Giorgia Pitteri
email: giorgia.pitteri@u-bordeaux.fr

Aurélie Bugeau
email: aurelie.bugeau@u-bordeaux.fr

Slobodan Ilic
email: sobodan.ilic@siemens.com

Vincent Lepetit
email: vincent.lepetit@enpc.fr

3D Object Detection and Pose Estimation of Unseen Objects in Color Images with Local Surface Embeddings

We present an approach for detecting and estimating the 3D poses of objects in images that requires only an untextured CAD model and no training phase for new objects. Our approach combines Deep Learning and 3D geometry: It relies on an embedding of local 3D geometry to match the CAD models to the input images. For points at the surface of objects, this embedding can be computed directly from the CAD model; for image locations, we learn to predict it from the image itself. This establishes correspondences between 3D points on the CAD model and 2D locations of the input images. However, many of these correspondences are ambiguous as many points may have similar local geometries. We show that we can use Mask-RCNN in a class-agnostic way to detect the new objects without retraining and thus drastically limit the number of possible correspondences. We can then robustly estimate a 3D pose from these discriminative correspondences using a RANSAClike algorithm. We demonstrate the performance of this approach on the T-LESS dataset, by using a small number of objects to learn the embedding and testing it on the other objects. Our experiments show that our method is on par or better than previous methods.

Introduction

Deep Learning (DL) provides powerful techniques to estimate the 6D pose of an object from color images, and impressive results have been achieved over the last years [START_REF] Kehl | SSD-6D: Making RGB-Based 3D Detection and 6D Pose Estimation Great Again[END_REF][START_REF] Rad | BB8: A Scalable, Accurate, Robust to Partial Occlusion Method for Predicting the 3D Poses of Challenging Objects Without Using Depth[END_REF][START_REF] Tekin | Real-Time Seamless Single Shot 6D Object Pose Prediction[END_REF][START_REF] Jafari | IPose: Instance-Aware 6D Pose Estimation of Partly Occluded Objects[END_REF][START_REF] Xiang | PoseCNN: A Convolutional Neural Network for 6D Object Pose Estimation in Cluttered Scenes[END_REF][START_REF] Zakharov | DPOD: Dense 6D Pose Object Detector and Refiner[END_REF], including in the presence of occlusions [START_REF] Oberweger | Making Deep Heatmaps Robust to Partial Occlusions for 3D Object Pose Estimation[END_REF][START_REF] Peng | PVNet: Pixel-Wise Voting Network for 6DoF Pose Estimation[END_REF][START_REF] Hu | Single-Stage 6D Object Pose Estimation[END_REF], in the absence of texture, and for objects with symmetries (which create pose ambiguities) [START_REF] Sundermeyer | Augmented Autoencoders: Implicit 3D Orientation Learning for 6D Object Detection[END_REF][START_REF] Park | Pix2pose: Pixel-Wise Coordinate Regression of Objects for 6D Pose Estimation[END_REF]. However, most of recent works focus on supervised approaches, which require that for each new object, these methods have to be retrained on many different registered images of this object. Even if domain transfer methods allow for training such methods on synthetic images instead of real ones at least to some extent, such training sessions take time, and avoiding them is highly appealing.

Fig. 1. Overview of our method. We detect and estimate the 3D poses of objects, given only an untextured CAD model, without having to retrain a deep model for these objects. Given an input RGB image, we predict local surface embeddings (LSEs) for each pixel that we match with the LSEs of 3D points on the CAD models. We then use a PnP algorithm and RANSAC to estimate the 3D poses from these correspondences. We use the predicted masks to constrain the correspondences in a RANSAC sample to lie on the same object, in order to control the complexity. The LSE prediction network is trained on known objects but generalizes well to new objects. Similarly, we train Mask-RCNN on known objects, and use mask R-CNN to segment the objects in the image. Because we train Mask-RCNN in a class-agnostic way, it also generalizes to new objects without retraining. Note that we use masks of different colors for visualization only.

Recently, a few methods were proposed for 3D pose estimation for objects that were not seen during training, only exploiting an untextured CAD model provided for the new objects. This is an important problem in industrial contexts, but also very challenging as aligning an untextured CAD model to a color image remains very difficult, especially without pose prior. In DeepIM [START_REF] Li | DeepIM: Deep Iterative Matching for 6D Pose Estimation[END_REF], the authors propose a pose refiner able to perform such alignment given some initial pose, however it has been demonstrated on very simple synthetic images with constant lighting. [START_REF] Pitteri | CorNet: Generic 3D Corners for 6D Pose Estimation of New Objects without Retraining[END_REF] proposes to learn to detect corners by using training images of a small set of objects, and estimates the object pose by robustly matching the corners of the CAD model with the corners detected in the input image. However, this requires the object to have specific corners and a skilled user to select the corners on the CAD model. Very recently, [START_REF] Sundermeyer | Multi-path Learning for Object Pose Estimation Across Domains[END_REF] proposes a single-encoder-multidecoder network to predict the 6D pose of even unseen objects. Their encoder can learn an interleaved encoding where general features can be shared across multiple instances. This leads to encodings that can represent object orientations from novel, untrained instances, even when they belong to untrained categories. Even if the idea is promising, to achieve competitive results, they need to use depth information and refine the pose with an ICP algorithm.

In this paper, we investigate 6D object pose estimation in an industrial scenario with the challenges this implies: We want to handle symmetrical, textureless, ambiguous, and unseen objects, given only their CAD models. By contrast with some previous works, we also do not assume that the ground truth 2D bounding boxes for the objects are available. As shown in Figure 1, our approach combines machine learning and 3D geometry: Like previous works [START_REF] Brachmann | Uncertainty-Driven 6D Pose Estimation of Objects and Scenes from a Single RGB Image[END_REF][START_REF] Zakharov | DPOD: Dense 6D Pose Object Detector and Refiner[END_REF][START_REF] Park | Pix2pose: Pixel-Wise Coordinate Regression of Objects for 6D Pose Estimation[END_REF][START_REF] Wang | Normalized Object Coordinate Space for Category-Level 6D Object Pose and Size Estimation[END_REF], we establish dense correspondences between the image locations and 3D points on the CAD model, as they showed that this yields to accurate poses. However, there is a fundamental difference between these works and ours: They can train a machine learning model in advance to predict the 3D coordinates of the pixels in a given image. In our case, we want to avoid any training phase for new objects. We therefore rely on a different strategy: We introduce an embedding capturing the local geometry of the 3D points lying on the object surface. Given a training set for a small number of objects, we learn to predict these embeddings per pixel for images of new objects. By matching these embeddings with the embeddings computed for 3D points on the object surface, we get 2D-3D correspondences from which we estimate the object's 6D pose using RANSAC and a PnP solver.

This approach is conceptually simple, robust to occlusions, and provides an accurate 6D pose. However, to be successful, some special care is needed. First, the embeddings need to be rotation invariant. Second, because of the symmetries and this rotation invariance, many correspondences between pixels and 3D points are possible a priori and the complexity of finding a set of correct correspondences can become exponential. We control this complexity in two ways. We focus on image locations with the most discriminative embeddings as they have less potential correspondences. We also observe that Mask R-CNN [START_REF] He | Mask R-CNN[END_REF] is able to predict the masks of new objects when trained without any class information, and thus can segment new objects without re-training. We use this to constrain the sets of correspondences in RANSAC to lie on the same mask, and thus drastically decrease the number of samples to consider in RANSAC.

In the remainder of the paper, we review the state-of-the-art on 3D object pose estimation from images, describe our method, and evaluate it on the T-LESS dataset [START_REF] Hodan | T-LESS: An RGB-D Dataset for 6D Pose Estimation of Texture-less Objects[END_REF], which is made of very challenging objects and sequences.

