Matthieu Puigt

Gilles Delmaire

Gilles Roussel

Environmental signal processing: new trends and applications

In the last years, environmental monitoring was shown to be a major application field of modern signal processing and machine learning techniques. In particular, it provides some interesting problems for which specific signal methods were proposed. In this session, we aim to review some recent advances in this topic. We propose a taxonomy of the major trends in environmental surveillance according to the characteristics of the sensing devices, i.e., (i) for a unique sensor or an array of sensors (e.g., bio-sensor, chemical sensor arrays), (ii) for remote observation, and (iii) using large-scale sensor networks.

Introduction

Environmental pollution is a major problem faced by the world. Indeed, studies show the effects of pollution levels-e.g., in air [START_REF] Lelieveld | The contribution of outdoor air pollution sources to premature mortality on a global scale[END_REF] or water [START_REF] Wright | The physical impacts of microplastics on marine organisms: A review[END_REF]-on health, and more generally on biodiversity [START_REF] Bobbink | The effects of air-borne nitrogen pollutants on species diversity in natural and semi-natural european vegetation[END_REF]. Human activity is also shown to be the main origin of climate change [START_REF]Climate change[END_REF], which particularly affects billions of lifes.

As a consequence, monitoring environmental phenomena has been investigated for several decades. Environmental surveillance not only consists of monitoring air, water, or soil pollutant levels or surface temperature, but also of observing noise and light pollution in urban areas, which have consequences on health and biodiversity [START_REF] Stephen | Noise pollution: non-auditory effects on health[END_REF][START_REF] Francis | Noise pollution changes avian communities and species interactions[END_REF][START_REF] Longcore | Ecological light pollution[END_REF]. Such a surveillance has been possible using sensors, and with their increasing number emerged specific signal processing and/or machine learning techniques. In this paper, we aim to review some of these techniques-and some associated open challenges-as well as to briefly introduce the papers of the special session on environmental signal processing of the 2017 ESANN conference.

The remainder of the paper reads as follows. In Section 2, we focus on approaches for single sensors and sensor arrays met in, e.g., source apportionment and bio-chemical sensors. Section 3 shows how remote sensing can be used in environmental monitoring. Section 4 covers some approaches for large-scale sensor networks. We finally conclude in Section 5.

Chemometrics for single sensor and sensor arrays

Processing single environmental sensors or sensor arrays arised as a major topic in environmental monitoring, known under the name of chemometrics. Indeed, while sensors used in source apportionment provide fine particulate analyzed by chemists which must then be numerically processed to enhance their interpretability, there has also been an increasing need to process data from chemical/bio-sensors which might have nonlinear responses. We review hereafter some of these aspects.

Source apportionment

Source apportionment aims at estimating the impacts of emissions from different sources of pollutants based on ambient data registered at monitoring sites. It has been used for many applications, to monitor indoor [START_REF] Ouaret | Indoor air pollutant sources using blind source separation methods[END_REF] or outdoor air [START_REF] Limem | Non-negative matrix factorization under equality constraints-a study of industrial source identification[END_REF], sea [START_REF] Luo | Distribution, source apportionment, and transport of PAHs in sediments from the Pearl River Delta and the northern South China Sea[END_REF], and soil pollution [START_REF] Zuo | Source apportionment of polycyclic aromatic hydrocarbons in surface soil in tianjin, china[END_REF], for example. The awareness of the increasing pollution and the necessity to find answers to these issues has led to some growing interest and increasing number of scientific studies, as recently reviewed in [START_REF] Hopke | Review of receptor modeling methods for source apportionment[END_REF].

Among the classical source apportionment techniques, the receptor model assumes that a chemical mass measure is a non-negative linear combination of the masses of source profiles, weighted by their associated contribution over time. This model relies on the law of conservation of mass and implicitly assumes that air suspended particles or sedimentary matter are not reacting along their path toward the sensor. The receptor model may then be written as a non-negative matrix factorization product, i.e.,

X ≈ G • F, (1)
where X accounts for the n × m concentration matrix (in ng/m 3 ), G is the n × p contribution matrix gathering the loadings of all the sources over time and F is the p × m profile matrix, where a profile is a source signature involving all the chemical species proportions (in ng/ng). In some studies, e.g., [START_REF] Ouaret | Indoor air pollutant sources using blind source separation methods[END_REF], the above matrices X and F are defined regarding the considered particle-size distribution. Principal Component Analysis (PCA) has been used in numerous receptor models for a long time [START_REF] Hopke | It's time to drop principal components analysis as a 'receptor model[END_REF]. However, the non-negativity property was lost and, as a consequence, other techniques-i.e., Positive (PMF) / Non-negative Matrix Factorization (NMF)-emerged as an alternative [START_REF] Paatero | Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values[END_REF] which became the standard receptor model. However, the unicity of their solution is not guaranteed [START_REF] Viana | Source apportionment of particulate matter in europe: a review of methods and results[END_REF] and data might be corrupted by outliers. Lastly, the profiles in F sum to one. As a consequence, some (possibly-robust) informed NMF techniqueswhich restrict the space of admissible solutions-were recently proposed to that end, e.g., [START_REF] Limem | Non-negative matrix factorization under equality constraints-a study of industrial source identification[END_REF][START_REF] Plouvin | Enhanced NMF initialization using a physical model for pollution source apportionment[END_REF][START_REF] Chreiky | Robust informed split gradient NMF using Alpha Beta-divergence for source apportionment[END_REF].

Chemical sensors

While linear chemometrics methods have been massively proposed during the last decades, an increasing number of problems require specific methods for processing chemical-or bio-sensors which may provide highly nonlinear readings. Such sensors are usually grouped into an array and aim to deliver the concentration of chemical species present in the air, rivers, or a given analyte. In some problems, the objective consists of estimating the concentrations of several chemical profiles, which may be achieved using some Ion-Selective Electrode arrays (ISE) or bio-sensors located at fixed positions. Usually, this step is achieved in a supervised way by learning neural networks-e.g., for electronic tongues [START_REF] González-Calabuig | High dimensionality voltammetric biosensor data processed with artificial neural networks[END_REF]-which is costly. As an alternative, some authors proposed unsupervised techniques to work in a plug and sense way. This turns out to be a nonlinear source separation problem, which is known to be a hard task [START_REF] Comon | Handbook of blind source separation : independent component analysis and applications[END_REF]Ch. 14].

