Andrea De Maio
email: andrea.de-maio@laas.fr

Simon Lacroix

Simultaneously Learning Corrections and Error Models for Geometry-based Visual Odometry Methods

Andrea De Maio 1 and Simon Lacroix 1

Abstract-This paper fosters the idea that deep learning methods can be used to complement classical visual odometry pipelines to improve their accuracy and to associate uncertainty models to their estimations. We show that the biases inherent to the visual odometry process can be faithfully learned and compensated for, and that a learning architecture associated with a probabilistic loss function can jointly estimate a full covariance matrix of the residual errors, defining an error model capturing the heteroscedasticity of the process. Experiments on autonomous driving image sequences assess the possibility to concurrently improve visual odometry and estimate an error associated with its outputs. V ISUAL odometry (VO) is a motion estimation process successfully applied in a wide range of contexts such as autonomous cars or planetary exploration rovers [START_REF] Scaramuzza | Visual odometry [tutorial[END_REF]. Seminal works largely resorted to stereovision. By tracking point features in images, 3D points correspondences are used to recover the motion between two stereoscopic acquisitions. The integration of elementary motions yields an estimate of the robot pose over its course. Continuous work on VO led to a well established processes pipeline, composed of feature extraction, matching, motion estimation, and finally optimization. This scheme has been extended to single camera setups, in which case motions are estimated up to a scale factor, retrieved e.g. by fusing inertial information. Direct methods for VO have also recently been proposed. They bypass the feature extraction process and optimize a photometric error [START_REF] Engel | Direct sparse odometry[END_REF]. These methods overcome the limits of sparse feature-based methods in poorly textured environments or in presence of low quality images (blurred), and they have proven to be on average more accurate.

The advent of convolutional neural networks (CNN) sprouted alternate solutions to VO. The full estimation process can be achieved by deep-learning architectures in an endto-end fashion (see e.g. [START_REF] Konda | Learning visual odometry with a convolutional network[END_REF], [START_REF] Li | Undeepvo: Monocular visual odometry through unsupervised deep learning[END_REF], and especially [START_REF] Wang | End-to-end, sequence-tosequence probabilistic visual odometry through deep neural networks[END_REF] the system has to learn the various information necessary to perform vision-based egomotion estimation, which can be a daunting task for a CNN. This paper builds upon existing work that exploits a CNN to predict corrections to classic stereo VO methods [START_REF] Peretroukhin | Dpc-net: Deep pose correction for visual localization[END_REF], aiming at improving their precision. We argue that complementing classical localization processes with learning-based methods can return better results than delegating the full pose estimation process to a CNN. On the one hand, classical localization processes do not output totally erroneous poses, as a learning approach could when confronted with features unseen in the training set. On the other hand, classical processes can be monitored by some explicit indicators (e.g. number of tracked points in VO, inliers, etc.), so as to detect erroneous cases. Our developments consider that visual odometry estimation errors do not have zero mean, as assessed in e.g. [START_REF] Dubbelman | Bias compensation in visual odometry[END_REF], [START_REF] Peretroukhin | Optimizing camera perspective for stereo visual odometry[END_REF], and provide corrections that improve the precision of VO. Furthermore, at the same time, they produce a full error model for each computed motion estimation (in form of a Gaussian model), akin to [START_REF] Liu | Deep inference for covariance estimation: Learning gaussian noise models for state estimation[END_REF]. This is a significant achievement, as it is generally complex to derive precise error models for geometrical VO methods.

II. PROBLEM STATEMENT AND RELATED WORK

Consider a robot moving in a three dimensional environment. Let x i ∈ R 6 be its pose (3 translations and 3 orientations) at time i in a given reference frame. The actual motion (ground truth) between time instants i and i + 1 is represented by a homogeneous transformation matrix i T i+1 .

A vision-based motion estimator uses raw image data I i ∈ R n to obtain an estimate i Ti+1 . In the VO case, the raw data I i is a pair of monocular or stereoscopic images captured at two different time instants i, i + 1 (i.e. 2 or 4 images). The error e i of VO is:

e i = i T i+1 • i T-1 i+1 (1)
We can create a dataset D = {I i , e i |∀i ∈ [1, d]}, where d is the size of the dataset. The literature provides two different approaches to leveraging this type of dataset. The two approaches enhance a classic VO process with learning to either estimate (i) a motion correction to apply to i T-1 i+1 , thus improving its accuracy [START_REF] Peretroukhin | Dpc-net: Deep pose correction for visual localization[END_REF], or (ii) an error model associated with i T-1 i+1 [START_REF] Liu | Deep inference for covariance estimation: Learning gaussian noise models for state estimation[END_REF], thus allowing its fusion with any other motion or pose estimation process. Alternatively, with the same semantic, substituting errors with actual motion transforms, it is possible to directly learn the motion estimate and associated error [START_REF] Wang | End-to-end, sequence-tosequence probabilistic visual odometry through deep neural networks[END_REF].