Related Work

In this section, we first review recent works on 3D object detection and pose estimation from color images. We also review methods for using synthetic images for training as it is a popular solution for 3D pose estimation. Finally, we review the few works that consider the same problem as us.

3D Object Detection and Pose Estimation from Color Images

The use of Deep Learning has recently significantly improved the performance of 6D pose estimation algorithms. Different general approaches have been proposed. One approach is to first estimate 2D bounding boxes for the visible objects, and predict the 6D pose of each object directly from the image region in the bounding box [START_REF] Kehl | SSD-6D: Making RGB-Based 3D Detection and 6D Pose Estimation Great Again[END_REF][START_REF] Rad | BB8: A Scalable, Accurate, Robust to Partial Occlusion Method for Predicting the 3D Poses of Challenging Objects Without Using Depth[END_REF][START_REF] Tekin | Real-Time Seamless Single Shot 6D Object Pose Prediction[END_REF][START_REF] Xiang | PoseCNN: A Convolutional Neural Network for 6D Object Pose Estimation in Cluttered Scenes[END_REF][START_REF] Sundermeyer | Augmented Autoencoders: Implicit 3D Orientation Learning for 6D Object Detection[END_REF]. The pose can be predicted directly using quaternions for the rotation example, or via 3D points or the reprojections of 3D points related to the object, or by learning a code book using AutoEncoders. This last method has the advantage to work well with symmetrical objects, which are common in industrial contexts. Another approach, aiming to be more robust to occlusions, is to predict for each pixel offsets to the reprojections of 3D points related to the object [START_REF] Oberweger | Making Deep Heatmaps Robust to Partial Occlusions for 3D Object Pose Estimation[END_REF][START_REF] Hu | Segmentation-Driven 6D Object Pose Estimation[END_REF][START_REF] Peng | PVNet: Pixel-Wise Voting Network for 6DoF Pose Estimation[END_REF].

Closer to our own approach, several works first predict for each pixel its 3D coordinates in the object's coordinate frame [START_REF] Taylor | The Vitruvian Manifold: Inferring Dense Correspondences for One-Shot Human Pose Estimation[END_REF][START_REF] Brachmann | Uncertainty-Driven 6D Pose Estimation of Objects and Scenes from a Single RGB Image[END_REF][START_REF] Jafari | IPose: Instance-Aware 6D Pose Estimation of Partly Occluded Objects[END_REF][START_REF] Wang | 3d pose estimation for fine-grained object categories[END_REF][START_REF] Zakharov | DPOD: Dense 6D Pose Object Detector and Refiner[END_REF][START_REF] Park | Pix2pose: Pixel-Wise Coordinate Regression of Objects for 6D Pose Estimation[END_REF][START_REF]cdpn: Coordinates-based disentangled pose network for real-time rgb-based 6-dof object pose estimation[END_REF][START_REF] Wang | Normalized Object Coordinate Space for Category-Level 6D Object Pose and Size Estimation[END_REF][START_REF] Hodan | EPOS: Estimating 6D Pose of Objects with Symmetries[END_REF]. This yields 2D-3D correspondences from which the object's 3D pose can be estimated using a PnP algorithm, possibly together with RANSAC for more robustness. In our case, we cannot directly predict the 3D coordinates of pixels as it can be done only for the objects or categories used for training. Instead, we learn to predict an embedding for the 3D local geometry corresponding to each pixel, and we rely on this embedding to match the pixel to its corresponding 3D point on the CAD models of new objects.

Training on synthetic images for 6D pose estimation

One popular approach to 6D pose estimation given only a CAD model and no, or few, real training images is to exploit synthetic images. There is however a domain gap between real and synthetic images, which has to be considered to make sure the method generalizes well to real images.

A very simple approach is to train a convolutional network for some problem such as 2D detection on real images and use the first part of the network for extracting image features [START_REF] Hinterstoisser | On Pre-Trained Image Features and Synthetic Images for Deep Learning[END_REF][START_REF] Kehl | SSD-6D: Making RGB-Based 3D Detection and 6D Pose Estimation Great Again[END_REF]. Then, a network taking these features as input can be trained on synthetic images. This is easy to do, but it is not clear how many layers should be used exactly. Generative Adversarial Networks (GANs) [START_REF] Goodfellow | Generative Adversarial Nets[END_REF] and Domain Transfer have been used to make synthetic images more realistic [START_REF] Bousmalis | Domain Separation Networks[END_REF][START_REF] Müller | GANerated Hands for Real-Time 3D Hand Tracking from Monocular RGB[END_REF][START_REF] Bousmalis | Unsupervised Pixel-Level Domain Adaptation with Generative Adversarial Networks[END_REF][START_REF] Zhu | Unpaired Image-To-Image Translation Using Cycle-Consistent Adversarial Networks[END_REF][START_REF] Ganin | Domain-Adversarial Training of Neural Networks[END_REF][START_REF] Long | Learning Transferable Features with Deep Adaptation Networks[END_REF][START_REF] Tzeng | Simultaneous Deep Transfer Across Domains and Tasks[END_REF][START_REF] Lee | Diverse Image-To-Image Translation via Disentangled Representations[END_REF][START_REF] Zakharov | Keep It Unreal: Bridging the Realism Gap for 2.5D Recognition with Geometry Priors Only[END_REF]. Another interesting approach is domain randomization [START_REF] Tobin | Domain Randomization for Transferring Deep Neural Networks from Simulation to the Real World[END_REF], which generates synthetic training images with random appearance by applying drastic variations to the object textures and the rendering parameters to improve generalization.

These works can exploit CAD models for learning to detect new objects, however they also require a training phase for new objects. In this work, we do not need such phase.

6D pose estimation without retraining

Very few recent works already tackled 6D pose estimation without retraining for new objects. One early approach targeting texture-less objects is to rely on templates [START_REF] Hinterstoisser | Gradient Response Maps for Real-Time Detection of Textureless Objects[END_REF]. Deep Learning has also been applied to such problem, by learning to compute a descriptor from pairs or triplets of object images [START_REF] Wohlhart | Learning Descriptors for Object Recognition and 3D Pose Estimation[END_REF][START_REF] Balntas | Pose Guided RGBD Feature Learning for 3D Object Pose Estimation[END_REF][START_REF] Zakharov | 3D Object Instance Recognition and Pose Estimation Using Triplet Loss with Dynamic Margin[END_REF][START_REF] Bui | When Regression Meets Manifold Learning for Object Recognition and Pose Estimation[END_REF]. Like ours, these approaches do not require re-training, as it only requires to compute the descriptors for images of the new objects. However, it requires many images from points of view sampled around the object. It may be possible to use synthetic images, but then, some domain transfer has to be performed. But the main drawback of this approach is the lack of robustness to partial occlusions, as the descriptor is computed for whole images of objects. It is also not clear how it would handle ambiguities, as it is based on metric learning on images. In fact, such approach has been demonstrated on the LineMod, which is made of relatively simple objects, and never on the T-LESS dataset, which is much more challenging.

More recently, DeepIM proposed a pose refiner able to refine a given initial pose. In [START_REF] Li | DeepIM: Deep Iterative Matching for 6D Pose Estimation[END_REF], this refiner was applied to new objects, but only on very simple synthetic images with constant lighting. [START_REF] Pitteri | CorNet: Generic 3D Corners for 6D Pose Estimation of New Objects without Retraining[END_REF] proposes to learn to detect corners by using training images of a small set of objects and estimates the object pose by robustly matching the corners of the CAD model with the corners detected in the input image. This method requires objects to have specific corners and to offline select corners on the CAD model. Even more recently, [START_REF] Sundermeyer | Multi-path Learning for Object Pose Estimation Across Domains[END_REF] proposes an extension of [START_REF] Sundermeyer | Augmented Autoencoders: Implicit 3D Orientation Learning for 6D Object Detection[END_REF] able to generalize to new objects. Thanks to the single-encodermulti-decoder architecture, they are able to learn an interleaved encoding where general features can be shared across multiple instances of novel categories. To achieve competitive results, they need to use depth information and refine the pose with an ICP algorithm.