Nonlinear source separation is mainly solved by assuming a known nonlinear mixture model where some parameters must be estimated. In chemical sensor arrays, the Nernst equation provides a scaled logarithmic dependency between the output voltage v i of the i-th sensor and the mixed ionic activity, i.e.,

v i (n) = e i + RT z i F log   s i (n) + j =i a ij s j (n) z i z j   , (2)
where s j (n) and z j account for the ionic activity and the valence of the j-th species, respectively, R and F are the gas and Faraday constants, respectively, and T is the temperature. This model is difficult to deal with, because of the power term appearing inside the logarithm. In the case of equal valences, the authors in [START_REF] Duarte | Source separation in chemical analysis : Recent achievements and perspectives[END_REF] describe this equation as a general post-nonlinear model andassuming that the source signals are independent-propose a simplified inverting process which reads

y k (n) = P i=1 w ki exp(f i + h i v i (n)), (3)
where the parameters w ki , f i , and h i have to be tuned according to an independence criterion. To prevent the algorithm to be stuck into a local minimum, Bayesian approaches [START_REF] Duarte | A bayesian nonlinear source separation method for smart ion-selective electrode arrays[END_REF]-enabling to describe the complete distribution of the source signals-may also be used, to the price of a high computational load. Despite the great number of studies in nonlinear unmixing for chemical sensing, there are many assumptions which govern the proposed solutions, which may be hard to fulfill in real case studies. Some global solutions in real conditions are still expected.

Environmental monitoring using hyperspectral imagery

Hyperspectral (HS) sensing technologies allowed many opportunities of new applications over the past two decades. Most HS sensors are embedded in airborne or spaceborne platforms. Spectral, spatial, and temporal resolution has reached an unprecedented level and enabled uncountable applications requiring fine identification of materials or estimation of physical parameters. HS imaging has been increasingly used for earth surveillance, e.g., for soil, sea, soil-water interfaces and also astrophysical observations [START_REF] Kaufmann | EnMAP a hyperspectral sensor for environmental mapping and analysis[END_REF]. Moreover, HSI can be used to monitor industrial discharges-e.g., atmospheric effluents, storage of solid residues, liquid residues and sludges-with might have an impact on environment and health [START_REF] Manolakis | Long-wave infrared hyperspectral remote sensing of chemical clouds: A focus on signal processing approaches[END_REF]. Such an increasing number of applications need sophisticated signal processing algorithms because of the complexity to extract relevant information, the high dimensionality and the size of the hyperspectral data, the (linear or nonlinear) spectral mixing, the measurement process. The framework used in these topics is rooted on signal and image processing, statistical inference, and machine learning fields.

Data and hypotheses

HS imaging (HSI) aims to acquire HS datacubes which consists of collected spectral vectors sorted as a tensor denoted X ∈ R n1×n2×n b . Each image of n = n 1 × n 2 pixels represents the radiance in n b spectral bands of the wavelength interval covered by the embedded sensor(s). Usually, the values in X smoothly vary along the spectral bands and the spatial pixels, and thus live in a low-dimensional manifold. Spectral information can then be denoised-using, e.g., PCA-while smooth spatial variations are well-suited for supervised classification [START_REF] Bioucas-Dias | Hyperspectral remote sensing data analysis and future challenges[END_REF]. When processing HS data, one must take into account some physical and optical considerations-e.g., the surface and ground-leaving reflectances, the effects of illumination, the angle of view, etc-and each interface between surface, atmosphere and sensor leads to a correction or a calibration process. Such considerations yield some signal processing techniques that we present below.

Hyperspectral processing techniques

As explained above, HSI gained a massive interest from the scientific community since two decades. Many data processing techniques were proposed to extract insightful information from hyperspectral data cubes, as recently reviewed in [START_REF] Bioucas-Dias | Hyperspectral remote sensing data analysis and future challenges[END_REF]. We briefly recall them below, with more emphasis on a few of them.

One main family of data processing techniques consists of enhancing the interpretability of the hyperspectral images by denoising (restoration) and fusing data cubes (spatial, spectral, spatial-spectral data fusion). In particular, several authors are interested in pansharpening [START_REF] Loncan | Hyperspectral pansharpening: A review[END_REF] (or sharpening [START_REF] Chen | Fusion of hyperspectral and multispectral images: A novel framework based on generalization of pan-sharpening methods[END_REF]), i.e., the fusion of a panchromatic (or a multispectral) image with an HS one, which provide a good spatial and spectral resolution, respectively. The pansharpened image then combines the good spatial and spectral accuracy of both images.

Classification is also an important topic met in HSI. It consists of assigning a unique label to each spatial pixel of an hyperspectral cube, corresponding to a given class. A high spatial resolution is a very desirable property for classification techniques-divided as supervised, unsupervised, and semi-supervised techniques [START_REF] Bioucas-Dias | Hyperspectral remote sensing data analysis and future challenges[END_REF]-so that most data only contain pure pixels. A pure pixel represents a predominant signature and is thus naturally linked to a class. However, a trade-off between spatial and spectral resolution must be chosen in HSI, as both cannot increase concurrently. As a consequence, hyperspectral unmixing (HU) techniques-a.k.a. blind source separation-were proposed to enhance classification performance.

In HU, the Linear Mixture Model (LMM) is widely used and most HU approaches unfold the above datacube X to derive a n × n b matrix, denoted Y hereafter. Each column of Y contains the spectral information located at a given spatial pixel of X and the LMM methods provides the following mixture model:

min M,A Y -M A F , s.t. A ≥ 0, 1 T p A = 1 n , (4)
where A contains the non-negative fractional abundance vectors of the LMMi.e., the mixing parameters which sum to 1-and M contains the spectral signatures of the materials-named endmembers-and 1 p stands for the p × 1 vector of ones. Actually, HU has some similarities with the source apportionment problem introduced in Subsection 2.1, except that the rows of F sum to one in source apportionment while the columns of A sum to one in HU, thus yielding different NMF solutions. Main unmixing strategies assume the existence of pure pixels for each endmember-and are based on sparsity and/or geometric properties-or the absence of a few endmembers in some pixels, using volume-based approaches to perform HU in that case [START_REF] Bioucas-Dias | Hyperspectral remote sensing data analysis and future challenges[END_REF]. Several problems are currently investigated, i.e., the estimation of the intrinsic dimensionality [START_REF] Chang | Estimation of number of spectrally distinct signal sources in hyperspectral imagery[END_REF]-i.e., the number of endmembers in Y -the endmember variability [START_REF] Revel | Impact of the initialisation of a blind unmixing method dealing with intra-class variability[END_REF] for both LMM but also nonlinear mixing models which are increasingly studied [START_REF] Heylen | A review of nonlinear hyperspectral unmixing methods[END_REF].

