arXiv subjects

Laurent Jacques

Publications and source records attributed to Laurent Jacques.

At least 91 records · Page 5Linked to original sources

arXiv2016

A Non-Convex Blind Calibration Method for Randomised Sensing Strategies

The implementation of computational sensing strategies often faces calibration problems typically solved by means of multiple, accurately chosen training signals, an approach that can be resource-consuming and cumbersome. Conversely, blind calibration does not require any training, but corresponds to a bilinear inverse problem whose algorithmic solution is an open issue. We here address blind calibration as a non-convex problem for linear random sensing models in which we aim to recover an unknown signal from its projections on sub-Gaussian random vectors, each of which is subject to an unknown multiplicative factor (gain). To solve this optimisation problem we resort to projected gradient descent starting from a suitable initialisation. An analysis of this algorithm allows us to show that it converges to the global optimum provided a sample complexity requirement is met, i.e., relating convergence to the amount of information collected during the sensing process. Finally, we present some numerical experiments in which our algorithm allows for a simple solution to blind calibration of sensor gains in computational sensing applications.

cs.IT↗

arXiv2016

Blind Deconvolution of PET Images using Anatomical Priors

Images from positron emission tomography (PET) provide metabolic information about the human body. They present, however, a spatial resolution that is limited by physical and instrumental factors often modeled by a blurring function. Since this function is typically unknown, blind deconvolution (BD) techniques are needed in order to produce a useful restored PET image. In this work, we propose a general BD technique that restores a low resolution blurry image using information from data acquired with a high resolution modality (e.g., CT-based delineation of regions with uniform activity in PET images). The proposed BD method is validated on synthetic and actual phantoms.

cs.CV↗

arXiv2016

Error Decay of (almost) Consistent Signal Estimations from Quantized Gaussian Random Projections

This paper provides new error bounds on "consistent" reconstruction methods for signals observed from quantized random projections. Those signal estimation techniques guarantee a perfect matching between the available quantized data and a new observation of the estimated signal under the same sensing model. Focusing on dithered uniform scalar quantization of resolution $δ>0$, we prove first that, given a Gaussian random frame of $\mathbb R^N$ with $M$ vectors, the worst-case $\ell_2$-error of consistent signal reconstruction decays with high probability as $O(\frac{N}{M}\log\frac{M}{\sqrt N})$ uniformly for all signals of the unit ball $\mathbb B^N \subset \mathbb R^N$. Up to a log factor, this matches a known lower bound in $Ω(N/M)$ and former empirical validations in $O(N/M)$. Equivalently, if $M$ exceeds a minimal number of frame coefficients growing like $O(\frac{N}{ε_0}\log \frac{\sqrt N}{ε_0})$, any vectors in $\mathbb B^N$ with $M$ identical quantized projections are at most $ε_0$ apart with high probability. Second, in the context of Quantized Compressed Sensing with $M$ Gaussian random measurements and under the same scalar quantization scheme, consistent reconstructions of $K$-sparse signals of $\mathbb R^N$ have a worst-case error that decreases with high probability as $O(\tfrac{K}{M}\log\tfrac{MN}{\sqrt K^3})$ uniformly for all such signals. Finally, we show that the proximity of vectors whose quantized random projections are only approximately consistent can still be bounded with high probability. A certain level of corruption is thus allowed in the quantization process, up to the appearance of a systematic bias in the reconstruction error of (almost) consistent signal estimates.

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arXiv2016

Multi-resolution Compressive Sensing Reconstruction

We consider the problem of reconstructing an image from compressive measurements using a multi-resolution grid. In this context, the reconstructed image is divided into multiple regions, each one with a different resolution. This problem arises in situations where the image to reconstruct contains a certain region of interest (RoI) that is more important than the rest. Through a theoretical analysis and simulation experiments we show that the multi-resolution reconstruction provides a higher quality of the RoI compared to the traditional single-resolution approach.

cs.CV↗

arXiv2016

Cell segmentation with random ferns and graph-cuts

The progress in imaging techniques have allowed the study of various aspect of cellular mechanisms. To isolate individual cells in live imaging data, we introduce an elegant image segmentation framework that effectively extracts cell boundaries, even in the presence of poor edge details. Our approach works in two stages. First, we estimate pixel interior/border/exterior class probabilities using random ferns. Then, we use an energy minimization framework to compute boundaries whose localization is compliant with the pixel class probabilities. We validate our approach on a manually annotated dataset.

cs.CV↗

arXiv2015

On The Exact Recovery Condition of Simultaneous Orthogonal Matching Pursuit

Several exact recovery criteria (ERC) ensuring that orthogonal matching pursuit (OMP) identifies the correct support of sparse signals have been developed in the last few years. These ERC rely on the restricted isometry property (RIP), the associated restricted isometry constant (RIC) and sometimes the restricted orthogonality constant (ROC). In this paper, three of the most recent ERC for OMP are examined. The contribution is to show that these ERC remain valid for a generalization of OMP, entitled simultaneous orthogonal matching pursuit (SOMP), that is capable to process several measurement vectors simultaneously and return a common support estimate for the underlying sparse vectors. The sharpness of the bounds is also briefly discussed in light of previous works focusing on OMP.

