Dr Fan Jiang

Dr Mathieu Achard
email: mathieu.achard@univ-rennes1.fr

Thierry Roisnel

Vincent Dorcet

Dr Christian Bruneau

Ruthenium(II) and Iridium(III

HAL is

Introduction

Transition metal complexes featuring polydentate donor ligands have found broad applications in homogeneous catalysis and a lot of efforts have been devoted to increase or alter stability/selectivity and activity by steric and electronic ligand modifications. [1,[START_REF] De Vries | The Handbook of Homogeneous Hydrogenation[END_REF] Among them, ancillary bidentate diphosphines represent an important part of this class of compounds. [START_REF] Tolman | Bite angle, electronic and steric effects: a)[END_REF][START_REF]Phosphorus Ligand in Asymmetric Catalysis[END_REF] Owing to the presence of distinct coordination sites, mixed/hybrid ligands represent another part of interesting compounds. [5][6][7][8] These prochelates can lead to the formation of stabilized transition metal complexes with unique properties. Depending on the nature of one σ-donor ligand or ligand geometry, partial decoordination can occur expressing hemilabile properties generating a vacant coordination site on the metallic center. [8a] While in all the examples cited above, catalytic transformations occurred at the metal center, ligands containing reactive functionality can cooperate during catalysis to afford multifunctional catalytic systems. [9] Among them, proton responsive/cooperative/non innocent [6f,10-12] and redox active [13,14] ligands represent an important class of ligands of increasing interest. [9][10][11][12][13][14] One of the most cited examples of bifunctional catalysis involve the use of ruthenium complex containing diphosphine diamine ligands [15] where an acidic proton is held by the ligand through the formation of a N-H functionality [16] and the hydride bound to the ruthenium center allowing the ionic hydrogenation of the carbonyl group through a concerted outer sphere mechanism. The 2-pyridone/2-hydroxypyridine (Hp) moiety have been at the center of research activities for the preparation of peculiar ligands. [17] 2-Hydroxypyridines can act as carboxylate surrogates but due to the ability of such a functional group to enable hydrogen bonding interaction, 2-pyridone-based ligands have also found broad application for the preparation of water soluble species or selfassembled systems such as the well-known 6-DPPon which has proven efficiency in hydrogenation and related transformations. [18][19] On the other hand, the proximity of the hydroxyl group in 2-hydroxypyridine derivatives can also generate active metal-ligand bifunctional catalytic systems in hydrogen transfer reactions and more interestingly dehydrogenation process. [20][21][22] However, to the best of our knowledge nothing is known on the synthesis of phosphinepyridonate derivatives keeping intact the hydroxyl functionality to enable bifunctional catalysis. Therefore, with these reports in mind, we postulate that similar strategy could be used to access corresponding five-membered proton responsive P,O and P,N chelates from 2-pyridone derivatives.

Here we report the synthesis of two new phosphine hybrid ligands featuring 6-methyl-2-pyridinol and 2-quinolinol moieties and the access to the corresponding well-defined ruthenium and iridium complexes. Preliminary investigation in hydrogenation reaction highlights their potential in hydrogen transfer processes.

Results and Discussion

The ligands L1-H and L2-H were directly prepared from 6-methyl-2-pyridinol and 2-quinolinol, respectively. According to a modified procedure of Brunner, [23] reaction of unprotected 6-methyl-2-pyridinol with 2.1 equivalent of n-butyl lithium resulted first in the deprotonation of the hydroxyl group followed by the formation of the dianionic species (Scheme 1). Addition of chlorodiphenylphosphine rapidly afforded a complicated mixture of phosphinites and phosphines which, after overnight stirring at room temperature cleanly afforded the crude L1-Li. Further treatment followed by protonation with a saturated ammonium chloride solution gave the expected L1-H in 40% isolated yield. The preparation of L2-H in turn, required the directed o-lithiation of 2quinolinol with n-butyl lithium to generate the corresponding o-lithiated quinolinolate as key intermediate followed by the addition of Ph2PCl to afford after treatment and selective protonation of the oxygen atom the ligand L2-H in 23% isolated yield.

Scheme 1. Preparation of the Phosphine-pyridonate L1-H and -quinolinolate L2-H.