Our approach is related to [START_REF] Pitteri | CorNet: Generic 3D Corners for 6D Pose Estimation of New Objects without Retraining[END_REF], but considers any 3D location on the objects to get matches, not only corners. We compare against [START_REF] Pitteri | CorNet: Generic 3D Corners for 6D Pose Estimation of New Objects without Retraining[END_REF] and [START_REF] Sundermeyer | Multi-path Learning for Object Pose Estimation Across Domains[END_REF] in the experimental section.

Method

We describe our approach in this section. We first explain how we compute the local surface embeddings and how we obtain correspondences between the CAD models and the images. We then describe our pose estimation algorithm.

Local Surface Embeddings

To match new images with CAD models, we rely on embeddings of the local surfaces of the objects. To be able to match these embeddings under unknown poses, they need to be translation invariant and rotation invariant. Achieving translation invariance is straightforward, since we consider the local geometry centered on 3D points. Achieving rotation invariance is more subtle, especially because of ambiguities arising in practice with symmetrical objects. This is illustrated in Figure 2(b): We need to compute the same embeddings for local geometries that are similar up to a 3D rotation.

More exactly, given a 3D point P on the surface of an object, we define the local geometry as the set of 3D points M n in a spherical neighborhood centered on P and of radius r. In practice, on T-LESS, we use r = 3cm. To compute a rotation-invariant embedding, we transform these points from the object coordinate system to a local patch coordinate system using a rotation matrix computed from the decomposition of the covariance matrix of the 3D points M n after centering on P [START_REF] Eggert | Estimating 3D Rigid Body Transformations: A Comparison of Four Major Algorithms[END_REF]: where v n = (M n -P) using a Singular Value Decomposition (SVD):

C = n v n • v n , (1)
C = L ΣR . (2)
R is an orthogonal matrix, but not necessarily a rotation matrix, and small differences in the local geometry can result in very different values for R. We therefore apply a transformation to R to obtain a new matrix R so that R is a suitable rotation matrix. It can be checked that applying R to the v i vectors will achieve rotation invariance for the local surface embeddings.

Let's denote by r 1 , r 2 , and r 3 the rows of R, and by r1 , r2 , and r3 the rows of R. Applying R to the normal n of the object's surface at P yields a 3-vector R • n close to either [0, 0, 1] or [0, 0, -1] , depending on the orientation of R selected for the SVD. For normalisation, we choose that R • n should always be closer to [0, 0, 1] . We therefore compute o = r 3 • n. If o is positive, we take r3 = r 3 , otherwise we take r3 = -r 3 . As a result, R • n is always closer to [0, 0, 1] that to [0, 0, -1] . Finally, we take r1 = r 1 and r2 = -r 1 ∧ r3 , where ∧ denotes the cross-product, which ensures that R is a rotation matrix.

We explain now how we define the local surface embeddings. For our experiments, we use the local moments of the local 3D points for simplicity but any other embeddings such as [START_REF] Deng | PPF-FoldNet: Unsupervised Learning of Rotation Invariant 3D Local Descriptors[END_REF] could also work. Let us denote by [x n , y n , z n ] the vectors Rv n , then local surface embeddings can be computed as:

LSE i,j,k (P) = n w n x i n y j n z k n , (3)
where

w n = exp(-v n 2 /σ 2
) is a weight associated to each point based on its distance from P (we use σ = 5 in practice) and i, j, k are exponents in the range

Input image i = 0, j = 2, k = 1 i = 0, j = 2 , k = 0 i = 0, j = 2, k = 2 i = 2, j = 0, k = 1 i = 2, j = 0, k = 0 i = 0, j = 0, k = 2 i = 2, j = 0, k = 2
Fig. 3. Visualization of some LSEs coordinates for an example image.

[0, 1, 2]. Theoretically it is possible to take all the combinations of exponents but we empirically found that the most discriminative values are computed using: i ∈ {0, 2}, j ∈ {0, 2}, k ∈ {0, 1, 2}, which gives 11 values for the full vector LSE(P) as taking i = j = k = 0 gives a constant value and is not useful. Finally, we normalize the values of LSE(P) to zero mean and unit variance so they have similar ranges. Figure 3 displays the embeddings for an example image.

Predicting the local surface embeddings for new images

Given a new CAD model, it is trivial to compute the local surface embeddings on points on its surface. Given a new input image, we would like to also compute the embeddings for the object points visible in this image. We use a deep network to perform this task. To do so, we create a training set by generating many synthetic images of known objects under various poses. We also compute the LSEs for all the pixels corresponding to a 3D point of one of the objects. We then train a U-Net-like architecture [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF] to predict the LSEs given a color image. More details on the architecture and its training are provided in the experimental section. This training is done once, on known objects, but because the embeddings depend only on the local geometry, the network generalizes well to new objects, as shown in Figure 4.

Pose Estimation Algorithm

The pseudocode for our detection and pose estimation algorithm is given as Algorithm 1. Given a new image, we compute the LSEs for each of its pixels using the network described in Section 3.2 and establish correspondences between image pixels and object 3D points. However, the number of possible correspondences can quickly become very large, which would yield a combinatorial explosion in the number of set of correspondences needed in RANSAC. We control the complexity in two different ways. First, we focus on the most discriminative embeddings. Points on planar regions are very common and would generate many correspondences. We discard them by thresholding the embedding values: Points with very low absolute embedding values for the LSEs are removed. Figure 5 (a) shows how pixels are selected.

Second, we force the correspondences in each sample considered by RANSAC to belong to the same object. Even when objects are not known in advance, it is possible to segment them: To do so, we use Mask-RCNN [START_REF] He | Mask R-CNN[END_REF] to predict the masks of the objects. We fine-tuned it on our synthetic images already used for training the LSE predictor, as described in Section 3.2 in a class-agnostic way since we want to generalize to new objects. We found out that it works very well with new objects even for cluttered backgrounds, as shown in Figure 6. This also allows us to easily discard pixels on the background from the possible correspondences.

We match the embeddings predicted for the pixels of the input image against the embeddings computed for the 3D points on the CAD model based on their Euclidean distances. In our implementation, we use the FLANN library [START_REF] Muja | Fast Approximate Nearest Neighbors with Automatic Algorithm Configuration[END_REF] to efficiently get the k nearest neighbors of a query embedding. In practice, we use k = 100. This usually returns points in several clusters, as close points tend to have similar embeddings. We therefore go through the list of nearest neighbors sorted by increasing distances. We keep the first 3D point and remove from the Algorithm 1 Pose estimation algorithm. list the other points that are also close to this point, and we iterate. This provides for each pixel a list of potential corresponding 3D points separated from each other.

When working on industrial objects like the ones in T-LESS, some pixels can be matched with several 3D points, as shown in Figure 5(b), because of the rotation invariance property of the local LSEs and the similarities between local parts of different objects.

We finally use LO-RANSAC (Locally Optimized RANSAC) [START_REF] Chum | Locally Optimized RANSAC[END_REF] with a PnP algorithm (we use [START_REF] Lepetit | EPnP: An Accurate o(n) Solution to the PnP Problem[END_REF] followed by a Levenberg-Marquardt optimization) to compute the poses of the visible objects. We take random n ∈ [6; 10] for each RANSAC sample. At each iteration, we compute a score for the predicted pose as a weighted sum of the Intersection-over-Union between the mask from Mask- RCNN and the mask obtained by rendering the model under the estimated pose, and the Euclidean distances between the predicted LSEs and the LSEs for the CAD model after reprojection. We keep the pose with the largest score and refine it using all the inlier correspondences to obtain the final 6D pose.