A. Directly learning VO and an error model

The work in [START_REF] Wang | End-to-end, sequence-tosequence probabilistic visual odometry through deep neural networks[END_REF] introduces an end-to-end, sequence-tosequence probabilistic visual odometry (ESP-VO) based on a recurrent CNN. ESP-VO outputs both a motion estimate i T-1

i+1 and an associated error. The learned error model is a diagonal covariance matrix, hence not accounting for possible correlations between the different motion dimensions. It is unclear how the probabilistic loss is mixed to the mean squared error of the Euclidean distance between the ground truth and the estimated motions. Finally, the authors make use of a hand-tuned scaling factor to balance rotation and translation. The article presents significant results obtained on a large variety of datasets, with comparisons to state-of-the-art VO schemes. The results show that ESP-VO is a serious alternative to classic schemes, all the more since it also provides variances associated with the estimations. Yet, they are analysed over whole trajectories, which inherit from the random walk effect of motion integration, and as such do not provide thorough statistical insights -e.g. on the satisfaction of the gaussianity of the error model or on the evaluation of the mean loglikelihood.

B. Learning corrections to VO

The work presenting DPC-Net [START_REF] Peretroukhin | Dpc-net: Deep pose correction for visual localization[END_REF] learns an estimate of e i , which is further applied to the VO estimate i T-1 i+1 to improve its precision. The authors introduce an innovative pose regression loss based on the SE(3) geodesic distance modelled with a vector in Lie algebra coordinates. Instead of resorting to a scalar weighting parameter to generate a linear combination of the translation and rotation errors, the proposed distance function naturally balances these two types of errors. The loss takes the following form:

L(ξ) = 1 2 g(ξ) T Σ -1 g(ξ) (2)
where ξ ∈ R 6 is a vector of Lie algebra coordinates estimated by the network, g(ξ) computes the equivalent of (1) in the Lie vector space, and Σ is an empirical average covariance of the estimator pre-computed over the training dataset. Such covariance matrix cannot be used as an uncertainty measure but only as a balancing factor between rotation and translation terms. The paper provides statistically significant results showing that DPC-Net improves a classic feature-based approach, up to the precision of a dense VO approach. In particular, it alleviates biases (e.g. due to calibration errors) and environmental factors. The system interlace low rate corrections with estimates produced by the underlying VO, which processes all the images, using a pose-graph relaxation approach.

C. Learning an error model of VO

Inferring an error model for VO comes to learn the parameters of a predefined distribution to couple VO with uncertainty measures. The work in [START_REF] Liu | Deep inference for covariance estimation: Learning gaussian noise models for state estimation[END_REF] introduces DICE (Deep Inference for Covariance Estimation), which learns the covariance matrix of a VO process as a maximum-likelihood for Gaussian distributions. Nevertheless, it considers the distribution over measurement errors as a zero-mean Gaussian N (0, Σ). Such model is acceptable for unbiased estimators, which unfortunately it is often not the case of VO. Yet, the authors show that their variance estimates are highly correlated with the VO errors, especially in case of difficult environmental conditions, such as large occlusions.

III. SIMULTANEOUSLY LEARNING CORRECTIONS AND

UNCERTAINTIES

To jointly estimate a correction to the VO process and a full error model after having applied the correction, we initially enhance the network output of [START_REF] Liu | Deep inference for covariance estimation: Learning gaussian noise models for state estimation[END_REF] adding a vector µ i ∈ R 6 to the output layer to account for biases in the estimator. Such vector is incorporated in the negative loglikelihood loss that is derived as follows. Given a dataset D of size d, where the observations {e 1 , . . . , e d } T of VO errors are assumed to be independently drawn from a multivariate Gaussian distribution, we can estimate the parameters of the Gaussian as arg max

µ 1:d ,Σ 1:d d i=1 p(e i |µ i , Σ i ) (3)
This is equivalent to minimize the negative log-likelihood arg min

µ 1:d ,Σ 1:d d i=1 -log (p(e i |µ i , Σ i )) (4)
= arg min

µ 1:d ,Σ 1:d d i=1 log |Σ i | + (e i -µ i ) T Σ -1 i (e i -µ i ) (5)
≈ arg min

fµ 1:d ,f Σ 1:d d i=1 log |f Σ (I i )| + (e i -f µ (I i )) T f Σ (I i ) -1 (e i -f µ (I i )) (6)
We split the output of the network in two parts: the mean vector f µ (I i ) and the covariance matrix f Σ (I i ), where f (I i ) represents the full output given a pair of stereo images.

A. Retrieving a valid covariance matrix

To enforce a positive definite covariance matrix we tested two different matrix decompositions.

1) LDL: The first one is the LDL matrix decomposition as in [START_REF] Liu | Deep inference for covariance estimation: Learning gaussian noise models for state estimation[END_REF], to which the reader can refer to for a complete description. The predicted covariance matrix f Σi (I i ) is generated through a vector

α i = [l i , d i ] T , with l i ∈ R ( n 2 2 -n 2 ) and d i ∈ R n . We have then Σ i ≈ f Σ (I i ) = L(l i )D(d i )L(l i ) T (7)
where l i and d i are the vectors containing the elements of the respective L and D matrices. The LDL decomposition is unique and exists as long as the diagonal of D is strictly positive. This can be enforced using the exponential function exp(d i ) on the main diagonal. By doing so, the computation of its log-determinant, i.e. the first term of [START_REF] Peretroukhin | Dpc-net: Deep pose correction for visual localization[END_REF], can be reduced to sum(d i ), that is the sum of the elements in the vector d i . In the second term f Σ (I i ) -1 is replaced by the LDL product.

