arXiv subjects

Laurent Jacques

Publications and source records attributed to Laurent Jacques.

At least 55 records · Page 3Linked to original sources

arXiv2020

Factorization over interpolation: A fast continuous orthogonal matching pursuit

We propose a fast greedy algorithm to compute sparse representations of signals from continuous dictionaries that are factorizable, i.e., with atoms that can be separated as a product of sub-atoms. Existing algorithms strongly reduce the computational complexity of the sparse decomposition of signals in discrete factorizable dictionaries. On another flavour, existing greedy algorithms use interpolation strategies from a discretization of continuous dictionaries to perform off-the-grid decomposition. Our algorithm aims to combine the factorization and the interpolation concepts to enable low complexity computation of continuous sparse representation of signals. The efficiency of our algorithm is highlighted by simulations of its application to a radar system.

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arXiv2020

Keep the phase! Signal recovery in phase-only compressive sensing

We demonstrate that a sparse signal can be estimated from the phase of complex random measurements, in a "phase-only compressive sensing" (PO-CS) scenario. With high probability and up to a global unknown amplitude, we can perfectly recover such a signal if the sensing matrix is a complex Gaussian random matrix and the number of measurements is large compared to the signal sparsity. Our approach consists in recasting the (non-linear) PO-CS scheme as a linear compressive sensing model. We built it from a signal normalization constraint and a phase-consistency constraint. Practically, we achieve stable and robust signal direction estimation from the basis pursuit denoising program. Numerically, robust signal direction estimation is reached at about twice the number of measurements needed for signal recovery in compressive sensing.

cs.IT↗

arXiv2020

Going Below and Beyond, Off-the-Grid Velocity Estimation from 1-bit Radar Measurements

In this paper we propose to bridge the gap between using extremely low resolution 1-bit measurements and estimating targets' parameters, such as their velocities, that exist in a continuum, i.e., by performing Off-the-Grid estimation. To that end, a Continuous version of Orthogonal Matching Pursuit (COMP) is modified in order to leverage the 1-bit measurements coming from a simple Doppler Radar. Using Monte-Carlo simulations, we show that although the resolution of the acquisition is dramatically reduced, velocity estimation of multiple targets can still be achieved and reaches performances beyond classic On-the-Grid methods. Furthermore, we show empirically that adding a random and uniform dithering before the quantization is necessary when estimating more than one target.

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arXiv2020

When compressive learning fails: blame the decoder or the sketch?

In compressive learning, a mixture model (a set of centroids or a Gaussian mixture) is learned from a sketch vector, that serves as a highly compressed representation of the dataset. This requires solving a non-convex optimization problem, hence in practice approximate heuristics (such as CLOMPR) are used. In this work we explore, by numerical simulations, properties of this non-convex optimization landscape and those heuristics.

cs.LG↗

arXiv2020

One Bit to Rule Them All : Binarizing the Reconstruction in 1-bit Compressive Sensing

This work focuses on the reconstruction of sparse signals from their 1-bit measurements. The context is the one of 1-bit compressive sensing where the measurements amount to quantizing (dithered) random projections. Our main contribution shows that, in addition to the measurement process, we can additionally reconstruct the signal with a binarization of the sensing matrix. This binary representation of both the measurements and sensing matrix can dramatically simplify the hardware architecture on embedded systems, enabling cheaper and more power efficient alternatives. Within this framework, given a sensing matrix respecting the restricted isometry property (RIP), we prove that for any sparse signal the quantized projected back-projection (QPBP) algorithm achieves a reconstruction error decaying like O(m-1/2)when the number of measurements m increases. Simulations highlight the practicality of the developed scheme for different sensing scenarios, including random partial Fourier sensing.