Both the classification and HU can be used for HS target detection, where one aims to detect and identify man-made or natural clutters from the HSI variations [START_REF] Manolakis | Long-wave infrared hyperspectral remote sensing of chemical clouds: A focus on signal processing approaches[END_REF].

Sensor networks for environmental monitoring

During the last decades, sensors have been massively spread around the world to provide fine grained yet accurate environmental analytics (e.g., temperature, air quality, etc) [START_REF] Rundel | Environmental sensor networks in ecological research[END_REF]. A taxonomy of such environmental sensor networks is provided in [START_REF] Hart | Environmental sensor networks: A revolution in the earth system science?[END_REF] and some insight about specific problems is provided. These problems have been studied in the literature since the work in [START_REF] Hart | Environmental sensor networks: A revolution in the earth system science?[END_REF] and we now summarize this work, together with some more recent open challenges.

Dealing with sensor inaccuracies

One of the first issues met with sensor networks is the possible lack of trust in the readings provided by the sensors. Indeed, the latter can be miscalibratedresulting in biased scientific analyzes [START_REF] Karl | Possible artifacts of data biases in the recent global surface warming hiatus[END_REF]-but their readings may also suffer from the presence of outliers, due to sensor faults for example, or to data loss. We discuss about both issues hereafter.

Sensor calibration is an important step in environmental monitoring which should be regularly performed. This is usually done in a laboratory where the sensor readings y(t) are mapped to a controlled physical input x(t), so that the the sensor calibration function F can be learned. Once learned and assuming that F is invertible, it is then possible to derive x(t) from the sensor readings.

However, such a calibration task is not possible in many environmental problems, either because the sensors are not accessible like, e.g., in remote sensing [START_REF] Carfantan | Statistical linear destriping of satellite-based pushbroom-type images[END_REF], or because they are so numerous that it would be too time consuming to do so. As a consequence, some blind calibration or self-calibration techniques were proposed. They aim to calibrate the sensors from their readings and might be possibly using other calibrated sensor readings for that purpose.

In order to perform blind calibration, it is first necessary to know the sensor model F in order to estimate its intrinsic parameters. Most authors-e.g., [START_REF] Balzano | Blind calibration of sensor networks[END_REF][START_REF] Wang | Moments based blind calibration in mobile sensor networks[END_REF][START_REF] Dorffer | Blind calibration of mobile sensors using informed nonnegative matrix factorization[END_REF]-assume it is affine, i.e., F is defined by an offset and a gain. A few ones consider an extended nonlinear model which might be piecewise linear [START_REF] Wang | Blindly calibrating mobile sensors using piecewise linear functions[END_REF] or polynomial [START_REF] Wang | Calibrating nonlinear mobile sensors[END_REF][START_REF] Dorffer | Nonlinear mobile sensor calibration using informed semi-nonnegative matrix factorization with a Vandermonde factor[END_REF]. As an alternative to both above models which are function of a single input x(t), many sensors-e.g., miniaturized gas sensor-are driven by multiple-say M -inputs, e.g., the sensed gas concentration, the ambient temperature and the humidity [START_REF] Piedrahita | The next generation of low-cost personal air quality sensors for quantitative exposure monitoring[END_REF]. In that case, the simplest model of F(.) is multilinear. Moreover, as the sensors must be regularly calibrated, this means that their response drifts with time. Some authors propose to incorporate this drift inside the sensor response model [START_REF] Piedrahita | The next generation of low-cost personal air quality sensors for quantitative exposure monitoring[END_REF]. Once the sensor model is known, calibration can be performed, either simultaneously for the whole sensor network (macro-calibration [START_REF] Whitehouse | Macro-calibration in sensor/actuator networks[END_REF]), or sequentially for each individual sensor (micro-calibration [START_REF] Saukh | Reducing multi-hop calibration errors in largescale mobile sensor networks[END_REF]). Many macro-calibration techniques are based on the statistical moments of the sensed phenomenon [START_REF] Carfantan | Statistical linear destriping of satellite-based pushbroom-type images[END_REF][START_REF] Wang | Moments based blind calibration in mobile sensor networks[END_REF], or assume the sensed phenomenon to be low-rank and project the readings on a previously learned subspace [START_REF] Balzano | Blind calibration of sensor networks[END_REF]. These approaches can be applied to mobile and fixed sensors. On the contrary, most micro-calibration methods were proposed for mobile sensors. While a few of them are based on moments [START_REF] Lee | A blind calibration scheme exploiting mutual calibration relationships for a dense mobile sensor network[END_REF], many assume the sensors to be in rendezvous [START_REF] Saukh | On rendezvous in mobile sensing networks[END_REF]. Two sensors in rendezvous are in the same spatio-temporal vicinity and should thus acquire the same phenomenon. As a consequence, assuming that some sensors are calibrated, it is possible to perform calibration by regression of both the calibrated and the uncalibrated sensor readings in rendezvous [START_REF] Saukh | Reducing multi-hop calibration errors in largescale mobile sensor networks[END_REF]. As an alternative to this strategy, the authors in [START_REF] Dorffer | Blind calibration of mobile sensors using informed nonnegative matrix factorization[END_REF][START_REF] Dorffer | Nonlinear mobile sensor calibration using informed semi-nonnegative matrix factorization with a Vandermonde factor[END_REF] revisited macro-calibration by discretizing along space and time an observed area, using the rendezvous definition. Within this framework, self-calibration can be tackled as a structured matrix factorization problem with missing entries.