2) LL: Alternatively, it is possible to resort to the classical Cholesky decomposition. In this case f Σ (I i ) is replaced by the LL * product, where L is a lower triangular matrix and L * is its conjugate transpose. We consider LL T as no complex number is involved in our work. The L matrix can be generated through a vector

l i ∈ R ( n 2 2 + n 2 )
. This decomposition also has nice properties around its log-determinant, as it is easy to prove log

|LL T | = 2 i log(L ii ).
We tested both decompositions and did not experience relevant changes in the network accuracy. Although the presence of the exponential term in the LDL can alter the numerical stability of the loss function, clamping its value mitigates this problem. Therefore, we decided to pursue training using the first method in order to introduce fewer variables that could affect the comparisons in the results (Sec.IV-C).

B. Optimization problem

We incorporate the LDL formulation for the covariance matrix in the negative log-likelihood. Replacing f µ (I i ) with the estimated mean output vector μi we finally obtain

L(I 1:d ) = arg min μ1:d ,α 1:d d i=1 sum(d i ) + (e i -μi ) T (L(l i )D(exp(d i ))L(l i ) T ) -1 (e i -μi ) (8)
Formulating the problem as in [START_REF] Peretroukhin | Optimizing camera perspective for stereo visual odometry[END_REF], the second term of the loss function loosely recalls the formulation of the Lie algebra loss in [START_REF] Engel | Direct sparse odometry[END_REF]. The covariance matrix in this case is learned in relation to the input, capturing the heteroscedastic uncertainty of each sample. The learned covariance matrix acts as in [START_REF] Kendall | Geometric loss functions for camera pose regression with deep learning[END_REF], weighing position and orientation errors. The main difference resides in the nature of the learned uncertainty, homoscedastic vs heteroscedastic: through back-propagation with respect to the input data, [START_REF] Kendall | What uncertainties do we need in bayesian deep learning for computer vision?[END_REF], we aim to learn a heteroscedastic error. Assuming that errors can be drawn from a distribution N (µ i , Σ i ), µ i matches the expected value for the predicted distribution. This corresponds to the desired correction in our case. At the same time, we estimate a covariance matrix Σ i , returning an uncertainty measure relative to each particular input and pose after the predicted correction. Inverting Eq.1, corrections are applied as

i T i+1 ≈ i T V O î+1 • î+1 T corr i+1 (9)
that is the composition of the pose estimate produced by VO and the estimated correction produced by the neural network.

IV. EXPERIMENTS

A. Setup 1) Dataset: We carry out experiments using the KITTI dataset, which provides various sequences of rectified images acquired while driving in urban areas [START_REF] Geiger | Are we ready for autonomous driving? the kitti vision benchmark suite[END_REF]. Depth images are generated using a semi global block matching algorithm to generate and filtered using a weighted least squares filter. We train the network splitting train and validation trajectories in different configurations: for all results shown here we trained using sequences 04 to 10 (which share the same calibration parameters), excluding one for testing and one for validation purposes. Most of our experiments are validated using four sequences (05, 06, 09, 10).

For the initial motion estimates, we use the open-source VO implementation libviso2 [START_REF] Kitt | Visual odometry based on stereo image sequences with ransac-based outlier rejection scheme[END_REF]. It is a feature-based approach, that uses a Kalman Filter in combination with a RANSAC scheme to produce SE(3) estimates using rectified stereoscopic pairs. The estimated corrections, and relative uncertainties, are expressed in camera frame (z axis pointing forward), and the Tait-Bryan angles are defined w.r.t. this reference frame (i.e. yaw encodes rotations around the optical axis z).

2) Evaluation metrics: To evaluate the precision of the motion estimates, we make use of the absolute trajectory error (ATE) metric [START_REF] Zhang | A tutorial on quantitative trajectory evaluation for visual(-inertial) odometry[END_REF], [START_REF] Geiger | Are we ready for autonomous driving? the kitti vision benchmark suite[END_REF]. It is defined as

AT E rot = 1 N N i=1 ||e R i || 2 AT E trans = 1 N N i=1 ||e t i || 2 (10)
where e t i is the difference between the estimated position and the ground truth, and e R i is the rotation angle, in angle-axis representation, of the product R i RT i . ATE comes with the advantage of returning a single scalar to evaluate rotation and translation errors, making it easy to compare them among multiple estimators. At the same time, its main disadvantage lies in the lack of robustness to isolated poor estimations and their relative position in the trajectory [START_REF] Zhang | A tutorial on quantitative trajectory evaluation for visual(-inertial) odometry[END_REF], [START_REF] Geiger | Are we ready for autonomous driving? the kitti vision benchmark suite[END_REF], [START_REF] Kümmerle | On measuring the accuracy of slam algorithms[END_REF]. We use ATE for early architectural choices (section IV-B), but in order to provide a more informative analysis, we use relative error statistics. The idea of relative error is to select segments of predefined lengths of the trajectory and compute the error on all the aligned sub-trajectories. This way, it is possible to obtain statistics on the tracking error (mean and standard deviation) and evaluate it for short or long-term accuracy [START_REF] Peretroukhin | Dpc-net: Deep pose correction for visual localization[END_REF], [START_REF] Zhang | A tutorial on quantitative trajectory evaluation for visual(-inertial) odometry[END_REF]. In our evaluations, we select sub-trajectories of 10, 20, 30, 40 and 50 % of the full trajectory length.

