Amélie Rouchon
email: amelie.rouchon@cea.fr

Walid Jarrah
email: walid.jarrah@cea.fr

Andrea Zoia
email: andrea.zoia@cea.fr

THE NEW NEUTRON NOISE SOLVER OF THE MONTE CARLO CODE TRIPOLI-4 R

Keywords: Neutron noise, Monte Carlo, TRIPOLI-4 R, Frequency domain, Complex statistical weights

Neutron noise appears as fluctuations of the neutron field induced by vibrations of fuel elements, control rods or any other mechanical structures in the core, as well as from global or local fluctuations in the flow, density or void fraction of the coolant. Neutron noise equations are obtained by assuming small perturbations of macroscopic cross-sections around a steady-state neutron field and by subsequently taking the Fourier transform in the frequency domain. Recently, a new Monte Carlo algorithm was proposed in order to solve the transport equation in neutron noise theory with complex-valued weights and a modified collision operator. This paper presents the new neutron noise solver based on these methods and implemented in the reference Monte Carlo code TRIPOLI-4 R developed at CEA. We illustrate the capabilities of the solver by considering the noise analysis of a UOX PWR assembly.

INTRODUCTION

Neutron noise analysis addresses the description of time-dependent flux fluctuations induced by small global or local perturbations of the macroscopic cross-sections, which may occur in nuclear reactors due to stochastic density fluctuations of the coolant, to vibrations of fuel elements, control rods, or any other structures in the core [START_REF] Pázsit | Noise Techniques in Nuclear Systems[END_REF]. Neutron noise techniques are adopted in the nuclear industry for non-invasive general monitoring, control and detection of anomalies in nuclear power plants [START_REF] Fry | Use of Neutron Noise of Diagnosis of In-vessel Anomalies in Light-Water Reactors[END_REF]. They are also applied to the measurement of the properties of the coolant, such as speed and void fraction.

The general noise equations are obtained by assuming small perturbations around a steady-state neutron flux and by subsequently taking the Fourier transform in the frequency domain. The outcome of the Fourier transform analysis is a fixed-source equation with complex operators for the perturbed neutron field, which can then be solved so as to predict noise measurements at detector locations.

Until recently, neutron noise equations have been only solved by analytical techniques [START_REF] Jonsson | Analytical inverstigation of the properties of the neutron noise induced by vibrationg absorber and fuel rods[END_REF] and by resorting to diffusion theory [START_REF] Demazière | CORE SIM: A multi-purpose neutronic tool for research and education[END_REF][START_REF] Malmir | Development of a 2-D 2-group neutron noise simulator for hexagonal geometries[END_REF]. It is therefore necessary to validate these approaches via Monte Carlo simulation. In 2013, a Monte Carlo algorithm was first proposed in order to solve the transport equation in neutron noise theory [START_REF] Yamamoto | Monte Carlo method with complex-valued weights for frequency domain analyses of neutron noise[END_REF][START_REF] Yamamoto | Implementation of a frequency-domain neutron noise analysis method in a production-level continuous energy Monte Carlo code: Verification and application in a BWR[END_REF]. Such algorithm is a cross-over between fixedsource and power iteration methods and adopts a weight cancellation technique. This method yields satisfactory results but has some shortcomings, such as the need of introducing a "binning procedure" for the weight cancellation: each fissile region must be divided into a large number of small regions where positive and negative weights are summed up and cancelled. In 2016, a second Monte Carlo algorithm was proposed [START_REF] Rouchon | A new Monte Carlo method for neutron noise calculations in the frequency domain[END_REF]: contrary to [START_REF] Yamamoto | Monte Carlo method with complex-valued weights for frequency domain analyses of neutron noise[END_REF], this method uses the conventional algorithm for fixed-source problems for all frequencies, does not need any weight cancellation technique, and is based on a modified collision kernel with a real total cross-section.

In this work, we present the new neutron noise solver based on this second Monte Carlo algorithm and implemented in the reference Monte Carlo code TRIPOLI-4 R developed at CEA [START_REF] Brun | TRIPOLI-4 R , CEA, EDF and AREVA reference Monte Carlo code[END_REF]. This paper is organized as follows. In Sec. 2, the general neutron noise theory will be briefly introduced and the Monte Carlo algorithm implemented in TRIPOLI-4 R will be presented. In Sec. 3, we will verify our Monte Carlo solver by comparison with some analytical solutions. In Sec. 4, we will illustrate the capabilities of the proposed Monte Carlo solver by performing the analysis of neutron noise for a standard UOX PWR assembly. Conclusions will be drawn in Sec. 5.