The coordination ability of these ligands to ruthenium and iridium metal centers was explored. We focused our attention on the synthesis of (η 6 -arene)ruthenium(II) and three legs piano stool iridium(III) complexes featuring five-membered P,N and P,O chelates. Treatment of the prochelate L1-H with [RuCl2(p-cymene)]2 precursor in methanolic solution (α = 0.93) [24] resulted in a red precipitate and afforded the corresponding stable half-sandwich cationic ruthenium(II) complex Ru-1 (Scheme 2). Ru-1 formation was monitored by NMR spectroscopy in CD2Cl2 (α=0.30) and the spectroscopic data are consistent with the formation of the five-membered chelate. NMR analyses revealed the presence of an acidic proton located at 14.18 ppm and the methylene bridge consists of a set of a doublet of doublet and one triplet located at 4.16 and 3.98 ppm, respectively. Confirmation was obtained in the 31 P NMR spectra highlighted by the formation of a sole metallic species with a chemical shift at 55.6 ppm. This result is in line with the known lability of Ru-Cl bond in a polar protic solvent such as methanol and demonstrates that the cationic form is sufficiently stable and did not require the use of stabilizing counteranion. [6e] In contrast to this result, the reaction of [IrCl2(Cp*)]2 bearing a bulky pentamethylcyclopentadienyl ligand with L1-H in methanolic solution afforded sensitive species in dynamic equilibrium. It is noteworthy that in these cases, a molecule of methanol was observed in a 1:1 ratio MeOH/complex demonstrating hydrogen bonding interactions. Analysis in CD2Cl2 revealed the formation of a sole complex Ir-1 characterized at 22.4 ppm in 31 P NMR and 1 H NMR exhibiting the presence of diastereotopic protons on the methylene bridge at 4.27 (JP-H = 13.7 Hz, JH-H = 16.5 Hz) and 4.06 ppm (JP-H = 11.5 Hz, JH-H = 16.5 Hz). However, slow decoordination of the nitrogen atom leading to the neutral iridium(III) complex Ir-1' was observed during these analyses demonstrated by the presence of a new signal in 31 P NMR at 1.7 ppm along with the remaining starting complex at 22.4 ppm. Comparison of NMR data with the piano stool iridium complex featuring a 2-(2diphenylphosphanylethyl)pyridine ligand by Pandey confirmed that reversible chloride displacement occurred accounting on Ir-1' complex formation and thus demonstrating the crucial influence of the solvent towards chloride displacement and Table 1. Selected bond lengths and bond angles for complexes of Figure 1. [a] [M]-Cl aromatization/dearomatization process. [25] In contrast to these result, prior formation of L1-K arising from the deprotonation of L1-H with potassium tert-butoxide cleanly afford after reaction with [IrCl2(Cp*)]2 the neutral piano stool iridium Ir-2 complex in 83% isolated yield. The formation of five-membered P,O chelate was next investigated starting from L2-H. Mixing L2-H in the presence of [RuCl2(p-cymene)]2 metal precursor, clean conversion to the corresponding Ru-2 complex was obtained in interaction with methanol molecules. In the 1 H NMR spectrum, the supposed O-H signal was shifted downfield at 15.60 ppm in CDCl3 (α = 0.44) with a sharp signal for the methanol whereas in CD2Cl2 an upfield signal at 8.50 ppm along with a broad signal at 3.50 ppm for methanol suggested that tautomerization via a proton shuttle to N-H occurred. [26] The corresponding neutral ruthenium(II) complex Ru-3 was obtained by mixing insitu prepared L2-K with [RuCl2(p-cymene)]2 in 91% isolated yield as a stable complex. Noteworthy that no concentration dependence of the chemical shifts was observed in these complexes. [27] Finally, formation of complexes Ru-1, Ir-1'.CH2Cl2, Ir-2.CH2Cl2, Ru-2.2(MeOH) and Ru-3.CH2Cl2 by using solvent diffusion technique (hexane:CH2Cl2) was confirmed by X-ray crystallography. ORTEP views at 50% thermal ellipsoids probability are described in Figure 1 and representative bond lengths and angles are outlined in Table 1. In the solid state, hydrogen bonding interactions were observed in the crystal structure of Ru-1 between the chloride counteranion and the hydroxyl group of the hydroxypyridine ring (OH•••Cl = 2.117 Å). In the case of Ir-1, only the minor neutral Ir-1' selectively crystallized. Interestingly, the crystal structure of Ir-2 suggests an interaction with the dichloromethane molecule and the oxygen atom of the pyridone ring (O•••HCHCl2 = 2.110 Å). The chloride counteranion in Ru-2 is stabilized by hydrogen bonding with a methanol molecule (Cl -•••HOMe = 2.266 Å). The carbonyl moiety presents in complexes Ir-1', Ir-2 and Ru-2 is supported by shorter bond lengths of 1.260( 5), 1.246(3) and 1.262(3) Å, respectively. In all these structures, the M-Cl bond are comparable with related reported complexes and M-P bonds in the respective complexes are also in the range of the reported values. [25,28] In light of the potential reversible protonation and hemilability of the chelates in the resulting ruthenium complexes and the crucial influence of solvent, appropriate for the heterolytic [a] Reaction were carried out at 50 °C , in 0.5 mL of methanol, acetophenone (0.5 mmol) under 50 bar of H2 with S/C ratio = 100:1. [b] 31 P NMR analysis of the crude reaction mixture after hydrogenation cleavage of hydrogen, we next examined their reactivity towards acetophenone hydrogenation. [6f,10d,29] The reactions were carried out at 50 °C in methanol under 50 bar of molecular hydrogen with a catalyst/substrate ratio of 1:100. As expected, no or very low activities were detected in the absence of additives (entries 1,3 and 5). [29] Analyses of the reaction mixtures revealed similar 31 P displacement as the starting complexes demonstrating that the precatalysts remained intact and the necessity of an external base to ensure the formation of hydride species. [15] Therefore, addition of DABCO as an organic base indicated that these ruthenium complexes could be activated and demonstrated activity in hydrogenation. In the presence of catalytic amount of Ru-1 along with DABCO, 90% of phenylethanol was formed (entry 2). Analysis after reaction also revealed that the arene remained bound to the ruthenium metallic center and the formation of another species at 53.7 ppm presumably corresponds to the deprotonated neutral ruthenium complex [RuCl(p-cymene)L1]. To confirm this hypothesis, deprotonated L1 was reacted with [RuCl2(p-cymene)]2 (0.5 eq.) to give a brown complex soluble in methanol. Further treatment gave [RuCl(p-cymene)L1] with a 54.38 ppm chemical shift in 31 P NMR(CDCl3) with characteristic dearomatized proton patterns in 1 H NMR. Addition of small amount of methanol resulted in a slight upfield signal at 54.11 ppm which tend to confirm our assumption. [30] During hydrogenation, no difference of reactivity should be observed between Ru-2 and Ru-3. As results these two complexes exhibited similar activities leading to phenylethanol in 64 and 65%, respectively (entries 4 and 6). From these data, 1 H and 31 P analyses revealed the side formation of a new complex at 27.6 ppm in 31 P NMR. Comparison with reported structures tend to suggest the formation of [RuCl2(p-cymene)L2-H] (27.7 ppm) along with [RuCl(p-cymene)L2] (36.8 ppm), thus demonstrating the partial chelate opening of L2 during catalysis. [5b,31]

[M]-P [M]-N/O C-N C-O (N)O- H P-[M]- N/O P-[M]- Cl Ru-1 2

Conclusions

In conclusion, we have reported the preparation of well-defined ruthenium and iridium complexes featuring proton responsive fivemembered P,N and P,O chelates. Preliminary evaluation of the ruthenium complexes in hydrogenation of unfunctionalized ketones revealed their promising catalytic activities. Preparation of the corresponding enantiopure ligands is currently underway to evaluate their activities in enantioselective hydrogenation/reduction of unsaturated substrates with various transition metal complexes as well as dehydrogenation processes.

