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International audience; Controlling the electromagnetic properties of dielectric objects is demanded in microwave applications such as radar cross section (RCS) studies, metamaterial design, and antennas prototyping. Additive manufacturing has made the fabrication of desired shapes easier, but there is still room for improvement in controlling the permittivity. This article proposes a novel approach to control the permittivity of 3-D printed objects, in particular the ones with low permittivity contrasts. The effective permittivity is set by locally varying the material density. The object is first meshed using tetrahedral meshing software. Then its air percentage is controlled by adjusting the diameter of cylinders, which are positioned at each edge of the mesh. By tuning the volume fraction, one can achieve the required effective permittivity. Design, manufacturing, and characterization steps are discussed for the specific case of spheres. With such a canonical shape, the effective permittivity can be directly retrieved by comparing far-field electromagnetic scattering measurements with Mie computations. Bistatic RCS measurements and simulations are provided and discussed. They enable us to assess the validity of the proposed methodology, in particular the good adequacy between the adequately chosen unstructured air/material distribution, the desired relative permittivity, and the good homogenization of the object

International audience; The amount of interplanetary dust impacting the Earth's atmosphere is still not well estimated, in terms of total mass, size distribution, and concentration of particles. We propose to apply the counting techniques used in the Earth's atmosphere for determining the concentration of liquid and solid aerosols, to the detection of these interplanetary particles. The main differences with the Earth's atmosphere measurements are the very low concentrations, the high speed of the particles (at least several km/s) and the space conditions. We have developed recently an innovative design of aerosols counter, called LOAC, which provides the concentrations for 19 size classes of particles in the 0.2-50 micrometer range, and also an estimate of their typology (or their light absorbing properties). The particles are injected through a laser beam via a pumping system, and two photodiodes record the light scattered. This instrument combines the measurements at two different angles, the first at around 15° being insensitive to the refractive index and porosity of the particles, and the second one around 60° being very sensitive to the nature of the particles. LOAC is used in routine since 5 years on the ground and from all kinds of balloons for the troposphere (mainly pollution) and stratosphere monitoring, and is also involved in various international campaigns. An updated version of LOAC is in development for space applications, essentially for in situ measurements planetary atmospheres. LOAC can also be modified for the detection of high velocity particles, by using a light source of several cm long instead of a laser beam. No pump is needed, since the particles will cross an open cell oriented at a constant angle from the motion of the instrument. This instrument could perform measurements in Earth orbit, onboard a micro-satellite or on the International Space Station. We will present this new concept of instrument, and how the measurements can be used to better constrain the nature and flux of the incoming interplanetary material.

Bistatic Radar Cross Section (RCS) measurement represents a particular challenge, compared to monostatic measurement. First, bistatic RCS measurement requires a background subtraction to compensate the direct incidence from the source to the receiver. Second, the measurement accuracy varies as function of the amplitude of the received signal which, by itself, varies as a function of the receiver’s position with respect to the source. In the forward scattering direction when the receiving and transmitting antennas are facing each other, the Signal to Noise Ratio (SNR) is usually satisfactory but the signal level must be maintained beyond the non-linear region of the receiving devices. However, in the backward scattering direction the SNR is quite low [1]. When measuring low RCS targets, these issues become very critical because the total field (measured with the presence of the target) and the incident field (background subtraction) become very similar, and the extraction of the RCS quantity from their subtraction becomes very vulnerable to the random noise. This problem has limited in the past the accurate measurement of low RCS targets, with dimensions smaller than the wavelength and/or with low relative permittivity, that may have RCS values down to −60 dBm2. Since the measurement setup of the CCRM was recently renewed, it became possible with the new apparatuses to apply a “smart” selection of the setup settings to enhance the measurement accuracy over all the bistatic range. In this study our goal is to evaluate the impact of the noise control, which can be obtained through the proposed setting adjustment, on the RCS measurement.

International audience; We propose an approach for probing the near-subsurface based on the use of a multistatic configuration. As in this reflexion configuration, few pieces of information on the target are accessible, it is of great importance to define properly the antenna positions allowing to measure as much as possible all available information. To that end, in this paper, we present a study of the optimization of the geometrical configuration specifically meant at imaging shallowly buried targets. Images of the subsurface reconstructed from experimental fields are shown to assess the well-behavior of the proposed methodology.