As explained above, sensor readings might also suffer from outliers [START_REF] Zhang | Outlier detection techniques for wireless sensor networks: A survey[END_REF] and missing data [START_REF] Giannopoulos | Application of tensor and matrix completion on water treatment data[END_REF]. Interestingly, both problems can be tackled by some approaches-denoted robust principle component analysis or matrix/tensor completion, respectively-which use the same intrinsic assumption. If the sensor network is dense enough, it should oversample the observed phenomenon, i.e., if the observed data are arranged into a matrix or a tensor, the latter should be low-rank. Low-rank modeling attracted a lot of interest in the last decade and recent reviews on the topic can be found in, e.g., [START_REF] Zhouchen | A review on low-rank models in data analysis[END_REF] (for sparse outlier removal) and [START_REF] Davenport | An overview of low-rank matrix recovery from incomplete observations[END_REF] (for low-rank completion). Specific techniques for environmental applications were proposed, to deal with outliers in ozone measurements [START_REF] Harrou | Ozone measurements monitoring using data-based approach[END_REF] and with missing information in seismic data [START_REF] Kumar | Efficient matrix completion for seismic data reconstruction[END_REF] or in wireless sensor networks [START_REF] Giannopoulos | Application of tensor and matrix completion on water treatment data[END_REF].

Physics-driven inverse problems

At one point, environmental monitoring aims to derive some accurate information about a sparsely sensed phenomenon. Several authors combine models with some signal processing tools. For example, the authors in [START_REF] Roussel | Estimation of a semi-physical GLBE model using dual EnKF learning algorithm coupled with a sensor network design strategy: Application to air field monitoring[END_REF][START_REF] Ranieri | Near-optimal sensor placement for linear inverse problems[END_REF][START_REF] Roy | Spatio-temporal sensor management for environmental field estimation[END_REF] proposed an approach to optimize the location of the sensors with respect to the sensed field. While one may assume the sensors to move in order to reduce the uncertainties of a physical model [START_REF] Roussel | Estimation of a semi-physical GLBE model using dual EnKF learning algorithm coupled with a sensor network design strategy: Application to air field monitoring[END_REF], the authors in [START_REF] Ranieri | Near-optimal sensor placement for linear inverse problems[END_REF][START_REF] Roy | Spatio-temporal sensor management for environmental field estimation[END_REF] propose to only use a subset of available sensors which minimize such uncertainties.

Similarly, some work was performed to estimate (i) the positions of the punctual sources of emission [START_REF] Ranieri | Sampling and reconstruction of time-varying atmospheric emissions[END_REF], (ii) the shape of non-punctual sources [START_REF] Murray-Bruce | Solving inverse source problems for sources with arbitrary shapes using sensor networks[END_REF], and (iii) the source emission wavefronts [START_REF] Martinez-Camara | The fukushima inverse problem[END_REF] as an inverse problem.

Lastly, some authors propose to decompose a sensed phenomenon as linear combinations of interpretable modes, e.g., using non-negative decomposition of latent variables [START_REF] Lopez-Radcenco | Non-negative decomposition of geophysical dynamics[END_REF].

Privacy-preserving issues in partipatory science

Nowadays, environmental sensor networks are heterogeneous as some sensors can be fixed or mobile [START_REF] Hart | Environmental sensor networks: A revolution in the earth system science?[END_REF]. In the latter case, they can be carried by robots [START_REF] Dunbabin | Robots for environmental monitoring: Significant advancements and applications[END_REF] or volunteers in participatory science [START_REF] Roy | Understanding citizen science & environmental monitoring[END_REF]. This last situation has been particularly studied in the literature. Indeed, with the increase use of smartphones and their abilities to connect to sensing devices, it is quite easy to propose a large-scale experiment but also to reverse enginneer the geolocated data to find who are the volunteers. As a consequence, privacy-preserving techniques were proposed [START_REF] Christin | A survey on privacy in mobile participatory sensing applications[END_REF] and mainly consists of modifying (i) the sensed values and (ii) the locations of these readings, which both might affect the performance of the tasks discussed in this section. A few methods, e.g., [START_REF] Markert | Private rendezvous-based calibration of low-cost sensors for participatory environmental sensing[END_REF][START_REF] Harrane | Toward privacy-preserving diffusion strategies for adaptation and learning over networks[END_REF], take into consideration such an issue by processing data at the sensor scale while sharing as few information as possible with neighbour sensing devices but such a hot topic is still open.

Conclusion and discussion

To conclude, environmental monitoring is a major application field for signal processing and machine learning. Due to the impact of climate change and pollution levels, together with the intrinsic interest from the politics and the population, environmental data processing techniques must be carefully driven to keep a high level of trust. We reviewed some current problems regarding the sensor scale (single sensor/sensor array, remote sensing, large-scale sensor networks) but could not cover other very interesting problems, such as the multimodal fusion of data issued from all these sensor technologies.

The remainder of the Environmental Signal Processing session follows the same structure as this paper: chemical sensor processing [START_REF] Ouaret | Indoor air pollutant sources using blind source separation methods[END_REF][START_REF] González-Calabuig | High dimensionality voltammetric biosensor data processed with artificial neural networks[END_REF], remote sensing HU [START_REF] Revel | Impact of the initialisation of a blind unmixing method dealing with intra-class variability[END_REF], and signal processing for/using sensor networks [START_REF] Murray-Bruce | Solving inverse source problems for sources with arbitrary shapes using sensor networks[END_REF][START_REF] Lopez-Radcenco | Non-negative decomposition of geophysical dynamics[END_REF][START_REF] Giannopoulos | Application of tensor and matrix completion on water treatment data[END_REF].


Clément Dorffer
email: clement.dorffer@univ-littoral.fr

Matthieu Puigt
email: matthieu.puigt@univ-littoral.fr

Gilles Delmaire
email: gilles.delmaire@univ-littoral.fr

Gilles Roussel
email: gilles.roussel@univ-littoral.fr

Outlier-Robust Calibration Method for Sensor Networks

In this paper, we aim to blindly calibrate the responses of a sensor network whose outputs are possibly corrupted by outliers. In particular, we extend some well-known nullspacebased blind calibration approaches, proposed for fixed sensors with affine responses-i.e., with unknown gain and offset for each sensor-to that difficult case. These state-of-the-art approaches assume that the true data lie in a known lower dimensional subspace, so that in practice sensors can be calibrated by projection of the uncalibrated observations to this subspace. A robust extension was recently proposed in order to provide less sensitivity to noise. In this paper, we show that such methods (including the robust extensions) are very sensitive to outliers and we propose new extensions able to deal with such issues. For that purpose, we assume the outliers to be rare events, which can be modeled as a sparse contribution to the low-rank observed data. Using such an assumption, we separate sparse outliers from the low-rank data, so that we can perform calibration. We show that the proposed approach is able to handle up to 10% of outliers in the data without major impact on the calibration accuracy while state-of-the-art methods are already sensitive to the presence of one unique outlier.