DESCRIPTION OF THE NEW TRIPOLI-4 R NEUTRON NOISE SOLVER

Neutron noise theory

The general noise equations are obtained by assuming small perturbations Ψ(r, Ω, E, t) = Ψ 0 (r, Ω, E)+ δΨ(r, Ω, E, t) (which allows for a linear theory) around a steady-state neutron flux Ψ 0 and by subsequently taking the Fourier transform in the frequency domain. The analysis is performed based on the neutron kinetic equations, including the coupling with neutron precursors. The outcome of the Fourier transform analysis is a fixed-source equation with complex operators for the perturbed neutron field δΨ, which can then be solved so as to predict noise measurements at detector locations. For each frequency, the perturbed neutron flux is a complex function having an amplitude and a phase. Imposing a periodic perturbation of the kinetic operator, the noise equation reads [START_REF] Pázsit | Noise Techniques in Nuclear Systems[END_REF]:

Ω • ∇ + Σ 0 (r, E) + i ω δΨ(r, Ω, E, ω) = Σ 0,s (r, Ω • Ω, E → E)δΨ(r, Ω , E , ω)dE dΩ + 1 k χ p (E) 4π ν p (E )Σ 0, f (r, E )δΨ(r, Ω , E , ω)dE dΩ + 1 k j χ j d (E) 4π ν j d,ω (E )Σ 0, f (r, E )δΨ(r, Ω , E , ω)dE dΩ + S (r, Ω, E, ω), ( 1
)
where i is the imaginary unit, ω = 2π f the angular frequency, and

ν j d,ω (E) =        λ 2 j -iλ j ω λ 2 j + ω 2        ν j d (E) (2)
for the precursor family j; all other notations are standard. Thus, because of the delayed neutrons, the production operator depends on the frequency ω. Equation (1) can be conceptually split into a system of two equations for the real and imaginary part of δΨ. The two equations are formally coupled by two terms: iω/ and the modified delayed production operator including the complex multiplicity ν j d,ω (E). The noise source S is defined by:

S (r, Ω, E, ω) = -δΣ(r, E, ω)Ψ 0 (r, Ω, E) + δΣ s (r, Ω • Ω, E → E, ω)Ψ 0 (r, Ω , E )dE dΩ + 1 k χ p (E) 4π ν p (E )δΣ f (r, E , ω)Ψ 0 (r, Ω , E )dE dΩ + 1 k j χ j d (E) 4π ν j d,ω (E )δΣ f (r, E , ω)Ψ 0 (r, Ω , E )dE dΩ , (3)
where δΣ x (r, E, ω) is the Fourier transform of the perturbed term of the macroscopic cross-section Σ x (r, E, t) = Σ 0,x (r, E) + δΣ x (r, E, t) with Σ 0,x the static macroscopic cross-section.

The Monte Carlo algorithm implemented in TRIPOLI-4 R

We briefly sketch the Monte Carlo algorithm that we have chosen to implement in the reference Monte Carlo code TRIPOLI-4 R ; a thorough description is given in [START_REF] Rouchon | A new Monte Carlo method for neutron noise calculations in the frequency domain[END_REF]. Similarly as in [START_REF] Yamamoto | Monte Carlo method with complex-valued weights for frequency domain analyses of neutron noise[END_REF], our method is based on the simulation of particles carrying complex statistical weights w(ω) = {w (ω), w (ω)}, where the signs of the real and imaginary parts of particle weights can be positive or negative. In [START_REF] Yamamoto | Monte Carlo method with complex-valued weights for frequency domain analyses of neutron noise[END_REF], the complex cross section Σ 0 + iω/ on the left-hand-side of Eq. ( 1) is dealt with explicitly by modifying the particle weights during flights. For our algorithm, we choose instead to work with a real cross section and we modify the collision kernel accordingly. This is achieved by following the strategy discussed in [START_REF] Zoia | Monte Carlo methods for reactor period calculations[END_REF]: we add a term ηω/ δΨ, η being a real constant having the same sign as ω, to both sides of Eq. ( 1), and we move the term iω/ δΨ to the right-hand-side. Then, the equation becomes:

Ω • ∇ + Σ 0 (r, E) + η ω δΨ(r, Ω, E, ω) = η -i η η ω δΨ(r, Ω, E, ω) + (...). (4)
In this case, we work with a real modified total cross-section Σ0 (r, E, ω) = Σ 0 (r, E) + Σ ω (E) > 0 where Σ ω (E) = ηω/ > 0. Hence, flight lengths are sampled as in standard Monte Carlo calculations, provided that Σ0 is used instead of Σ 0 .

Because of the structure of Eq. ( 4), the collision operator is now different from that of the regular Boltzmann equation, and we have to treat two types of particle productions: regular fission with probability Σ 0, f / Σ0 , and a special ω-production associated to a copy operator with probability Σ ω / Σ0 . We treat the regular fission as customary, i.e., the number of prompt or delayed fission neutrons is determined as Int(ν q Σ 0, f / Σ0 +ξ) where Int(.) denotes the integer part, ξ is a uniform random number, and q = p, d. Because of the factor ν j d,ω appearing in the delayed fission production term, the weight w d of each new delayed neutron created by fission is modified to:

w d = w k        λ 2 j -iλ j ω λ 2 j + ω 2        , ( 5
)
where w is the particle weight before the fission event. The term representing the ω-fission production consists of a copy of the incident neutron with a new weight w ω given by:

w ω = w η -i η . (6)
Implicit capture (with forced fission) and Russian roulette can be used as in standard Monte Carlo method. Similarly as in [START_REF] Yamamoto | Monte Carlo method with complex-valued weights for frequency domain analyses of neutron noise[END_REF], Russian roulette is applied separately to the absolute value of the real and imaginary parts of the particle weight: the particle is killed only if the real and the imaginary parts are both killed.