International audience; Light scattering by optically soft particles is being theoretically investigated in many radiative studies. An interest is growing up to develop approximate methods when the resolution of Maxwell's equations is impractical due to time and/or memory size problems with objects of complex geometries. The participation of experimental studies is important to assess novel approximations when no reference solution is available. The microwave analogy represents an efficient solution to perform such electromagnetic measurements in controlled conditions. In this paper, we take advantage of the particular features of our microwave device to present an extensive experimental study on the electromagnetic scattering by spheroidal particles analogs with low refractive indices, as a first step toward the assessment of micro-organisms with low refractive index and heterogeneities. The spheroidal analogs are machined from a low density material and they mimic soft particles of interest to the light scattering community. The measurements are confronted to simulations obtained with Finite Element Method and T-Matrix method. A good agreement is obtained even with refractive index as low as 1.13. Scattered signals of low intensities are correctly measured and the position of the targets is precisely controlled. The forward scattering measurements show high sensitivity to noise and require careful extraction. The configuration of the measurement device reveals different technical requirements between forward and backward scattering directions. The results open interesting perspectives about novel measurement procedures as well as about the use of high prototyping technologies to manufacture analogs of precise refractive indices and shapes.

International audience; The calibration of the multistatic scattering matrix plays an important part in the construction of a quantitative microwave imaging system. For scattering measurement applications, the calibration must be performed on the amplitude and on the phase of the fields of interest. When the antennas are not completely identical, as for example with a multiplexed antennas array, a specific calibration procedure must be constructed. In the present work, we explain how a complex calibration matrix can be defined which takes advantage of the geometrical organization of the antennas. Indeed, for arrays of antennas positioned on a circle, the inherent symmetries of the configuration can be fully exploited by means of an adequate reorganization of the multistatic scattering matrix. In addition, the reorganization permits to detect antenna pairs which are not properly functioning and to estimate the signal-to-noise ratio. Experimental results obtained within a cylindrical cavity enclosed by a metallic casing are provided to assess the performance of the proposed calibration procedure.This calibration protocol, which is described here in detail, has already been applied to provide quantitative images of dielectric targets [1, 2].

International audience; The Surface Wave Scattering-Microwave Scanner (SWS-MS) is a device that allows the measurement of the electromagnetic fields scattered by objects totally or partially submerged in surface waves. No probe is used to illuminate the sample, nor to guide or scatter the local evanescent waves. Surface waves are generated by total internal reflection and the amplitude and phase of the fields scattered by the samples are measured directly, both in the far-field and the near-field regions. The device’s principles and their practical implementation are described in details. The surface wave generator is assessed by measuring the spatial distribution of the electric field above the surface. Drift correction and the calibration method for far-field measurements are explained. Comparison of both far-field and near-field measurements against simulation data shows that the device provides accurate results. This work suggests that the SWS-MS can be used for producing experimental reference data, for supporting a better understanding of surface wave scattering, for assisting in the design of near-field optical or infrared systems thanks to the scale invariance rule in electrodynamics, and for performing nondestructive control of defects in materials.

International audience; Probing the near-subsurface in the presence of absorbing media is a very challenging problem. Within that framework, we analyse the capabilities of a mono-frequency/multistatic set-up for detecting shallowly buried targets. As the antennas constitute an important part of the probing device, an accurate method for modelling the antennas behaviour is proposed. This modelling, performed thanks to a correct balanced set of elementary sources, is then incorporated in the calculation of the scattered field, performed with a home-made Finite Element Method software. Efforts have also been put into the measurement procedure. The measured fields are thus post-processed with an efficient method which takes profit of the spectral bandwidth properties of the scattered field. These fields serve as input data for the inversion algorithm, an extension of the DORT method to elongated targets. This qualitative and fast imaging procedure, which exploits the spectral properties of the multistatic scattering matrix, has been adapted to the present stratified configuration. Imaging results of shallowly buried targets embedded in a high losses medium are presented to assess the well-behaviour of the proposed methodology.