I. INTRODUCTION

While being increasingly cheap, sensors are now part of our daily lives. Indeed, deployed in large-scale networks, they provide massive quantities of data and allow many applications for, e.g., smart cities [START_REF] Zanella | Internet of things for smart cities[END_REF], agriculture [START_REF] Rehman | A review of wireless sensors and networks' applications in agriculture[END_REF], or environment monitoring [START_REF] Puigt | Environmental signal processing: new trends and applications[END_REF]. In such scenarios, the high number of measurements provided by the numerous sensors compensate the poor sensitivity and quality of an individual sensor (which is due to the fact that such sensors are low-cost) [START_REF] Cochran | Comparison between low-cost and traditional MEMS accelerometers: a case study from the M7.1 Darfield, New Zealand, aftershock deployment[END_REF].

However, an inherent problem to any sensor network is the sensor calibration step which is necessary to get interpretable and consistent data. Calibration usually consists of estimating the unknown parameters that link the sensors outputs to the corresponding physical inputs. Traditionally, calibration can be performed in a laboratory by determining the calibration parameters to fit a known and controlled input to which sensors are exposed to. However, when the sensors are too numerous or inaccessible, calibrating them in laboratory is no longer possible and specific signal processing methods have been proposed to remotely calibrate the sensors of the network from the collected measurements.

In the current literature, these calibration approaches can be classified into two main families, namely micro-and macrocalibration [START_REF] Whitehouse | Macro-calibration in sensor/actuator networks[END_REF]. Micro-calibration consists of calibrating one single sensor of a network at a time while macro-calibration operates on the whole set of sensors. Depending whether or not the sensors are mobile in the network, different assumptions have been proposed to solve such problem. When the sensors are mobile, they can be in rendezvous, i.e., they are in the same spatio-temporal neighborhood, thus sensing the same phenomenon [START_REF] Saukh | On rendezvous in mobile sensing networks[END_REF]. Such an assumption was recently used 1 in both micro- [START_REF] Saukh | Reducing multi-hop calibration errors in large-scale mobile sensor networks[END_REF], [START_REF] Miluzzo | CaliBree: A self-calibration system for mobile sensor networks[END_REF], [START_REF] Lee | A blind calibration scheme exploiting mutual calibration relationships for a dense mobile sensor network[END_REF] and macro-calibration2 [START_REF] Dorffer | Blind calibration of mobile sensors using informed nonnegative matrix factorization[END_REF], [START_REF] Dorffer | Blind mobile sensor calibration using a nonnegative matrix factorization with a relaxed rendezvous model[END_REF], [START_REF] Dorffer | Nonlinear mobile sensor calibration using informed semi-nonnegative matrix factorization with a Vandermonde factor[END_REF]. However, in the case of fixed sensors, other assumptions are needed. For fixed sensors, state-of-the-art calibration approaches assume to known the low-rank subspace where the sensed phenomenon lies [START_REF] Balzano | Blind calibration of sensor networks[END_REF], [START_REF] Lipor | Robust blind calibration via total least squares[END_REF] or consider a compressed sensing framework [START_REF] Schulke | Blind calibration in compressed sensing using message passing algorithms[END_REF], [START_REF] Bilen | Convex optimization approaches for blind sensor calibration using sparsity[END_REF]. Other approaches use statistical properties 3 of the sensed phenomenon to derive estimates of the calibration parameters [START_REF] Carfantan | Statistical linear destriping of satellitebased pushbroom-type images[END_REF].

Another issue with sensor networks may happen if the sensors are faulty. In such a case, the readings result in outliers which must be taken into account [START_REF] Zhang | Outlier detection techniques for wireless sensor networks: A survey[END_REF]. In this paper, we investigate the influence of outliers on blind calibration. In particular, we focus on the subspace-based approach [START_REF] Balzano | Blind calibration of sensor networks[END_REF] and its robust extension [START_REF] Lipor | Robust blind calibration via total least squares[END_REF]. We show that these methods are very sensitive to the presence of even a unique outlier and we propose an outlier-robust extension, which assumes that the number of outliers is low with respect to the mass of generated data, i.e., the outliers can be modeled as a sparse contribution in the observed low-rank data.

The remainder of the paper reads as follows. We introduce the sensor calibration problem in Section II. In Section III, we briefly recall the principles of state-of-the-art methods [START_REF] Balzano | Blind calibration of sensor networks[END_REF] and we show how sensitive to outliers these methods are. In Section IV, we propose a novel outlier-robust calibration method which combines a preprocessing stage-which finds and removes outliers from the sensor readings-to the nullspace-based strategies. In Section V, we experimentally show the enhancement provided by our proposed method. In particular, the preprocessing stage is shown to be able to filter up to 10% of outliers in the data. Lastly, we conclude and discuss the possible extension of the work in Section VI.

For the sake of readability, the notations used in this paper are defined in Table I.

II. PROBLEM STATEMENT

In this paper, we assume to observe the sensed signals z(t) ∈ R n provided by a synchronized network composed of n fixed sensors. Moreover, we also assume that the sensors are possibly defecting. Sensor faults might be due to, e.g., communication errors between sensors, signal subsampling due to desynchronization (assuming that missing samples are set to zero), sparse or continuous spurious sensor readings (see Fig. 1). In that case, the corrupted sensor readings z(t), observed at time t read

z(t) = y(t) + o(t), (1)
where y(t) and o(t) are the vectors of outlier-free sensor readings and outliers, respectively. In this paper, we assume the outliers to be sparse, i.e., most values in o(t) are null.

If one consider that sensor readings are outlier-free, i.e., o(t) = 0 for any time t, then one can directly observe y(t). If we assume that the calibration functions for each sensor of the network to be affine, then the true input phenomenondenoted x i (t)-reads

x i (t) = y i (t) • α i + β i , (2)
where y i (t) is the signal observed by the i-th sensor at time t, and α i and β i are the unknown sensor gain and offset associated with the sensor i, respectively. Then, by defining x(t) ∈ R n as the column vector containing the input phenomenon observed by the n sensors and Y(t) the diagonal matrix of y(t), one can write

x(t) = Y(t) • α + β, (3)
where α and β are the vectors containing the sensor gains and offsets, respectively. Calibrating the sensor network then consists of estimating α and β.