3) Network structures: We initially compared the results produced using the architectures presented in DPC-Net [START_REF] Peretroukhin | Dpc-net: Deep pose correction for visual localization[END_REF] and DICE [START_REF] Liu | Deep inference for covariance estimation: Learning gaussian noise models for state estimation[END_REF]. The first trial has been to adapt the loss defined in Eq. 8 to DPC-Net. We noticed that the mean output vector was still rather constant throughout entire trajectories, regardless of the dataset, and the same behavior was experienced using the loss defined in Eq. 2. Similar tests were conducted with DICE. We experienced problems in reducing the average mean error along all the six dimensions at the same time, and an increase in its standard deviation. Alleging these issues as being caused by the shallow architecture of DICE, we modified its network structure, first removing the max pool layers to preserve spatial information [START_REF] Handa | gvnn: Neural network library for geometric computer vision[END_REF], and achieving dimension reduction by setting the stride to 2 in early layers. We also increased the number of convolutional filters to tackle the estimation of both corrections and error models, adding dropout after each layer to prevent over-fitting. For the rest of the paper, we refer to this network as Deeper-DICE (D-DICE, Table I).

The convolutional layers are followed by two fully connected layer, respectively composed of 256 and 27 output units. In the six-dimensional case, we need 21 values for the LDL decomposition and 6 for the mean vector. We trained the network using both monocular images and stereo images. Additionally, we explored if pairing monocular images to their corresponding disparity maps could be beneficial to the training process, even if the disparities were produced separately from VO.

We used the Adam optimizer with a learning rate of 1e-04 and halted the learning when test and train loss started diverging. All the experiments have been carried out using an Nvidia GeForce RTX 2080 Ti with a batch size of 32. 4) Influence of input data: We tested the proposed architecture and loss using three different types of input data: monocular images, monocular images with associated disparity images, and stereo image pairs. Table II shows the ATEs on two sequences for the three input data: the stereo setup outperforms the two others in both sequences. Ideally, one would expect the best results with monocular images associated with disparity images: indeed, with such data the network does not have to infer depth information. However, with respect to monocular images, this input data does not improve the ATE metric as much as stereo pairs. It is likely that efficiently exploiting disparity images would require a specific convolutional architecture, as their nature differs significantly from intensity images.

Throughout the remainder of the paper, we will provide results with D-DICE obtained using stereo images, as for DPC-Net.

ATE

B. Qualitative Evaluation

To validate the choice of the proposed architecture and loss, we discuss preliminary results obtained exploring different possibilities in this regard.

1) Loss comparison: To show the impact of dynamically estimating a covariance matrix for each transform, we adapt the loss in Eq. 2 to our proposed architecture. Table III compares the ATEs obtained with this loss and the one we proposed in Eq. 8. As we are going to showcase in further results, the negative log-likelihood loss generally outperforms the loss associated with DPC-Net. At the same time, we experience a more stable improvement in translation than in rotation. This is especially true in sequences 09 and 10, where VO presents smaller errors (see figure 3). We associate this behavior to the need for tailored large uncertainties to poor estimations, as opposed to sequences where the necessity of corrections is lower due to a better tracking by VO. Besides, our loss yields the prediction of an error model, which can be used to further reduce trajectory errors, as shown in section IV-C2. 2) Architecture comparison: Selecting an appropriate architecture was driven by a few factors. A first issue was represented by the significantly higher incidence of numerical instability when pairing it with a NLL loss, particularly in the matrix inversion and in the exponential. We also noticed that the DPC-Net system (architecture + loss) tends to output rather constant corrections throughout a whole sequence, certainly compensating biases. This behavior was less prominent when training D-DICE with the same loss. The two systems behave quite differently, as can be seen in Fig. 2, with D-DICE exhibiting more data-dependent corrections. The architectures of DICE and D-DICE are thoroughly compared in Section IV-C2.

C. Quantitative Evaluation

We evaluate D-DICE performances in two schemes. First, we make use of the estimated mean vector as corrections for each image pair (Sec. IV-C1). This approach does not use the uncertainty model and ensures a fair comparison with DPC-Net systems producing corrections outside a probabilistic context. With respect to uncertainty models (Sec. IV-C2), we test the Gaussian assumption and compare log-likelihood values. Additionally, we use the covariance information to further reduce trajectory errors. In a first step, we correct the trajectory as in Eq. 9. Subsequently, we solve a pose-graph optimization problem, weighting errors on the inverse of the covariance matrix and manually adding a ground-truth loop closure.

1) Trajectory correction: We evaluate the accuracy of the correction produced by D-DICE with respect to the chosen baseline VO solution (Fig. 3). We also compare to DPC-Net, trained on the same data split as D-DICE. In these comparisons, both systems estimate transforms with consecutive stereo pairs, and the uncertainty estimates provided by D-DICE are not used. D-DICE outperforms DPC-Net in all the selected sequences (Table IV). While both systems constantly improve libviso2, we noticed a larger improvement in the translation errors especially on the y-axis (upwards) -the vertical drift is one of the most prominent weaknesses of this VO implementation using the KITTI dataset. For more in-depth results, we show mean relative segment errors with associated standard deviations in Table V.

2) Uncertainty estimation: Since we assume a Gaussian error model ∼ N (µ, Σ), we can measure its relevance by checking the fraction of samples that do not respect the following inequality:

µ i -nσ i ≤ e i ≤ µ i + nσ i (11)
where e i is the error along the dimension i after correction, and µ i , σ i are respectively the mean and standard deviation predicted for the input associated with e on the i-th dimension. The parameter n is the number of considered standard deviations. We test against the three-sigma interval (n = 3) that, in case of samples drawn from a normal distribution, should cover approximately 99.7% of the samples. Statistics for a pair of KITTI sequences using D-DICE are displayed in Table VI. On average, more than 99% of samples falls within the three-sigma interval.