The spatial depolarization of light emitted by heterogeneous bulks is predicted with exact electromagnetic theories. The sample microstructure and geometry is connected with partial polarization.

link to the publisher version: https://ieeexplore.ieee.org/document/6912945; International audience; In this paper, we focus on an imaging procedure which retrieves the complex permittivity map of an investigation domain from electromagnetic scattered field measurements . The transmitter and the receiver are independently moved along two parrallel lines of finite extent. In such configuration, the antenna effects are of drastic importance. We propose thus a quantitative modeling of the antennas which does not require additional calibration process. The antennas model is completely incorporated in the inversion procedure and allows to recover tubes whose diameters are of the order ofAbstractIn this paper, we focus on an imaging procedure which retrieves the complex permittivity map of an investigation domain from electromagnetic scattered field measurements. The transmitter and the receiver are independently moved along two parrallel lines of finite extent. In such configuration, the antenna effects are of drastic importance. We propose thus a quantitative modeling of the antennas which does not require additional calibration process. The antennas model is completely incorporated in the inversion procedure and allows to recover tubes whose diameters are of the order of [(λ)/5].

International audience; An accurate knowledge of the complex permittivity value of materials is compulsory when performing experimental electromagnetic applications. Unfortunately, these values are not so obvious to determine in practice. In this letter, we propose a novel approach for determining the complex dielectric constant of materials. This method combines free-space far-field scattering pattern measurements with a Bayesian procedure, which fully exploits the measurement uncertainties. Therefore, the measured values weighted according to their experimental accuracy are incorporated in the permittivity determination algorithm. In this letter, the samples are all shaped as spheres in order to benefit from efficient Mie scattered field computations. The dielectric properties of typical plastic samples are first determined and compared to values found in the literature in order to assess the validity and the accuracy of the proposed methodology. A more “exotic” sample extracted from a microwave absorber, which is a polyurethane foam charged with carbon particles, is also analyzed.

In this paper, we address the problem of the reconstruction of the shape of a 3D perfectly conducting object. This object is assumed to be immersed in a homogeneous lossy embedding medium in a cross borehole configuration. The method is based on an optimization process performed with the help of a conjugate gradient method. A boundary integral formalism and an adjoint state method are used in order to compute the field scattered by an estimated object and its derivatives with respect to small deformations of the shape. In this paper we also extend to the 3D case a regularization process in which we optimize a growing number of Fourier coefficients describing the shape of the estimation. The method is tested against the reconstruction of an ellipsoid, and the influence of noise is also discussed.

In near-subsurface imaging, the antenna constitutes an important part of the probing device. Unfortunately, during the data processing sequence, its radiation pattern is often neglected, inappropriately approximated, or accurately modeled, but with heavy procedure. In the present work, we propose an approach for accurately modeling the antenna behavior. The methodology was derived keeping in mind that it must remain simple to implement and that it should not require a complete description of the antenna geometry. Instead, a correctly balanced set of elementary sources, acquired through free-space transmission measurements, are introduced to mimic the antenna radiation characteristics. Comparisons with experimental fields acquired in controlled laboratory conditions show that this procedure allows a correct implementation of the antenna effect. © 2014 Wiley Periodicals, Inc. Microwave Opt Technol Lett 56:2516–2522, 2014

The scattering process occurring when a complex medium is illuminated with a monochromatic light is well known to generate polarization effects on the scattered field which are directly related to the scattering origins, and the scattering regime.

The Finite-element method (FEM) is well suited for solving problems involving inhomogeneous and arbitrary shaped objects. Unfortunately, for large-scale problems, this method yields a huge coefficient matrix system whose inversion may be time and memory consuming. Many techniques have been reported to increase the computational efficiency. Among them, the domain decomposition-based methods are especially appealing and in particular, the dual-primal finite element tearing and interconnecting method (FETI-DP) has been recognized as one of the most scalable parallel solver in computational mechanics and acoustics. Over the past years, it has also been successfully applied to the simulation of two- and three-dimensional (2D and 3D) large-scale electromagnetic (EM) problems (J.-M. Jin and D. J. Riley, IEEE, 2009).

In most of the cases, the soil is probed by a monostatic or bistatic device which emits an ultra-wideband electromagnetic wave either via temporal pulses (A. Yarovoy, T.G. Savelyev, P. Aubry, P.E. Lys, L.P. Ligthart, IEEE Trans. On Microwave Theory and Techniques, 55:1288–1295, 2007) or in a step-frequency mode (J. Minet, P. Bogaert, M. Vanclooster, S. Lambot, Journal of Hydrology, 424–425:112–123, 2012).

