Antoine Caillot
email: caillot@esigelec.frouerghi@esigelec.fr

Safa Ouerghi

Pascal Vasseur
email: pascal.vasseur@u-picardie.fr

Yohan Dupuis
email: ydupuis@cesi.fr

Rémi Boutteau
email: remi.boutteau@univ-rouen.fr

Multi-Agent Cooperative Camera-Based Evidential Occupancy Grid Generation

About a decade ago the idea of cooperation has been introduced to self-driving with the aim to enhance safety in dangerous places such as intersections. Infrastructure-based cooperative systems emerged very recently bringing a new point of view of the scene and more computation power. In this paper, we want to go beyond the framework presented in the vehicle-to-infrastructure (V2I) cooperation by including the vehicle's point of view in the perception of the environment. To keep the cost low, we decided to use only two-dimensional bounding boxes, thus depriving ourselves of depth information that contrasts with state-of-the-art methods. With this in-thescene point-of-view, we propose a new framework to generate a cooperative evidential occupancy grid based on the Dempster-Shafer Theory and which employs a Monte Carlo framework to incorporate position noise in our algorithm. We also provide a new cooperative dataset generator based on the CARLA simulator. Finally, we provide an extended review of our new cooperative occupancy grid map generation method which improves the state-of-the-art techniques.

I. INTRODUCTION

Nowadays, navigation and traffic management in intersections and roundabouts is still a complex task [START_REF] Bento | Intelligent traffic management at intersections supported by V2V and V2I communications[END_REF], [START_REF] Khayatian | A Survey on Intersection Management of Connected Autonomous Vehicles[END_REF]. The perception of the environment aims at providing an accurate and robust estimation of the state of other road users in order to make safe navigation decisions both for humans and autonomous vehicles.

To perceive the environment, we decided to use data from the infrastructure but also data from the in-the-scene users' points of view. The inclusion of vehicle data in the scene enhances the intersection's safety while allowing reducing the number of sensors on the edge of the road and, consequently, the cost of the infrastructure. Currently, to tackle this challenge, many systems propose the implementation of infrastructure allowing the observation of a scene [START_REF] Lv | Lidar-enhanced connected infrastructures sensing and broadcasting high-resolution traffic information serving smart cities[END_REF], [START_REF] Li | Proof-ofconcept of a sdn based mmwave v2x network for safe automated driving[END_REF], [START_REF] Bosch | Conduite automatisée : comment les voitures et les infrastructures communiquent en milieu urbain[END_REF], [START_REF] Annkathrin Krämmer | Providentia -a large scale sensing system for the assistance of autonomous vehicles[END_REF] but none of these solutions take into account the vehicles' point of view.

In this paper, we use only two-dimensional bounding box information from cameras to keep the cost of our cooperative system low and to reduce the required data rate and thus the communication burden of the cooperative systems. We noted that the state of the art systems uses range sensors such as LiDAR [START_REF] Lv | Lidar-enhanced connected infrastructures sensing and broadcasting high-resolution traffic information serving smart cities[END_REF], [START_REF] Li | Proof-ofconcept of a sdn based mmwave v2x network for safe automated driving[END_REF], [START_REF] Bosch | Conduite automatisée : comment les voitures et les infrastructures communiquent en milieu urbain[END_REF] or RADAR [START_REF] Annkathrin Krämmer | Providentia -a large scale sensing system for the assistance of autonomous vehicles[END_REF], which are more expensive than cameras. Bounding box information extraction is out of the scope of this paper and is therefore considered given either by an algorithm such as YOLO [START_REF] Redmon | Yolov3: An incremental improvement[END_REF] or by an ADAS such as the Mobileye's solution used in [START_REF] Baek | Vehicle Trajectory Prediction and Collision Warning via Fusion of Multisensors and Wireless Vehicular Communications[END_REF].

In this work, we take advantage of the in-scene point of view from the vehicles in order to enrich a common map of the scene in the form of an evidential occupancy grid [START_REF] Camarda | Fusion of evidential occupancy grids for cooperative perception[END_REF] and to build confidence while decreasing the costs. Usually, classical occupancy grids as defined in [START_REF] Elfes | Using occupancy grids for mobile robot perception and navigation[END_REF] are used [START_REF] Nuss | A random finite set approach for dynamic occupancy grid maps with real-time application[END_REF], [START_REF] Steyer | Grid-based environment estimation using evidential mapping and particle tracking[END_REF]. However, the latter use joint probabilities based method to merge the grids which do not take into account the unobserved cells.

We also present a new dataset generator with available source code 1 based on the CARLA simulator [START_REF] Dosovitskiy | CARLA: An open urban driving simulator[END_REF] allowing to have several instrumented agents in the scene.

In the remainder of this section, we review publications related to our work. Then, we start in section II by presenting the global architecture of the framework made available. The computation methods used are presented in the section III. Finally, we present our results in section IV, in which we present the cooperative vehicle-infrastructure dataset that has been generated and used to test our system, before bringing a conclusion in section V.

A. Related work

Cooperative systems are more and more present in the state of the art of perception in the autonomous driving context [START_REF] Caillot | Survey on cooperative perception in an automotive context[END_REF]. Today, two main categories of cooperative systems exist; vehicle-to-vehicle (V2V) systems [START_REF] Chen | Cooper: Cooperative perception for connected autonomous vehicles based on 3d point clouds[END_REF] and vehicle-toinfrastructure (V2I) systems [START_REF] Lv | Lidar-enhanced connected infrastructures sensing and broadcasting high-resolution traffic information serving smart cities[END_REF], [START_REF] Li | Proof-ofconcept of a sdn based mmwave v2x network for safe automated driving[END_REF], [START_REF] Bosch | Conduite automatisée : comment les voitures et les infrastructures communiquent en milieu urbain[END_REF], [START_REF] Annkathrin Krämmer | Providentia -a large scale sensing system for the assistance of autonomous vehicles[END_REF]. Other systems also exist and are generally grouped under the name of vehicles-to-everything (V2X). During the last years, the V2I paradigm has been significantly developed with the aim of obtaining an omniscient point of view. However, with the development of the V2I paradigm, more interest has been devoted to the data at the infrastructure's level while the data from the vehicles to the infrastructure has been left aside which could have been used to refine the detections made by the infrastructure. This motivated us to take into consideration these data to propose a framework that is able to exploit multiple points of view.

In order to make a cooperative system, the question of the data sharing level arises. Several strategies have been investigated in the state of the art [START_REF] Zhang | Distributed dynamic map fusion via federated learning for intelligent networked vehicles[END_REF]. Either raw data, directly provided by sensors can be shared [START_REF] Chen | Cooper: Cooperative perception for connected autonomous vehicles based on 3d point clouds[END_REF], [START_REF] Li | Proof-ofconcept of a sdn based mmwave v2x network for safe automated driving[END_REF] or data in the form of labels where the agents do most of the processing locally before sharing them as in [START_REF] Bosch | Conduite automatisée : comment les voitures et les infrastructures communiquent en milieu urbain[END_REF], [START_REF] Annkathrin Krämmer | Providentia -a large scale sensing system for the assistance of autonomous vehicles[END_REF], [START_REF] Lv | Lidar-enhanced connected infrastructures sensing and broadcasting high-resolution traffic information serving smart cities[END_REF]. Sharing raw data has the advantage of densifying the measurements and thus having an impact on the accuracy of the detection whereas, for label sharing, the majority of the detections are done locally on each of the agents which reduce the impact of the cooperation on the system performances. However, raw data sharing comes at the cost of higher network pressure and bandwidth requirements. In our paper, we have decided to work with pre-processed data where only bounding boxes of other detected vehicles in the scene are shared without the use of the whole image. This relies on performing an early detection at the sensor's level. This can be generated by a detection algorithm in order to satisfy the requirements of a cooperative environment in terms of bandwidth as presented in [START_REF] Lv | Lidar-enhanced connected infrastructures sensing and broadcasting high-resolution traffic information serving smart cities[END_REF], [START_REF] Kim | Cooperative perception for autonomous vehicle control on the road: Motivation and experimental results[END_REF].