A common way of evaluating uncertainties is to inspect the mean log-likelihood value. Even though it is not possible to use it alone as a measure of fitness, it can be used to compare different distribution parameters. The log-likelihood directly describes how well the estimated distributions capture the errors in the dataset and is easily obtained as it is the function minimized by our neural network. Table VII shows benchmarks for the predicted distributions obtained by DICE and D-DICE with different losses. While it is true that a higher log-likelihood does not systematically translate into a reduced trajectory error, we found this trend generally confirmed in our experiments.

Finally, to study the utility of covariances in diminishing tracking errors, we set up an error minimization problem using a typical 3D pose-graph formulation. The optimizer seeks to minimize the function

i∈D e T i,i+1 Σ -1 i e i,i+1 + e T d,1 Σ -1 gt e d,1 (12)
where e i,i+1 is the error function between the nodes i, i + 1 and the corresponding constraint z i,i+1 . A loop closure is manually triggered adding the ground truth measurement z d,1 and a small covariance Σ gt (practically, forcing the last point of the trajectories to fit the ground truth). We use the framework g2o to formulate and solve the problem using a Levenberg-Marquardt optimizer [START_REF] Kümmerle | g 2 o: A general framework for graph optimization[END_REF]. The results of the inclusion of the predicted covariances after pose-graph optimization are summarized in Tables VIII and IX. To provide a complete overview we show D-DICE and DICE errors obtained considering biases or zero-mean Gaussians. Additionally, we use DPC-Net corrections with a fixed covariance. D-DICE consistently shows smaller errors compared to the other methods. It is worth to notice that without the corrections, particularly for long trajectories, the initial problem state given by VO may lock the optimizer in a local minimum. This mainly happens with DICE (Seq 05, 09) but also with D-DICE when a zero-mean normal distribution is considered (Seq 09). Despite using only the latest left image available to VO, DICE is able to have good results on all datasets considered. Still, on average, D-DICE outperforms it. While it is hard to assess the exact reason for the more accurate results, we argue that such improvements are due to two major factors. First of all, our network has access to a larger spectrum of information, the stereo images, which are also available to the VO algorithm. Such information can be exploited to retrieve absolute scale for the translation part. This assumption is backed up by experiencing an average larger standard deviation for the translation part when using monocular images instead of stereo pairs. Additionally, even in the monocular case, having both left images gives the information necessary to extract the robot motion. In this case, the derived uncertainty will not be based only on contextual information (e.g. lack of texture, blurred images) but also on the type/magnitude of motion. Secondly, accounting for non-zero mean allows a further minimization of the negative log-likelihood. D-DICE can rely on estimating both the Gaussian parameters to minimize the same loss.

V. CONCLUSIONS

We presented an insight into the learning of errors in visual odometry. Relying on existing state-of-the-art techniques, we iterated on analysing what type of error and uncertainty can be learned by deep neural networks. We concentrated our efforts on approaches that complement classical visual odometry pipelines in order to ease the work done by the network and exploiting a robust and well established feature-based processes. We demonstrated that it is possible to assimilate the distribution over visual odometry errors to Gaussians, and proceeded to cast the error prediction to a full maximum likelihood for normal distributions case. Knowing that the errors are biased, we have modelled such Gaussians as nonzero mean distributions, showing the beneficial aspects of this Fig. 3: Boxplots of the relative segment errors (top two rows), side view and top view of the trajectories (bottom two rows), for KITTI sequences 05, 06, 09, 10 (left to right). D-DICE (blue) and DPC-Net (green) corrections are used to reduce tracking errors using libviso2 as the baseline estimator (red). The ground truth trajectory is in black. approach compared to works that rely only on the estimation of the covariance matrix. Additionally, we have integrated visual odometry corrections with a more precise error model, inferred thanks to the assumption of biased distributions. To build on this matter, it can result interesting to adopt the same approach with dense estimators which have access to full disparity information. Finally, we plan to explore similar approaches with different perception processes that are yet to be associated with precise error models, e.g. iterative closest points algorithm based on LIDAR scans [START_REF] Pomerleau | A review of point cloud registration algorithms for mobile robotics[END_REF].

  Index Terms-Deep Learning for Visual Perception, Visual-Based Navigation, Localization I. INTRODUCTION

Fig. 1 :

 1 Fig.1: D-DICE produces corrections to classic visual odometry methods in a probabilistic framework. The system generates full covariance matrices that can be used to minimize errors in pose-graph optimization.

Fig. 2 :

 2 Fig. 2: Estimated translation and rotation corrections using DPC-Net and D-DICE over time.

Fig. 4 :

 4 Fig. 4: Results after pose-graph minimization. Boxplots of the relative segment errors (top two rows), side view and top view of the trajectories (bottom two rows), for KITTI sequences 05, 06, 09, 10 (left to right). D-DICE (blue), DPC-Net (green) and DICE (red) use corrections and uncertainty data (fixed variances for DPC-Net). Cyan and yellow are used for D-DICE and DICE versions that only learned error models. The ground truth trajectory is in black.

TABLE I :

 I

	Layer Kernel size Stride Number of channels
	conv1	5x5	2	64
	conv2	5x5	2	128
	conv3	3x3	2	256
	conv4	3x3	2	512
	conv5	3x3	1	1024

D-DICE convolutional architecture.

TABLE II

 II

: Mean Absolute Trajectory Error (ATE) on two validation sets for three different input data types.