As described in [START_REF] Rouchon | A new Monte Carlo method for neutron noise calculations in the frequency domain[END_REF], at low and very high frequencies, i.e., far from the plateau region of the zeropower reactor transfer function of the system (typically below 0.01 Hz and above 1 kHz), Monte Carlo methods for noise propagation lead to the production of a very large number of particles and the calculation time explodes. To perform simulations at low and very high frequencies, in [START_REF] Yamamoto | Monte Carlo method with complex-valued weights for frequency domain analyses of neutron noise[END_REF] a weight cancellation technique is applied; the method proposed [START_REF] Rouchon | A new Monte Carlo method for neutron noise calculations in the frequency domain[END_REF] does not need any weight cancellation technique and turns out to be rather robust. For the frequency region of interest for most applications in reactor noise analysis, i.e., approximatively between 0.01 Hz and 100 Hz, the algorithm described above converges safely with η = 1. Extreme cases beyond this frequency region have been also tested and calculations converge by suppressing implicit capture (and possibly tuning the η value as well at very high frequencies).

To summarize, our method (i) uses the conventional algorithm for fixed-source problems for all frequencies, (ii) does not need any weight cancellation technique, and (iii) is based on a real total cross-section and a modified collision kernel. The major advantage of this algorithm is that it introduces minimal modifications to standard Monte Carlo algorithms for particle transport (the key modification concerning the weight modification at delayed fission events and the special "copy" event) and can thus be implemented in continuous-energy production Monte Carlo codes such as TRIPOLI-4 R . Scores such as flux and reaction rates over volumes and meshes have been extended in order to decompose the complex Monte Carlo estimators into modulus and phase, or equivalently real and imaginary part.

The noise source sampling method

In the general case, the noise source S is a complex function depending on the stationary flux Ψ 0 and the sign of its real and imaginary part can thus be space and/or energy-dependent. In order to sample arbitrary noise sources by Monte Carlo methods, we implemented in TRIPOLI-4 R the following method: a standard power iteration is first run in order to determine Ψ 0 exactly. Convergence is achieved after a sufficient number of inactive cycles; then, each term in Eq. ( 3) is sampled by weighting the 'noise particles' generated by each component of S by the stationary collision density associated to Ψ 0 . For instance, for the prompt fission term we would have

1 k χ p (E) 4π ν p (E ) δΣ f (r, E , ω) Σ 0, f (r, E ) Σ 0, f (r, E )Ψ 0 (r, Ω , E )dE dΩ , (7)
which means that at fission events we would instantiate ν p /k 'noise particles' with spectrum χ p (E) and with a statistical weight δΣ f (r, E , ω)/Σ 0, f (r, E ) depending on the specific perturbation. By analogy with the strategy discussed in [START_REF] Faucher | New kinetic simulation capabilities for TRIPOLI-4 R : Methods and applications[END_REF], a complete power iteration is performed once in the first batch: during the final cycle noise source particles are sampled so that the noise simulation can begin, while the generated fission neutrons are transferred to the second batch. This latter runs a few additional inactive cycles in order to ensure reasonable decorrelation between cycles before sampling the noise source for the second replica. The third batch gets the fission neutrons generated by the second one, and so on. In a parallel run, each processor would apply this strategy.

The proposed simulation scheme is illustrated in Fig. 1. The critical eigenvalue k that is required for the noise equations can be estimated during the first critical calculation, or be obtained from a separate calculation.

NUMERICAL VERIFICATIONS OF TRIPOLI-4 R NEUTRON NOISE SOLVER

In this section, we briefly discuss some numerical verifications of the TRIPOLI-4 R neutron noise solver against analytical solutions for an infinite homogeneous medium in the case of single-speed and continuous-energy problems, for an infinitely homogeneous cylindrical core, and for onedimensional homogeneous core surrounded by a reflector. All Monte Carlo results (track length estimator) are plotted with their 1-σ error bars (barely visible in the figures).

Infinite homogeneous medium

For an infinite homogeneous medium in the case of single-speed problem, the results of the TRIPOLI-4 R neutron noise solver are verified against analytical results (see [START_REF] Yamamoto | Monte Carlo method with complex-valued weights for frequency domain analyses of neutron noise[END_REF] for the problem description and the analytical solutions). Here, the noise source is defined by S = -1 + i at each frequency. Results are shown in Fig. 2: an excellent agreement is found. For an infinite homogeneous medium in the case of continuous-energy problem, the result of our neutron noise solver is again verified against analytical results (see Fig. 3). Here, the noise source is defined by S = 2 -3i between 1 and 2 MeV and S = -0.75 + 5i between 0.0001 and 0.0005 MeV at 3 Hz. For the sake of simplicity, for our tests we have modelled scattering by using a Maxwell spectrum and fission by a Watt spectrum. This example illustrates the capability of the neutron noise solver to correctly simulate continuous-energy problems.