On the other hand, occupancy grids have been extensively used for mapping but generally using either multiple range sensors [START_REF] Elfes | Using occupancy grids for mobile robot perception and navigation[END_REF], [START_REF] Moravec | High resolution maps from wide angle sonar[END_REF] or with cameras but from a single point of view [START_REF] Richter | Semantic evidential grid mapping using monocular and stereo cameras[END_REF]. This motivated us to use vision-based occupancy grid mapping in a cooperative context using bounding boxes detection. We project bounding boxes corresponding to the detected vehicles onto the ground using back-projection as presented in [START_REF] Kim | Multivehicle cooperative driving using cooperative perception: Design and experimental validation[END_REF]. In the latter, the back-projected images on the ground are merged by using the union method. However, our merging method is based on Dempster Shaffer Theory [START_REF] Sentz | Combination of evidence in Dempster-Shafer theory[END_REF], [START_REF] Shafer | Dempster-shafer theory[END_REF] as presented in [START_REF] Camarda | Fusion of evidential occupancy grids for cooperative perception[END_REF]. We have also used the average occupancy fusion method based on the intersection between detections to assess the efficiency of each fusion method in our vision-based cooperative context.

II. SYSTEM ARCHITECTURE

In this section, we present our cooperative V2I bidirectional framework for creating occupancy grids from camera data based on ROS [START_REF] Quigley | Ros: an open-source robot operating system[END_REF]. Our framework is made of two types of elements: agents and a Road Side Unit (RSU).

A. Agents

The agents can be intelligent roadside sensors or connected vehicles and can be of an arbitrary amount in the scene. They are equipped with an image sensor and a system to identify vehicles in their field of view that extracts bounding boxes. Every agent publishes their messages on a global topic containing every bounding box of every agent and will be read by the RSU as illustrated in Fig. 1.

In our work, we consider that the extraction of bounding boxes is derived from off-the-shelf solutions and is therefore not a topic covered here. We consider that timestamps are generated at the time of shooting from a GPS clock and thus the sensors are roughly synchronized. Therefore, we used the synthetic data from the ground truth to which we added random Gaussian noise.

B. Road Side Unit

The Road Side Unit (RSU) is the central element of our framework. It aggregates every agents' messages and compiles them to form an occupancy grid of the mobile objects in the scene. It is composed of different blocks as illustrated in Fig. 2. We explain the role of each of the blocks in the following subsections, the mathematical details will be given later.

Monte Carlo Uncertainties Sampler:

This block takes the bounding boxes and the sensor pose from which they are extracted and models the uncertainties by applying noise to the parameters on N samples created from each original measurement, with N the larger possible.

Back Projector:

This block uses the bounding box parameters for each of the N samples, finds the 4 corners of the bounding box, and projects them on the ground by ray tracing.

Rasterizer:

This block takes the 4 projected points on the ground of each bounding box and N samples and rasterizes them on the N occupation grid.

Sample Merger:

This block merges the N occupancy grids forming a local occupancy grid (LOG) for a sensor.

Stack:

This block keeps the LOGs until the next block empties it.

Basic Belief Assignment (BBA):

This block assigns, from the observations, the masses to the different classes used with the DST method to each cell of the occupation grids. This block appears only for the DST fusion. In other cases, it is bypassed.

Combiner:

This block merges the occupancy grids of the stack either based on the DST and the BBA values or directly with the probabilities contained in the occupancy grids.

III. METHODS

In this section, we provide more details about the functioning of the blocks composing the RSU. We start with the basic principle of our system: the Back Projection, which allows us to obtain the footprints of the bounding boxes. We also present the methods allowing us to generate the local occupancy grids (LOG). Finally, we present the details of the methods for merging the LOG. As we go along, we also give details on the use of Monte Carlo methods.

A. Back Projection

The 2D bounding boxes are given in the camera's image frame in pixel coordinates. Therefore, we need to backproject them into the 3D world space. The projection of a point P W = (X W , Y W , Z W ) expressed in the world's reference frame into image point p = (u, v) , where u and v are pixel coordinates, involves two steps: The first one considers transforming the point into the camera's reference frame, given by the position of the optical center of the camera t c in the world frame and the rotation R c from the world back to the camera frame.

The second step consists in transforming the 3D point into the 2D image plane which requires the intrinsic parameters matrix K of the camera. The pinhole camera model can be expressed using homogeneous coordinates as:

s   u v 1   =   f 0 u c 0 f v c 0 0 1   [R c |t c ]     X W Y W Z W 1     = KT W C P W 1
(1) where s is an unknown scale factor, f the focal length, the parameters (u c , v c ) represent the camera's principal point coordinates in pixels and T W C the extrinsic transformation from the world's reference frame to the camera's reference frame.

It is therefore possible to compute the inverse transformation to estimate the 3D position of a point in 3D space up to a scale factor s.

In fact, for each bounding box, we have a set of five points including the camera's pose and the four corners of the bounding box and we need to compute their projection onto the ground. We start, first, by finding the corresponding 3D points up to a scale that belong to rays passing through both the optical center and the 2D points in front of the camera using the inverse transformation of the pinhole model. Second, we compute the projection of these points onto the ground and we use the Plücker coordinates which is a convenient representation for directed lines in affine 3D space, in our case the rays from the center to the corners.

The ground is assumed to be a plane π that can be represented in 3-space as in (2),

π 1 X + π 2 Y + π 3 Z + π 4 = 0 (2)
where the vector π = (π 1 , π 2 , π 3 , π 4 ) refers to its homogeneous representation [START_REF] Hartley | Multiple view geometry in computer vision 2nd ed[END_REF]. Using Plücker coordinates, a ray is represented using a 4x4 Plücker matrix defined according to Equation (3),

L = AB -BA (3)
where

A = [x 1 , y 1 , z 1 , w 1 ] and B = [x 2 , y 2 , z 2 , w 2 ]
are two points in the 3D space expressed in homogeneous coordinates.

Given the ground plane and the set of four rays corresponding to the four corners of each bounding box, the intersection between each ray L and the ground plane π can be found according to equation (4),

X gnd = Lπ (4)
where X gnd is a four-dimensional vector in homogeneous coordinates. Thus, we obtain for each bounding box a set of four projected points on the ground plane forming a polygon that represents a footprint on the ground of the detected object at a given point of view.

B. Local Occupancy Grids generation

We consider a local occupancy grid (LOG) as an occupancy grid containing the information provided by a single sensor. Let M be the occupancy grid map over a region of interest divided into square cells M x,y where x, y correspond to the position of the cell M x,y within M as defined in [START_REF] Thrun | Learning occupancy grid maps with forward sensor models[END_REF]. Thus, the problem addressed is the determination of the probability of occupancy of each grid cell given the measurements. The assigned values to M x,y are {m x,y ∈ Z| -1 ≤ m x,y ≤ 100} where -1 denotes a cell of unknown occupation, 0 denotes a free cell and 100 an occupied cell as given in ROS documentation.

1) From Ground coordinates to Occupancy Grid (OG) coordinates : Let δ be the length of the square cell in meters, (O x , O y ) correspond to the position of the origin in the occupancy grid. The position of a footprint is obtained in cell coordinates from metric coordinates according to [START_REF] Bosch | Conduite automatisée : comment les voitures et les infrastructures communiquent en milieu urbain[END_REF],

x grid = 1/δ 0 0 O x 0 1/δ 0 O y X gnd (5)
where x grid is a 2-vector and X gnd a 4-vector.

2) Rasterisation: As explained earlier, detected objects are projected on the ground plane as polygons forming footprints which now need to be rasterized in order to fill the corresponding cells of the occupancy grid. To do so, we considered the occupancy grid as an image and used the fillpoly function from the OpenCV library [START_REF] Bradski | The OpenCV Library[END_REF]. We used the 8-connected line parameter, also known as the Moore neighborhood, for the polygon edges. At this point, the occupancy of a cell is not proportional to the area occupied by the polygon but is assigned in a binary manner. Therefore, the cells in the camera frustum take the value 0, those in the bounding box frustum takes the value 100 and all the others of unknown state take the value -1.