TABLE III :

 III Comparison of ATEs obtained with the Lie loss and the negative log-likelihood.

		0.004 0.006			Correction Correction (x) (y) (z) Correction
		0.002
	xyz(m)	-0.002 0
		-0.004
		-0.006
		-0.008
		500	1000	1500 frames	2000	2500

TABLE IV :

 IV Mean Absolute Trajectory Error (ATE) before and after applying corrections to libviso2 VO

TABLE V :

 V Relative segment errors (mean error ± standard deviation, computed on all relative segments for each sequence) before and after applying corrections to libviso2 VO.

		σ	2σ	3σ
	x	85.55%	97.45%	98.64%
	y	83.82%	98.00%	99.36%
	z	76.27%	95.82%	99.27%
	roll	77.55%	96.00%	99.18%
	pitch	85.64%	97.27%	98.91%
	yaw	74.27%	96.64%	99.27%
	mean	80.51%	96.86%	99.10%
		σ	2σ	3σ
	x	87.24%	98.91%	99.63%
	y	91.15%	99.42%	99.67%
	z	79.49%	96.12%	98.94%
	roll	75.72%	94.89%	98.69%
	pitch	72.97%	95.14%	98.76%
	yaw	70.57%	93.55%	98.69%
	mean	79.52%	96.34%	99.07%

TABLE VI :

 VI Sequence 06 (top) and 05 (bottom). Percentages of samples that lie in the various sigma-intervals around the mean. Mean and standard deviations are produced by D-DICE.

	Estimator	05	06	09	10	mean
	DICE N (0, Σ)	41.53	44.42	41.77	42.72	42.61
	D-DICE N (0, Σ)	43.15	44.24	41.77	42.14	42.82
	DICE N (µ, Σ)	42.04	44.67	42.27	41.86	42.71
	D-DICE N (µ, Σ)	44.08	44.67	41.66	41.60	43.00

TABLE VII :

 VII Mean log-likelihood for different network architectures and losses.

TABLE VIII :

 VIII ATE for 5 different estimators. Networks paired with N (0, Σ) do not use bias estimation to minimize the NLL. Values tagged with † point to cases where the optimizer noticeably got stuck in a local minimum.

TABLE IX :

 IX Relative segment errors for 5 different estimators (mean error ± standard deviation, computed on all relative segments for each sequence). Network paired with N (0, Σ) does not use bias estimation to minimize the NLL. Oppositely, N (µ, Σ) results are obtained employing the loss in Eq. 8. Values tagged with † denote cases where the optimizer noticeably got stuck in a local minimum.


Adrien Laffont
email: adrien.laffont@isae-supaero.fr

Théo Delage

Romain Pascaud

Marjorie Grzeskowiak

Cyril Cailhol

Thierry Callegari

Laurent Liard

Olivier Pascal

Jean-Pierre Adam

Design of a Tunable Subwavelength Plasma-Based Design of a Tunable Subwavelength Plasma-Based Resonator for Electrically Small Antenna Applications in L-band

Keywords: antennas, electrically small antennas, epsilon negative material, localized surface plasmon resonance, plasma antennas, plasma devices

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Reducing the size of radio communication systems is a major contemporary challenge. In particular, a lot of effort is put forward to develop electrically small antennas (ESAs). ESAs are defined as antennas with an electrical size ka of less than 1, where k is the free space wave number, and a the minimum radius of a sphere that circumscribes the antenna [START_REF] Wheeler | Fundamental limitations of small antennas[END_REF]. Reducing the electrical size of an antenna has a significant impact on its performances as shown by several theoretical limits [START_REF] Chu | Physical limitations of omni-directional antennas[END_REF]- [START_REF] Harrington | Effect of antenna size on gain, bandwidth, and efficiency[END_REF]. Miniaturization techniques have thus been proposed to develop efficient ESAs [START_REF] Volakis | Small Antennas: Miniaturization Techniques & Applications[END_REF].

Among these miniaturization techniques, the use of a negative permittivity resonator coupled to an electrically small dipole was proposed in 2006 [START_REF] Stuart | Electrically small antenna elements using negative permittivity resonator[END_REF], [START_REF] Ziolkowski | Metamaterial-based efficient electrically small antennas[END_REF]. For instance, the ESA proposed by Stuart et al consists of a subwavelength homogeneous hemispherical ε-negative resonator on a ground plane fed with a coaxial probe. It is numerically shown that for a relative permittivity around -2, a localized surface plasmon resonance (LSPR) can arise [START_REF] Maier | Plasmonics: Fundamentals and Applications[END_REF]. This resonance leads to a significant enhancement of the radiated electromagnetic field that could be used for antenna applications. As suggested by the authors of [START_REF] Stuart | Electrically small antenna elements using negative permittivity resonator[END_REF] and [START_REF] Ziolkowski | Metamaterial-based efficient electrically small antennas[END_REF], such a negative permittivity resonator could be obtained with plasma discharges at microwave frequencies.

Plasma discharges are globally neutral mixtures of electrons, ions, and neutrals. They have already shown promising features to design stealth, tunable and steerable antennas [START_REF] Borg | Plasmas as antennas: Theory experiment and applications[END_REF]- [START_REF] Sokoloff | Non-thermal plasma potentialities for microwave device reconfigurability[END_REF]. For a low pressure, non-equilibrium, and non-magnetized plasma, its complex relative permittivity can be calculated using the Drude model:

ε p (ω) = 1 - ω p 2 ω(ω + iν) , with ω p = e 2 n e ε 0 m e (1)
where ν, ω p , e, n e , m e , and ε 0 are respectively the electron collision frequency, the plasma angular frequency, the Coulomb charge, the electron density, the electron mass, and the vacuum permittivity. It is clear from (1) that the real part of the dispersive relative permittivity can be negative. According to this model, it only depends on the electron density n e and the collision frequency ν at a given wave angular frequency ω. The electron density is mainly controlled by the power delivered to the discharge, the gas type, and the pressure, while the collision frequency depends on the type of gas and its pressure.