For that purpose, we briefly recall the assumptions made in [START_REF] Balzano | Blind calibration of sensor networks[END_REF], upon which we build our proposed method. We first assume that the sensor network is oversampling the observed signal, i.e., x(t) lies in a low dimensional subspace S which is assumed to be spanned by a set of r basis vectors, where r n is the rank of S. These basis vectors are assumed to be known. Such an assumption might look strong. However, it arises in several situations, e.g., for bandlimited signals with sensor distances lower than required by the Shannon-Nyquist sampling rate [START_REF] Balzano | Blind calibration of sensor networks[END_REF]. It is also valid in a machine learning framework-where the sensors are calibrated in their early life-so that S can be learned from early sensor readings. Then, while sensors are aging, their responses drift with time, i.e., their calibration parameters vary. This implies that at a given time t, the sensor readings follow the calibration model (3). Besides, as S is known, one can compute P, the linear projection on its orthogonal subspace. The key idea used in [START_REF] Balzano | Blind calibration of sensor networks[END_REF] (and in its robust extension [START_REF] Lipor | Robust blind calibration via total least squares[END_REF]) consists of noticing that x (t) is in the nullspace of P, i.e.,

P(x(t)) = P(Y(t) • α + β) = 0. (4)
In practice, we assume to collect synchronized snapshots y(t 1 ), y(t 2 ), . . . , y(t k ) from the sensor network, where t j denotes the time of the j-th snapshot. In order to perform calibration, the number k of snapshots satisfies [START_REF] Balzano | Blind calibration of sensor networks[END_REF] k

≥ n -1 n -r + 1. (5)
Defining Y j Y(t j ) , the nullspace blind calibration problem thus aims to solve {α, β} = arg min

α,β P Y j • α + β , ∀j ∈ [[1, k]] . (6)
In the next sections, we briefly introduce the principles of the nullspace-based calibration methods [START_REF] Balzano | Blind calibration of sensor networks[END_REF], [START_REF] Lipor | Robust blind calibration via total least squares[END_REF]. We then show that they are very sensitive to the presence of outliers in the observed signals-i.e., when o(t) = 0-and we propose a novel nullspace calibration method which introduces a preprocessing stage to remove outliers.

III. NULLSPACE-BASED SENSOR CALIBRATION

We now briefly recall the principles of the nullspace-based calibration methods. As P is linear and defining

Ȳ 1 k k j=1 Y j , (7)
the centered version of Eq. ( 4) reads

P Y j -Ȳ • α = 0, ∀j ∈ [[1, k]] , (8)
which yields

P Y j -Ȳ • α = 0, ∀j ∈ [[1, k]] . (9)
The authors in [START_REF] Balzano | Blind calibration of sensor networks[END_REF] then defined a matrix C by stacking the projection in each snapshot, i.e.,

C    P Y 1 -Ȳ . . . P Y k -Ȳ    . (10)
Combining Eqs. ( 9) and [START_REF] Lee | A blind calibration scheme exploiting mutual calibration relationships for a dense mobile sensor network[END_REF] implies that the unknown gain vector α is solution of

α = arg min α ||C • α|| 2 , (11)
which can be solved using ordinary Least Square (LS) or SVD-noticing that the last eigenvector is proportional to αas proposed in [START_REF] Balzano | Blind calibration of sensor networks[END_REF]. However, the solutions of Eq. ( 11) are not unique: indeed, if α is a solution to Eq. ( 11), then λ • α is also a solution. The authors in [START_REF] Balzano | Blind calibration of sensor networks[END_REF], [START_REF] Lipor | Robust blind calibration via total least squares[END_REF] removed the scale ambiguity by setting the first element 4 of α to 1. The authors in [START_REF] Lipor | Robust blind calibration via total least squares[END_REF] then proposed an extension of [START_REF] Balzano | Blind calibration of sensor networks[END_REF] which uses Total Least Square (TLS) to solve Eq. [START_REF] Dorffer | Fast nonnegative matrix factorization and completion using nesterov iterations[END_REF]. TLS mainly consists of a weighted SVD where the weights allow to take into account the accuracy of each sensor.

Once the gain vector is estimated, the authors in [START_REF] Balzano | Blind calibration of sensor networks[END_REF] estimate the offsets 5 , assuming the observed phenomenon is zero-mean (or at least its average value is known). In that case, one easily derive

β = -Ȳ • α. (12)
Finally, the calibration process is summarized in Algorithm 1 and provides an interesting performance while remaining easy to implement. However, the projection step makes Algorithm 1 Nullspace-based calibration methods [START_REF] Balzano | Blind calibration of sensor networks[END_REF], [START_REF] Lipor | Robust blind calibration via total least squares[END_REF].

the measurement matrices Y j (see Eq. ( 9)) ; Construct C (see Eq. ( 10)) ; Estimate α from Eq. ( 11) using LS, SVD or TLS ; Estimate β (see Eq. ( 12)).

the method very sensitive to outliers. Indeed, in a realistic framework, the measurements provided by the sensor network can be corrupted by outliers, as explained in Section II. The outlier-free vectors y(t j ) are thus inaccessible and one must deal with corrupted sensor readings z(t j ) instead. If 4 Please note that other strategies-e.g., assuming to know the sum of the values of the unknown calibration parameters [START_REF] Bilen | Convex optimization approaches for blind sensor calibration using sparsity[END_REF]-have been proposed in the literature and can be also applied with these methods instead. In the remainder of the paper, we follow the same strategy as in [START_REF] Balzano | Blind calibration of sensor networks[END_REF], [START_REF] Lipor | Robust blind calibration via total least squares[END_REF]. 5 An approach for estimating β when n m offsets are known was also proposed in [START_REF] Balzano | Blind calibration of sensor networks[END_REF]. one directly applies the above nullspace-based calibration techniques on z(t j ), then

P Z j • α + β = P (Y j + O j ) • α + β , (13)
= P Y j • α + β + P (O j • α) , (14)
where Z j diag (z(t j )) and O j diag (o(t j )). It turns out that the estimation of α is perturbated by O j , which also implies a poor estimation accuracy of β. Please note that the TLS method should be robust to outliers if the latter are present in very few sensors. Indeed, by lowering the weights associated to the relevant sensors, it should be still possible to perform accurate calibration on the outlier-free sensors. However, TLS should provide a very limited enhancement (or even no enhancement at all) if the outliers are distributed over all the sensors readings, which is more likely to happen in practice.