Experimental 6-((diphenylphosphino)methyl)pyridin-2-ol L1-H

6-methyl-2-pyridinol (1.0 g, 9.16 mmol, 1.0 equiv.) was dissolved in 5.0 mL THF, cooled at 0 o C followed by the slow addition of n-BuLi (12 mL, 19.2 mmol, 2.1 equiv.). After one hour stirring, the mixture was cooled to -78 o C, which was added to another solution containing diphenylphosphine chloride (1.6 mL, 9.16 mmol, 1.0 equiv.) in 2.5 mL THF solution at -78 o C. The temperature was allowed to increase naturally to ambient temperature, then the solution was stirred at this temperature overnight. Cloudy ora nge solution was obtained, which was evaporated, solubilize in acetone/dichloromethane followed by the addition of 20 mL of satur ated ammonium chloride solution to precipitate the ligand. Filtration and washings with acetone afforded a white solid as L1-H. .00 mmol, 1.0 equiv.) was dissolved in 5.0 mL THF, cooled at 0 o C followed by the slow addition of n-BuLi (9.4 mL, 14.70 mmol, 2.1 equiv.) over 10 minutes. The resulting mixture was warmed up to ambient temperature, heated to 50 o C for 10 minutes and then cooled down to room temperature for 20 minutes. Finally the mixture was cooled to -78 o C and was cannulated to a cooled solution of diphenylphosphine chloride (1.2 mL, 6.37 mmol, 0.91 equiv.) in 2.5 mL THF solution at -78 o C. The temperature was allowed to increase naturally to ambient temperature, then the solution was stirred at this temperature overn ight. Dark orange to pale brown clear solution was obtained, which was evaporated, quenched with 20 mL degassed water and dissol ved with dichloromethane 40 mL, then acidified with 1N HCl until pH was around 3. After separation and extraction, the combined organic layers were dried over MgSO4. Filtration and evaporation afforded a yellow sticky solid. Addition of degassed acetone 30 mL and diethylether 20 mL followed by cooling at -30 o C resulted in the apparition of white precipitates which were collected by filtration. The resulting white solid was further washed with acetone and diethylether to afford L2-H.

Scheme 3 .Figure 1 .

 31 Scheme 3. Reactivity of L1-H with ruthenium and iridium metallic precursors.

3 -

 3 (diphenylphosphino)quinolin-2-ol L2-H (0.48 g, 23%) 1 H NMR (400 MHz, DMSO-d 6 ): δ 11.94(d, 1H, J = 1.6 Hz), 7.51-7.47 (m, 1H), 7.43-7.41 (m, 7H), 7.34-7.28 (m, 5H), 7.13-7.11 (m, 2H);31 P { 1 H} NMR (162 MHz, DMSO-d 6 ): δ -16.80 (s); 13 C { 1 H} NMR (125 MHz, DMSO-d 6 ): δ 161.91 (d, JP-C = 20.8 Hz, Cquat), 143.17 (d, JP-C = 2.1 Hz), 138.91 (s, Cquat), 135.4 (d, JP-C = 10.6 Hz, Cquat), 133.54 (d, JP-C = 20.5 Hz), 132.57 (d, JP-C = 9.9 Hz, Cquat), 130.70 (s), 129.11 (s), 128.78 (d, JP-C = 7.3 Hz), 127.81 (s), 121.94 (s), 118.99 (s, Cquat), 115.04 (s); HRMS-ESI: HRMS-ESI: (C21H16NONaP), Calcd: 352.08617; Found: 352.0815 (2 ppm).

Table 2 .

 2 Activity of well-defined ruthenium complexes in hydrogenation of acetophenone

	Entry [a]	Cat.	Additive	Yield	31 P [b]
	1	Ru-1	None	6%	55.5 (100%)
	2	Ru-1	DABCO	90%	53.7 (100%)
	3	Ru-2	None	1%	37.1 (100%)
	4	Ru-2	DABCO	64%	36.8 (77%) 27.6 (23%)
	5	Ru-3	None	7%	37.0 (100%)
	6	Ru-3	DABCO	65%	36.8 (84%) 27.7 (16%)

Acknowledgements

The authors thank the European Union and Synflow consortium (FP-7 integrated project Synflow, NMP-2009-3.2-1 no. 246461; http://synflow.eu) for financial support of this project. Thanks are also due to Prof. C. Claver, Prof D. Heller and Dr. A. Igau, for helpful comments.


Cyprien Ruffino
email: cyprien.ruffino@insa-rouen.fr

Rachel Blin

Samia Ainouz

Gilles Gasso

Romain Hérault

Fabrice Meriaudeau

Stéphane Canu

Physically-admissible polarimetric data augmentation for road-scene analysis

Polarimetric imaging, along with deep learning, has shown improved performances on different tasks including scene analysis. However, its robustness may be questioned because of the small size of the training datasets. Though the issue could be solved by data augmentation, polarization modalities are subject to physical feasibility constraints unaddressed by classical data augmentation techniques. To address this issue, we propose to use CycleGAN, an image translation technique based on deep generative models that solely relies on unpaired data, to transfer large labeled road scene datasets to the polarimetric domain. We design several auxiliary loss terms that, alongside the CycleGAN losses, deal with the physical constraints of polarimetric images. The efficiency of this solution is demonstrated on road scene object detection tasks where generated realistic polarimetric images allow to improve performances on cars and pedestrian detection up to 9%. The resulting constrained CycleGAN is publicly released, allowing anyone to generate their own polarimetric images.

Introduction

Road scene analysis is a vital task for driving assistance systems. Robust road users detection including vehicles, pedestrians, animals, is a precondition for a secure navigation. The complexity and variability of outdoor illumination conditions make RGB-based detection algorithms poor and limited, especially when they are faced to strong reflections, noises or bad weather conditions. In contrast, polarization encoded images are a non-conventional modality where each reflected wave light from a pixel is strongly linked to the physical properties of the surface. The significant interest resides in the fact that polarimetric imaging is a rich modality that enables to characterize an object by its reflective properties. In a polarimetric image, each pixel encodes information regarding the object's roughness, its orientation and its reflection [START_REF] Lawrence | Polarization camera sensors[END_REF]. Applications of polarimetric imaging range from indoor autonomous navigation [START_REF] Berger | Depth from stereo polarization in specular scenes for urban robotics[END_REF], depth map estimation [START_REF] Zhu | Depth From a Polarisation + RGB Stereo Pair[END_REF], 3D objects reconstruction [START_REF] Morel | Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging[END_REF] to differentiation of healthy and unhealthy cervical tissues in order to detect cancer at an early stage [START_REF] Rehbinder | Ex vivo Mueller polarimetric imaging of the uterine cervix: a first statistical evaluation[END_REF]. Also, recently, polarization imaging was exploited in autonomous driving applications either to enhance car detection [START_REF] Fan | Polarization-Based Car Detection[END_REF], road mapping and perception [START_REF] Todd M Aycock | Polarization-based mapping and perception method and system[END_REF] or to detect road objects in adverse weather conditions [START_REF] Blin | Road scenes analysis in adverse weather conditions by polarization-encoded images and adapted deep learning[END_REF].