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arXiv2015

Discriminative and Efficient Label Propagation on Complementary Graphs for Multi-Object Tracking

Given a set of detections, detected at each time instant independently, we investigate how to associate them across time. This is done by propagating labels on a set of graphs, each graph capturing how either the spatio-temporal or the appearance cues promote the assignment of identical or distinct labels to a pair of detections. The graph construction is motivated by a locally linear embedding of the detection features. Interestingly, the neighborhood of a node in appearance graph is defined to include all the nodes for which the appearance feature is available (even if they are temporally distant). This gives our framework the uncommon ability to exploit the appearance features that are available only sporadically. Once the graphs have been defined, multi-object tracking is formulated as the problem of finding a label assignment that is consistent with the constraints captured each graph, which results into a difference of convex (DC) program. We propose to decompose the global objective function into node-wise sub-problems. This not only allows a computationally efficient solution, but also supports an incremental and scalable construction of the graph, thereby making the framework applicable to large graphs and practical tracking scenarios. Moreover, it opens the possibility of parallel implementation.

cs.CV↗

arXiv2015

Robust 1-Bit Compressive Sensing via Binary Stable Embeddings of Sparse Vectors

The Compressive Sensing (CS) framework aims to ease the burden on analog-to-digital converters (ADCs) by reducing the sampling rate required to acquire and stably recover sparse signals. Practical ADCs not only sample but also quantize each measurement to a finite number of bits; moreover, there is an inverse relationship between the achievable sampling rate and the bit depth. In this paper, we investigate an alternative CS approach that shifts the emphasis from the sampling rate to the number of bits per measurement. In particular, we explore the extreme case of 1-bit CS measurements, which capture just their sign. Our results come in two flavors. First, we consider ideal reconstruction from noiseless 1-bit measurements and provide a lower bound on the best achievable reconstruction error. We also demonstrate that i.i.d. random Gaussian matrices describe measurement mappings achieving, with overwhelming probability, nearly optimal error decay. Next, we consider reconstruction robustness to measurement errors and noise and introduce the Binary $ε$-Stable Embedding (B$ε$SE) property, which characterizes the robustness measurement process to sign changes. We show the same class of matrices that provide almost optimal noiseless performance also enable such a robust mapping. On the practical side, we introduce the Binary Iterative Hard Thresholding (BIHT) algorithm for signal reconstruction from 1-bit measurements that offers state-of-the-art performance.

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arXiv2015

Consistent Basis Pursuit for Signal and Matrix Estimates in Quantized Compressed Sensing

This paper focuses on the estimation of low-complexity signals when they are observed through $M$ uniformly quantized compressive observations. Among such signals, we consider 1-D sparse vectors, low-rank matrices, or compressible signals that are well approximated by one of these two models. In this context, we prove the estimation efficiency of a variant of Basis Pursuit Denoise, called Consistent Basis Pursuit (CoBP), enforcing consistency between the observations and the re-observed estimate, while promoting its low-complexity nature. We show that the reconstruction error of CoBP decays like $M^{-1/4}$ when all parameters but $M$ are fixed. Our proof is connected to recent bounds on the proximity of vectors or matrices when (i) those belong to a set of small intrinsic "dimension", as measured by the Gaussian mean width, and (ii) they share the same quantized (dithered) random projections. By solving CoBP with a proximal algorithm, we provide some extensive numerical observations that confirm the theoretical bound as $M$ is increased, displaying even faster error decay than predicted. The same phenomenon is observed in the special, yet important case of 1-bit CS.