Evaluation

In this section, we present and discuss the results of our pose estimation algorithm on the challenging T-LESS dataset [START_REF] Hodan | T-LESS: An RGB-D Dataset for 6D Pose Estimation of Texture-less Objects[END_REF], made of texture-less, ambiguous and symmetrical objects with no disciminative parts. It is well representative of the problems encountered in industrial context.

Dataset

To train our LSE prediction network, we generate synthetic images using the CAD models provided with T-LESS for a subset of objects in this dataset. The exact subset depends on the experiment, and we will detail them below. Similar to the BlenderProc4BOP [START_REF] Denninger | Blenderproc[END_REF] introduced in the BOP challenge [START_REF] Hodan | BOP: Benchmark for 6D Object Pose Estimation[END_REF], these images are created with Cycles, a photorealistic rendering engine of the open source software Blender by randomly placing the training objects in a simple scene made of a plane randomly textured and randomly lighted. We used both these synthetics and real images for training the network combined with data augmentation to take care of the domain gap between our synthetic images and the real test images. More specifically, we use 15K synthetic images and ∼ 7K real images-all the training images provided by T-LESS for the objects that are used for training the LSE prediction and Mask-RCNN. To create the ground truth embeddings, for each training image, we backproject the pixels lying on the objects to obtain their corresponding 3D points and their LSEs. Neither the embedding prediction network nor Mask-RCNN see the test objects during training.

LSE prediction network architecture and training

The architecture of the network predicting the LSEs for a given input image is a standard U-Net-like [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF] encoder-decoder convolutional neural network taking a 720 × 540 RGB image as input. The encoder part is a 12-layer ResNet-like [START_REF] He | Deep Residual Learning for Image Recognition[END_REF] architecture; the decoder upsample the feature maps up to the original size using bilinear interpolations followed by convolutional layers. We train the network with the Adam optimizer and a learning rate set to 10 -4 . We also use batch normalization to ensure good convergence of the model. Finally, the batch size is set to 8 and we train the network for 150 epochs.

Metrics

We evaluate our method using several metrics from the literature. Analogously to other related papers [START_REF] Zakharov | DPOD: Dense 6D Pose Object Detector and Refiner[END_REF][START_REF] Rad | BB8: A Scalable, Accurate, Robust to Partial Occlusion Method for Predicting the 3D Poses of Challenging Objects Without Using Depth[END_REF][START_REF] Pitteri | CorNet: Generic 3D Corners for 6D Pose Estimation of New Objects without Retraining[END_REF][START_REF] Tekin | Real-Time Seamless Single Shot 6D Object Pose Prediction[END_REF], we consider the percentage of correctly predicted poses for each sequence and each object of interest, where a pose is considered correct based on the ADD metric (or the ADI metric for symmetrical objects) [START_REF] Hinterstoisser | Gradient Response Maps for Real-Time Detection of Textureless Objects[END_REF]. This metric is based on the average distance in 3D between the model points after applying the ground truth pose and the estimated one. A pose is considered correct if the distance is less than 10% of the object's diameter. Following the BOP benchmark [START_REF] Hodan | BOP: Benchmark for 6D Object Pose Estimation[END_REF], we also report the Visible Surface Discrepancy (VSD) metric. The VSD metric evaluates the pose error in a way that is invariant to the pose ambiguities due to object symmetries. It is computed from the distance between the estimated and ground truth visible object surfaces in the following way:

err VSD ( Ŝ, S, S I , V , V , τ ) = Mean p∈ V V 0, if p ∈ V V ∧ | Ŝ(p) -S(p)| < τ 1, otherwise (4)
where Ŝ and S are distance maps obtained by rendering the object model in the estimated and ground-truth poses respectively. The distance maps are compared [START_REF] Pitteri | CorNet: Generic 3D Corners for 6D Pose Estimation of New Objects without Retraining[END_REF]. We report results also for Objects #9 in Scenes #11 and #12 and for Object #26 in Scene #15 even though [START_REF] Pitteri | CorNet: Generic 3D Corners for 6D Pose Estimation of New Objects without Retraining[END_REF] does not. See text for details.

with the distance map S I of the test image I to obtain the visibility masks V and V , i.e. the sets of pixels where the object model is visible in image I. We report the mean VSD recall of 6D object poses at err VSD < 0.3 with tolerance τ = 20mm and > 10% object visibility.

Results

The complexity of the test scenes in T-LESS varies from several isolated objects on a clean background to very challenging ones with multiple instances of several objects with a high amount of occlusions and clutters. We compare our method against the two works that already consider 6D object detection and pose estimation for unknown objects on T-LESS, CorNet [START_REF] Pitteri | CorNet: Generic 3D Corners for 6D Pose Estimation of New Objects without Retraining[END_REF] and the MP-Encoder method of [START_REF] Sundermeyer | Multi-path Learning for Object Pose Estimation Across Domains[END_REF]. As the codes for these two methods are not available at the time of writing, we use the same protocols as in these works and report the results from the papers.

Comparison with CorNet [START_REF] Pitteri | CorNet: Generic 3D Corners for 6D Pose Estimation of New Objects without Retraining[END_REF]. We use here the same protocol as in [START_REF] Pitteri | CorNet: Generic 3D Corners for 6D Pose Estimation of New Objects without Retraining[END_REF]: We split the objects from T-LESS into two sets: One set of known objects (#6, #19, #25, #27, and #28) and one set of unknown objects (#7, #8, #20, #26, and #29), and we compare the 3D detection and pose estimation performance of our method and CorNet for the unknown objects in T-LESS test scenes #02, #03, #04, #06, #08, #10, #11, #12, #13, #14, and #15. We use synthetic images of the known objects for training the LSE prediction network. Results are reported in Table 1. We outperform CorNet on most of the objects, except on objects #7 and #8. This is because these objects have some 3D points with local geometry very different from the training objects (at the connections of the different parts). As a result, the predicted LSEs for these parts are not very accurate, generating wrong matches. Figured 8 shows some qualitative results for the unknown objects in the test images.

Comparison with MP-Encoder [START_REF] Sundermeyer | Multi-path Learning for Object Pose Estimation Across Domains[END_REF]. We use here the same protocol as in [START_REF] Sundermeyer | Multi-path Learning for Object Pose Estimation Across Domains[END_REF]:

The objects from T-LESS are split into a set of known objects (#1-#18) and one set of unknown objects (#19-#30), and we compare the 3D detection and pose estimation performance of our method and MP-Encoder for the unknown objects in T-LESS test scenes following the BOP benchmark [START_REF] Hodan | BOP: Benchmark for 6D Object Pose Estimation[END_REF]. We use synthetic images of the known objects for training the LSE prediction network. Note that VSD recall MP-Encoder [START_REF] Sundermeyer | Multi-path Learning for Object Pose Estimation Across Domains[END_REF] 20.53 Ours 23.27 Table 2. Mean Visible Surface Discrepancy (VSD) recall using the protocol of [START_REF] Sundermeyer | Multi-path Learning for Object Pose Estimation Across Domains[END_REF]. This metric evaluates the pose error in a way that is invariant to the pose ambiguities due to object symmetries. It is computed from the distance between the estimated and ground truth visible object surfaces.

we report here the number of Table 3 from the [START_REF] Sundermeyer | Multi-path Learning for Object Pose Estimation Across Domains[END_REF] paper as the other reported results assume that the ground truth bounding boxes, the ground truth masks, or depth information are provided. Results are reported in Table 2. While our method performs slightly better, the performances are close and tells us that both methods are promising. The main difference is that the MP-Encoder relies on an embedding completely learnt by a network while our method incorporates some geometrical meaning that makes our approach more appealing for industrial purposes. Note that, as shown in Fig. 7, our LSE network can handle objects with rounded shape without being limited to objects with prominent corners as [START_REF] Pitteri | CorNet: Generic 3D Corners for 6D Pose Estimation of New Objects without Retraining[END_REF].