The key element of this work comes from the combination of polarization richness and deep learning-1 arXiv:submit/4357013 [cs.CV] 15 Jun 2022 based detection. However, these applications are still characterized by the reduced size of the available training databases which restrains them from using deep neural networks. To overcome this limitation, one possible solution is the generation of polarimetric encoded images.

Generative Adversarial Networks (GAN) [Goo+14; WSW19] are powerful deep generative models used to implicitly learn complex data distributions and to generate realistic samples from them. In its standard form, a GAN consists of two models: a generator which maps samples drawn from a latent lowdimensional distribution (usually uniform or Gaussian distributions) to high-dimensional points expected to follow the sought data distribution, and a discrimination model which discriminates the real samples from the generated ones [START_REF] Goodfellow | Generative Adversarial Nets[END_REF]. GAN have proven remarkable for various application domains including image generation [START_REF] Arjovsky | Wasserstein Generative Adversarial Networks[END_REF], image-toimage translation [Iso+17; Zhu+17; Hof+18] or image attribute manipulation [START_REF] Antipov | Face aging with conditional generative adversarial networks[END_REF] to name a few.

Arguably most of the impressive achievements of GAN were obtained for color images. A body of work attempted to extend GAN architectures to other uncommon imaging domains. For instance, some existing methods rely on CycleGAN [START_REF] Zhu | Unpaired image-toimage translation using cycle-consistent adversarial networks[END_REF], an image-to-image translation network, to generate infrared road scenes from RGB counterpart images [START_REF] Zhang | Synthetic data generation for end-to-end thermal infrared tracking[END_REF], to produce thermal images for person reidentification [START_REF] Kniaz | ThermalGAN: Multimodal Color-to-Thermal Image Translation for Person Re-Identification in Multispectral Dataset[END_REF] or for infrared image colorization [START_REF] Mehri | Colorizing Near Infrared Images through a Cyclic Adversarial Approach of Unpaired Samples[END_REF]. In the same vein, data augmentation in the field of medical imaging was achieved [START_REF] Nie | Medical image synthesis with context-aware generative adversarial networks[END_REF] by transforming MRI inputs into pseudo-CT images. Following the previous stream of work, this paper contributes to generative models for non-conventional imaging techniques. Specifically we propose a generative model framework to produce realistic polarimetric images.

Unlike RGB, LiDAR or infrared image generation which mostly responded to visual qualitative constraints, unless some learnable knowledge constraints are enforced (see [START_REF] Hu | Deep generative models with learnable knowledge constraints[END_REF] for pose conditional person image generation), sampling polarization images is more challenging. Indeed, this imaging technique comes with physical admissibility constraints on the pixels of an image [START_REF] Bass | Handbook of optics[END_REF]. To be physically fea-sible, each pixel entry of such an image should satisfy some physical constraints related to light polarization principle and to the calibration setup of the acquisition devices. Therefore, we propose a set of constraints that, when satisfied, ensure that the generated images are physically feasible. Based on these constraints, we formulate our problem of polarimetric image generation as a domain-transfer learning problem under physical constraints to ensure that the generated images are valid. We propose a learning framework based on CycleGAN [START_REF] Zhu | Unpaired image-toimage translation using cycle-consistent adversarial networks[END_REF], which enables unpaired image-to-image translation with relatively few images, to which we add constraints for handling the physical polarization constraints during training. This allows for circumventing the expensive labelling step by transferring a source labelled RGB dataset to the polarimetric domain while keeping the shapes and contents of the source image unchanged.

We demonstrate the effectiveness of our constrained-output CycleGAN on the KITTI dataset [START_REF] Geiger | Are we ready for autonomous driving? the kitti vision benchmark suite[END_REF] as well as on the Berkeley Deep Drive dataset (BDD100K) [START_REF] Xu | End-to-end learning of driving models from large-scale video datasets[END_REF], which are commonly used for object detection in road scenes. Using the generated polarization-encoded images to train a deep object detector, we witness an improvement of the detection performances of cars and pedestrians which are of great interest for autonomous driving applications.

To summarize, the contributions of this paper are:

• as far as our knowledge goes, we propose the first framework for generating physically admissible polarization-encoded images starting from RGB images,

• we propose an extension of CycleGAN which allows to generate polarimetric-encoded images while handling the physical constraints that should be satisfied by the pixels of the generated images,

• we show that, when plugged into the training procedure of an object detector for pre-training, the generated images help improving the detection performances.

The remainder of the paper is organized as follows: the polarization formalism and the physical constraints it involves are first presented. Then, the CycleGAN approach is described and a way to take into account these physical constraints during the training process of the CycleGAN for generating polarimetric images is investigated. Experimental evaluations are conducted; they aim to translate RGB images from KITTI and BDD100K datasets into polarimetric images. Finally, the generated images are exploited to boost the performances of an object detection network. The code for the experiments and the trained models are available at: https://anonymous.4open.science/ r/4a83820e-9c65-417c-af3a-ab2979d6e2e8/

Framework

This section introduces the polarization formalism, the GAN approach and the CycleGAN principles. It focuses on the formulation of the proposed modelling framework, namely the learning of a CycleGAN with output constraints. A solution approach and the related learning principle are presented.

Physical principles of polarimetry

Polarization is the direction of propagation of the electrical field of the light wave. The electrical field of the wave propagating in z direction at time t is characterized by the scalar components of the vector

E =   E x (z, t) E y (z, t) E z (t)   =   E 0x cos(ωt -kz) E 0y cos(ωt -kz + φ) 0   , (1)
with E 0x and E 0y the maximum amplitudes of each component. φ is the phase shift between the two components E x (z, t) and E y (z, t), ω is the angular frequency, k represents the wave number, directly related to the wavelength λ. Figure 1 depicts the different parameters.

The two components of the electrical field could be combined by eliminating the temporal term ωt in the following way :

E x (t) E 0x 2 + E y (t) E 0y 2 -2 E x (t)E y (t) E 0x E 0y cos(φ) = sin 2 (φ)
(2) If φ is constant over the time, Equation 2 defines an ellipse meaning that the electrical field of a propagating wave has an elliptical behaviour as illustrated in Figure 2.

According to sin(φ), three states of polarization could be observed. If sin(φ) > 0 the polarization is said to be right-handed, otherwise, if sin(φ) < 0, the polarization is left handed. If sin(φ) = 0, the trajectory of the wave is linear and the polarization is known as linear polarization. If sin(φ) = ±1, the polarization of the wave is respectively right and left circular. For these three cases the wave is totally polarized, otherwise, if φ is not constant over the time, the wave is unpolarized [START_REF] Bass | Handbook of optics[END_REF]. Usually, in an uncontrolled environment, a light wave is a combination of a totally polarized part and an unpolarized one, this is the reason why the natural light is known to be partially polarized.