Infinitely-long homogeneous cylindrical core

We have further probed the noise solver in the case of an infinitely homogeneous cylindrical core with two energy groups (see [START_REF] Yamamoto | Monte Carlo method with complex-valued weights for frequency domain analyses of neutron noise[END_REF] for all details of the problem data and analytical results in diffusion theory). Here, the noise source is defined by S = -1 for the thermal group at 1 Hz and at the center of the core. Results are shown in Fig. 4: some differences are observed at the center of the core, due to the expected slight discrepancies between solutions derived in diffusion theory and our transport results. Overall, the agreement is very satisfactory. The final verification test concerns a one-dimensional homogeneous core surrounded by a reflector with two energy groups (see [START_REF] Demazière | CORE SIM: A multi-purpose neutronic tool for research and education[END_REF] for all details of the problem data and analytical results). Results are shown in Fig. 5. Here, the noise source is defined by S = -3 -0.5i for the fast group and S = -1.2 + 2i for the thermal group at 1Hz and at the core/reflector interface. Again, an excellent agreement is found with respect to exact results.

ANALYSIS OF NEUTRON NOISE IN A UOX PWR ASSEMBLY

We conclude our analysis by illustrating the capabilities of the TRIPOLI-4 R neutron noise solver on a more realistic configuration, a standard UOX fuel assembly (see Fig. 6). We will first compare the induced neutron noise to the standard zero-power transfer function for the case of a simple delta-like noise source; then, we will consider the case of a vibration-induced noise source. We impose a simple isotropic noise source S at position r 0 within the fuel assembly and energy E 0 , with a given frequency ω 0 . We compare the induced neutron noise δΨ(ω) integrated over the whole assembly to the zero-power transfer function G 0 (ω) = 1/ iω Λ eff + β eff iω+λ eff , corresponding to the limit case of point-kinetics, i.e., a system responding to the perturbation insensitive of the space or energy effects. The position r 0 of the noise source varies from the center to the top right corner of the system, occupying the center of the fuel rods (positions R0, R4 and R8 in Fig. 6). We tested different values for the energy E 0 of the noise source, i.e. 1 MeV, 1 KeV and 0.025 eV. Here for the sake of conciseness we show the results for E 0 = 1 keV. The angular frequency ω 0 takes values that are located in the 'plateau' region of G 0 : physical perturbations in experimental and commercial reactors generally take place approximatively between ω low = λ eff ≈ 0.1 Hz and ω high = λ eff + β eff /Λ eff ≈ a few tens of Hz.

In Fig. 7, we show the comparison between the modulus and the phase of δΨ(ω) and G 0 as a function of the noise source position (R0, R4 and R8, see Fig. 6) at E 0 = 1 keV. As expected, we note that the computed neutron noise δΨ(ω) slightly deviates from the point kinetics behaviour. As an illustration, Fig. 8 presents the spatially-resolved modulus and phase of δΨ(ω) for the case of a noise source at position R4, energy 1 keV and frequency 30 Hz. We consider next the spatially-resolved modulus and phase of δΨ(ω) induced by a fuel pin vibration along x-axis at 1 Hz with an amplitude of 0.1 cm (the position of the perturbed fuel pin is detailed in Fig. 6). The sinusoidal vibration model has been described in [START_REF] Rouchon | Analysis of Vibration-induced Neutron Noise using Onedimension Noise Diffusion Theory[END_REF]. Results are shown in Fig. 9 for a noise source corresponding to a critical state Ψ 0 computed as detailed above.

CONCLUSIONS

In this paper, we have presented the new TRIPOLI-4 R solver for the neutron noise analysis in the frequency domain. Future work will concern the comparison with neutron noise simulations in the time and frequency domain performed with APOLLO3 R , the multi-purpose deterministic transport code under development at CEA [START_REF] Gammicchia | Neutron kinetics equations in APOLLO3 R code for application to noise problems[END_REF][START_REF] Rouchon | The new 3-D multigroup diffusion neutron noise solver of APOLLO3 R and a theoretical discussion of fission-modes noise[END_REF], and the validation of the developed solvers based on the new experimental campaigns carried out in the framework of the COLIBRI experimental program at the Crocus reactor (operated by EPFL, Lausanne, Switzerland) [START_REF] Rais | Towards the validation of neutron noise simulators: qualification of data acquisition systems[END_REF].

Figure 1 :

 1 Figure 1: Neutron noise TRIPOLI-4 R simulation process with M batches.

  (a) Modulus of the neutron noise. (b) Phase of the neutron noise.

Figure 2 :

 2 Figure 2: Modulus and phase of δΨ(ω) versus frequency in an infinite homogeneous medium (single-speed case).

  (a) Modulus of the neutron noise. (b) Phase of the neutron noise.