Robustness to Texture

Our focus is on untextured objects, as industrial objects typically do not exhibit textures like the T-LESS objects. However, our LSEs can be predicted for textured objects as well. To show this, we retrained our LSE prediction network on synthetic images of the T-LESS objects rendered with random textures. The LSEs prediction for some test images are shown in the bottom row of Fig. 7.

Conclusion

We introduced a novel approach for the detection and the 3D pose estimation of industrial objects in color images. It only requires the CAD models of the objects and no retraining is needed for new objects. We introduce a new type of embedding capturing the local geometry of the 3D points lying on the object surface and we train a network to predict these embeddings per pixel for images of new objects. From these local surface embeddings, we establish correspondences and obtain the pose with a PnP+RANSAC algorithm. Describing the local geometries of the objects allows to generalize to new categories and the rotation invariance of our embeddings makes the method able to solve typical ambiguities that raise with industrial and symmetrical objects. We believe that using local and rotation invariance descriptors is the key to solve the 6D pose of new textureless objects from color images.

Fig. 2 .

 2 Fig. 2. (a): Computation of the LSEs for a given point P on a CAD model. We transform the 3D points in the neighborhood of P into a rotation-invariant local system and weight them before computing their moments. (b): Visualization of the rotationinvariance property on different parts of the same object. Similar local geometries yield similar LSEs. Through this paper, we represent the LSEs using only their 3 first values mapped to the red, green, blue channels except for Figure 3 that shows all the values.

Fig. 4 .

 4 Fig. 4. Generalization of the LSE prediction network to new objects. (a) Input RGB image with objects seen during the training of the network (blue boundaries) and new objects (red boundaries). The LSE predictions (c) are close to the LSE Ground truth (b) for both the known and new objects.

1 :

 1 C ← CAD models for the new objects 2: E(C) ← LSECAD(C), the LSEs of 3D points for each CAD model C 3: I ← input image 4: F ← LSE pred (I), the predicted LSEs for the input image 5: O ← Mask-RCNN(I), the masks predicted by Mask-RCNN 6: M ← {mi}i, the set of 2D-3D matches based on E(C) and F. Each match mi is made of an image location p and 3D points on the CAD models: (p, [P1, P2, ...

Fig. 5 .

 5 Fig. 5. Focusing on the most discriminative pixels. (a): In green, pixels with discriminative LSEs. We only consider them for correspondences with the CAD models. (b): A pixel can be matched with multiple 3D points on symmetrical objects because of the rotation invariance property of the LSEs.

Fig. 6 .

 6 Fig. 6. Generalization of Mask-RCNN to unknown objects. We train Mask-RCNN in a class-agnostic way on a set of known objects. It generalizes well to new objects, and we use these masks to constrain the pose estimation. Note that we use masks of different colors for visualization only. Mask-RCNN cannot identify the new objects individually as it was not trained on them, it can only detect objects in a class-agnostic way.

Fig. 7 .

 7 Fig. 7. Top row: Image with objects with rounded shapes (a). Ground truth and predicted LSEs (b) and (c). Bottom row: Image with random textures applied to some T-LESS objects (d). Ground truth and predicted LSEs (e) and (f).

Fig. 8 .

 8 Fig. 8. Qualitative results on the unknown objects of the test scenes from T-LESS. Green bounding boxes denote ground truth poses, blue bounding boxes correspond to our predicted poses.

Table 1 .

 1 Our quantitative results on T-LESS test Scenes #02, #03, #04, #06, #08, #10, #11, #12, #13, #14, #15 as used in

	Scene 02 03 04 04 06 08 10 11 11 12 12 13 15 15 14
	Obj	7	8	26	8	7	20 20	8	9	7	9	20 29 26 20	Avg
	[13] 68.3 57.9 28.1 21.2 36.8 10.0 27.8 58.8 -23.1 -26.6 48.0 -10.0 34.7(±18.5)
	Ours 61.0 44.1 55.6 39.1 44.8 38.2 38.3 40.8 46.1 41.2 45.8 39.5 77.0 63.6 24.9 46.7 (±12.0)


Danièle Fournier-Prunaret

Pascal Chargé
email: pascal.charge@univ-nantes.fr

Route to chaos in a circuit modeled by a 1-dimensional piecewise linear map

Keywords: piecewise linear bimodal map, chaos generator, border collision bifurcation

A very simple hybrid circuit proposed as chaos generator is studied. It is modeled using a 3-slopes piecewise linear map defined on [0,1] and depending upon three parameters. The parameter space is investigated in order to classify regions of existence of stable periodic orbits and regions associated with chaotic behaviors. Bifurcation curves are obtained numerically and analytically. Border collision bifurcations and homoclinic bifurcations occurring in cyclical chaotic regions leading to chaos in one-piece are detected.

Introduction

Many circuits including switches have been considered during the last decade. Such systems are of interest for two reasons: first, they belong to the class of hybrid systems that have attracted much interest those last ten years, secondly, they permit to obtain chaos in a very easy way. Hybrid systems have been much considered in many kinds of applications during the last decade. Such systems appear for example in Electronics and Electrotechnics and the understanding of their behaviour is of great interest. Hybrid systems evolve in continuous time, but it is possible to model them using discrete time maps, by introducing a discretization similar to the building of a Poincaré map. This is the case for the circuit we consider in this paper. Concerning the second point, chaotic signals have appeared as very useful in many kinds of applications in the last twenty years, particularly in telecommunications and image processing. Indeed, for some kind of applications, it is necessary to consider robust chaos [START_REF] Banerjee | Robust chaos[END_REF], which can endure, even if parameter values are slightly modified. A way to obtain robust chaotic signals is to consider systems where border collision bifurcations appear [START_REF] Banerjee | Bifurcations in 2D piecewise smooth maps -theory and applications in switching circuits[END_REF][START_REF] Mandal | An integrated CMOS chaos generator[END_REF][START_REF] Kousaka | Analysis of Border-Collision Bifurcation in a Simple Circuit[END_REF][START_REF] Kousaka | Experimental realization of controlling chaos in the periodically switched nonlinear circuit[END_REF][START_REF] Sushko | Degenerate bifurcations and border collisions in piecewise smooth 1D and 2D maps[END_REF], that generally constitute a class of hybrid systems. Chaotic generators can also be obtained using continuous time models as systems based on the Chua's circuit (analogical circuit), while others are discrete time systems which directly iterate a chaotic map (digital circuit). The problem with these systems is that chaos is not necessarily as robust.

In this paper, we propose a chaos generator obtained from a simple RC circuit including switches managed using a clock (impulse waveform) and the charging/discharging of the capacitor. We have previously proposed such a kind of chaos generator, where the model was a two-dimensional nonlinear map [START_REF] Fournier-Prunaret | Border collision bifurcations and chaotic sets in a two-dimensional piecewise linear map[END_REF][START_REF] Gardini | Border collision bifurcations in a twodimensional PWS map from a simple circuit[END_REF][START_REF] Chargé | Two-dimensional piecewise defined maps realized with a simple switching circuit[END_REF]. Our aim in considering a one-dimensional case is to analyze rigorously the bifurcations and to prove the existence of chaos in an even more simple circuit. The section 2 is devoted to the description of the circuit and its modeling. The model under a one-dimensional piecewise linear map is given in section 3. The dynamical behaviour of the map, with the study of bifurcations permitting to obtain periodic orbits and chaos is explained for some parameter values in section 4.