Mathematical representation

There are different ways to describe the polarization of the light and its interaction with the media. The Stokes parameters are one of them to describe the polarization state of an electromagnetic wave, especially in the general case of a partially polarized wave form. The Stokes vector [S 0 , S 1 , S 2 , S 3 ] contains four polarization parameters where each component brings us information on the reflected light wave as follows:

• S 0 represents the total intensity of the light which is not a polarization property. S 0 > 0 is always positive, which means that every natural object, when illuminated, reflects a light,

• S 1 defines the amount of the polarized light in the wave horizontally or vertically polarized, meaning that the electrical field represented by figure 2 moves in a horizontal or a vertical way,

• S 2 is the amount of the polarized light oriented at 45 • and -45 • and,

• S 3 represents the amount of the circular polarization.

In the case of outdoor scenes, the sunlight is unpolarized and it is physically proved that when an unpolarized light wave is being reflected, it becomes partially linearly polarized. Its polarization depends on the normal surface and the refractive index of the material it impinges [START_REF] Lawrence | Polarization camera sensors[END_REF]. The circular component is thus neglected and set to zero in all applications concerning non controlled environments [START_REF] Morel | Active lighting applied to three-dimensional reconstruction of specular metallic surfaces by polarization imaging[END_REF]. We focused in our work on the only linearly polarized light represented by the three first components of the Stokes vector, namely [S 0 , S 1 , S 2 ].

In order to access to observable optical field, the two components E x (t) and E y (t) are averaged over the time. The polarization ellipse of Equation 2 average in case of linear polarization (sin(φ) = 0) is given by the following equation:

E x (t) 2 E 2 0x + E y (t) 2 E 2 0y -2 E x (t)E y (t) E 2 0x E 2 0y = 0. (3)
The averaged values of the electrical components are:

E 2 x (t) = 1 2 E 2 0x (4) E 2 y (t) = 1 2 E 2 0y
(5)

E x (t)E y (t) = 1 2 E 0x E 0y . (6)
By replacing the time averaged terms in Equation 3, the elliptical equation can be rewritten in the following form

(E 2 0x + E 2 0y ) -(E 2 0x -E 2 0y ) -(2E 0x E 0y ) = 0. (7)
The linear Stokes vector is finally derived from Equation 7 as

S =   S 0 S 1 S 2   =   E 2 0x + E 2 0y E 2 0x -E 2 0y 2E 0x E 0y .   (8)
One salient physical property, obtained from the Stokes parameters, is the degree of linear polarization (DOP) [START_REF] Ainouz | Adaptive processing of catadioptric images using polarization imaging: towards a pola-catadioptric model[END_REF] defined by:

DOP = S 2 1 + S 2 2 S 0 . (9)
The DOP ∈ [0, 1] (or equivalently between 0 and 100% ). It refers to the rate of the polarized part in the light wave. It is equal to 1 for a totally polarized light, 0 for unpolarized light and between 0 and 1 for partially polarized light. Equation 7 describes the totally polarized wave. By construction of the Stokes vector, the total energy S 2 0 in this case is equal to the partial energy S 2 1 + S 2 2 meaning that the light is totally composed of polarized waves linearly oriented. In case of partially polarized wave, the wave contains an unpolarized part, the equality is transformed to an inequality meaning that S 2 0 S 2 1 + S 2 2 . From this property, the Stokes vector is said to be physically admissible if and only if the two following conditions are met:

S 0 > 0 and S 2 0 S 2 1 + S 2 2 . ( 10
)
This condition means also that the DOP never exceeds 1, otherwise the partial energy of the wave may exceed its total energy, which is, physically, an impossible phenomenon.

Stokes imaging

I αi = 1 2 1 cos(2α i ) sin(2α i )   S 0 S 1 S 2   , ∀i = 1, 4
(11) that is:

I = AS , (12)
where I = I 0 I 45 I 90 I 135 refers to the four intensities according to each angle of the polarizer (α i ) i=1:4 and A ∈ R 4×3 , to the calibration matrix of the polarization camera, defined as:

A = 1 2     1 cos(2α 1 ) sin(2α 1 ) 1 cos(2α 2 ) sin(2α 2 ) 1 cos(2α 3 ) sin(2α 3 ) 1 cos(2α 4 ) sin(2α 4 )     = 1 2     1 1 0 1 0 1 1 -1 0 1 0 -1     . (13)
An example of the different intensities for the same scene is shown1 in Figure 3. To get the unknown Stokes parameters from the measured intensities (see Equation ( 12)), we require à = (A A) -1 A ∈ R 3×4 the pseudoinverse of the matrix A. The relationship between S and I is then defined by:

S = ÃI =   1 0 1 0 1 0 -1 0 0 1 0 -1       I 0 I 45 I 90 I 135     =   I 0 + I 90 I 0 -I 90 I 45 -I 135   . (14)
Combining Equations ( 12) and ( 14), we have:

I = A ÃI , (15)
which is satisfied if and only if:

I 0 + I 90 = I 45 + I 135 . (16)
Stokes images should then satisfy two main conditions: the physical admissibility constraints in (10) and the calibration constraint given by (16). The generated polarimetric images have to comply with these essential constraints.

Unpaired image-to-image translation with CycleGAN

Given two domains X and Y , unpaired image-toimage translation is the task of learning the mapping functions M XY : X → Y and M Y X : Y → X using unpaired samples x i ∈ X with i ∈ [1.

.N ] and y j ∈ Y with j ∈ [1..M ]. An effective approach to achieve the task is CycleGAN [START_REF] Zhu | Unpaired image-toimage translation using cycle-consistent adversarial networks[END_REF]. It consists in learning the two mapping models M XY and M Y X by combining the objective function of the standard Generative Adversarial Network (GAN) [START_REF] Goodfellow | Generative Adversarial Nets[END_REF] with a Cycle-Consistency loss function. The adversarial cost related to the GAN serves for training the models to generate samples that will match the target domain distribution, while the Cycle-Consistency cost ensures that the learned models are able to correctly reconstruct an original image (of the source domain) from a generated one.