The discrete dipole approximation with surface interaction (DDA-SI) is analyzed and expanded for the modeling of far-field scattering by objects on a surface illuminated by an evanescent wave generated by total internal reflection. More specifically, the electric field scattered in the far zone predicted via the DDA-SI is compared against scaled microwave experiments; additional comparisons are also performed using results from a Finite Element Method (FEM). Three cases are considered: a lossless cube with a side length of λ/1.79 (size parameter x=1.76), a lossless sphere with a diameter of λ/1.92 (x=1.63) and an absorbing sphere with a diameter of λ/0.87 (x=3.63), where λ is the wavelength. For lossless scatterers, a good agreement between the DDA-SI, the FEM and scaled microwave analog experiments is observed, especially when modified Fresnel reflection coefficients are used for computing the surface interaction in the DDA-SI. For an absorbing sphere, the experimental and FEM results are in reasonable agreement, while the DDA-SI exhibits a different trend. This behavior might be due to the fact that the accuracy of the DDA decreases as the permittivity and the size parameter increase. This work suggests that the DDA-SI could be used as a forward model in an evanescent wave-based characterization framework given that a thorough convergence and accuracy analysis is carried on in order to improve the performance when dealing with objects having large permittivity and/or size parameter.

International audience; Due to the increasing number of applications in engineering design and optimization, more and more atention has been paid to full-wave simulations based on computational electromagnetics. In particular, the finite-element method (FEM) is well suited for problems involving inhomogeneous and arbitrary shaped objects. Unfortunately, solving large-scale electromagnetic problems with FEM may be time consuming. A numerical scheme, called the dual-primal finite element tearing and interconnecting method (FETI-DPEM2), distinguishes itself through the partioning on the computation domain into non-overlapping subdomains where incomplete solutions of the electrical field are evaluated independently. Next, all the subdomains are ''glued'' together using a modified Robintype transmission condition along each common internal interface, apart from the corner points where a simple Neumann-type boundary condition is imposed. We propose an extension of the FETI-DPEM2 method where we impose a Robin type boundary conditions at each interface point, even at the corner points. We have implemented this Extended FETI-DPEM2 method in a bidimensional configuration while computing the field scattered by a set of heterogeneous, eventually anistropic, scatterers. The results presented here will assert the efficiency of the proposed method with respect to the classical FETI-DPEM2 method, whatever the mesh partition is arbitrary defined.

International audience; We propose an approach for probing the nearsubsurface based on the use of a single frequency combined with a multistatic configuration. Our goal is to retrieve the near-subsurface permittivity spatial distribution from measured electromagnetic fields. To that end, efforts have been done on the calibration procedure. In particular, the radiation pattern of the antenna is taken into account in the modeling of the scattering problem. We also preprocess the measured fields with an efficient method which takes profit of the spectral bandwidth properties of the scattered field allowing the removal of the residual interface contribution. Imaging results of shallowly buried target embedded in a high-losse medium are presented to assess the well-behavior of the proposed methodology.

A new setup for characterizing shallowly buried targets is proposed. It corresponds to a multistatic configuration working at a single frequency. This microwave device is coupled with an accurate calibration procedure detailed herein. First qualitative imaging results from experimental fields are presented.

International audience; In this work, we present an extension of the classical DORT method for time-harmonic inverse medium scattering problems. This tool enables one to estimate the shape of the unknown scatterers even if they are not necessarily point-like scatterers. This method is described and compared with existing sampling methods. Moreover, a mathematical derivation is provided for its validation, either in a free-space environment or in a metallic enclosure setup. Two-dimensional examples, based on measured data sets acquired in the circular microwave scanner developed at Institut Fresnel, are presented, which show that the method is computationally efficient and robust to noise.

International audience; A microwave imaging problem is considered where a soil column is surrounded by a metallic circular enclosure. This system is presently being developed to demonstrate the potentiality of non-invasive microwave imaging systems for volumetric water content monitoring. The final goal is to retrieve soil moisture information as it is an important parameter for understanding fluid flows in the soil as well as the water uptake by plant roots. Here, we address the qualitative part of the problem with the help of a rapid and rather robust sampling methods: the Linear Sampling Method (LSM) and the MUSIC method. These methods are tested for the following configuration: one water source diffuses in a heterogeneous background. We show that, to a certain extent, the evolution of the soil water content can be qualitatively monitored with the LSM. Moreover, we propose a quick way for detecting the position of the water source inside the soil column.