Description of the circuit

The 1D proposed chaos generator is a quite simple circuit given in Fig. 1, very similar to those discussed in [START_REF] Mandal | An integrated CMOS chaos generator[END_REF][START_REF] Kousaka | Analysis of Border-Collision Bifurcation in a Simple Circuit[END_REF]. The analog state variable is V (t), the voltage across the capacitor. We can see that the switch position is ruled by the logic subsystem output p. In the same time inputs of the logic subsystem are provided by comparison of the analog state variable V (t) with constant voltages V U and V D . Analog and logic subsystems then form a hybrid system. Behavior of the logic subsystem can be described by the following finite state diagram. At any state transition, the condition is given by the boolean expression above the line, whereas the action at the transition is indicated below the same line. As an example, the transition from the state C 1 to the state D occurs when the expression ((t<T)&( V (t) ≥ V U )) is true. At this transition instant, the system output p is set to the logic value 1. Actions are then described with variables allocations. The variable t denotes the time. When the time reaches the value t = T , a state transition is expected and the time variable is reset (allocation t := 0;).

From this diagram observation, it comes that the logic subsystem is an asynchronous sequential system, that is periodically forced to return to the state C 1 by an external periodic event. In [START_REF] Fournier-Prunaret | Bifurcation structure in a circuit modeled by a 1dimensional piecewise linear map[END_REF][START_REF] Fournier-Prunaret | Chaos generation from 1D or 2D circuits including switches[END_REF], we proposed a very simple implementation of this logic subsystem based on a few R-S latches and logic operators. The periodic event is then realized thanks to a pulse generator.

A switch occurs at every T -periodic event or when the state variable V (t) reaches the value of V D or V U . The capacitor charges (p=0) when V (t) reaches V D or discharges (p=1) when V (t) reaches V U . Let us call V the reference voltage, V A the borderline value corresponding to V (t)=V U , V B the borderline value corresponding to V (t)=V D and τ = RC. Let us call V the reference voltage, namely the constant voltage supply in the circuit. Then, we can propose a model for our system.

Modeling

By considering the classical equations of a RC circuit, we obtain

V A = V -(V -V U )e T τ , V B = V -V D V U (V -V U )e T τ .
Let us call V n , the state value at each time t = T when the time is initialized again, we obtain the following relations:

0 <V n ≤ V A ,V n+1 = V -(V -V n )e -T τ V A <V n <V B ,V n+1 =(V -V n ) V U V -V U e -T τ V B <V n ≤ 1 ,V n+1 = V -(V -V n ) V U V D V -V D V -V U e -T τ (1)
We normalize the state variables and the parameters in order to deal with dimensionless values and a normalized phase space equal to [0, 1]:

x n = V n V ∈ [0, 1] ,x n+1 = V n+1 V ∈ [0, 1] β = V U V ∈]0, 1[ , 0 <m= V D V U < 1 ,δ = e -T τ ∈]0, 1] x b = V B V ,x a = V A V (2)
The following switching values in [0, 1] are:

x a =1-1-β δ x b =1-m(1-β) δ (3)
Finally, the circuit is modeled by the map x n+1 = F (x n ) (cf. Fig. 3), which is defined as follows:

if x n ∈ [0,x a [,F (x n )=F 1 (x n )=1-(1 -x n )δ if x n ∈ [x a ,x b [,F (x n )=F 2 (x n )=(1-x n ) δβ 1-β if x n ∈ [x b , 1],F (x n )=F 3 (x n )=1-(1 -x n ) δ m 1-βm 1-β (4)
It is easy to see that the map is well defined as F is continuous and maps the interval [0, 1] (the phase space of interest) into itself.

Dynamical behaviour of the 1-D model

The map ( 4) is a 1-dimensional piecewise linear map with three pieces. In this section, we study the fixed points, order k periodic orbits and their bifurcations. We also put in evidence the homoclinic bifurcations giving rise to chaotic behaviour. First, let us give the slopes of the map, depending where is located the point in the interval [0, 1]. These values will be useful for the rest of our study.

if x ∈ [0,x a [,p 1 = δ>0 if x ∈ [x a ,x b [,p 2 = -δβ 1-β < 0 if x ∈ [x b , 1],p 3 = δ m 1-βm 1-β > 0 (5)

Bifurcations of fixed points and order 2 periodic orbits

This study has been previously done in [START_REF] Fournier-Prunaret | Border collision bifurcations and chaotic sets in a two-dimensional piecewise linear map[END_REF][START_REF] Fournier-Prunaret | Chaos generation from 1D or 2D circuits including switches[END_REF]. We recall the results. Two fixed points X * 1 and X * 2 exist and are given by:

X * 1 = δβ 1-β+δβ ∈ [x a ,x b ] X * 2 =1 (6)
X * 1 is stable when the slope p 2 = -δβ 1-β ∈] -1, 0[, that means β< 1 1+δ . X * 2 is stable when the slope p 3 = δ m 1-βm 1-β ∈]0, 1[, that means β< m-δ
m(1-δ) and m>δ . So, we can define the bifurcation curves related to fixed points bifurcations:

FB1a : β = 1 1+δ ,m < δ, FB1b : β = 1 1+δ ,m > δ FB1c : β = m-δ m(1-δ) ,m > δ (7)
These curves correspond to degenerate flip bifurcations, FB1a corresponds to the appearance of a stable order 2 cycle after the flip bifurcation of X * 1 , FB1b corresponds to the appearance of an order 2 cyclic chaotic attractor (C 1 ,C 2 ), after X * 1 has changed its stability (see Fig. 6); for instance, the order 2 periodic orbit (Y 1 ,Y 2 ), which appears at the flip bifurcation FB1b, undergoes in the same time a border collision bifurcation (Y 2 is merging with x b ) and becomes unstable (see Fig. 5). FB1c corresponds to a border collision bifurcation for X * 1 , which disappears when merging with x b (β below FB1b in the plane (m, β), δ being fixed); at the same bifurcation value, X * 2 becomes a stable fixed point (β below FB1b in the plane (m, β), δ being fixed). The curve FB1c (β above FB1b) also corresponds to a degenerate bifurcation for X * 2 with the appearance of an order 2 cyclic chaotic attractor, after X * 2 has become unstable. Then, the order 2 cyclic chaotic attractor after an homoclinic bifurcation, gives rise to an one-piece chaotic attractor (see Fig. 7). Such chaos can be considered as robust in the sense of remaining chaotic even if the parameters or the initial conditions are slightly changed [START_REF] Banerjee | Robust chaos[END_REF].

When there exists a stable order 2 cycle, this cycle can undergo a degenerate flip bifurcation, which gives rise to an order 4 cyclic chaotic attractor. The bifurcation curve is given by the following equation, corresponding to the slope of F 2 that becomes equal to -1( s e eF i g .4 ) :

FB2a : β = 1 1+δ 2 (8)
FB2a corresponds to the slope of F 2 equal to p 1 p 2 (F 2 = F 1 F 2 ). It is impossible to obtain a flip bifurcation for F 2 = F 1 F 3 , because p 1 p 3 can never be negative.

Other bifurcation curves correspond to border collision bifurcations. It is possible to have F 2 (x a )= x a or F 2 (x b )=x b . The corresponding curves are respectively denoted BC2a and BC2b. Their equations are given by:

BC2a : β = 1 1+δ BC2b : β = m-δ 2 m(1-δ 2 ) ,m > δ 2 (9)

Bifurcations of order k periodic orbits

Using the same method as for fixed points and order 2 cycles, we can obtain the bifurcation curves for order k cycles. In this paragraph, we do not intend to study and write the bifurcation curves equations for all the categories of order k cycles that can be defined, but only for those that we have obtained in numerical simulations. Previous studies concerning bimodal piecewise linear maps have been done in a more general way [START_REF] Panchuk | Three segmented piecewise linear map[END_REF][START_REF] Yu | On period-adding sequences of attracting cycles in piecewise linear maps[END_REF]. In this work, we only intend to give the analytical forms of bifurcation curves equations with the aim of a practical use for our model.