Formally a GAN is composed of a generative model G : Z → X which maps a known distribution p Z , usually normal or uniform, to the unknown distribution p X of the samples and a discrimination model D : X → [0, 1]. The generator G attempts to fool the discriminator D, which in turn tries to distinguish a real sample from a sample generated by the model G. Learning a GAN amounts to solve the following problem:

G * , D * = arg min G max D L GAN (D, G) , (17)
with:

L GAN (D, G) = E x∼p X log(D(x)) (18) + E z∼p Z log(1 -D(G(z))) , (19)
.

CycleGAN trains the two models M XY and M Y X by using solely unpaired real samples x ∈ X and y ∈ Y respectively drawn according to the (unknown) distributions p X and p Y as input. It also learns two discrimination networks D X : X → [0, 1] and D Y : Y → [0, 1] able to detect generated samples from real ones in the domains X and Y respectively. CycleGAN relies on the Least-Squares variant of GAN [START_REF] Mao | Least squares generative adversarial networks[END_REF] and considers the following adver-sarial costs:

L GAN (D Y ,M XY ) = E y∼p Y (D Y (y) -1) 2 + E x∼p X D Y (M XY (x)) 2 , (20) L GAN (D X ,M Y X ) = E x∼p X (D X (x) -1) 2 + E y∼p Y D X (M Y X (y)) 2 . ( 21
)
In order to ensure the cyclic consistency, i.e. both the compositions M XY • M Y X and M Y X • M XY are identity functions, a 1 reconstruction error term is devised for the mapping models:

L reco (M XY , M Y X ) = E y∼p Y ||y -M XY (M Y X (y))|| 1 + E x∼p X ||x -M Y X (M XY (x))|| 1 . (22)
Gathering all these elements leads to the objective function:

L CycleGAN (D X ,D Y , M XY , M Y X ) = L GAN (D Y , M XY )+ L GAN (D X , M Y X ) + λL reco (M XY , M Y X ) , (23)
where λ > 0 is an hyper-parameter that controls the reconstruction term. Training a CycleGAN consists in solving, via alternate gradient descent, the following minmax problem:

M * XY ,M * Y X , D * X , D * Y = arg min M XY M Y X max D X D Y L CycleGAN (D X , D Y , M XY , M Y X ) .
(24)

Proposed approach

As discussed above, our main goal is to learn a generative model able to produce realistic polarizationbased images starting from RGB images. As our proposed solution to this problem, we adopt the imageto-image translation framework and extend it to account for the constraints a polarimetric image must fulfill.

To generate a polarimetric image from an RGB one, we propose to use the CycleGAN approach to learn the translation models M XY and M Y X between X the domain of the polarimetric images and Y the RGB image domain. Let Î ∈ R 4 be the intensity vector associated to a pixel of a generated polarimetric image. To be physically admissible, each pixel has to satisfy the admissibility (10) and the calibration (16) constraints. We refer to these polarimetric constraints by C 1 , C 2 and C 3 as follows:

C 1 : I = AS , C 2 : S 2 0 S 2 1 + S 2 2 , (25)
C 3 : S 0 > 0 .
By design, S 0 is always positive as it represents the total intensity reflected from an object. As the last layer of the generation models customary uses the hyperbolic tangent as activation function, each output intensity Î is within the range ] -1, 1[ which we scale to ]0, 255[. Hence Ŝ0 = Î0 + Î90 (see ( 14)) is ensured to be strictly positive. Therefore, constraint C 3 can be deemed satisfied for the generated polarimetric images. To handle the remaining constraints C 1 and C 2 , one could resort to the Lagrangian dual of CycleGAN optimization problem (24) subject to these constraints. However, this may be computationally expensive, as it requires to optimize four neural networks (respectively the discrimination and the mapping network models) in an inner loop of a dual ascent algorithm. Moreover the overall optimization procedure may not be stable because of the minmax game involved in the CycleGAN learning.

In order to derive an efficient algorithm to learn CycleGAN under output constraints, we introduce a relaxation of the problem. Instead of strictly enforcing the constraints, we measure how far the generated image pixels are from the feasibility domain through additional cost functions we attempt to minimize. For the constraint C 1 , we propose to use a 2 distance between the generated image M Y X and A Ŝ as:

L C1 = E y∼p Y ||M Y X (y) -A Ŝ|| 2 , (26)
with Ŝ = Ŝ0 Ŝ1 Ŝ2 the Stokes vector computed from the generated image by M Y X (y) using equation ( 14). Similarly, to enforce the constraint C 2 , a rectified linear penalty L C2 is considered, defined as:

L C2 = E y∼p Y max Ŝ2 1 + Ŝ2 2 -Ŝ2 0 , 0 . (27)
The loss L C1 enforces the respect of the acquisition conditions according to the calibration matrix A while L C2 pushes the generated images towards respecting the physical admissibility constraint on the Stokes vectors obtained from the generated image. Gathering all these elements, we train our Cycle-GAN by optimizing the following objective function:

L f inal = L CycleGAN + µL C1 + νL C2 . ( 28
)
The full training algorithm is summed up in Algorithm 1

Algorithm 1 Constrained CycleGAN training algorithm Require: X and Y two unpaired datasets, M XY and M Y X the mapping networks, D X and D Y the discrimination models, m the mini-batch size repeat sample a mini-batch {x i } m i=1 from X sample a mini-batch

{y i } m i=1 from Y update D X by stochastic gradient descent of m i=1 (D X (x i ) -1) 2 + (D X (M Y X (y i ))) 2 update D Y by stochastic gradient descent of m i=1 (D Y (y i ) -1) 2 + (D Y (M XY (x i ))) 2 sample a mini-batch {x i } m i=1 from X sample a mini-batch {y i } m i=1 from Y update M XY by stochastic gradient descent of n i=1 (D Y (M XY (x i )) -1) 2 + λ(||x i - M Y X (M XY (x i ))|| 1 +||y i -M XY (M Y X (y i ))|| 1 ) update M Y X by stochastic gradient descent of n i=1 (D X (M Y X (y i )) -1) 2 + λ ||x i - M Y X (M XY (x i )|| 1 +||y i -M XY (M Y X (y i ))|| 1 +µ(||M Y X (y i ) -A Ŝ|| 2 ) + ν max( Ŝ2 1 + Ŝ2 2 -Ŝ2 0 , 0) until a stopping condition is met
The non-negative hyper-parameters µ and ν ∈ R + control respectively the balance of calibration and admissibility constraints according to the CycleGAN loss L CycleGAN (see ( 23)). As the values of L C1 and L C2 are computed pixel-wisely, we consider their averages over the whole image in the objective function. The training principle of the proposed generative model is illustrated in Figure 5.

Experimental evaluation

Hereafter, the experimental setup, including the image generation procedure and its evaluation, is presented.