International audience; To overcome the ill-posedness of the inverse scattering problem, we introduce an appropriate way of representing the unknown permittivity profile which benefits from two a priori pieces of information which are commonly encountered. First, the unknown permittivity map has a limited spatial support which can be contained within a circular investigation area. Second, from physical reasoning based on the scattering operator, there is only a limited number of independent parameters that can be retrieved from the measured fields. Both pieces of a priori information can be adequately introduced by representing the unknown permittivity profile in terms of a Zernike polynomial expansion, correctly truncated according to the number of independent parameters. To investigate the effectiveness of such a Zemike representation, the reconstructions obtained from experimentally acquired data in the circular microwave scanner developed at the Institut Fresnel are analyzed.

[1] We present a 3D spherical bistatic experimental setup developed in the anechoic chamber of Institut Fresnel in Marseille, France, for scattering measurements of targets in the microwave domain, both in copolarization and cross-polarization. We first describe the geometry of the setup, the measurement protocol, and the calibration process. The results obtained for different metallic and dielectric targets are then compared to numerical simulations, computed either using the coupled dipole method or Mie theory. These comparisons are used to validate the algorithms and the setup, as well as to provide information on the accuracy of the measurements.

Soil moisture is an important parameter for understanding fluid flow modelling as well as the water uptake by plants roots. A microwave scanner is presently being designed to demonstrate the potentiality of a non-invasive microwave imaging system for volumetric water content monitoring. After briefly describing the setup, the numerical tools which are needed for the modelling and the inversion of the electromagnetic fields are introduced. In particular, a finite element method has been adapted to the specific setup geometry. An iterative inversion scheme, based on a Lagrangian formalism has been implemented to provide dielectric permittivity maps of the domain under investigation.

International audience; A two-step electromagnetic detection procedure is proposed to characterize a dielectric obstacle buried at low depth under a rough surface from single-frequency and multistatic data. First, we have developed, in the framework of the small perturbation theory, a correlation procedure of the scattered field, which enables us to recover an estimation of the roughness profile. This method is tested for various cases with synthetic data provided by a rigorous boundary integral solver. Second, the obtained surface profile is introduced into the numerical simulation thanks to a finite-element code. An iterative process is then used, based on a level-set formulation, to obtain the shape of the buried target. The influence of the prior step on the accuracy of the reconstruction of the target is studied via various criteria and for different configurations.

International audience; This paper deals with the detection of a buried object beneath a rough surface in a two-dimensional configuration. A new method for detecting buried objects from wide-band, near-field, multi-static data is suggested. The so- called near-field configuration corresponds to a set of transmitters and receivers located along a line segment in the vicinity of the surface. The detection method proposed here is based on a correlation of the scattered fields from two sets of data associated with two transmitters. A theoretical study in the framework of a low-frequency approximation predicts an enhancement of the correlation peak resulting from cross-terms combining surface and target contributions, as confirmed by rigorous computations. The performance of the detector is tested with respect to both polarization and frequency through computation of the receiver operating characteristic curves.

The method of decomposition of the time reversal operator has been introduced with the aim of focusing acoustic waves on a selected target. It is based on the derivation of the invariants of the time reversal operator, which is linked to the transfer matrix of the array of transducers insonifying a time invariant scattering medium. The theoretical and numerical results concern independent point-like scatterers. Here, we apply numerically the same technique to electromagnetic waves, and all the interactions between the various scatterers are taken into account. This method may be very useful to detect and localize buried objects. With this goal, we have investigated the role of polarization and of the geometrical or electromagnetic parameters, as well as the ability of accurately focusing on a scatterer. The sensitivity of the results to the initial set of data, namely the number of measurements or the accuracy, is also studied.