eess.SP↗

arXiv2020

Compressive Learning of Generative Networks

Generative networks implicitly approximate complex densities from their sampling with impressive accuracy. However, because of the enormous scale of modern datasets, this training process is often computationally expensive. We cast generative network training into the recent framework of compressive learning: we reduce the computational burden of large-scale datasets by first harshly compressing them in a single pass as a single sketch vector. We then propose a cost function, which approximates the Maximum Mean Discrepancy metric, but requires only this sketch, which makes it time- and memory-efficient to optimize.

cs.LG↗

arXiv2020

The importance of phase in complex compressive sensing

We consider the question of estimating a real low-complexity signal (such as a sparse vector or a low-rank matrix) from the phase of complex random measurements. We show that in this "phase-only compressive sensing" (PO-CS) scenario, we can perfectly recover such a signal with high probability and up to global unknown amplitude if the sensing matrix is a complex Gaussian random matrix and if the number of measurements is large compared to the complexity level of the signal space. Our approach proceeds by recasting the (non-linear) PO-CS scheme as a linear compressive sensing model built from a signal normalization constraint, and a phase-consistency constraint imposing any signal estimate to match the observed phases in the measurement domain. Practically, stable and robust signal direction estimation is achieved from any "instance optimal" algorithm of the compressive sensing literature (such as basis pursuit denoising). This is ensured by proving that the matrix associated with this equivalent linear model satisfies with high probability the restricted isometry property under the above condition on the number of measurements. We finally observe experimentally that robust signal direction recovery is reached at about twice the number of measurements needed for signal recovery in compressive sensing.

cs.IT↗

arXiv2020

Sparsity-Driven Moving Target Detection in Distributed Multistatic FMCW Radars

We investigate the problem of sparse target detection from widely distributed multistatic \textit{Frequency Modulated Continuous Wave} (FMCW) radar systems (using chirp modulation). Unlike previous strategies (\emph{e.g.}, developed for FMCW or distributed multistatic radars), we propose a generic framework that scales well in terms of computational complexity for high-resolution space-velocity grid. Our approach assumes that \emph{(i)} the target signal is sparse in a discrete space-velocity domain, hence allowing for non-static target detection, and \emph{(ii)} the resulting multiple baseband radar signals share a common support. By simplifying the representation of the FMCW radar signals, we propose a versatile scheme balancing complexity and detection accuracy. In particular, we design a low-complexity, factorized alternative for the Matching Pursuit algorithm leveraging this simplified model, as well as an iterative methodology to compensate for the errors caused by the model simplifications. Extensive Monte-Carlo simulations of a K-band radar system show that our method achieves a fast estimation of moving target's parameters on dense grids, with controllable accuracy, and reaching state-of-the-art performances compared to previous sparsity-driven approaches.

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arXiv2019

(l1,l2)-RIP and Projected Back-Projection Reconstruction for Phase-Only Measurements

This letter analyzes the performances of a simple reconstruction method, namely the Projected Back-Projection (PBP), for estimating the direction of a sparse signal from its phase-only (or amplitude-less) complex Gaussian random measurements, i.e., an extension of one-bit compressive sensing to the complex field. To study the performances of this algorithm, we show that complex Gaussian random matrices respect, with high probability, a variant of the Restricted Isometry Property (RIP) relating to the l1 -norm of the sparse signal measurements to their l2 -norm. This property allows us to upper-bound the reconstruction error of PBP in the presence of phase noise. Monte Carlo simulations are performed to highlight the performance of our approach in this phase-only acquisition model when compared to error achieved by PBP in classical compressive sensing.

eess.SP↗

arXiv2019

Jointly Low-Rank and Bisparse Recovery: Questions and Partial Answers

We investigate the problem of recovering jointly $r$-rank and $s$-bisparse matrices from as few linear measurements as possible, considering arbitrary measurements as well as rank-one measurements. In both cases, we show that $m \asymp r s \ln(en/s)$ measurements make the recovery possible in theory, meaning via a nonpractical algorithm. In case of arbitrary measurements, we investigate the possibility of achieving practical recovery via an iterative-hard-thresholding algorithm when $m \asymp r s^γ\ln(en/s)$ for some exponent $γ> 0$. We show that this is feasible for $γ= 2$, and that the proposed analysis cannot cover the case $γ\leq 1$. The precise value of the optimal exponent $γ\in [1,2]$ is the object of a question, raised but unresolved in this paper, about head projections for the jointly low-rank and bisparse structure. Some related questions are partially answered in passing. For rank-one measurements, we suggest on arcane grounds an iterative-hard-thresholding algorithm modified to exploit the nonstandard restricted isometry property obeyed by this type of measurements.