Figure 3 :

 3 Figure 3: Modulus and phase of δΨ(ω) versus energy at 3 Hz in an infinite homogeneous medium (continuous-energy case).

  (a) Modulus of the fast neutron noise. (b) Modulus of the thermal neutron noise.

Figure 4 :

 4 Figure 4: Modulus of δΨ(ω) versus the radial position from the core center at 1 Hz in an infinitely-long homogeneous cylindrical core.

3. 3 .

 3 One-dimensional homogeneous core surrounded by a reflector

  (a) Modulus of the fast neutron noise. (b) Modulus of the thermal neutron noise.

Figure 5 :

 5 Figure 5: Modulus of δΨ(ω) versus position at 1 Hz in an one-dimensional homogeneous core surrounded by a reflector.

Figure 6 :

 6 Figure 6: Top view of the UOX assembly.

  (a) Modulus of neutron noise (A.U.). (b) Phase of neutron noise (degrees).

Figure 7 :

 7 Figure 7: Comparison between G 0 and δΨ(ω) (modulus and phase) at 1 keV versus noise source position (R0, R4 and R8).

  (a) Modulus of neutron noise (A.U.). (b) Phase of neutron noise (degrees).

Figure 8 :

 8 Figure 8: Modulus and phase of δΨ(ω) induced by a simple isotropic noise source at position R4, energy 1 keV and frequency 30 Hz.

4. 2 .

 2 An example of a vibration-induced noise source

  (a) Modulus of neutron noise (A.U.). (b) Phase of neutron noise (degrees).

Figure 9 :

 9 Figure 9: Modulus and phase of δΨ(ω) induced by a fuel pin vibration along x-axis at 1 Hz with an amplitude of 0.1 cm.

ACKNOWLEDGEMENTS

TRIPOLI-4 R is a registered trademark of CEA. The authors thank Électricité de France (EDF) for partial financial support. The research leading to these results has received funding from the Euratom research and training programme 2014-2018 under grant agreement No 754316 (CORTEX project [16]).


D Ȃnut ¸ovidiu Pop

DETECTION OF PEDESTRIAN ACTIONS BASED ON DEEP LEARNING APPROACH

Keywords: Pedestrian Detection, Pedestrian Action Recognition, Deep Learning

The pedestrian detection has attracted considerable attention from research due to its vast applicability in the field of autonomous vehicles. In the last decade, various investigations were made to find an optimal solution to detect the pedestrians, but less of them were focused on detecting and recognition the pedestrian's action. In this paper, we converge on both issues: pedestrian detection and pedestrian action recognize at the current detection time (T=0) based on the JAAD dataset, employing deep learning approaches. We propose a pedestrian detection component based on Faster R-CNN able to detect the pedestrian and also recognize if the pedestrian is crossing the street in the detecting time. The method is in contrast with the commonly pedestrian detection systems, which only discriminate between pedestrians and non-pedestrians among other road users.

INTRODUCTION

Pedestrian detection is a significant problem for computer vision, which involves several applications, including robotics, surveillance, and the automotive industry. It is one of the main interests of transport safety since it implies reducing the number of traffic collisions and the protection of pedestrians (i.e., children and seniors), who are the most vulnerable road users.

There are almost 1.3 million persons die in road traffic collisions each year, and nearly 20-50 million are injured or disabled due to human errors inherited in the usual road traffic. Moreover, the clashes between cars and pedestrians are the leading cause of death among young people, and it could be effectively reduced if such human errors were eliminated by employing an Advanced Driver Assistance System (ADAS) for pedestrian detection.

Over the last decade, the scientific community and the automobile industry have contributed to the development of different types of ADAS systems in order to improve traffic safety. The Nissan company has developed a system which recognizes the vehicle's environment, including pedestrians, other vehicles, and the road. Lexus RX 2017 has a self-driving system which is linked up to a pedestrian detection system. More recently, the Audi ADAS system accumulates the data of the camera and/or radar sensor to determine the possibility of a collision by detecting pedestrians or cyclists and warns the driver with visual, acoustic and haptic alerts if a crash is coming.

These current ADAS systems still have difficulty distinguishing between human beings and nearby objects. In recent research investigations, deep learning neural networks have frequently improved detection performance. The impediment for those patterns is that they require a large amount of annotated data.

This paper proposes a pedestrian detection system which not only discriminates the pedestrians among other road users but also able to recognize the pedestrian actions at the current detection time (T=0).

The contribution of this paper concerns detecting and classifying pedestrian actions. To do so, we develop the following methodology relying on a deep learning approach:

• Train, all pedestrian samples with the Faster R-CNN-Inception version 2 for pedestrian detection, proposes [START_REF] Huang | Speed/accuracy trade-offs for modern convolutional object detectors[END_REF]; • Train all pedestrian samples also using the pedestrian actions tags (cross/not cross) with the CNN as mentioned above for detection and action recognition based on the Joint Attention for Autonomous Driving (JAAD) [START_REF] Rasouli | Are they going to cross? a benchmark dataset and baseline for pedestrian crosswalk behavior[END_REF] dataset;

The paper is organized as follows: Section 2 outlines sever existing approaches from the literature and supplies our main contribution. Section 3 presents an overview of our system. Section 4 describes the experiments and the results on the JAAD dataset. Finally, Section 5 presents our conclusions.