3) Modeling uncertainties: The position estimation of the camera in the scene is subject to noise as well as the bounding box position and dimension determination on the image. To model these uncertainties, we created N samples from each original measurement, with N the larger possible. Then, we applied noise to the pose estimation and bounding box estimation parameters for each of the sample. The noise follows a Gaussian distribution with parameters µ the original measurement and σ the standard deviation presented in [START_REF] Lecrosnier | Camera pose estimation based on pnl with a known vertical direction[END_REF] and in [START_REF] Abdulmajed | Accuracy comparison between gps only and gps plus glonass in rtk and static methods[END_REF]. Each of the N samples is projected N sample grids and then merged by averaging the cells.

C. Local Occupancy Grids Merging

Since each sensor provides a local occupancy grid, these latter have to be merged in order to create a global one. The LOG is already created with respect to a global frame reference and can therefore be directly merged without frame transformations. In fact, two main paradigms have been investigated in the state of the art to perform the merging namely the probabilistic approach and the Evidential approach.

Let M be a global occupation grid. Let's consider a local occupation grid M l and M i x,y a given cell of M l where x, y refer to the location of the cell and i to the index of the agent 1 ≤ i ≤ N A with N A the number of the available agents.

1) Probabilistic merging method: The probabilistic method is based on making the intersection between the probabilities of occupation of M i x,y , i ∈ {1, 2, ..., N } to erode detections to get the final shape at the intersection of the point of views. We consider the probability of each cell as independent, thus P (M i x,y ∩ M j x,y ) = P (M i x,y ) × P (M j x,y ), i = j ∈ {1, 2, ..., N }. Therefore, we propose two methods that perform a product between cells to determine its occupancy probability as given in (6) [START_REF] Franco | Fusion of multiview silhouette cues using a space occupancy grid[END_REF]. The former, named inter1, considers the cells having an unknown state (-1) as having a probability of 0.5 before performing the product of the cells. The latter, named inter2, ignores the cells having an unknown value (-1) in the product. For both of them, values between 0 and 100 are divided by 100.

M x,y = N i=1 M i x,y (6) Algorithm 1 Basic Belief Assignment Require: C ∈ M m(∅), m(O), m(F ), m(Ω) ← 0 if C = -1 then m(Ω) ← 1 else if C is from an infrastructure sensor then m(F ) ← 1.0 -C 100 m(Ω) ← C 100 else m(O) ← C 100 m(Ω) ← 1.0 -C 100 end if end if
2) Evidential merging method: We can also use the Dempster-Shafer Theory (DST) [START_REF] Sentz | Combination of evidence in Dempster-Shafer theory[END_REF] to merge the LOGs as attempted in [START_REF] Camarda | Fusion of evidential occupancy grids for cooperative perception[END_REF]. We give two possible statuses forming the universe given by the equation [START_REF] Redmon | Yolov3: An incremental improvement[END_REF]. O describes the status of an occupied cell and F that of a free cell. The elements of the power set 2 Ω represent the status of the cell and are formed by [START_REF] Baek | Vehicle Trajectory Prediction and Collision Warning via Fusion of Multisensors and Wireless Vehicular Communications[END_REF].

Ω = {O, F } (7)
2 Ω = {∅, {O}, {F }, Ω} (8)
X∈2

Ω m(X) = 1 (9)
Thus, Ω represents an unknown state. For example, if a cell has the value -1, we know that it has not been observed and, consequently, it can be either free or occupied without being able to choose one state more than the other. Each state has a mass m corresponding to the probability of the cell being in that state respecting the distribution of the equation ( 9). Since a cell is either occupied or free, m(∅) = 0.

The association of a mass with a 2 Ω status is performed by a function named basic belief assignment (BBA). Our basic belief assignment function is given by the Algorithm 1. We distinguish two distinct points of view, that of a vehicle with a grazing view on the scene giving an estimation of the occupation and that of the infrastructure with a quasi bird eye view giving an estimation of the absence of occupation of a cell. Once each cell of each LOG has had its masses assigned, it is possible to merge them one by one with Dempster's rule of combination given in the equation [START_REF] Elfes | Using occupancy grids for mobile robot perception and navigation[END_REF] to merge two cells where X ∈ 2 Ω is defined by equation [START_REF] Nuss | A random finite set approach for dynamic occupancy grid maps with real-time application[END_REF]

with K = Y ∩Z=∅ m 1 (Y )m 2 (Z). m f (X) = m 1 (X) ⊕ m 2 (X) (10) m f (X) = 1 1 -K Y ∩Z=X =∅ m 1 (Y )m 2 (Z) (11)
m out (X) = N i=0 m i (X) (12)
Dempster's rule of combination being commutative and associative, it is, therefore, possible to combine N masses as expressed in equation [START_REF] Steyer | Grid-based environment estimation using evidential mapping and particle tracking[END_REF]. Thus, we combine the masses associated with cells of the same coordinate in each layer.

We propose two methods to associate the values to the cells from the masses of the final grid. The former, named dst1, directly assigns to the cell the mass of the set O while the latter one, named dst2, assigns to the cell the value of O if the mass of the set F is bigger than the mass of the set O and Ω, the value of -1 if Ω is bigger than O and F and the value of 100 in every other cases.

IV. RESULTS

A. Carla dataset

To the best of our knowledge, we have not identified any dataset that delivers a vehicle-infrastructure cooperative experimental framework. Therefore, we created a dataset generator based on the CARLA simulator [START_REF] Dosovitskiy | CARLA: An open urban driving simulator[END_REF] allowing the generation of datasets with one or more viewpoints from infrastructure and vehicles. For the works of this paper, we generated a dataset with 4 agents: 3 connected vehicles and an infrastructure. The vehicles pass through the roundabout and are in the field of view of the infrastructure. Also, some vehicles will enter the field of view of one or more other vehicles and have their fields of view overlapping as shown in Fig. 3. Each agent can have different sensors:

• 1× RGB camera (90 • fov, 1384 × 1032 pixels) • 1× Depth camera (90 • , 1384 × 1032) • 1× Semantic segmentation camera (90 • , 1384 × 1032) • 1× LiDAR (32 layers, 40 • vertical fov)
For the different agents, the rigid transformation between each onboard sensor and the attached reference frame is the same. For the infrastructure, the sensors are positioned at 13m altitude at the center of the roundabout (located at the scene's center) and with a pitch of -20 • . For the vehicles, the sensors are located at 1.9m above the chassis. In addition to the raw data, the vehicle's state is stored in a JSON file for each frame. This latter contains the sensor's transformation matrix with respect to the world's reference frame as well as the vehicle's transformation matrix, linear velocity, angular velocity, acceleration, forward vector, and 3D bounding box.

In order to generate the ground truth, we used the JSON files. We retrieve the 4 points forming the bottom plan of the bounding box and place them in the scene with the given transformation matrix to express their coordinates in the world reference frame. We get a perfect polygon forming the footprint of the vehicle which is then rasterized on the grid. Fig. 4c illustrates at the frame 155 of the dataset an example of the map saved where black color corresponds to a value of 100 and white color corresponds to a value of 0. Alongside, in Fig. 4, the outputs of each above-mentioned algorithm are featured.

B. Qualitative Evaluation

We evaluate our algorithm within 280 frames in which we can distinguish 6 sequences as described below: Seq 0: This sequence corresponds to the best coverage from the cars. Each car sees at least one other vehicle.

C. Quantitative Evaluation

We based our quantitative evaluation on Intersection over Union (IoU) and F1-score which are two common metrics for occupancy grid evaluation. Both of them are based on the number of True Positive (T P ), False Positive (F P ), and False Negative (F N ). To define if a cell is positive, we compare its value to a threshold. IoU is defined as in equation ( 13) and F1-score is defined in [START_REF] Caillot | Survey on cooperative perception in an automotive context[END_REF].

IoU = T P T P + F P + F N (13)
F 1 = T P T P + 1 2 (F P + F N ) (14)
Table I gives an overview of the IoU and the F1-score for each sequence and each algorithm. These results were given with a noise applied following a normal distribution with a standard deviation of σ = 0.00243m on the lateral and longitudinal position and of σ = 0.0518m on the altitude as we can find in [START_REF] Abdulmajed | Accuracy comparison between gps only and gps plus glonass in rtk and static methods[END_REF]. For the rotations, the noise follows a normal distribution with a standard deviation of σ = 0.1 • on all axes as we can find in [START_REF] Lecrosnier | Camera pose estimation based on pnl with a known vertical direction[END_REF]. The bounding boxes have a normal distribution noise with a standard deviation of 5 pixels applied on each edge of the bounding box. The threshold was set at 0.5 but requires in-depth research to determine its impact on the results.