math.NA↗

arXiv2019

Close Encounters of the Binary Kind: Signal Reconstruction Guarantees for Compressive Hadamard Sampling with Haar Wavelet Basis

We investigate the problems of 1-D and 2-D signal recovery from subsampled Hadamard measurements using Haar wavelet sparsity prior. These problems are of interest in, e.g., computational imaging applications relying on optical multiplexing or single-pixel imaging. However, the realization of such modalities is often hindered by the coherence between the Hadamard and Haar bases. The variable and multilevel density sampling strategies solve this issue by adjusting the subsampling process to the local and multilevel coherence, respectively, between the two bases; hence enabling successful signal recovery. In this work, we compute an explicit sample-complexity bound for Hadamard-Haar systems as well as uniform and non-uniform recovery guarantees; a seemingly missing result in the related literature. We explore the faithfulness of the numerical simulations to the theoretical results and show in a practically relevant instance, e.g., single-pixel camera, that the target signal can be obtained from a few Hadamard measurements.

cs.IT↗

arXiv2019

STIM map: detection map for exoplanets imaging beyond asymptotic Gaussian residual speckle noise

Direct imaging of exoplanets is a challenging task as it requires to reach a high contrast at very close separation to the star. Today, the main limitation in the high-contrast images is the quasi-static speckles that are created by residual instrumental aberrations. They have the same angular size as planetary companions and are often brighter, hence hindering our capability to detect exoplanets. Dedicated observation strategies and signal processing techniques are necessary to disentangle these speckles from planetary signals. The output of these methods is a detection map in which the value of each pixel is related to a probability of presence of a planetary signal. The detection map found in the literature relies on the assumption that the residual noise is Gaussian. However, this is known to lead to higher false positive rates, especially close to the star. In this paper, we re-visit the notion of detection map by analyzing the speckle noise distribution, namely the Modified Rician distribution. We use non-asymptotic analysis of the sum of random variables to show that the tail of the distribution of the residual noise decays as an exponential distribution, hence explaining the high false detection rate obtained with the Gaussian assumption. From this analysis, we introduce a novel time domain detection map and we demonstrate its capabilities and the relevance of our approach through experiments on real data. We also provide an empirical rule to determine detection threshold providing a good trade off between true positive and false positive rates for exoplanet detection.

astro-ph.IM↗

arXiv2019

Identifiability Conditions for Multi-channel Blind Deconvolution with Short Filters

This work considers the multi-channel blind deconvolution problem under the assumption that the channels are short. First, we investigate the ill-posedness issues inherent to blind deconvolution problems and sufficient and necessary conditions on the channels that guarantee well-posedness are derived. Following previous work on blind deconvolution, the problem is then reformulated as a low-rank matrix recovery problem and solved by nuclear norm minimization. Numerical experiments show the effectiveness of this algorithm under a certain generative model for the input signal and the channels, both in the noiseless and in the noisy case.