RELATED WORK

A wide variety of methodologies have been proposed with optimization in performance, resulting in the development of detection methods using deep learning approaches [START_REF] Dȃnut ¸ovidiu Pop | Incremental cross-modality deep learning for pedestrian recognition[END_REF][START_REF] Zhang | How far are we from solving pedestrian detection?[END_REF][START_REF] Schlosser | Fusing lidar and images for pedestrian detection using convolutional neural networks[END_REF] or combination of features followed by a trainable classifier [START_REF] Zhang | How far are we from solving pedestrian detection?[END_REF][START_REF] Benenson | Ten years of pedestrian detection, what have we learned?[END_REF].

A deep, unified pattern that conjointly learns feature extraction, deformation handling, occlusion handling, and classification evaluated on the Caltech and the ETH datasets for pedestrian detection was proposed in [START_REF] Ouyang | Jointly learning deep features, deformable parts, occlusion and classification for pedestrian detection[END_REF]. An investigation focused on the detection of small scale pedestrians on the Caltech data set connected with a CNN learning of features with an end-to-end approach was presented in [START_REF] Bunel | Detection of pedestrians at far distance[END_REF]. A combination of three CNNs to detect pedestrians at various scales was introduced on the same monocular vision data set [START_REF] Eisenbach | Cooperative multi-scale convolutional neural networks for person detection[END_REF]. A cascade Aggregated Channel Features detector is used in [START_REF] Chen | Pedestrian Detection with Deep Convolutional Neural Network[END_REF] to generate candidate pedestrian windows followed by a CNN-based classifier for verification purposes on monocular Caltech and stereo ETH data sets.

In [START_REF] Lan | Pedestrian detection based on yolo network model[END_REF] is presented a pedestrian detection based on a variation of YOLO network model, (three layers were added to the original one) in order to join the shallow layer pedestrian features to the deep layer pedestrian features and connect the high, and low-resolution pedestrian features.

A mixture of CNN-based pedestrian detection, tracking, and pose estimation to predict the crossing pedestrian actions based on JAAD dataset is addressed in [START_REF] Fang | Is the pedestrian going to cross? answering by 2d pose estimation[END_REF]. The authors utilize the Faster R-CNN object detector based on VGG16 CNN architecture for the classification task, use a multi-object tracking algorithm based on Kalman filter, apply the pose estimation pattern on the bonding box predicted by the tracking system and finally handle the SVM/Random Forest to classify the pedestrian actions (Crossing /Not Crossing). This approaches mentioned above use standard pedestrian detection pattern which only discriminates the pedestrian and non-pedestrian among other road users. To our knowledge, there is no prior research linked to pedestrian detection, which involves various pedestrian behavior tags.

We investigate the pedestrian detection issue in two ways, using the pedestrian and non-pedestrian tags, we call the Classical Pedestrian Detection Method (CPDM), and various pedestrian tags the: pedestrian, pedestrian crossing which we call Crossing Pedestrian Detection Method (CrPDM).

PEDESTRIAN DETECTION METHOD

This paper concerns solve the problem of pedestrian detection and pedestrian recognition actions using deep learning approach.

For developing a pedestrian detection system is mandatory to take into account three main components: the sensors employed to capture the visual data, the modality image processing elements, and the classification parts.

In general, all these components are synchronically developed to achieve a high detection performance, but seldom specific item could be investigated independently according to the target application. We concurrently exam in the detection part by applying a generic object detector based on the public Faster R-CNN [START_REF] Shaoqing Ren | Faster R-CNN: towards real-time object detection with region proposal networks[END_REF]. We handle the Inception CNN architecture versions 2 for the classification task with the TensorFlow public open-source implementation described in [START_REF] Huang | Speed/accuracy trade-offs for modern convolutional object detectors[END_REF]. All the training process is based on JAAD [START_REF] Rasouli | Are they going to cross? a benchmark dataset and baseline for pedestrian crosswalk behavior[END_REF] dataset. This dataset has different pedestrian tags. It has an annotation of pedestrians with behavioral tags and pedestrians without behavior tags.

EXPERIMENTS

This section presents the set of experiments, including setups and performance assessment of our approaches. 4.1. Data Setup. The experiments are completed on the JAAD dataset [START_REF] Rasouli | Are they going to cross? a benchmark dataset and baseline for pedestrian crosswalk behavior[END_REF] because of its data set in typical urban road traffic environments for various locations, times of the day, road and weather conditions. This dataset supplies pedestrian bounding boxes (BB) for pedestrian detection, tagged as non-occluded, partially occluded and heavily occluded BBs. Moreover, it includes the pedestrian actions annotations for several of them, the pedestrian attributes for estimating the pedestrian behavior and traffic scene elements.