We note that inter1 obtains a global result 0%, either on IoU and F1-score, showing that the basic probability-based occupancy grid fusion method as presented in [START_REF] Franco | Fusion of multiview silhouette cues using a space occupancy grid[END_REF], [START_REF] Thrun | Probabilistic robotics[END_REF] is not usable in this situation. However, offers slightly better results with a maximum IoU of 21.45% in sequence 4 as shown in Fig. 5 but a global IoU of 1.35%.

This behavior is explainable by the fact that inter2 is a modified version of inter1. Indeed, although these two algorithms follow the same merging rule, inter2 excludes the unobserved cells during the merging. This avoids that individual detections be removed from the final map because of the successive multiplications by 0.5 that inter1 would perform in individual detections. However, this approach shows its limits since beams appear when projecting the frustums as shown in Fig. 4b, thus increasing the number of false negatives. In sequence 4, the position of the vehicles offers viewpoints to the agents allowing them to reduce the beams due to frustum and thus to reduce the number of false positives. Nevertheless, this situation disappears when moving to sequence 5, and the false-positive rate increases dramatically.

Regarding the results obtained with our method, the algorithms based on the Dempster-Shafer theory (DST) offers much better results than the standard method cited in the previous paragraph. The fusion algorithm dst1 offers a maximum IoU of 30.52% in the sequence 1 and a global IoU of 22.35% while dst2 offers a global IoU of 18.8%.

The fusion algorithms dst1 and dst2, based on the DST, show much better results since the DST allows the management of cells with an unknown state. We can consider that the distribution of masses can give a hint on the confidence of a measurement. Thus, when a cell is not observed, the confidence associated with this measurement is null. The consequence of this behavior is the elimination of the beams as observed in the methods inter1 and inter2 and thus the reduction of the false positives. We notice a more erratic behavior on Fig. 5 until frame 140. This is due to the fact that a vehicle is too far away to be detected which corresponds to a false negative. Moreover, the vehicles are distant from each other, which has the consequence of amplifying the observation errors. As for the last sequences, the vehicles move away from each other and leave the field of view of the infrastructure, thus increasing the measurement errors. Therefore, we can conclude that the results are given at the beginning and the end of the traffic circle transit as given in Fig. 5 are due to measurement errors. To conclude, we note that dst1 and dst2 do not seem to be affected by the arrangement of the vehicles as is inter2 and therefore dst1 and dst2 are more robust than the state of the art methods while providing better results.

V. CONCLUSION AND FUTURE WORK

In this paper, we presented a new approach for cooperative perception in to create an evidential occupancy grid map. We used the vehicle point of view in addition to the infrastructure point of view in order to build confidence at low cost.

In addition, we propose a method for cooperative generation of evidential occupancy grid using only the twodimensional bounding boxes given by an image sensor as well as the position of that sensor with the aim of keeping the system's cost low as well as reducing the load on the communication system.

Finally, we propose a study on different data fusion methods based either on a Bayesian approach or on a Dempster-Shafer based approach on which we observe much better results. We have validated our results on a cooperative dataset that we have created from the CARLA simulator that we provide and to which we have added measurement noise.

In future work, we will explore decision taking approaches for the generation of the occupancy grid map more advanced than thresholding as well as the impact of the number of agents in the scene.

Fig. 1 :

 1 Fig. 1: Macro organisation of the agents and the RSU.

Fig. 2 :

 2 Fig. 2: Illustration of bounding box processing and fusion framework in the RSU.

Fig. 3 :

 3 Fig. 3: Synchronous video frames from each camera of our multi-agent dataset made with CARLA.

Fig. 4 :

 4 Fig. 4: Occupancy grid map for different methods (frame 155).

Fig. 5 :

 5 Fig. 5: Example of the evolution of the IoU with a threshold of detection of 0.50 (normalized) with 3 vehicles transiting in a roundabout.

TABLE I :

 I Example of the evolution of the IoU and F1 scores with a threshold of detection of 0.50 (normalized) with 3 vehicles transiting in a roundabout.

	Algorithm Metric	Seq 0	Seq 1	Seq 2	Seq 3	Seq 4	Seq 5	Total
	inter1	IoU	0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
	inter1	F1 Score 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
	inter2	IoU	0.052524 0.005595 0.004103 0.022008 0.214580 0.019923 0.013490
	inter2	F1 Score 0.099806 0.011128 0.008172 0.043068 0.353341 0.039067 0.026620
	dst1	IoU	0.055993 0.305245 0.287551 0.233977 0.174009 0.162757 0.223513
	dst1	F1 Score 0.106048 0.467721 0.446663 0.379224 0.296435 0.279951 0.365362
	dst2	IoU	0.175572 0.217005 0.213857 0.175982 0.172789 0.142080 0.188038
	dst2	F1 Score 0.298700 0.356621 0.352359 0.299293 0.294663 0.248810 0.316551

https://github.com/caillotantoine/ carla-V2X-dataset-generator


Haoming Yang

Olivier Mentré

Tianyu Zhu

Claire Minaud

Clemens Ritter

Xinan Zhang

Yong Jin

Minfeng Lü

Fragile magnetic ordering between robust 2D-ferrimagnets in the AFe 3 (SeO 3 ) 2 F 6 (A=K, Rb, Cs) series

Mixed anion compounds AFe3(SeO3)2F6 (A=K, Rb) have been synthesized by hydrothermal reactions. Prior results obtained for the isostructural Cs analog have revealed original 2D-ferrimagnetic (FI) blocks, antiferromagnetically (AFM) ordered around 120 K. Surprisingly, the whole series orders at a nearly constant TN temperature with similar field-induced magnetization steps, despite significant changes of the interlayer thickness. The metamagnetic transition from AFM to FI on oriented single crystals and Cp measurements prove an in-plane easy-magnetic axis confirmed by neutron powder diffraction (NPD). The ferrimagnetic alignment [Fe(1)↑-Fe(2)↓-Fe(1)↑] in each 2D subunit is robust, but the AFM-ordering between them is broken under weak field leaving field-aligned 2D-macrospins, with weak magnetocrystalline anisotropy. DFT+U calculations validate the exchange ratio Jinter/Jintra of 5.10 -3 for all compounds and the relatively high TN value is well predicted by a modified random-phase approximation (RPA) like relation, established for Quasi-Low-Dimensional Heisenberg Antiferromagnets. However, our calculations of intralayer magnetic dipole-dipole (MDD) interactions indicate a non-neglectable values and suggest the latter to assist the in-plane spin orientation below TN. At TN, under field a 3D-Heisenberg model was deduced for the AFM →FI transition from the critical behavior using modified Arrott plots.

Introduction

Besides the well-known eleven bidimensional (2D) Archimedean tilings in which all vertices are equivalent , 1 there has been intensive search on the magnetism of more complex lattices, extending the dimensionality from 1D to 3D. It gives rise to exotic magnetic states and a variety of critical points. In the 3D cases, one good example concerns the 3D-frustrated pyrochlore lattice which can lead to the emergence of spin-ice fundamental states, such as found in Ln2Ti2O7 compounds. 2 Although the theory on incrementally complex lattices is developed with application of various parameters, such as moment values (and quantum versus classical spins), magnetic anisotropy etc .., the realization of specific lattices in real materials is often challenging. Recently, Lu and Kageyama have prepared the perfect emulation of the so-called 2D-cubic tiling of S=5/2 spins in the CsFe3(SeO3)2F6 inorganic mixed anion compound. 3 Although built on three staggered layers of Fe 3+ ions arranged in isolated blocks, the projection perpendicular to the stacking axis corresponds perfectly to the in-planar representation of the cubictilling, as shown Figure 1. Indeed, the experimental characterization of CsFe3(SeO3)2F6 suggests a ferrimagnetic (FI) ordering within the 2D cube-tiles, similar to what has been theoretically predicted for this lattice with S=1/2, obtained as one of the variants after distortion of the triangular lattice. 3 Despite the absence of significant magnetic exchange paths between individual 2D-lattices in the CsFe3(SeO3)2F6, 3D antiferromagnetic ordering was observed below TN120K, easily broken into a field-induced ferrimagnetic state at relatively low field (0.H<1T). Such "high" TN in a system built on mainly uncoupled FI 2D-subunits is somewhat intriguing. Here we prepared and characterized the A= K and Rb analogs and investigate the full series on the basis of neutron powder diffraction (NPD), anisotropic magnetic measurements using oriented single crystals, heat capacity DFT+U and magnetic dipole-dipole (MDD) interactions calculations,. We paid special attention to the ingredients driving the metamagnetic transition, returning a common model valid for the full series.