First, we give the bifurcation curves corresponding to the degenerate flip bifurcations. Indeed, if we consider an order k cycle, each point X of this cycle verifies F k (X)=X and F k is obtained by a combination of its three determinations F 1 , F 2 and F 3 . We can write that, in F k , F 1 appears i times, F 2 j times and F 3 l times, with i + j + l = k. So regarding the degenerate flip bifurcation, it is obtained by considering the three slopes p 1 , p 2 and p 3 such that p i 1 p j 2 p l 3 = -1. This curve is denoted FBk ijl and is given by the following equation:

FBk ijl : δ i ( δβ β-1 ) j ( δ m 1-βm 1-β ) l = -1,i+ j + l = k, (10)
which can also be written:

δ k = (1-β) j+l m l (1-βm) l β j ,i+ j + l = k (11)
All the degenerate flip bifurcation curves are known analytically and can be plotted in the parameter planes (β, δ), m being fixed or (m, δ), β being fixed.

It is also possible to find the border collision bifurcation curves. The collision can occur with any of both points x a or x b . So, we will have to look the parameter values for which one has F k (x a )=x a or F k (x b )=x b . Anyhow, we have to look at the way of exchanging the points of cycles by F .

The simplest case occurs when an order k cycle has its points exchanged k -1 times by F 1 and one time by F 2 or k -1 times by F 1 and one time by F 3 . It is possible to obtain the border collision bifurcation curves, which are denoted BCka if the collision occurs with x a , that means F k (x a )=x a , or BCkb if the collision occurs with x b , that means F k (x b )=x b . The corresponding equations are:

BCka : β = 1-δ k-1 1-δ k ,m < δ BCkb : β = m-δ 2 m(1-δ 2 ) ,m > δ 2 (12)
On Fig. 4 are plotted the bifurcation curves of an order 3 cycle whose exchange of points is given by F 2 1 F 2 . Figure 8 gives the areas of stability of order k periodic orbits, k =1, ...5, it is very easy to observe that these areas are limited by the bifurcation curves obtained in Fig. 4. An enlargment of these figures is provided in Fig. 9 and Fig. 10, which permits to put in evidence order 6 and 7 cycles. Bifurcation curves of order k cycles, k =1 , ...7 are plotted on Fig. 10. All these cycles are cycles of

F k = F k-1 1 F 2 .
Let us remark that we obtain a period-adding sequences of periodic orbits. Between two stability areas of periodic orbits, chaos appears (k-pieces chaotic attractor), which undergoes a succession of homoclinic bifurcations giving rise to an one-piece chaotic attractor.

On Fig. 8, we consider the parameter plane (δ, β), m being fixed and equal to 0.4, we can see the stability areas of some order k cycles, k =1 , ...5. On Fig. 12 and Fig. 13, an enlargment permits to obtain order k cycles, k =1 , ..., 7. Those cycles have their points exchanged as given below (we consider the cycles from left to right in the figures). It is also possible to use Symbolic Dynamics of a bimodal map [START_REF] Lampreia | Symbolic dynamics of bimodal maps[END_REF][START_REF] Martins | Isentropic real cubic maps[END_REF], we have chosen to represent a cycle by beginning with the point at the left of the interval [0, 1] (generally, this point is located inside I 1 and F 1 is applying). The use of symbolic dynamics corresponds to give a code to every order k cycle by considering the location of the points in the intervals I 1 , I 2 or I 3 ; generally, the following rule is used: a point of the cycle is coded by L when it belongs to I 1 , M when it belongs to I 2 and R when it belongs to I 3 (sometimes, the codes A or B can be used if one point of the cycle is merging with x a or x b , in this paper, we will not take this possibility into account). Here are the obtained coding for the cycles of Fig. 12 from left to right in the figure: • order 3 cycle, • order 7 cycle,

• order 6 cycle, F 2 F 4 3 F 1 , LRRRRM • order 5 cycle, F 2 F 3 3 F 1 , LRRRM • order 4 cycle, F 2 F 2 3 F 1 , LRRM • order 7 cycle, F 2 (F 2 3 F 1 ) 2 , LRRLRRM
F 2 F 3 F 1 , LRM • order 5 cycle, F 2 (F 3 F 1 ) 2 , LRLRM • order 7 cycle, F 2 (F 3 F 1 ) 3 , LRLRLRM • order 2 cycle, F 2 F 1 , LM • order 7 cycle, F 1 F 2 F 1 (F 3 F 1 ) 2 , LRLRLM L • order 5 cycle, F 1 F 2 F 1 F 3 F 1 , LRLM L • order 3 cycle, F 2 F 2 1 , LLM
F 2 F 2 1 F 3 F 3 1 , LLLRLLM • order 4 cycle, F 2 F 3 1 , LLLM
All the stability areas in Fig. 12 are bounded by degenerate flip bifurcation curves DF Bk ijl (10) or border collision bifurcation curves. The border collision bifurcation curves must be calculated by considering the exchange of the points of the cycle and the point x a or x b with which there is a collision. The cycles above can be classified in different categories, related to the kind of symbols which represent them or regarding the way of exchanging their points. We can propose the following classification and give the equations of the border collision bifurcation curves for each of them (let us recall that the degenerate flip bifurcation curves are given by [START_REF] Chargé | Two-dimensional piecewise defined maps realized with a simple switching circuit[END_REF] [START_REF] Fournier-Prunaret | Bifurcation structure in a circuit modeled by a 1dimensional piecewise linear map[END_REF]):

• The order 2 cycle (F 2 F 1 , LM ), the order 3 cycle (F 2 F 2 1 , LLM ) and the order 4 cycle (F 2 F 3 1 , LLLM ) have already been studied and their bifurcation curves are given in [START_REF] Chargé | Two-dimensional piecewise defined maps realized with a simple switching circuit[END_REF] [START_REF] Fournier-Prunaret | Bifurcation structure in a circuit modeled by a 1dimensional piecewise linear map[END_REF] [START_REF] Yu | On period-adding sequences of attracting cycles in piecewise linear maps[END_REF].

• The order 6 cycle (F 2 F 4 3 F 1 , LRRRRM ), the order 5 cycle (F 2 F 3 3 F 1 , LRRRM ), the order 4 cycle (F 2 F 2 3 F 1 , LRRM ) and the order 3 cycle (F 2 F 3 F 1 , LRM ) belong to the same class of order k cycle, which can be denoted: F 2 F k-2 3 F 1 , LR...RM , where the symbol R appears (k -2) times. The border collision bifurcation curves are given by:

BCka : δ k (1-βm) k-2 β+(1-β) k-1 m k-2 δ(1-β) k-2 m k-2 =1, BCkb : δ k =( (1-β)m (1-βm) ) k-1 (13)
• The order 5 cycle (F 2 (F 3 F 1 ) 2 , LRLRM ) and the order 7 cycle (F 2 (F 3 F 1 ) 3 , LRLRLRM ) belong to the same category of order k cycle, k odd, which can be denoted:

F 2 (F 3 F 1 ) (k-1) 2
, LRLR...M , where the symbol LR appears ( k-1

2 ) times. The border collision bifurcation curves are given by:

BCka : δ k-2 (1-βm) k-3 2 m k-4 (1-β) k-1 2 -δ k β(1-βm) k-1 2 m k-3 (1-β) k+1 2 =1, BCkb : δ k =( (1-β)m (1-βm) ) k+1 2 (14)
• Concerning the order 7 cycle (F 2 (F 2 3 F 1 ) 2 , LRRLRRM ), we only give its own bifurcation curves:

BCka : -δ 7 β(1-βm) 4 m 4 (1-β) 5 + δ 4 (1-βm) 2 m 2 (1-β) 3 =1, BCkb : δ 7 =( (1-β)m (1-βm) ) 5 (15)
Fig. 14. A chaotic attractor is located in the three intervals I 1 , I 2 and I 3 and is issued from an order 3-cycle whose points are exchanging using the three determinations of F .