It should be noticed that the above methods were also extended to a partially blind calibration problem in [START_REF] Balzano | Blind calibration of sensor networks[END_REF], [START_REF] Lipor | Robust blind calibration via total least squares[END_REF]. In that case, some sensor calibration parameters are assumed to be known, thus improving the calibration performance. However, even these methods are sensitive to outliers as they are also based on nullspace projection.

As a consequence, a novel approach able to handle outliers must be proposed, which we introduce in the next section.

IV. PROPOSED OUTLIER-ROBUST CALIBRATION METHOD

The outliers that may corrupt the data can be seen as sparse noisy data. In sensor networks, it usually corresponds to spurious readings or transmission faults.

At this stage, it should be noticed that Z j • α + β in Eq. ( 13) can also be expressed as

Z j • α + β = A • z(t j ) + B, (15)
where A diag (α) and B diag β , respectively. We now define the matrix Z [z(t 1 ), . . . , z(t k )], i.e., whose (i, j)-th element reads (Z) ij z i (t j ).

From Eq. ( 1) and Eq. ( 16), we can derive

Z = Y + O, (17)
where Y and O contain the entries of y(t j ) and o(t j ), respectively.

Using the same formalism as in Eq. ( 15), Eq. (3) provides

X = A • Y + B, (18)
where X [x(t 1 ), . . . , x(t k )]. Since x(t) lies in a low dimensional subspace S defined by r min(n, k) basis functions, it turns out that X is a low-rank matrix, and as a consequence, Y is also low-rank. As O is assumed to be sparse, it turns out that the sensor readings Z are written as the sum of a low-rank and a sparse matrix, which has been intensively studied under the name of Robust Principal Component Analysis (RPCA) since the pioneering work in [START_REF] Fazel | Matrix rank minimization with applications[END_REF], [START_REF] Candès | Robust principal component analysis?[END_REF]. RPCA aims to solve

min Y,O ||Y || 2 + λ ||O|| 1 , s.t. Z = Y + O, (19)
where ||•|| is the nuclear norm of a matrix-i.e., the sum of its eigenvalues-||.|| 1 is the 1 norm, and λ a positive weighting parameter. RPCA became a popular tool as it finds numerous applications, e.g., in video surveillance, face recognition, or ranking and collaborative filtering [START_REF] Candès | Robust principal component analysis?[END_REF]. Numerous RPCA algorithms have been proposed in the literature and can thus be applied to Z, in order to estimate Y and O. In some preliminary tests, we found the Inexact Augmented Lagrange Multiplier (IALM) method [START_REF] Lin | The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices[END_REF] to be fast and accurate for the considered application. As most RPCA algorithms, the IALM-RPCA technique is iterative and computes partial SVDs of the size of Z to estimate O and to derive Ŷ , an estimation of Y .

Then, we can easily extract the vectors ŷ(t j ) and build the diagonal matrices Ŷj , ∀j ∈ [ [1, k]] . At this stage, it is straigthforward to perform blind calibration by applying any of the above projection-based calibration techniques to Ĉ, the matrix defined in Eq. ( 10) using the estimated Ŷj . The structure of the proposed methods is provided in Algorithm 2. Let us stress that using RPCA as a preprocessing stage of Algorithm 2 Proposed outlier-robust calibration methods.

Construct Z (see Eq. ( 16)) Estimate Ŷ using IALM-RPCA [START_REF] Lin | The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices[END_REF] Center the matrices Ŷj (see Eq. ( 9)) ; Construct Ĉ (see Eq. ( 10)) ; Estimate α from Eq. ( 11) using LS, SVD or TLS ; Estimate β (see Eq. ( 12)). the nullspace-based calibration does not involve any extraassumption. Moreover, in a machine learning framework, it can also be applied to early readings-where the sensors are assumed to be calibrated-as sensor faults or communication errors might happen. In that case, the estimation of S might be improved as well. Lastly, such a preprocessing can also be applied to the partially blind extensions of the above nullspacebased blind calibration methods, as proposed in [START_REF] Balzano | Blind calibration of sensor networks[END_REF], [START_REF] Lipor | Robust blind calibration via total least squares[END_REF].

In the next section, we investigate the enhancement provided by our proposed method with respect to the state-of-the-art ones, in both blind and partially blind scenarios.

V. EXPERIMENTAL VALIDATION AND DISCUSSION

In this section, we investigate the performance of the proposed outlier-robust calibration methods. To that end, we simulate a low-dimensional subspace S of rank r = 20 and we derive a set of n = 100 sensors by randomly picking 100 gains and offsets. We then simulate k = 3 • r • log(n) signals from the subspace S-as done in [START_REF] Lipor | Robust blind calibration via total least squares[END_REF]-and we derive the outlier-free sensor responses associated to each signal-using Eq. ( 2)-thus providing the vectors y j , ∀j ∈ [ [1, k]] .

In the tests below, we explore the effects of both the subspace error and the number of outliers on the calibration performance. Indeed, as the subspace is assumed to be learned prior to the calibration, it may be not perfectly known. Its estimation error is simulated by adding random zero-mean Gaussian perturbations to the theoretical subspace with increasing variance [START_REF] Balzano | Blind calibration of sensor networks[END_REF]. Such an error is then estimated as

S 1 k • k i=1 ||P (x i )|| 2 ||x i || 2 . (20)
In the tests reported below, the values of S range from 0 to 0.45. A few outliers are then randomly added to the observed data matrix Y . Their proportion ρ o ranges between 0 and 20% of the number of data points in Z. For each pair of parameters ( S , ρ o ), we apply the LS [START_REF] Balzano | Blind calibration of sensor networks[END_REF] and TLS [START_REF] Lipor | Robust blind calibration via total least squares[END_REF] nullspace-based calibration methods to the sensor readings corrupted by outliers. The weights used in TLS are all set to 1, so that TLS behaves as SVD. We also apply our proposed extension, where IALM-RPCA is applied to Z before applying LS or TLS calibration. As a performance criterion, we compute the error between the estimated parameters α and the true ones α using