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arXiv2015

Non-parametric PSF estimation from celestial transit solar images using blind deconvolution

Context: Characterization of instrumental effects in astronomical imaging is important in order to extract accurate physical information from the observations. The measured image in a real optical instrument is usually represented by the convolution of an ideal image with a Point Spread Function (PSF). Additionally, the image acquisition process is also contaminated by other sources of noise (read-out, photon-counting). The problem of estimating both the PSF and a denoised image is called blind deconvolution and is ill-posed. Aims: We propose a blind deconvolution scheme that relies on image regularization. Contrarily to most methods presented in the literature, our method does not assume a parametric model of the PSF and can thus be applied to any telescope. Methods: Our scheme uses a wavelet analysis prior model on the image and weak assumptions on the PSF. We use observations from a celestial transit, where the occulting body can be assumed to be a black disk. These constraints allow us to retain meaningful solutions for the filter and the image, eliminating trivial, translated and interchanged solutions. Under an additive Gaussian noise assumption, they also enforce noise canceling and avoid reconstruction artifacts by promoting the whiteness of the residual between the blurred observations and the cleaned data. Results: Our method is applied to synthetic and experimental data. The PSF is estimated for the SECCHI/EUVI instrument using the 2007 Lunar transit, and for SDO/AIA using the 2012 Venus transit. Results show that the proposed non-parametric blind deconvolution method is able to estimate the core of the PSF with a similar quality to parametric methods proposed in the literature. We also show that, if these parametric estimations are incorporated in the acquisition model, the resulting PSF outperforms both the parametric and non-parametric methods.

cs.CV↗

arXiv2015

A Quantized Johnson Lindenstrauss Lemma: The Finding of Buffon's Needle

In 1733, Georges-Louis Leclerc, Comte de Buffon in France, set the ground of geometric probability theory by defining an enlightening problem: What is the probability that a needle thrown randomly on a ground made of equispaced parallel strips lies on two of them? In this work, we show that the solution to this problem, and its generalization to $N$ dimensions, allows us to discover a quantized form of the Johnson-Lindenstrauss (JL) Lemma, i.e., one that combines a linear dimensionality reduction procedure with a uniform quantization of precision $δ>0$. In particular, given a finite set $\mathcal S \subset \mathbb R^N$ of $S$ points and a distortion level $ε>0$, as soon as $M > M_0 = O(ε^{-2} \log S)$, we can (randomly) construct a mapping from $(\mathcal S, \ell_2)$ to $(δ\mathbb Z^M, \ell_1)$ that approximately preserves the pairwise distances between the points of $\mathcal S$. Interestingly, compared to the common JL Lemma, the mapping is quasi-isometric and we observe both an additive and a multiplicative distortions on the embedded distances. These two distortions, however, decay as $O(\sqrt{(\log S)/M})$ when $M$ increases. Moreover, for coarse quantization, i.e., for high $δ$ compared to the set radius, the distortion is mainly additive, while for small $δ$ we tend to a Lipschitz isometric embedding. Finally, we prove the existence of a "nearly" quasi-isometric embedding of $(\mathcal S, \ell_2)$ into $(δ\mathbb Z^M, \ell_2)$. This one involves a non-linear distortion of the $\ell_2$-distance in $\mathcal S$ that vanishes for distant points in this set. Noticeably, the additive distortion in this case is slower, and decays as $O(\sqrt[4]{(\log S)/M})$.

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arXiv2015

Compressive Imaging and Characterization of Sparse Light Deflection Maps

Light rays incident on a transparent object of uniform refractive index undergo deflections, which uniquely characterize the surface geometry of the object. Associated with each point on the surface is a deflection map (or spectrum) which describes the pattern of deflections in various directions. This article presents a novel method to efficiently acquire and reconstruct sparse deflection spectra induced by smooth object surfaces. To this end, we leverage the framework of Compressed Sensing (CS) in a particular implementation of a schlieren deflectometer, i.e., an optical system providing linear measurements of deflection spectra with programmable spatial light modulation patterns. We design those modulation patterns on the principle of spread spectrum CS for reducing the number of observations. The ability of our device to simultaneously observe the deflection spectra on a dense discretization of the object surface is related to a Multiple Measurement Vector (MMV) model. This scheme allows us to estimate both the noise power and the instrumental point spread function. We formulate the spectrum reconstruction task as the solving of a linear inverse problem regularized by an analysis sparsity prior using a translation invariant wavelet frame. Our results demonstrate the capability and advantages of using a CS based approach for deflectometric imaging both on simulated data and experimental deflectometric data. Finally, the paper presents an extension of our method showing how we can extract the main deflection direction in each point of the object surface from a few compressive measurements, without needing any costly reconstruction procedures. This compressive characterization is then confirmed with experimental results on simple plano-convex and multifocal intra-ocular lenses studying the evolution of the main deflection as a function of the object point location.

cs.CV↗

arXiv2015

Post-Reconstruction Deconvolution of PET Images by Total Generalized Variation Regularization

