splitting method.

Novel methods for quantitative, transient-state multiparametric imaging are increasingly being demonstrated for assessment of disease and treatment efficacy. Here, we build on these by assessing the most common Non-Cartesian readout trajectories (2D/3D radials and spirals), demonstrating efficient anti-aliasing with a k- space view-sharing technique, and proposing novel methods for parameter inference with neural networks that incorporate the estimation of proton density. Our results show good agreement with gold standard and phantom references for all readout trajectories at 1.5 T and 3 T. Parameters inferred with the neural network were within 6.58% difference from the parameters inferred with a high-resolution dictionary. Concordance correlation coefficients were above 0.92 and the normalized root mean squared error ranged between 4.2 and 12.7% with respect to gold-standard phantom references for T1 and T2. In vivo acquisitions demonstrate sub-millimetric isotropic resolution in under five minutes with reconstruction and inference times < 7 min. Our 3D quantitative transient-state imaging approach could enable high-resolution multiparametric tissue quantification within clinically acceptable acquisition and reconstruction times.

Regularization plays a pivotal role when facing the challenge of solving ill-posed inverse problems, where the number of observations is smaller than the ambient dimension of the object to be estimated. A line of recent work has studied regularization models with various types of low-dimensional structures. In such settings, the general approach is to solve a regularized optimization problem, which combines a data fidelity term and some regularization penalty that promotes the assumed low-dimensional/simple structure. This paper provides a general framework to capture this low-dimensional structure through what we coin partly smooth functions relative to a linear manifold. These are convex, non-negative, closed and finite-valued functions that will promote objects living on low-dimensional subspaces. This class of regularizers encompasses many popular examples such as the 1 norm, 1 − 2 norm (group sparsity), as well as several others including the ∞ norm. We also show that the set of partly smooth functions relative to a linear manifold is closed under addition and pre-composition by a linear operator, which allows to cover mixed regularization, and the so-called analysis-type priors (e.g. total variation, fused Lasso, finite-valued polyhedral gauges). Our main result presents a unified sharp analysis of exact and robust recovery of the low-dimensional subspace model associated to the object to recover from partial measurements. This analysis is illustrated on a number of special and previously studied cases, and on an analysis of the performance of ∞ regularization in a compressed sensing scenario.We also recall the fundamental first-order minimality condition of a convex function: x is the global minimizer of a convex function f if, and only if, 0 ∈ ∂ f (x). 1.5.3Operator norm. Let J 1 and J 2 be two finite-valued gauges defined on two vector spaces V 1 and V 2 , and A : V 1 → V 2 a linear map. The operator bound |||A||| J 1 →J 2 of A between J 1 and J 2 is given byNote that |||A||| J 1 →J 2 < +∞ if, and only if A Ker(J 1 ) ⊆ Ker(J 2 ). In particular, if J 1 is coercive (i.e. Ker J 1 = {0} from Lemma 4(iv)), then |||A||| J 1 →J 2 is finite. As a convention, |||A||| J 1 →||·|| p is denoted as |||A||| J 1 → p . An easy consequence of this definition is the fact that for every x ∈ V 1 , J 2 (Ax) |||A||| J 1 →J 2 J 1 (x).

In this paper we propose a novel and efficient model for compressed sensing of hyperspectral images. A large-size hyperspectral image can be subsampled by retaining only 3% of its original size, yet robustly recovered using the new approach we present here. Our reconstruction approach is based on minimizing a convex functional which penalizes both the trace norm and the TV norm of the data matrix. Thus, the solution tends to have a simultaneous low-rank and piecewise smooth structure: the two important priors explaining the underlying correlation structure of such data. Through simulations we will show our approach significantly enhances the conventional compression rate-distortion tradeoffs. In particular, in the strong undersampling regimes our method outperforms the standard TV denoising image recovery scheme by more than 17dB in the reconstruction MSE.