The experiment employs all the pedestrian samples, including the partially and heavily occluded pedestrians for all training and testing process.

4.2.

Training, Testing and Evaluation Protocol. We used the first 70% of samples from the whole JAAD dataset for the learning process. The training set consists of first 190 video sequence training samples, and includes even the partially occluded and heavily occluded BBs, in contrast with [START_REF] Rasouli | Are they going to cross? a benchmark dataset and baseline for pedestrian crosswalk behavior[END_REF] where the authors used just a part of the dataset, omitting samples with partially and heavy occlusion. Moreover, the authors do not give a detailed explanation of how the datasets were divided into training and testing sets; hence, it does not allow a fair comparison.

The validation set represents 10% of the learning set. We used the holdout validation method, which held back from training the model. The evaluation of a model skill on the training dataset would result in a biased score. Therefore The JAAD dataset has three main pedestrian actions tags: • pedestrian completes crossing the street;

• pedestrian does not cross the street;

• pedestrian does not have any intention of crossing. The first pedestrian action we tag as pedestrian cross and others as pedestrian considering his/her intention is ambiguous or does not cross the street.

We adopt two approaches for the training stage (the main architecture is described in Fig 1 ): • using the Classical Pedestrian Detection Method (CPDM) where we consider all pedestrian samples without any specific tag (see Fig 2); • using the Crossing Pedestrian Detection Method (CrPDM) where we use various pedestrian tags: Pedestrian is crossing the street (PCr), and Pedestrian (P) for all other pedestrians who do not cross the street or their action is obscure (see Fig 3). We perform the CNN training process on 200000 iterations, using an initial learning rate value to 0.00063 with ADAM learning algorithm and momentum at 0.9 [START_REF] Rasouli | Are they going to cross? a benchmark dataset and baseline for pedestrian crosswalk behavior[END_REF]. We used the pre-trained weights from COCO dataset with the default Faster RCNN loss function which is optimized for a multi-task loss function.

The testing set used to assess the CNN model performance is independent of the training dataset. It contains 110 video samples, the last 30% of video samples from the whole JAAD dataset.

The evaluation process for all the CNN models is performed with Tensorflow Deep Neural Network Framework. The performances are assessed by the mean average precision (mAP) for the detection part using the TensorFlow metrics tool.

We calculate the Confidence Interval (CI) to evaluate whether one model is statistically better than another one.

(1) CI = 2 * 1.96 P (100 -P ) N %.

In this formulation, P represents the performance system (e.g., mAP) and N represents the number of video testing. 4.3. Results and Discussions. The detection results are presented in Table 1. We observed that the detection performance achieved with the CPDM method (using all samples as pedestrians) a good performs on Jaad dataset since it has to identify the pedestrian among other road users. The CrPDM approach (using multiple pedestrian tags), although it detects the pedestrians, cannot be associated with the first method because it also instantly classifier the action of the pedestrians during the detection step. Therefore its performance is less than the first detection approach. On the other hand, pedestrian detection using the multiple tags approach could be a start point for a deep investigation. This approach estimates the pedestrian actions on the current time (T=0) and could be beneficial for developing a pedestrian prediction system. We can not compare our detection models with JAAD approaches [START_REF] Rasouli | Are they going to cross? a benchmark dataset and baseline for pedestrian crosswalk behavior[END_REF] as our results are not directly comparable since the authors made a classification for a specific pedestrian action based on pedestrian attention information and only used the non-occluded pedestrian samples [START_REF] Rasouli | Are they going to cross? a benchmark dataset and baseline for pedestrian crosswalk behavior[END_REF]. Their approach is based on a variation of AlexNet-imagenet CNN where the input data are cropped beforehand.

Contrariwise we should evaluate our model using approximately the same non-occluded pedestrian training samples as in [START_REF] Rasouli | Are they going to cross? a benchmark dataset and baseline for pedestrian crosswalk behavior[END_REF] for a fair comparison, but we did not do that in this previous work since we assume that in the all traffic congestion even exist pedestrians who are partially and heavily occluded.

Nevertheless, we compare our detection models with another method [START_REF] Kim | Pedestrian detection with simplified depth prediction[END_REF] which is close to our first one. This approach is based on a variation of SSD-Inception CNN based on SvDPed dataset. The method merged the RGB images, low-resolution Lidar and the distance between the camera and object detected. Notwithstanding the [START_REF] Kim | Pedestrian detection with simplified depth prediction[END_REF] approach used many sensors for pedestrian detection, our methods outperforms this approach significantly (please see Table 1).

CONCLUSIONS

This paper presents a classical pedestrian detection system and a pedestrian detection system able to recognize even the pedestrian actions based on deep learning approaches using JAAD dataset.