Improving the quality of positron emission tomography (PET) images, affected by low resolution and high level of noise, is a challenging task in nuclear medicine and radiotherapy. This work proposes a restoration method, achieved after tomographic reconstruction of the images and targeting clinical situations where raw data are often not accessible. Based on inverse problem methods, our contribution introduces the recently developed total generalized variation (TGV) norm to regularize PET image deconvolution. Moreover, we stabilize this procedure with additional image constraints such as positivity and photometry invariance. A criterion for updating and adjusting automatically the regularization parameter in case of Poisson noise is also presented. Experiments are conducted on both synthetic data and real patient images.

cs.CV↗

arXiv2014

Quantization and Compressive Sensing

Quantization is an essential step in digitizing signals, and, therefore, an indispensable component of any modern acquisition system. This book chapter explores the interaction of quantization and compressive sensing and examines practical quantization strategies for compressive acquisition systems. Specifically, we first provide a brief overview of quantization and examine fundamental performance bounds applicable to any quantization approach. Next, we consider several forms of scalar quantizers, namely uniform, non-uniform, and 1-bit. We provide performance bounds and fundamental analysis, as well as practical quantizer designs and reconstruction algorithms that account for quantization. Furthermore, we provide an overview of Sigma-Delta ($ΣΔ$) quantization in the compressed sensing context, and also discuss implementation issues, recovery algorithms and performance bounds. As we demonstrate, proper accounting for quantization and careful quantizer design has significant impact in the performance of a compressive acquisition system.

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arXiv2014

Robust Phase Unwrapping by Convex Optimization

The 2-D phase unwrapping problem aims at retrieving a "phase" image from its modulo $2π$ observations. Many applications, such as interferometry or synthetic aperture radar imaging, are concerned by this problem since they proceed by recording complex or modulated data from which a "wrapped" phase is extracted. Although 1-D phase unwrapping is trivial, a challenge remains in higher dimensions to overcome two common problems: noise and discontinuities in the true phase image. In contrast to state-of-the-art techniques, this work aims at simultaneously unwrap and denoise the phase image. We propose a robust convex optimization approach that enforces data fidelity constraints expressed in the corrupted phase derivative domain while promoting a sparse phase prior. The resulting optimization problem is solved by the Chambolle-Pock primal-dual scheme. We show that under different observation noise levels, our approach compares favorably to those that perform the unwrapping and denoising in two separate steps.

math.OC↗

arXiv2013

Compressive Optical Deflectometric Tomography: A Constrained Total-Variation Minimization Approach

Optical Deflectometric Tomography (ODT) provides an accurate characterization of transparent materials whose complex surfaces present a real challenge for manufacture and control. In ODT, the refractive index map (RIM) of a transparent object is reconstructed by measuring light deflection under multiple orientations. We show that this imaging modality can be made "compressive", i.e., a correct RIM reconstruction is achievable with far less observations than required by traditional Filtered Back Projection (FBP) methods. Assuming a cartoon-shape RIM model, this reconstruction is driven by minimizing the map Total-Variation under a fidelity constraint with the available observations. Moreover, two other realistic assumptions are added to improve the stability of our approach: the map positivity and a frontier condition. Numerically, our method relies on an accurate ODT sensing model and on a primal-dual minimization scheme, including easily the sensing operator and the proposed RIM constraints. We conclude this paper by demonstrating the power of our method on synthetic and experimental data under various compressive scenarios. In particular, the compressiveness of the stabilized ODT problem is demonstrated by observing a typical gain of 20 dB compared to FBP at only 5% of 360 incident light angles for moderately noisy sensing.

cs.CV↗

arXiv2013

Quantized Iterative Hard Thresholding: Bridging 1-bit and High-Resolution Quantized Compressed Sensing

In this work, we show that reconstructing a sparse signal from quantized compressive measurement can be achieved in an unified formalism whatever the (scalar) quantization resolution, i.e., from 1-bit to high resolution assumption. This is achieved by generalizing the iterative hard thresholding (IHT) algorithm and its binary variant (BIHT) introduced in previous works to enforce the consistency of the reconstructed signal with respect to the quantization model. The performance of this algorithm, simply called quantized IHT (QIHT), is evaluated in comparison with other approaches (e.g., IHT, basis pursuit denoise) for several quantization scenarios.

cs.IT↗

arXiv2013

On the optimality of a L1/L1 solver for sparse signal recovery from sparsely corrupted compressive measurements

This short note proves the $\ell_2-\ell_1$ instance optimality of a $\ell_1/\ell_1$ solver, i.e a variant of \emph{basis pursuit denoising} with a $\ell_1$ fidelity constraint, when applied to the estimation of sparse (or compressible) signals observed by sparsely corrupted compressive measurements. The approach simply combines two known results due to Y. Plan, R. Vershynin and E. Candès.

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