The interaction between an impinging electromagnetic wave and a target results in ascattered wave, which depends upon the frequency, the size, shape and electromagnetic propertiesof the target. The measurement of a scattered signal can be used to detect or characterizea target, and the applications range from radar systems at large scale to optical tomography atthe microscale. When the targets have very large or very small dimensions, the microwaveanalogy appears as a useful approach to experimentally investigate a scattering problem. Theexperiment is therefore scaled to the microwave range and the target is mimicked by a centimeter-sized analog, while maintaining the same initial wavelength over target dimension ratioand conserving the same initial permittivity. The microwave scattering experiments are realizedin the anechoic chamber of the Centre Commun de Ressources en Microondes at thefrequencies between 2 GHz and 18 GHz.The objective of this thesis is to widen the application of the microwave analogy byadopting the appropriate techniques to create objects of controlled shapes and electromagneticproperties using special manufacturing technologies, as well as by developing a versatile setupproviding accurate measurements. A special interest is given on low scattering targets, i.e.with low permittivity contrasts and/or small dimensions compared to the wavelength. An extensivestudy on the random noise characterization of the measurement setup was made andthe undesired reflections within the anechoic chamber were investigated. As a result, a noveloptimization technique was proposed which allow having a flexible control of the setup parametersaccording to the investigated target. It consists on angular zones decompositions ofthe bistatic region with different power profiles. In addition, A Hardgating system utilizingtwo RF switches was installed to the measurement setup, allowing to make pulsed signalsmeasurements and to filter out the stray signals.Thanks the new setup optimization, it became possible to accurately measure low scatteringlevels down to -60 dBm2. Moreover, additive manufacturing technologies were involvedin the fabrication of analogs and a novel technique was proposed to obtain targets with“on-demand” value complex permittivity and shape. It became possible manufacture low scatteringanalogs of the real part of permittivity for any value between 1 and 3 by creating wellcontrolled porous structures.Three main targets are studied within the scoop of this thesis: low permittivity spheroids,analogs of photosynthetic microalgae, soot aggregates analogs with complex shape, andscaled forest scene models composed of tree analogs with some metal vehicle analogs., La diffraction est un phénomène issu de la l’interaction d’une onde électromagnétiqueincidente avec un objet. En électromagnétisme, le champ diffracté, qui en résulte, est fonctionde la fréquence, de la taille de l’objet et de ses caractéristiques diélectriques et magnétiques.La mesure du signal ainsi diffracté peut être mise à profit pour caractériser un objet. On peutciter les radars comme des applications qui concernent les grands objets et la tomographieoptique pour les échelles micrométriques. Dans le cadre des études expérimentales, l’analogiemicroonde peut s’avérer bénéfique autant pour les grands que les petits objets. Son principeest de transposer à l’échelle centimétrique les objets dont nous souhaitons étudier la diffraction.Pour être pertinente, cette translation d’échelle doit être réalisée en créant un analoguerespectant le même rapport dimension sur longueur d’onde, tout en conservant la, sur unefréquence donnée, la même permittivité. Toutes les mesures de diffraction de cette thèse ontété réalisées dans la chambre anéchoïque du Centre Commun de Ressources en Microondesavec des fréquences variant entre 2 GHz et 18 GHz.Les objectifs de cette thèse sont d’élargir le champ des applications de l’analogie microondeen adaptant les techniques de fabrication des analogues pour créer des objets de caractéristiquesélectromagnétiques et de géométrie contrôlées d’une part, et d’autre part de développerun outil versatile et précis pour réaliser les mesures de diffraction. Un intérêt particuliera été porté aux objets faiblement diffractant, ce qui peut être dû à leurs faibles dimensionsaussi bien qu’à leurs faibles permittivités. A la suite de la caractérisation des réflexionsparasites et de celle du bruit aléatoire perturbant les mesures, une nouvelle technique d’optimisationdu paramétrage des appareils de mesure, qui tient compte des objets étudiés, a puêtre proposée. Elle comporte notamment un réglage des puissances de source en fonction desangles de bistatisme et un filtrage temporel, par switch, qui a été mis en place et paramétrépour filtrer les signaux parasites mesurés.Les bénéfices de ces diverses optimisations des paramètres de mesures ont été démontréset ils ont permis de mesurer des niveaux de section efficace radar aussi bas que -60 dBm2.De plus, les avancées sur le contrôle de la permittivité permettent de réaliser des analogues degéométrie maîtrisée avec des permittivités relatives à la carte dans la gamme de 1 à 3 ; lapermittivité étant ajustée par contrôle de la porosité.Les trois principales études présentées concernent : des sphéroïdes de faible permittivité,analogues de microalgues photosynthétiques, des agrégats de suies de forme complexes,et des scènes forestières composées d’analogues d’arbres et de véhicules.