Error = ||α -α|| 2 ||α|| 2 . (21)
Figure 2 shows the median performance of the tested methods obtained over 100 trials. Let us first focus on the enhancement reached by the state-of-the-art methods. When the proportion of outliers is set to 0, the TLS calibration approach outperforms the LS one, which is consistent with [START_REF] Lipor | Robust blind calibration via total least squares[END_REF]. While the median calibration error reached using LS is above 0.9 when the subspace error S is above 0.26, the TLS-based technique provides median calibration errors always below 0.30. However, its spread (shown by considering minimum and maxium errors) is higher than with LS. As an example, the errors reached when S = 0.45 are between 0.17 and 0.92 with TLS calibration while they are between 0.94 and 0.98 with LS calibration. When some outliers are added to the data, the calibration error reached by both methods significantly increases. However, their behaviour is not similar. On the one hand, the median error reached with LS calibration remains close to 1 for almost any tested value S of the subspace error (it is always above 0.9 when only ρ o = 0.2% of outliers are present in the data, for any initial subspace error). On the contrary, the median TLS calibration error remains always below 0.6 when ρ o ≤ 0.2%. However, for higher values of ρ o , the median TLS calibration error drastically increases (it is always above 10 when ρ o ≥ 2%, for any tested value of present in the data. Indeed, in that case, the performance reached by the state-of-the-art calibration methods and their respective proposed extensions are almost equal. When the outlier proportion is higher than 20% (the situation is not plotted on Fig. 2), the outlier matrix O is not very sparse and the IALM-RPCA method fails in accurately estimating it, thus leading to a degradation of the calibration performance. We also investigated the case of partially blind (PB) calibration. We reproduced the same experiment as above, except that we considered a set of 5 sensors to be perfectly calibrated, i.e., their calibration parameters are assumed to be known. Due to space consideration, we cannot show the plots on the paper but the performance reached by the state-of-theart and the proposed methods are consistent with the blind case. Indeed, the PB state-of-the-art techniques using LS and TLS show the same sensitivity to outliers than the blind LS and TLS approached tested above. The use of well calibrated sensors is then not sufficient to make these methods robust to outliers. Using RPCA as a preprocessing for PB calibration methods allows some robustness to the presence of up to 10% of outliers while it does not affect the performance of the methods in outlier-free simulations.

It should be noticed that in the nullspace-based calibration problem, the rank of Y is assumed to be known, as it was estimated with S. This knowledge is not fully taken into account in the IALM-RPCA algorithm, as it aims to find a matrix of minimal rank but it does not take into account the known rank r + 1 of matrix Y . Keeping the rank equality constraint in RPCA might allow to filter additive noise in the sensor readings, and thus to improve the performance reached by the proposed techniques.

VI. CONCLUSION AND PERSPECTIVES

In this paper, we proposed a robust extension of nullspacebased blind calibration methods. We reordered the collected data in a low-rank matrix that we filtered using an RPCA algorithm. While not adding any extra-assumption, such a preprocessing stage allows removing up to 10% of outliers without any loss of calibration performance. This work opens many perspectives. We recently proposed some mobile sensor calibration techniques based on informed matrix factorization [START_REF] Dorffer | Blind calibration of mobile sensors using informed nonnegative matrix factorization[END_REF], [START_REF] Dorffer | Blind mobile sensor calibration using a nonnegative matrix factorization with a relaxed rendezvous model[END_REF], [START_REF] Dorffer | Nonlinear mobile sensor calibration using informed semi-nonnegative matrix factorization with a Vandermonde factor[END_REF]. In future work, we will investigate some outlier-robust extensions of these approaches, using a similar low-rank modeling. We will compare such a formalism to robust informed matrix factorization using parametric divergences [START_REF] Limem | Non-negative matrix factorization using weighted beta divergence and equality constraints for industrial source apportionment[END_REF], [START_REF] Chreiky | Robust informed split gradient nmf using alpha betadivergence for source apportionment[END_REF] or the Huber norm [START_REF] Chreiky | Informed split gradient non-negative matrix factorization using huber cost function for source apportionment[END_REF], that we recently proposed for another application [START_REF] Limem | Nonnegative matrix factorization under equality constraints-a study of industrial source identification[END_REF]. Moreover, the proposed techniques need to know the low-rank subspace where the sensed phenomenon lie. We will investigate the use of machine learning to learn S and to perform sensor calibration.

Fig. 1 .

 1 Fig. 1. Examples of sensor faults: (top) outlier-free sensor reading, (middle) presence of 2 local sensor faults (e.g., constant or bottom-noise fault) , and (bottom) fault due to defective sensor or communication lost.

Fig. 2 .

 2 Fig.2. Performance of the nullspace-based calibration methods with respect to the subspace estimation error S and the outlier proportion ρo. The nullspace in Eq. (11) is estimated using: (a) LS[START_REF] Balzano | Blind calibration of sensor networks[END_REF], (b) RPCA preprocessing + LS, (c) TLS[START_REF] Lipor | Robust blind calibration via total least squares[END_REF], and (d) RPCA preprocessing + TLS.

Some approaches, e.g.,[START_REF] Wang | Moments based blind calibration in mobile sensor networks[END_REF], do not use the rendezvous assumption.

Please note that in that case, macro-calibration has similarities with lowrank matrix completion[START_REF] Dorffer | Fast nonnegative matrix factorization and completion using nesterov iterations[END_REF].

[START_REF] Puigt | Environmental signal processing: new trends and applications[END_REF] Moment-based calibration methods were also proposed for mobile sensor networks, e.g., in[START_REF] Wang | Moments based blind calibration in mobile sensor networks[END_REF].

S ), thus showing that such a robust approach is actually very sensitive to the presence of outliers.On the contrary, both our proposed methods-which combine the IALM-RPCA preprocessing with the above calibration techniques-provide almost constant performance up to ρ o = 10% of outliers. Interestingly, the preprocessing does not affect the calibration performance when no outlier is

ACKNOWLEDGEMENT

This work was funded partly by the "OSCAR" project within the Région Hauts-de-France "Chercheurs Citoyens" Program, and partly by the CNRS MASTODONS project under the grant "DoMaSQ'Air 2017".