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arXiv2019

Reference-less algorithm for circumstellar disks imaging

Circumstellar disks play a key role in the understanding of stellar systems. Direct imaging of such extended structures is a challenging task. Current post-processing techniques, first tailored for exoplanets imaging, tend to produce deformed images of the circumstellar disks, hindering our capability to study their shape and photometry in details. We address here the reasons of this shortcoming and propose an algorithm that produces more faithful images of disks taken with ground-based telescopes. We also show that our algorithm is a good candidate for exoplanets imaging. We then explain how our approach can be extended in the form of a regularized inverse problem.

astro-ph.IM↗

arXiv2019

Compressive Single-pixel Fourier Transform Imaging using Structured Illumination

Single Pixel (SP) imaging is now a reality in many applications, e.g., biomedical ultrathin endoscope and fluorescent spectroscopy. In this context, many schemes exist to improve the light throughput of these device, e.g., using structured illumination driven by compressive sensing theory. In this work, we consider the combination of SP imaging with Fourier Transform Interferometry (SP-FTI) to reach high-resolution HyperSpectral (HS) imaging, as desirable, e.g., in fluorescent spectroscopy. While this association is not new, we here focus on optimizing the spatial illumination, structured as Hadamard patterns, during the optical path progression. We follow a variable density sampling strategy for space-time coding of the light illumination, and show theoretically and numerically that this scheme allows us to reduce the number of measurements and light-exposure of the observed object compared to conventional compressive SP-FTI.

cs.CV↗

arXiv2019

Quantized Compressive Sensing with RIP Matrices: The Benefit of Dithering

Quantized compressive sensing (QCS) deals with the problem of coding compressive measurements of low-complexity signals with quantized, finite precision representations, i.e., a mandatory process involved in any practical sensing model. While the resolution of this quantization clearly impacts the quality of signal reconstruction, there actually exist incompatible combinations of quantization functions and sensing matrices that proscribe arbitrarily low reconstruction error when the number of measurements increases. This work shows that a large class of random matrix constructions known to respect the restricted isometry property (RIP) is "compatible" with a simple scalar and uniform quantization if a uniform random vector, or a random dither, is added to the compressive signal measurements before quantization. In the context of estimating low-complexity signals (e.g., sparse or compressible signals, low-rank matrices) from their quantized observations, this compatibility is demonstrated by the existence of (at least) one signal reconstruction method, the projected back projection (PBP), whose reconstruction error decays when the number of measurements increases. Interestingly, given one RIP matrix and a single realization of the dither, a small reconstruction error can be proved to hold uniformly for all signals in the considered low-complexity set. We confirm these observations numerically in several scenarios involving sparse signals, low-rank matrices, and compressible signals, with various RIP matrix constructions such as sub-Gaussian random matrices and random partial discrete cosine transform (DCT) matrices.

cs.IT↗

arXiv2019

One-Bit Sensing of Low-Rank and Bisparse Matrices

This note studies the worst-case recovery error of low-rank and bisparse matrices as a function of the number of one-bit measurements used to acquire them. First, by way of the concept of consistency width, precise estimates are given on how fast the recovery error can in theory decay. Next, an idealized recovery method is proved to reach the fourth-root of the optimal decay rate for Gaussian sensing schemes. This idealized method being impractical, an implementable recovery algorithm is finally proposed in the context of factorized Gaussian sensing schemes. It is shown to provide a recovery error decaying as the sixth-root of the optimal rate.

cs.IT↗

arXiv2018

Proceedings of the fourth "international Traveling Workshop on Interactions between low-complexity data models and Sensing Techniques" (iTWIST'18)

The iTWIST workshop series aim at fostering collaboration between international scientific teams for developing new theories, applications and generalizations of low-complexity models. These events emphasize dissemination of ideas through both specific oral and poster presentations, as well as free discussions. For this fourth edition, iTWIST'18 gathered in CIRM, Marseille, France, 74 international participants and featured 7 invited talks, 16 oral presentations, and 21 posters. From iTWIST'18, the scientific committee has decided that the workshop proceedings will adopt the episcience.org philosophy, combined with arXiv.org: in a nutshell, "the proceedings are equivalent to an overlay page, built above arXiv.org; they add value to these archives by attaching a scientific caution to the validated papers." This means that all papers listed in the HTML page of this arxiv publication (see the menu on the right) have been thoroughly evaluated and approved by two independent reviewers, and authors have revised their work according to the comments provided by these reviewers.

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