• The order 7 cycle (F 1 F 2 F 1 (F 3 F 1 ) 2 , LRLRLM L) and the order 5 cycle (F 1 F 2 F 1 F 3 F 1 , LRLM L) belong to the same category of order k cycle, k odd, which can be written under the following general form:

F 1 F 2 F 1 (F 3 F 1 ) (k-2) 3
, LRLR...LRLM L, where the symbol LR appears k-3 2 times. The border collision bifurcation curves are given by:

BCka : δ 2 (1-β) -δ k β(1-βm) k-3 2 m k-3 2 (1-β) k-1 2 =1, BCkb : δ k =( (1-β)m (1-βm) ) k-3 (16)
• Concerning the order 7 cycle (F 2 F 2 1 F 3 F 3 1 , LLLRLLM ), we only give its own bifurcation curves:

BCka : δ 3 (1-β) -δ 7 β(1-βm) m(1-β) 2 =1, BCkb : δ 7 =( (1-β)m (1-βm) ) 2 (17)
All the bifurcation curves for order k cycles, k =1, ..., 7 are plotted on Fig. 13. They clearly limit the stability areas of order k cycles of Fig. 12. The curves of degenerate flip bifurcations and border collision with x b are given in an explicit form and can be directly plotted, the curves related to the border collision with x a are given under an implicit form and have to be plotted by using a numerical method (Newton-Raphson).

Between two stability regions, there exists a chaotic attractor. The chaotic attractor issued from a degenerate flip bifurcation is a cyclic one, then after a succession of homoclinic bifurcations, it becomes an one-piece chaotic one.

For some parameter values, chaos can be located on the three domains I 1 , I 2 and I 3 (Fig. 14). This chaos is obtained from border collision bifurcations of order k-cycles whose points exchange by F 1 , F 2 and F 3 . This is the case around the stability area of the order 3-cycle (F 2 F 3 F 1 , LRM )(see Fig. 12). In this case, chaos can be considered as robust; indeed its existence domain in the parameter space is large enough.

Conclusions

We have proposed a very simple circuit under the form of a hybrid system, which presents two major interests. First, this circuit is modeled by a one-dimensional piecewise bimodal linear map, such map exhibits very rich signals properties and dynamics. Secondly, such a circuit allows to produce chaotic signals, which can be robust in the sense of remaining chaotic even if the parameters or the initial conditions are slightly changed; such chaotic signals can be very useful for different kinds of applications. The model is very simple and depends on three parameters, we have been able to derive the exact analytical expressions of degenerate flip bifurcation and border collision bifurcation curves in the parameter space for some periodic orbits. The border collision bifurcations give rise to chaos, which can be robust, depending upon parameter values and configurations of periodic orbit sequences in parameter space. We intend to continue our study, in order to have a more general classification of all periodic orbits of the map. This will help to have a good understanding of the dynamical behaviour of the circuit and, depending on the intended application, to propose suitable sets of parameters values.

Fig. 1 .

 1 Fig. 1. The hybrid circuit including the analog and logic subsystems.

Fig. 2 .

 2 Fig. 2. The finite state diagram, which describes the logic subsystem.

Fig. 3 .

 3 Fig. 3. The 3-pieces piecewise linear map.

Fig. 4 .

 4 Fig. 4. In the (m, β) plane, δ =0.4, bifurcation curves of the map F related to fixed points and order 2 and 3 cycles.

Fig. 5 .

 5 Fig. 5. Degenerate flip bifurcation (FB1b) in the plane (x n ,x n+1 )forδ =0.4, m =0.6a n dβ =0.7142857.

Fig. 6 .

 6 Fig. 6. An order 2 cyclic chaotic attractor (C 1 ,C 2 )existsforδ =0.4, m =0.6 and β =0.75. Let us remark that we have plotted the chaotic attractor in the plane (x n ,x n+1 ) for a better visualization.

Fig. 7 .

 7 Fig. 7. An one-piece chaotic attractor C appears after an homoclinic bifurcation. Parameter values are δ =0.4, m =0.6a n dβ =0.8.

Fig. 8 .

 8 Fig. 8. Stability areas of k periodic orbits in the parameter plane (m, β)f o r δ =0.4. They are limited by bifurcation curves obtained in Fig. 4.

Fig. 9 .

 9 Fig. 9. Enlargment of Fig. 8, order k cycles are observed, k =1, ..., 7.

Fig. 10 .

 10 Fig. 10. Bifurcation curves corresponding to the limit of stability regions of Fig. 9 in the parameter plane (m, β)f o rδ =0.4.

Fig. 11 .

 11 Fig. 11. Stability areas of k periodic orbits, k =1 , 2, 3 in the parameter plane (δ, β)f o rm =0.4.

Fig. 12 .

 12 Fig. 12. Enlargment of Fig. 11. Stability areas of k periodic orbits, k = 1, ..., 7 in the parameter plane (δ, β)f o rm =0.4. They are limited by bifurcation curves obtained in Fig. 13.

Fig. 13 .

 13 Fig. 13. Bifurcation curves of k periodic orbits, k =1 , 2, 3, 4, 5, 6, 7i nt h e parameter plane (δ, β)f o rm =0 .4. They limit the stability areas of Fig. 12. In order to clarify the figure, only bifurcation curves for k =1, ..., 5 have been indicated.

Acknowledgments

We thank Professor Laura Gardini for interesting and helpful discussions and advices.


Keywords: Overland flow, soil drainage, drift deposition, direct application river, groundwaters Retaining pesticides : 3 à 99% sorption, biotic/abiotic degradation, sedimentation, dilution, (infiltration) Ditches

Large variability of ditches types

• Which maintenance operations favor (long term?) retention ?

• Can the design of ditch networks increase buffer effect ?

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Goals

Can the contamination of water bodies be limited by managing ditches?

Conclusions and outlook

Ditches as a buffer zone • Confirmation of potentials and limits of the buffer effect of dicth and ditches network. • Elucidate the "pattern-process" link between ditch properties and ditch retention by identifying types of ditches with contrasting physical and retention properties. • Ditch classification is compound dependant.

Maintenance operation as a management lever

• Burning is the maintenance operation that favors the most retention according to the compound. • Management strategies must be reasoned in time and space Ditches must be considerd within their landscape....

• Plant protection pratices are heterogeneous within a catchment. Need to analyze succession of fields (properties, pratices) and ditch. • Ditches fullfill other landscape function (e.g. green corridor). What about the impact of maintenance operations on other ditch functions ?

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  Network model = f(t,x,y) • Reactive transfer model -ditch and network NUMERICAL SIMULATIONS • Retentions variability / ditch • Maintenance effect on retention • Spatial distribution effect PITCH : A simplified pesticide transfer model for ditches network Q i (t), C i (t) type (sand, clay, loam) • Ditch type (with substrates types and abundance, ...) • Hydrogram (Inflow) • Kd soil • Contamination types : • New contamination (input of concentrated water) • Old contamination (Initial pesticide stock in soil) according to the compound Histogram of simulated retentions (18,000 simulations) -Hydrophobic case molecule (diuron), contaminating flood