Experimental Synthetic procedures

Reagents:FeCl2•4H2O (Acros Organics, 99%), HF (Energy Chemical, 48%-55% wt in H2O), KOH (11M ) RbOH (Alfa Aesar, 50% w/w), and SeO2 (Alfa Aesar, 99.4%) were used as received.

KFe3(SeO3)2F6: Hexagonal shaped crystals were obtained from the hydrothermal reactions. A stoichiometric of KOH / FeCl2•4H2O /SeO2 with the molar ratio of 1:1:3 ( 1 mmol of KOH(0.096 ml), 1 mmol (0.1988 g) of FeCl24H2O, 3 mmol (0.3329 g) of SeO2) was mixed with 1 mmol of HF (0.4 ml) and 0.7mL of deionized water. RbFe3(SeO3)2F6: Hexagonal plate-like single crystals were obtained from the similar hydrothermal reactions using a mixture of RbOH / FeCl2•4H2O /SeO2 with the molar ratio of 1.6:1:3 (1.6 mmol of RbOH (0.15 ml), 1 mmol (0.1988 g) of FeCl24H2O, 3 mmol (0.3329 g) of SeO2) and 1.1 mmol of HF (0.4 ml), 0.8mL of deionized water). Please do not adjust margins Please do not adjust margins

Each mixture was transferred into a stainless-steel autoclave (23 mL) with a Teflon liner, and was heated to 220 °C for 72 hours and cooled down to room temperature with a rate of 2.5 °C/h. The products were recovered by filtration and rinsed thoroughly with water.

Characterization

The single crystal XRD (SCXRD) data of reported materials have been collected using a Bruker SMART APEX II. The diffraction data at room temperature were obtained via a narrow-frame method with scan widths of 0.30° in ω and an exposure time of 10 s/frame. The program SAINT [START_REF]Area-Detector Integration Software[END_REF] was used for integration of the the diffraction profiles. The intensities of the obtained data were amended for air absorption, polarization, Lorentz factor etc. Semi-empirical absorption corrections were applied using the program SADABS. [START_REF]Area-Detector Absorption Correction[END_REF] The data collection and pertinent data of the refinements for AFe3(SeO3)2F6 (A=K, Rb) are summarized in Table 1. Deposition number CCDC 2086420, 2086421 for KFe3(SeO3)2F6 and RbFe3(SeO3)2F6, respectively. The refined anisotropic displacement parameters and atomic coordinates are summarized in the supplementary information (SI Table S1,S2, S4, S5).

Powder XRD (PXRD) patterns were collected using a Bruker D8 Advance X-ray powder diffractometer with Cu-Kα radiation (λ = 1.54056 Å, 40 kV/40 mA). The XRD powder patterns shown in Figure S1, S2, match well the calculated patterns using the model obtained from the single crystal data reported.

SEM/EDX of the isolated transparent crystals was carried out to determine approximate ratios of elements using a Phenom ProX desktop analyzer for the reported materials. Semi-quantitative EDX for AFe3(SeO3)2F6 (A=K, Rb) reveal A:Se:Fe:F ratios of 1.03:2.28:3.15:5.28 (K) and 1.27:2.20:3.03:5.84 (Rb) respectively (Figure S3, S4) approaching the refined stoichiometry.

Infrared spectra of the reported materials were recorded on a Varian 670-IR FTIR spectrometer in the 400-4000 cm -1 spectral range, with the well ground samples intimately contacted by an attenuated total reflection (ATR) accessory (Figure S5). It shows the absence of any hydroxyde groups in the prepared samples. TGA experiments was performed using a thermal gravimetric analyzer (Model: TGA5500, TA Instruments). Powdered samples for the reported compounds were mounted in alumina crucibles and heated at a rate of 10 °C min -1 from room temperature to 800 °C under flowing argon gas (Figure S6,S7).

The measurements of dc magnetic susceptibilities were conducted on a commercial Quantum Design Physical Properties Measurement System (DynaCool PPMS-9, Quantum Design) using VSM detection applying zero-field cooled (ZFC) and Field cooled (FC) cycles between 2 and 400K. A furnace was also used until 600 K. Isothermal magnetization curves were collected at the specified temperatures between 0 and 9 T. The specific heat measurements were performed using the heat capacity option of the PPMS.

Neutron powder diffraction data were collected on a 700 mg powder sample of KFe3(SeO3)2F6 on the high intensity powder diffractometer D20 at the Institut Laue Langevin (ILL), Grenoble, France. The sample was placed in a 5 mm Vanadium cylinder, cooled in a standard ILL Orange cryostat and measured at 1.5 K and 150 K using a wavelength of  = 2.41 Å.

DFT+U calculations

DFT calculations were performed using the projector augmented-wave (PAW) method [START_REF] Blöchl | [END_REF] implemented in the Vienna ab initio simulation package (VASP), [START_REF] Kresse | Vienna Ab-initio Simulation Package[END_REF] with the full-potential local-orbital scheme (FPLO9.00-33). [START_REF] Koepernik | [END_REF] We used the generalized gradient approximation (GGA) for electron exchange and correlation corrections 9 applying the local density approximation (LDA ) and LDA+U, U = 5 eV, for Fe atoms, this Hubbard term giving reliable magnetic parameters for Fe 3+ oxides 10 (The cutoff energy for the plane wave expansion was 550 eV and reciprocal-space integration was performed on 8x8x2 k-points Monkhorst-Pack meshes. For the calculations of the magnetic exchanges identified in the plain text, we mapped the total energies for a number of collinear spin configurations onto a classical Heisenberg model, see Figure S8, to yield individual exchange couplings.

Results and discussion

The two reported compounds AFe3(SeO3)2F6 (A= K, and Rb) are perfect structural analogues to the A= Cs compound. 3 The full series adopt the trigonal centrosymmetric space group, P-3m1. In few words, there is one unique alkali metal site, one unique Se site, but two distinct Fe sites. Main crystal and bond distances and Bond valence sum (BVS) calculations 11 for the three compounds are given in Table S7, which confirmed Fe 3+ and Se 4+ oxidation states.

The crystal structure consists of AO12 polyhedra, FeF6 and trans-FeO3F3 octahedra sharing their F corners while the pending oxygen belong to SeO3 groups in trigonal pyramidal geometries, as shown in Figure 1a. The structure of AFe3(SeO3)2F6 consists of two-dimensional [Fe3(SeO3)2F6] -3-octahedra thick layers, previously assigned to cubic tiles, and isolated by A + cations. Each layer contains three Fe sublayers, capped with SeO3 linkers whereas the lone pairs on the SeO3 point out towards their interiors. Figure 1a highlights the conservation of nearly unchanged layers while mainly the interleave 15)°( see Table S3).

KFe3(SeO3)2F6 RbFe3(SeO3)2F6 CsFe3(SeO3)2F6
As discussed in ref. [3] such isolated cubic-tiling lattices were never reported in the perovskite-based structural series such as Ruddlesden-Popper, Dion-Jacobson, Aurivillius etc, because they generally do not result from stacking along the [111] direction of the cubic-protype. However, such octahedral connectivity exists as 3C fragments of some hexagonal perovskites after removal of the face sharing octahedra, such as in the 10 H (cccch)2 Ba5W3Li2O15 where the cubic-tile contains both diamagnetic Li + and W 6+ cations 12 , see Figure 1b. In AFe3(SeO3)2F6, Fe(2) is connected to six Fe(1) and Fe(1) is connected to three Fe(2) through Fe(2)-F-Fe(1) super-exchanges (from Fe-Fe 3.92 Å K-case-to 3.94 Å Cs-case, Fe-F-Fe 178°) expected to be significantly antiferromagnetic, see Figure 1c. The competing Fe(2)-F-F-Fe(2) and Fe(1)-F-F-Fe(1) super-exchanges mediated across F-F contacts of 2.72Å are expected to be weak compared to the linear Fe(2)-F-Fe(1) super-exchanges, despite Fe-F-F-Fe dihedral angles close to 0°( see Figure S9).