Polarimetric images generation using CycleGAN

To conduct the experiments, we rely on a combination of two datasets presented in [Bli+20; Bli+19], composed of polarimetric and RGB images. From these datasets we select 2485 unpaired images from each domain (RGB and polarimetry). Example instances are shown in Figure 6. The polarimetric images are of dimension 500 × 500 × 4. The latter dimension is due to the four intensities acquired by the camera, namely I 0 , I 45 , I 90 and I 135 . The RGB images are of dimension 906 × 945 × 3. Our constrained CycleGAN is trained for 400 epochs on randomly cropped patches of size 200×200. As for the constraints, we found experimentally that setting the hyper-parameters µ = 1 and ν = 1 in (28) provides the best performances. The hyperparameter λ, controlling the reconstruction cost, is set to λ = 10. The learning rate is decreased linearly from 2 × 10 -4 to 2 × 10 -6 during the 400 training epochs.

To evaluate the effectiveness of our trained generative model, we consider KITTI and BDD100K (only using daytime images since polarimetry fails to characterize objects during nighttime) which often serve as testbed in applications related to road scene object detection. The constrained-output Cy-cleGANs we train, are used to transfer RGB images from KITTI and BDD100K to the polarimetric do-main. The resulting datasets are denoted respectively as Polar-KITTI and Polar-BDD100K. Since the Cy-cleGAN architecture is fully convolutional, it has no requirement on the input image's size. Therefore, even though the model was trained on 200 × 200 patches, it scales straightforwardly to the images of size 1250 × 375 from KITTI and of size 1280 × 720 from BDD100K datasets. We also randomly horizontally flip, as flipping the image does not alter the physical properties of a polarimetric image [START_REF] Blanchon | Polarimetric Image Augmentation[END_REF].

To assess whether or not fulfilling the physical constraints is paramount, we investigate a variant of Polar-KITTI and Polar-BDD100K: we learn a standard unconstrained CycleGAN based on the same unpaired RGB/polarimetric images.

Evaluation of the generated images

In order to assert the ability of the generated Polar-KITTI and Polar-BDD100K datasets to preserve the relevant features for road scene applications, we train a detection network following the setup in Figure 4. For this experiment, a RetinaNet-50 [START_REF] Lin | Focal loss for dense object detection[END_REF] pretrained on the MS COCO dataset [START_REF] Lin | Microsoft coco: Common objects in context[END_REF] is finetuned in three different settings. In the first setup, the detection model is fine-tuned on the original RGB KITTI (or BDD100K) while the second one considers fine-tuning on the generated polarimetric images from KITTI (Polar-KITTI) or BDD100K (Polar-BDD100K) datasets. The third one uses the unconstrained variant of the generated images from KITTI or BDD100K datasets. After the first fine-tuning, the three final detection models are fine-tuned one last time on the real polarimetric dataset (described in Table 1). Overall, the trained CycleGAN and detection networks under these settings are evaluated in qualitative and quantitative ways. The end goal is to check: (i) the ability of the generated images to help learning polarimetry-based features for object detection, and (ii) the influence of respecting the polarimetric feasibility constraints on detection performances.

We measure the visual quality of the generated images by computing the classical Fréchet Inception Distance [START_REF] Heusel | Gans trained by a two time-scale update rule converge to a local Nash equilibrium[END_REF]. Computing this distance requires Figure 4: Full setup of our approach. For each experiment, we 1): train the generator M Y X ; 2): pre-train the RetinaNet-50 detection network on the MS-COCO dataset; 3): generate a full polarimetric dataset using the trained generator M Y X ; 4): perform a first training on the generated polarimetric KITTI dataset; 5): fine-tune on the real polarimetric dataset; and 6): evaluate the detection model on the test set of the real polarimetric dataset. Note that while the procedure is illustrated with the KITTI dataset, it straightforwardly extends to the BDD100K dataset. to extract visual features from each set of images (real and generated) using a pre-trained deep neural network (usually an Inception v3 [START_REF] Szegedy | Rethinking the inception architecture for computer vision[END_REF] network pretrained on ImageNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF]) and to evaluate the Fréchet (or Wasserstein) distance between the distributions of these features, which are assumed to be Gaussian distributions. We evaluate this distance using 509 images from each generated polarimetric dataset and from the test set as described in Table 1. As for the feature extractor, since the classical Inception v3 network is not adapted to polarimetric images, we use the convolutional part of a polarimetry-adapted RetinaNet detection network [START_REF] Blin | Road scenes analysis in adverse weather conditions by polarization-encoded images and adapted deep learning[END_REF], which has been trained on the MS-COCO dataset and fine-tuned on the same real polarimetric dataset that we used in our CycleGAN experiments. In order to evaluate the improvements in the detection, we compute the error rate evolution ER o . The improvement ER o on the detection of the object o is Table 1: Polarimetric dataset features. From the third row, the total number of instances within each class are indicated.

given by:

ER o = (1 -AP p o ) -(1 -AP RGB o ) 1 -AP RGB o , (29)

Results and discussion

First the qualitative coherence of the generated images is evaluated. The polarimetric images generated from their RGB equivalent are shown in Figure 7.

As for the constraints, Table 3 shows how including them to the CycleGAN's loss helps generating images which better fulfill the physical polarimetric properties at the pixel scale. The errors related to the constraints C 1 and C 2 on generated images are consistent with the observed measurement errors on the real images, whereas the unconstrained approach yields poor results. Constraint C 3 is met for all generated images thanks to the use of hyperbolic tangent as activation function for the last layer of the generative models. Additionally, the obtained Fréchet Inception Distances are of 6022.7 for the unconstrained CycleGAN and 4485.1 for our approach. Note that the scale of the FID scores computed with the pretrained RetinaNet is larger than when using a pretrained Inception v3 network. Thus, these scores are to be interpreted as relative metrics. This indicates that taking the constraints into account improves visual and physical quality of the generated samples. ). The mean and the median of the percentage of pixels in an image that do not fulfill the constraints C 2 is computed. Regarding the constraint C 1 , we normalize it by computing the mean and the median of ||I -AS||/(||I|| + ||AS||). Next, as an example, we show the benefits of the generated images through an object detection task. This enables to check if objects contained in the generated scene are physically coherent. A RetinaNetbased detection model is learned according to the setups described in Section 3.2 and the obtained detection performances in term of mean average precision (mAP ) are summarized in Table 2. We chose not to evaluate the bike and motorbike detection performances as the polarimetric dataset does not contain enough objects of those two classes.