Recently, a plasma-based ESA has been demonstrated in the low UHF band [START_REF] Laquerbe | Towards antenna miniaturization at radio frequencies using plasma discharges[END_REF]. It consists of a small coaxial probe placed above a ground plane and surrounded by a hemispherical glass shell that confines an inductively coupled plasma discharge. It has been shown that the plasma allows impedance matching and frequency agility from 310 to 390 MHz by controlling the power delivered to the plasma discharge. Meanwhile, significant radiation enhancement up to 20 dB at 300 MHz has been obtained when compared to the case without discharge. In [START_REF] Kong | Experimental studies on radiation intensification in gigahertz radio frequency band by subwavelength plasma structures[END_REF], the authors demonstrated that a cylindrical subwavelength plasma discharge could be used to enhance the electromagnetic field radiated by an electrically small elliptical dipole antenna. However, such an implementation does not allow an effective integration of the antenna with the plasma source. Colón Quiñones et al also presented an experimental validation of a plasma resonator supporting a LSPR in the Ku band using laser-induced plasma [START_REF] Colón Quińones | Experimental validation of tunable features in laser-induced plasma resonators[END_REF]. Still, this technology does not allow yet the creation of a stationary plasma for antenna applications.

In this paper, we present a preliminary study to develop a plasma-based resonator for ESA applications between 1 and 1.5 GHz. Its main purpose is to determine the key parameters of the plasma discharge to obtain the LSPR above 1 GHz. Then, we propose a practical implementation of this resonator that takes into account the problem of plasma ignition and sustainment.

II. PLASMA REQUIREMENTS

This section aims to define the plasma parameters, electron density and collision frequency, required to obtain the LSPR of a plasma discharge between 1 and 1.5 GHz.

As plasma is a gaseous medium, it should be confined in an enclosure. To mimic the original ESA concept from Stuart et al, we choose to confine the plasma in a hemispherical dielectric shell laying on a ground plane. The glass shell diameter is fixed to 2a = 3 cm to ensure an electrical size ka of less than 0.5 between 1 and 1.5 GHz. It consists of a dielectric layer with a thickness of 1.4 mm and a dielectric constant of 4.6 as it will be shown in the next section.

In order to evaluate the required plasma electron density, one has to study the LSPR of the plasma-based resonator. Assuming spherical symmetry and homogeneity of the plasma discharge, the original resonator can be viewed as an electrically small plasma hemisphere on an infinite ground plane. According to image theory, this problem is equivalent to a plasma sphere in free space. The resonant behavior of this spherical plasma is finally studied as a scattering problem using Mie theory [START_REF] Laquerbe | Towards antenna miniaturization at radio frequencies using plasma discharges[END_REF], [START_REF] Shore | Scattering of an electromagnetic linearly polarized plane wave by a multilayered sphere[END_REF].

The Mie model consists of a lossy homogeneous plasma sphere in a glass shell illuminated by a linearly polarized plane wave. Numerical results from Mie analysis are used to determine the required electron density corresponding to the resonant frequency of the LSPR. To do this, we consider the scattering efficiency Q sca that determines how much an incident radiation is scattered by an object in all directions relatively to its electrical size, that is to say to the frequency if we assume its size constant [START_REF] Bohren | Absorption and Scattering of Light by Small Particles[END_REF]. For a given electron density, it is assumed that the resonant frequency f 0 of the LSPR occurs when Q sca is maximum.

Fig. 1 shows the scattering efficiency Q sca as a function of the frequency for three different values of electron density considering a collisionless plasma, ν = 0 sec -1 . We observe that changing the electron density leads to a shift in the resonant frequency of the LSPR. For example, for n e = 6×10 10 cm -3 , the resonance is at f 0 = 1.05 GHz. According to [START_REF] Wheeler | Fundamental limitations of small antennas[END_REF], it corresponds to a real part of the relative permittivity of -3.36 for the plasma discharge. This value is different from -2 due to the electrical size of the plasma sphere and the dielectric glass shell that modifies the resonance condition [START_REF] Sihvola | Character of surface plasmons in layered spherical structures[END_REF]. We also observe that the smaller the electrical size of the resonator, or the frequency, the higher the magnitude of the scattering efficiency at resonance is. Finally, this curve also shows the frequency agility of the plasma resonator. Indeed, by changing the electron density n e of the plasma discharge, it is possible to change its resonant frequency f 0 . From these results, it is possible to compute the electron density required to obtain a given resonant frequency f 0 of the LSPR as shown in Fig. 2. Three cases are presented for different plasma losses, namely ν/ω = 0, ν/ω = 0.01, and ν/ω = 0.1. We notice that the required electron density varies between 0.5 × 10 11 and 1.4 × 10 11 cm -3 to obtain a resonant frequency between 1 and 1.5 GHz. Fig. 3 presents the corresponding required real part of the relative permittivity of the plasma as a function of the resonant frequency f 0 according to (1). Between 1 and 1.5 GHz, the required value for Re( p ) slightly varies with the resonant frequency and, therefore, the electrical size ka of the resonator [START_REF] Mei | A study of a negative permittivity and permeability for a small sphere[END_REF]. As long as the ratio ν/ω remains lower than 0.1, the resonance condition is almost unchanged. However, plasma losses have a greater impact on the magnitude of the LSPR.