It is worth recalling the main magnetic properties reported for A=Cs in the preliminary work. 3 This compound was announced to order antiferromagnetically below TN  130 K. The dominating AFM Fe(2)-F-Fe(1) super-exchanges suggests FI arrangements between the three layers of the 2D-cube lattices, in a staggered manner [Fe(1)-Fe(2)-Fe(1)] leaving a net moment per block of approximately 1/3 rd of its saturated value. A Curie-Weiss law was not established even well above TN , i.e 400K, which suggests strong magnetic exchanges inside the 2D-blocks. Below TN a metamagnetic transition was announced from M(H) plots at ∼0.05 T, typical of the Please do not adjust margins Please do not adjust margins breaking of AFM exchanges between the individual-blocks returning a 1/3 rd magnetization plateau (M5.2 B/f.u.) as expected for the easy alignment of the individual block-FI net moments. An antiferromagnetic structure with spins lying in the (a,b) plane was proposed.

For A =K and Rb, the thermal dependence of the magnetic susceptibility χ(T) for H= 1000 Oe, is shown in Figure 2a with a evidence of a significant ZFC/FC divergence below the sharp transition at 120 K in both cases.

A poor linearization of χ -1 (T) up to 400 K was observed for A=K and Rb, reminiscent of the A=Cs case. 3 In fact, a reliable χ = C/(T-θCW) Curie-Weiss fit was achieved only between 380 and 550 K. Above this temperature Fe3+ may initiate reduction in dynamical vacuum. It returns μeff = 5.40(1) μB per Fe 3+ slightly lower than the expected spinonly value of 5.92 μB for high-spin Fe 3+ and θCW = -337 (1) K. Using the mean-field (MF) model, θCW = zJS(S+1)/3kB, where z=4 is the average number of neighboring Fe spins taking in consideration the predominant Fe(1)-F-Fe(2) super-exchanges only, yields antiferromagnetic JMF/kB ≈ -29 K for S = 5/2. Although this value should be considered with precaution, dealing with the hightemperature susceptibility setup, they comfort the occurrence of strong low-dimensional magnetic correlations, well above roomtemperature.

Although slight, the divergence between ZFC and FC validates the development of a net magnetic contribution increasing with the field. This field-induced FI state is well evidenced by isothermal M(H) plots shown in Figure 2b,2c,S10. The maximal magnetization drastically depends on the temperature. At 2K it reaches 4.70 and 4.84 μB/f.u. at 9 T for A=K and Rb, respectively, nearly reaching the "ferrimagnetic" 1/3 rd of the saturated magnetization. As shown in the insets of Figure 2b,2c, the metamagnetic transition is pronounced around 0H = 0.2T at 2K for both the K and Cs analogs.

It is reversible between the different M(H) branches. This behavior denotes a fragile AFM ground state already in the low field region.

Below TN, the M/H (T) plots for A= K and Rb depend on the field values, see Figure 3. Below  0.1 T, they are nearly constant below the sharp TN peak. At higher fields, the TN peak progressively vanishes and M/H(T) smoothly increases on cooling until it reaches a "noisy" transitory regime. This latter is assigned to the relaxation of the 2D magnetic domains under the field, with marked ZFC/FC divergences.

The Neel temperatures can be roughly estimated from the sharp maxima in the thermal dependence of the product χT at low field, e.g. 10 Oe, followed by a subsequent drop. Even at such low field the maxima depend on the sample magnetic history and hysteretic χT(T) peaks are observed in ZFC/FC sequences, well understood by a 1 st order magnetic transition. We found the maxima at TN (K) 119cooling-122warming, 117cooling-120warming K and 117warming for A=K, Rb, Cs 3 respectively, as shown on the Figure 4a. It coincides with the weak kink in Cp and Cp/T at 119 K observed for A=K on cooling (Figure 4b), supporting the occurrence of a thermodynamic magnetic transition with poor contribution to the magnetic entropy at TN due to strong magnetic exchanges in-plane. The Cp peak rapidly vanishes under magnetic field, see the inset of the Figure 4b. It follows a singular situation above the metamagnetic transition which, at this stage resembles to a field-aligned paramagnet. Cp under zero field and 1T are shown on Figure S11 for A=K and Rb. For comparison, standard behaviors along both spin-flip and spin-flop transitions preserve the Please do not adjust margins Please do not adjust margins Cp lambda-type anomalies at TN at relatively high field, see recent cobalt arsenates compounds. 13 Hence, our data suggest the loss of uniaxial spins together with a relatively weak magneto-crystalline under applied field. It is noteworthy that the experimental TN values are very similar (120 K) for all compounds, a counter-intuitive behavior if one considers the expected variation of the Fe-Fe spinexchanges (SEs) across the [Alkali] interleaves with a versatile size, see Figure 1a. Indeed, the shortest super-super exchanges Fe-O-O-Fe (Jinter) paths between two next blocks along c involve Fe-Fe/O-O distances of 6.13/3.22 Å (A=K), 6.29/3.39 Å (A=Rb) and 6.61/3.76 Å (A=Cs), with O-O separations much above the sum of the oxygen Van der Waals radii. Such features refute the setting of significant AFM interactions by SEs, especially around TN120 K, a rather high ordering temperature. Indeed, SEs are more generally short-ranged (dropping exponentially with inter-spin distance in ferromagnetic insulators for instance). However, the situation resembles the high TN 3D-antiferromagnetic ordering of some cuprate-related superconductors such as (Ca0.85Sr0.15)CuO2 with high TN=537(5) K. It results from the crossover from large intraplanar spin-spin correlations at high temperature to a 3D ground state driven by very weak out-of-plane couplings (Jinter/Jintra = 0.016 ) between Cu 2+ ions . 14 The Néel of such Quasi-Low-Dimensional Heisenberg antiferromagnets was rationalized by Quantum Monte-Carlo (QMC) simulations giving empirical formulas describing TN for a wide range of exchange ratio 15 . This point will be discussed later on the light of the refined magnetic structure and calculated exchange values.

The magnetic structure of KFe3(SeO3)2F6 at zero field was solved and refined from NPD data collected at 1.5K on the D20 diffractometer (λ=2.41Å) at the Institut Laue Langevin (ILL), Grenoble, France. The magnetic contributions growing below TN can be indexed with the magnetic propagation vector k= [0,0, ½ ]. The magnetic symmetry analysis performed using Basireps 16 indicates a decomposition of the Г magnetic representation into four irreducible representations (Г1-4) of dimension 1 and two (Г5-6) of dimension 2. For Fe(2), the decomposition yields 1Г3 + 1Г6, but 1Г2+1Г3 +1Г5+ 1Г6 for Fe (1). The representation Г6 for both sites leads to the best agreement, which involves in-plane components only with spins parallel to [110] or trigonally-equivalent directions, see Table 2. Within isolated blocks, the two bordering Fe1 sites are parallel, but antiparallel to Fe2. Rmagn was significantly improved from 6.94% to 6.28% relaxing individual Fe1 and Fe2 moments with a better matching of most of the magnetic satellites. It returns MFe1 =3.72(3)µB and MFe2=4.25(5)µB,. We note the presence of preferred orientations mainly occurring along [001] related to the hexagonalcrystallite shape. They have been refined on the data collected at 150 K using the March-Dollase function and applied on the data collected at 1.5K. However, it does not perfectly correct the nuclear profile, as shown on the Figure 5, plausibly due to multi-directional preferred orientations. In essence, our refined magnetic structure matches rather well earlier prediction for A=Cs. 3 The spin structure is compatible with a 3D-XY model, with the coupling between the layers hold by either weak Jinter exchanges or by spin-anisotropy able to freeze the magnetic moments in specific directions.