As we can see in Table 2, using the generated polarimetric images improves the detection performance in real polarimetric images. The improvement is substantial for car and pedestrian detection. We achieve a 4% improvement of the error rate (see Equation 29) for car detection and of 12% for pedestrian detection which leads to a global improvement of 9% in the detection, using Polar-KITTI with constraints. Similarly, for the Polar-BDD100K dataset, we notice an improvement of 10% for pedestrian detection which leads to an increased mAP of 5% (pedestrians and cars). However, we notice that, for BDD100K, similar detection performances are obtained either for RGB or polarimetric images. This is due to the fact that generated images using CycleGAN tends to lack precision on small objects. In order to asses the impact of this effect, we compare the evolution of the detection scores when varying the minimal area of the bounding boxes from which objects are taken into account for the detection task. The obtained results are shown for the Polar-BDD100K and the RGB BDD100K datasets are shown in Figure 8 and illustrate that, when the minimal area of bounding boxes increases, the AP of car regarding the training including Polar-BDD100K, overcomes the one including RGB BDD100K. Even if the results are limited by the lower quality of the small objects in the generated images in this specific case, we can conclude that the generated polarimetric images help improving the overall detection results.

Conclusion and future work

In this work, we propose an efficient way to generate realistic polarimetric images subject to physical admissibility constraints. We design two loss terms that help enforcing these constraints and propose to adapt the CycleGAN algorithm to achieve the generation of physically admissible images. To train the proposed output-constrained CycleGAN, we combine the standard CycleGAN's objective function with two designed cost functions in order to handle the feasibility constraints related to each polarization-encoded pixel in the image. With the proposed generative model, we successfully translate RGB images from road scenes to polarimetric images showing an enhancement of the detection performances. Future work would consist in improving the quality of the small objects in generated images. Another extension could be the generation of polarimetric images to other domains such as medical and Synthetic-Aperture Radar (SAR) imaging. Additionally, to strictly ensure the physical feasibility constraints, solutions such as directly addressing the optimization task involved by the genuine constrained CycleGAN problem, instead of its proposed relaxation, using methods such as proximal gradient descent method could be explored. Finally, since the constraints we develop are not limited to the generative methods, they might be applicable to some other augmentation techniques, such as rotation or noise.

Acknowledgements

This work is supported by the ICUB project 2017 ANRprogram : ANR-17-CE22-0011. We also thank our colleagues at CRIANN, who provided us with the computation resources necessary for our experiments.

Figure 1 :

 1 Figure 1: Propagation principle of the electromagnetic wave, with E 0x and E 0y the amplitudes, λ the wavelength and φ the phase shift of the wave

  Polarimetric images are obtained by computing the Stokes vector related to each pixel. The acquisition principle is based on a device composed of a polarizer oriented at an angle α between the object and the sensor[START_REF] Wang | Polarimetric Camera Calibration Using an LCD Monitor[END_REF]. At least three acquisitions with three different angles are required to get the Stokes parameters. The reflected light from the object, represented by the unknown Stokes vector, passes through the polarizer before reaching the camera. In our work, we use a Polarcam 4D Technology polarimetric camera, providing simultaneously four images respectively obtained with four different linear polarizers oriented at (α i ) i=1:4 = (0°, 45°, 90°, 135°). The polarimetric camera measures an intensity I αi of the scene for each angle α i . The relationship between the Stokes vector S and the intensities I(α i ) i=1:4 reaching the camera is given by:

Figure 3 :

 3 Figure 3: Example of a polarimetric image. From left to right, the intensities corresponding to the polarizer rotation angles 0°, 45°, 90°and 135°.

Figure 5 :

 5 Figure 5: Training of the M Y X model using the Cy-cleGAN training process extended to our constraints. The full training objective consists of a GAN cost using the discriminator network D Y , a cycle-consistent reconstruction cost which ensures that M XY (X) remains consistent with X, and our custom objectives L C1 and L C2 (in red).

Figure 6 :

 6 Figure 6: Examples of images used to train the adapted CycleGAN. First row are the I 0 channels of some polarimetric images and second row are some RGB ones.

Figure 7 :

 7 Figure 7: Examples of generated polarimetric images. From left to right: I 0 , I 45 , I 90 , I 135 (ground truth), RGB image and I 0 , I 45 , I 90 , I 135 generated from RGB.

Figure 8 :

 8 Figure 8: Evolution of the average precision (AP) when setting a minimal area of the detected bounding boxes. Here green, blue and red lines respectively refer to the evolution of cars', mAP and person's detections. The dashed and the solid lines respectively refer to the use of BDD100K RGB and Polar-BDD100K during training.

Table 2 :

 2 Comparison of the detection performances for the person and car classes, as well as the mean average precision (mAP). RetinaNet-50 pre-trained on MS COCO is the baseline of all the experiments.

	Dataset	Class	Test	ER o	Dataset	Class	Test	ER o
	KITTI RGB	person 0.663 N/A BDD100K RGB person 0.736 N/A
		car	0.785 N/A		car	0.821 N/A
		mAP	0.724 N/A		mAP	0.778 N/A
	unconstrained person 0.673 -0.03	unconstrained	person 0.720	0.06
	Polar-KITTI	car	0.786 -0.01 Polar-BDD100K	car	0.816	0.03
		mAP	0.730 -0.02		mAP	0.768	0.05
	Polar-KITTI	person 0.704 -0.12 Polar-BDD100K person 0.762 -0.10
	with contraints	car	0.794 -0.04 with constraints	car	0.815	0.03
		mAP	0.749 -0.09		mAP	0.789 -0.05
	The first row is the fine-tunning on the original RGB KITTI/BDD100K. The second row is the fine-tuning
	on Polar-KITTI/Polar-BDD100K without constraints and the bottom row is the detection models fine-tuned
	on Polar-KITTI/Polar-BDD100K with the constraints. For a given dataset (KITTI or BDD100K), the best
	results overall are shown in boldface

Table 3 :

 3 Evaluation of the constraint fulfillment using the designed losses L C1 and L C2 at the pixel scale. Here, we compare the real polarimetric dataset to the generated polarimetric KITTI datasets, constrained and unconstrained. The column C indicates the evaluated constraint, with C 1 referring to the calibration constraint (I = AS) and C 2 to the optical constraint

	(S 2 0	S 2 1 + S 2 2

Since the polarimetric images can be very dark, we manually adjust their brightness in the Figures3, 4, 5, 6 and 7