Fig. 4 shows the scattering efficiency as a function of the frequency for three values of collision frequency and considering n e = 6 × 10 10 cm -3 . It is observed that the increase in the collision frequency results in a decrease in the quality factor of the LSPR. The magnitude of the scattering efficiency as a function of the resonant frequency for three values of collision frequency is also presented in Fig. 5. When ν/ω = 0.1, there is clearly a significant decrease in Q sca due to plasma losses.

From this preliminary study it can be seen that the plasma electron density n e must be between 0.5 × 10 11 and 1.4 × 10 11 cm -3 to obtain the LSPR between 1 and 1.5 GHz. Since plasma losses have a significant impact on the LSPR, the collision frequency ν has to be reduced. For this study, ν/ω = 0.01 is targeted, that is to say ν is between 63 × 10 6 and 94 × 10 6 sec -1 from 1 to 1.5 GHz, respectively.

III. DISCUSSION ON PRACTICAL IMPLEMENTATION

The collision frequency in a plasma discharge depends mainly on the type of gas and its pressure. For our application, neon is an interesting candidate since its collision frequency is relatively low. It can be roughly approximated using the following formula [START_REF] Raizer | Gas Discharge Physics[END_REF]:

ν = 1.2 × 10 9 P ( 2
)
where P is the gas pressure in Torr. For neon gas, the condition ν/ω = 0.01 then corresponds to a pressure of 50 mTorr (around 6.66 Pa) at 1 GHz. The thickness and material of the shell have been designed to withstand such a pressure difference stress. It consists of a boroscilicate glass with a 1.4 mm thickness and a dielectric constant of 4.6.

Regarding the electron density requirement, a plasma source must be designed to ignite and sustain a plasma with an electron density between 0.5 × 10 11 and 1.4 × 10 11 cm -3 at a pressure around 50 mTorr. Such plasma discharges have already been obtained with miniature inductively coupled plasma (mICP) sources [START_REF] Hopwood | Review of inductively coupled plasmas for plasma processing[END_REF], [START_REF] Yu | Miniaturization of inductively coupled plasma sources[END_REF]. A mICP source consists of a microstrip coil printed on a dielectric substrate and powered by a generator of a few hundred MHz with less than 10 W. In the inductive coupling mode, the time-varying magnetic field created by the coil induces an electric field in the plasma discharge. This electric field moves the electrons that will collide other species and sustain the discharge [START_REF] Hopwood | Electromagnetic fields in a radio-frequency induction plasma[END_REF].

In order for the mICP source to supply the plasma discharge through the ground plane required for our hemispherical resonator in our application, electrically small radial slots are machined in the ground plane. Practically, the operation of the coil is not purely inductive and capacitive coupling is present. The slots are machined to design a Faraday shield that also helps in reducing the capacitive coupling. A series capacitance and a parallel capacitance are used to tune the ICP coil at resonance with a 50 Ω coaxial line [START_REF] Yu | Miniaturization of inductively coupled plasma sources[END_REF].

Finally, Fig. 6 shows a schematic of the proposed design. Such an implementation allows to separate the plasma source with the plasma discharge operating as an electrically small resonator. For antenna applications, a possible electromagnetic coupling of this resonator would be to insert a small coaxial probe through the center of the Faraday shield [START_REF] Laquerbe | Towards antenna miniaturization at radio frequencies using plasma discharges[END_REF].

IV. CONCLUSION

A tunable electrically small plasma-based resonator using a localized surface plasmon resonance (LSPR) between 1 and 1.5 GHz has been studied. Such a resonator may be interesting for developing electrically small antennas with frequency agility.

As shown by this preliminary study, the final plasma-based resonator must be confined in a hemispherical glass shell 3 cm in diameter above a ground plane in order to obtain an electrical size of less than 0.5. The simulated electron density required for the plasma is between 0.5 × 10 11 and 1.4 × 10 11 cm -3 for having LSPR from 1 to 1.5 GHz, respectively. Finally, to reduce plasma losses and their impact on the quality factor of the LSPR, it is proposed to consider a neon gas at a pressure of 50 mTorr (around 6.66 Pa).

Currently, our work is focused on the realization and characterization of this L-band plasma-based resonator by considering an implementation using a miniature inductively coupled plasma (mICP) discharge.

= 6 × 3 Fig. 1 . 11 1 Fig. 2 .

 6311112 Fig. 1. Scattering efficiency Qsca as a function of the frequency f for three values of electron density ne considering ν = 0 sec -1 .

Fig. 3 .

 3 Fig. 3. Real part of the relative permittivity of the plasma Re( p) required for LSPR as a function of the resonant frequency f 0 for three values of collision frequency ν.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Scattering efficiency Qsca as a function of the frequency f for three values of collision frequency ν considering ne = 6 × 10 10 cm -3 .

Fig. 6 .

 6 Fig. 6. Design of the subwavelength plasma-based resonator: (a) top view and (b) Bottom view.

ACKNOWLEDGMENT

The authors would like to thank the Direction Générale de l'Armement (DGA), the Agence de l'Innovation de Défense (AID), and the Région Occitanie for their financial support.