To unravel this point, single crystals of KFe3(SeO3)2F6 have been Fe( 1)

Fe(2)
Fe( 1) Please do not adjust margins Please do not adjust margins aligned in epoxy resin at room temperature under an applied field of 9T. After drying, the observation of the crystals by transparency in the resin validates in-plane easy magnetic-axis active already in the paramagnetic domain, all of them being oriented with their c-axis normal to the field. The M(H) plots parallel and perpendicular to the c-axis are shown on Figure 6. They are very similar and both show the 1/3 rd saturation plateau against a robust in-plane spin direction.

It returns an easiest alignment in the (a,b) plane at low field, in good agreement with the refined magnetic structure. Note on the Figure 6, that the weak decrease of M//c after saturation is due to our not perfectly aligned sample. Above 20 KOe, the magnetization is nearly isotropic. In essence, when H is applied parallel to c, the moments rapidly flip following the field direction and M//c reaches the 1/3 rd plateau. It comforts the robust FI topology within the layers hold by strong Jintra superexchanges, returning a net 4.5B/f.u moment. The anisotropy field Hanis, can be quantified by the field at which the magnetization perpendicular to the easy-axis saturates. However, for the title series, dealing with a AFM → FI metamagnetic transition, it sounds more realistic to consider Hanis as the H difference to reach saturation between the two directions, i.e. 1 KOe. This value is much weaker that what is found in several other 2D-compounds such as the layered ferromagnetic CrI3 in bulk ( Hanis 19 KOe) with easy magnetic c-axis. 17,18 Despite very weak van der Waals inter-layer interactions in CrI3 , the situation is different, the magnetic properties being thickness-dependent, with the crossover between a mean-field type interactions in the bulk (Tc =61K) 19 and an Ising 2D-regime for single flakes (Tc=45 K). 20 The results of our DFT+U calculations of the magnetic exchanges in the layer Jintra (= JFe1-F-Fe2) and between the blocks Jinter (=JFe1-O-O-Fe1) are listed for the three compounds in Table 3. It shows nearly constant Jinter (93-96 K) and Jintra (0.21-0.44 K) exchange values. Note that Jintra are found inversely scaled by the intermediate O---O distances, as expected. The very weak Jintra/Jinter ratio of 3.10 -3 validate a strongly anisotropic distribution of J's in our compounds. Phenomenologically the Fe 3+ single-ion anisotropy , which even if expected very weak for (L=0) ions, mat also contribute to the spin-freezing at TN, hampering significantly the spin fluctuations and participate to the AFM ordering between the layers. 21 While genuinely 2D models cannot display long-range order (LRO) above zero temperature, apart the 2D-Ising exception (see BaFe2(PO4)2 with Tc 65 K 22 , the weak estimated Jinter couplings, lead to a finite Néel temperature TN 15 . The TN dependence was analyzed by a modified random-phase approximation (RPA) giving the simple formula TN=4•n•Jintra/ [b -ln(Jinter/Jintra)], with n and b constant estimated for S= ½ and S = 1. Besides the theory behind this empirical expression, it highlights the possibility for high ordering temperature, as verified in the (Ca0.85Sr0.15)CuO2 case mentioned above 14 . Using the parameters fitted for S=1 (n=0.68 and b=3.12) and the Js from DFT, we calculated the values listed in Table 3, and although underestimated (using parameters refined for S=1 Heisenberg spins), they show a correct correlation with the experimental results.

Here we discuss the ingredients governing the spinstructure below TN and the easy spin-reversal of 2D-macrospins in an applied magnetic field. Besides the well-known influence on the spin orientations by spin-orbit coupling (SOC), spinexchange (SE), magnetic dipole-dipole interactions (MDD), in the title series, the occurrence of AFM ordering around 120 K mediated by very weak exchanges across the alkali interleaves Table 3 The magnetic exchanges calculated for AFe3(SeO3)2F6 (A= K, Rb, Cs) by DFT +U (U=5 eV) and experimental versus estimated TN (see Ref. 15), and calculated MDD.

question about the relative role of the various ingredients at TN. First, similarly to the layered Sr3Fe2O5, 23 SOC being a single site interaction, it cannot determine the ordering between FI layers in the title series. Contrarily MDD are long-range multisite interactions, which are very weak between individual spins in an AFM state, but may become sizeable between 2Dmacrospins, and may influence the orientation of the spins in the layers. For instance, MDD interactions are responsible for the formation of domains at high temperature in ferromagnets, [START_REF] Ashcroft | Solid State Physics[END_REF] and strongly involved in the 3D magnetic ordering of highly separated layers in lamellar double hydroxides or chain systems. [START_REF] Ostrovsky | [END_REF] MDD interactions are scaled by 1/r 3 (where r is the interlayer distance given by the unit cell parameter c ) and were estimated between the external Fe(1) and Fe(1), Fe(2) atoms of the next block in a sphere delimited by the cutoff distance Fe-Fe  12Å, which correspond to the sum of >40 individual contributions. We use the spin orientations from the refined magnetic structures and the following formula recently applied to BaCoAs2O7 multiferroics 26 for individual contributions: Where 𝑚 𝑗,𝑘 ⃑⃑⃑⃑⃑⃑⃑ are refined moments in μB. ao is the Bohr radius (0.529177 Å). rjk is the distance between the spin sites j and k. and 𝑒 𝑗𝑘 ⃑⃑⃑⃑⃑ is the unit vector along the intersite distance. (μB 2 / ao 3 )= 0.181meV. Please do not adjust margins Please do not adjust margins

Our results are listed in Table 3 and show negative (i.e. stabilizing) energies of around 10% of Jinter which cannot be neglected. It strongly suggests that the collective MDDs stabilize the antiferromagnetic in-plane spin ground state, see Figure 7a. In addition, taking the fragile spin alignment broken at Hc = 100 Oe at 2 K, one can estimate the energy to flip individual spins in a magnetic field, given as E= g.B.Hc.S. We found E/Kb =0.03K meV, which is typically the order of magnitude of our calculated inter-layer MDD exchanges. In an antagonist manner, MDDs in a ferromagnetic order between weakly coupled ferromagnetic 2D units is favored by perpendicular spins and could be a plausible scenario for CrI3 in its mean field regime (Tc=61K), i.e. above the temperature of 45K of the Ising setting, see Figure 7b. All together, i) the contribution of MDDs and weak Jinter exchanges, ii) the Néel ordering at relatively high temperature, iii) the metamagnetic destabilization under field into a FI for any field direction, question about the critical behaviors at TN but at high field. Clearly we aim to probe the critical exponents associated with the field induced transition into a ferrimagnetic state, and the validity of our analysis makes sense at finite field, while critical exponents of ferromagnets can be defined only in the limit of H=0 and T= Tc.

In that purpose, we have used the modified Arrott-plot treatment detailed for the 3D-Cr11Ge9, 27 2D-CrI3, 19 2D-CrGeTe3, 28 all of them displaying a relatively weak magneto-crystalline anisotropy with easy spin reversal perpendicular to the easy-magnetic axis under moderate magnetic field, similarly to the title-series. They allow to experimentally estimate the critical exponents from the data at finite H around Tc. In our case dealing with a low field spin floplike transition, the validity of the Arrott-analysis, adapted to ferromagnets is questionable. However, if one assumes an effect of the low-field AFM ordering similar to the domain structure of an ideal ferromagnet, significant issues can be anticipated. For RbFe3F6(SeO3)2, the M(H) plots near Tc (interval T=1K) are shown in Figure S12a after correction for the demagnetizing field, using the low-H magnetization. Generally, the critical exponents obey the Arrott-Noakes equation of states M 1/β = A(T-TC)+B(H/M) 1/γ . 29 We first checked that the mean-field where =1/2 and =1.0 30 was not verified as M 2 vs. H/M should be a bunch of parallel lines, at least in the high field range, passing through the origin at Tc, see Figure S12b. Their positive slopes validate a 2 nd order magnetic transition, but their downward curvature suggests the use of modified Arrott plots to extract realistic critical exponents, besides the mean-field model. The four plausible models in agreement with the refined 3D-ordering and the alignment of 2D-macrospins, namely the 3D-Heisenberg ( = 0.365,  = 1.386), 3D-Ising ( = 0.325,  = 1.24), tricritical mean-field ( = 0. 25,  =1.0) and 3D-XY ( = 0.345,  = 1.316), have been used, as shown in Figure S13. The most appropriate models returning parallel lines (with a slope S(T) =dM 1/ /d(H/M) 1/ ) was determined by calculating the deviation of the normalized slope NS =S(T)/S(Tc) to 1. Tc is chosen for M 1/β vs. (H/M) 1/γ crossing the origin, i.e. around 121-123 K depending on the model and refined latter. NS(T) are plotted in Figure 8a, which returns the 3D-Heisenberg and 3D-XY as the best Please do not adjust margins Please do not adjust margins ones. It discards the 3D-Ising case, which is inconsistent with the 2Ddegeneracy of the refined magnetic structure. The 3D-tricritical model is also refuted, giving the worst agreement, contrarily to the CrGeTe3 case, in which a 2D to 3D magnetic phase transition locates near a tricritical point. 28 The theoretical  and  values are poorly discriminating between the 3D-Heisenberg and XY models. However, keeping in mind that the critical temperature concerns the ordering between individual layers with intrinsic FI correlations much above TN, the best-matching 3D-Heisenberg is relevant, assisted by i) the threeaxial (x,y,z) components of anisotropic exchanges and the MDD interactions across the interblock gap. Using the plot of Figure 8b, the linear extrapolation from the high field region ( between 0.H =3.6 and 9T , i.e. well above the AFM → FI deviation) to the intercepts with the axes M 1/ below Tc returns reliable spontaneous magnetization Ms(T,0), while their intercepts with the (H/M) 1/ above Tc returns χ0 -1 (T, 0) , see figure 8c. Then Tc, , and  have been fitted to the critical exponents expressions, defined near Tc as 31 :

Ms(T) =

Conclusions

We report here the interactions between magnetic ions leading to a periodic lattice of 2D spin-clusters already at room temperature, themselves interacting in an antiferromagnet at TN 120 K, and somixing two different energy scales. There is a recent interest in materials straddling this spin/macrospin border because the interplay of different dimensionalities and interaction scales within and between clusters can lead to unusual physics, see for example, the formation of hexameric Cu6 S=1 macrospins and their interactions in 2D-square lattices in A2Cu3(SO4)3 32 or Cu2OSeO3 with a chiral lattice built by inner-Cu4 tetrahedra which hosts several skyrmion phases. 33 However such magnetic systems are rarely achieved in real materials. In the AFe3F6(SeO3)2 family of compounds strong intralayer ferrimagnetic correlations are active up to 400 K, induced by strong Fe-F-Fe superexchanges between S=5/2 spins. The 2D-spin clusters order antiferromagnetically around TN =120 K assisted by weak exchanges inter the layers ( Jinter 0.5K) in the whole series, where MDD interactions support their in-plane spin orientations as refined from NPD. It is striking that weak Jinter/Jintra of 10 -3 ratio generate a magnetic structure at so "high" temperatures. In an applied magnetic field, the robust 2D-ferrimagnetic spinclusters progressively align to the external field, up to the 1/3 rd magnetization ferrimagnetic-saturation. This 2 nd order transition is associated with critical exponents suggesting a 3D Heisenberg topology,. Given, the relatively weak magnetic anisotropy deduced from M(a,b) and M//c measurements, the resulting spin configuration above Hc is expected directionless, forming a field induced paramagnet of 2D-macrospins in good agreement with the vanishing of the anomaly right above Hc and with respect to the relatively weak anisotropy field of 12K Oe measured on single crystals at 2K.

  space filled by A + varies along the K, Rb, Cs series with respect to the c-lattice parameter elongation. Typical distances are given for A=K as follows: the equidistant Fe(2)-F bond lengths are 1.919(3) Å with the F-Fe(2)-F bond angles of 89.76(12)-90.24(10)°. In the slightly distorted [Fe(1)O3F3] unit, Fe(1)-F and Fe(1)-O distances are 2.005(3) and 1.933(3) Å, respectively with O/F-Fe(1)-O/F bond angles of 85.35(12)-94.50(

Figure 1 aFeFigure 2

 12 Figure 1 a) Cube tile lattice of AFe3(SeO3)2F6 (A= Cs, Rb, K, etc) along the [110] direction, [Fe(2)F6] octahedra are drawn in dark blue, [Fe(1)O3F3] octahedra are highlighted in pale blue and grey.b) Evidence of similar three-octahedra-thick blocks in the 10 H (cccch)2 Ba5W3Li2O15. c) 2D [Fe3(SeO3)2F6] -block along the[00-1] axis with the tile-backbone and Js exchanges.

Figure 3

 3 Figure 3 ZFC (dotted-lines) and FC (full-lines) magnetic susceptibility of AFe3(SeO3)2F6 (A=K(a), Rb(b)) under different magnetic fields between 2 and 200K.

Figure 4

 4 Figure 4 (a)χ(T)T vs. T curve for AFe3(SeO3)2F6 (A= Cs (reference 3), Rb, K) at 10 Oe.(b) The temperature dependence of Cp/T under 0T for KFe3(SeO3)2F6.The inset shows the temperature dependence of Cp under different magnetic fields.

Figure 5 a

 5 Figure 5 a) PND Rietveld refined patterns at 1.5 K and b) refined magnetic models for KFe3(SeO3)2F6.

Figure 6 M

 6 Figure 6 M(H) curves of single crystals of KFe3(SeO3)2F6 in-plane (a) and parallel to the c-axis(b).

Figure 7

 7 Figure 7 scheme of the MDD interactions, macrospins and idealized field lines for a) the AFM AFe3F6(SeO3)2 compounds (only greenish and grey spheres are represented for Fe and Ba atoms), b) the FM CrI3 (brown and grey spheres are Cr and I atoms).

Figure 8 (

 8 Figure 8(a) Temperature dependence of normalized slope (NS) for RbFe3F6(SeO3)2, (b)The isotherms of M 1/β vs. (H/M) 1/ with parameters of 3D-Heisenberg model, (c) the MS (left) and χ0 -1 (right) as a function of temperature for RbFe3F6(SeO3)2, (d)isothermal M(H) at TC(all red solid curves are fitted).

  M0 ((Tc-T)/Tc)  T< Tc (1) H/M(T)=h0/m0((T-Tc)/Tc)  T> Tc (2) It returns =0.357(3), Tc=123.21(3)K in the low temperature region and =1.40(3), Tc =123.2(4) K in the high temperature region, close to theoretical values of the 3D-Heisenberg model (=0.365, =1.386).

Finally, the

  exponent  defined at Tc by M=DH 1/ was directly estimated from the critical isotherm at Tc, see Figure 8d. It yields  =4.795(5) for H> 2.5 T, close to its theoretical value of 4.8. It nearly verifies the Widom scaling relation =1 + / giving 4.92 from our fits, which demonstrate the reliability of our set of critical exponents.

Table 1

 1 Crystal data, measurement and structural refinement parameters of AFe3(SeO3)2F6 (A=K, Rb and Cs).

Table 2

 2 Mx,My, Mz according to the 6 irreducible representation for the magnetic structures of KFe3F6(SeO3)2compounds.

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 3

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 7

Acknowledgements

H. Y.,T. Z., X. Z., Y. J., and M.L. acknowledge support from the National Natural Science Foundation of China (Award No.21671185). O.M., acknowledge support from the ANR (Grant ANR-16-CE08-0023) and the FEDER Région Hauts-de-France, they also acknowledge the ILL for the allocated beamtime (DOI 10.5291/ILL-DATA.EASY-654) and the ILL Instrument Control Service for their support before and during the experiment. Finally dealing with mixed anion compounds, the JSPS Core-to-Core Program (A) Advanced Research Networks (JPJSCCA20200004) is also acknowledged as a source of motivation.

Notes and references

Conflicts of interest

There are no conflicts to declare.

Graphical Abstract

Weak magnetocrystalline anisotropy and breaking under field of the interlayer AFM exchanges in AFe3(SeO3)2F6 (A=K, Rb, Cs).