We evaluated the pedestrian detection approach (we called Classical Pedestrian Detection Method (CPDM)), where all sample are tagged as pedestrian and not pedestrian and a pedestrian detection approach using multiple tags (pedestrian, pedestrian cross), which we call Crossing Pedestrian Detection Method (CrPDM). The first method achieved better performance since it has only to distinguish the pedestrians among other road users in contrast with the second one who has to recognize even the pedestrian actions. The second detection approach returned a weaker performance than the classical one. Contrariwise, we deem this approach could be a start point for a deep investigation. The experiments were carried out on the common object detector based on the public Faster R-CNN merged with Inception version 2 architecture for the classification part.

Future work will be concerned with improving and benchmarking of pedestrian detection using multiple pedestrian action tags and extending the method to a pedestrian prediction system.

Figure 1 .

 1 Figure 1. Pedestrian detection system architecture. Label represents the pedestrian tags which could be pedestrian, pedestrian crossing; BB cord represents the bounding box coordinates.

Figure 2 .

 2 Figure 2. Pedestrian detection using the pedestrian tag for all pedestrians.

Figure 3 .

 3 Figure 3. Pedestrian detection using multiple tags.

Table 1 .

 1 Our detection performances using one or multiple output labels. One label represents that all samples are tagged as a pedestrian. Multiple labels represent: P=Pedestrian, PCr=Pedestrian Crossing.

	Method	Train on	Output	mAP% ± CI
	Faster R-CNN Inception v2	All pedestrian samples tagged as P	Pedestrian BB Label	70.91 ± 1.61
	Faster R-CNN Inception v2	All Pedestrian with P and PCr Tags	Pedestrian BB+ Action Label	64.31 ± 1.70
	SSD Fusion Inception [14]	RGB, Lidar, Distance	Pedestrian BB Label	51.88


Cyril Banderier

Florian Luca

On the period mod m of polynomially-recursive sequences: a case study

Many polynomially-recursive sequences have a periodic behavior mod m. In this paper, we analyze the period mod m of a class of second-order polynomially-recursive sequences. Starting with a problem originally coming from an enumeration of avoiding pattern permutations, we give a generalization which appears to be linked with nice number theory notions (the Carmichael function, algebraic integers, Wieferich primes).

Introduction

In his analysis of sorting algorithms, Knuth introduced the notion of forbidden pattern in permutations, which later became a field of research per se [START_REF] Knuth | The Art of Computer Programming[END_REF]. By studying the basis of such forbidden patterns for permutations reachable with k right-jumps from the identity permutation, the authors of [START_REF] Banderier | Right-jumps and pattern avoiding permutations[END_REF] discovered that the permutations of size n in this basis were enumerated by the sequence of integers (b n ) n≥0 given by b 0 = 1, b 1 = 0,

b n+2 = 2nb n+1 + (1 + n -n 2 )b n for all n ≥ 0. ( 1
)
This is sequence A265165 in the OEIS 1 ; it starts like 0, 1, 2, 7, 32, 179, 1182, 8993, 77440, 744425, 7901410, 91774375. . . Such a sequence defined by a recurrence with polynomial coefficients in n is called P-recursive (for polynomially recursive). Some authors also call such sequences holonomic, or D-finite (see, e.g., [START_REF] Flajolet | Analytic Combinatorics[END_REF][START_REF] Gessel | Symmetric functions and P-recursiveness[END_REF]13,[START_REF] Stanley | Enumerative Combinatorics[END_REF]). The D-finite (for differentially finite) terminology comes from the fact that a sequence (f n ) n≥0 satisfies a linear recurrence with polynomial coefficients in n if and only if its generating function F (z) = n≥0 f n z n satisfies a linear differential equation with polynomial coefficients in z. Accordingly, P-recursive sequences and D-finite functions satisfy many closure properties: this contributes to make them ubiquitous in combinatorics, number theory, analysis of algorithms, computer algebra, mathematical physics, etc. It is not always the case that such sequences have a closed form. In our case, the generating function of (b n ) n≥0 has in fact a nice closed form involving the golden ratio. Indeed, putting

α := 1 + √ 5 2 and β := 1 - √ 5 2
for the two roots of the quadratic equation x 2 -x -1 = 0, it was shown in [START_REF] Banderier | Right-jumps and pattern avoiding permutations[END_REF] that the exponential generating function of (b n ) n≥0 , namely

B(x) = n≥0 b n x n n! , satisfies B(x) = β β -α (1 -x) α + α α -β (1 -x) β -1. ( 2
)
It should be stressed here that our sequence (b n ) n≥0 is an instance of a noteworthy phenomenon: it is one of the rare combinatorial sequences exhibiting an irrational exponent in its asymptotics:

b n n! ∼ α √ 5Γ(α -1) n α-2 (1 + o(1)) as n → ∞,
where Γ(z) = +∞ 0 t z-1 exp(-t)dt is the Euler gamma function. We refer to the wonderful book of Flajolet and Sedgewick [START_REF] Flajolet | Analytic Combinatorics[END_REF] for a few other examples of such a phenomenon in analytic combinatorics, and to [START_REF] Banderier | Right-jumps and pattern avoiding permutations[END_REF]Section 4] for further comments on the links between G-functions and (ir)rational exponents in the asymptotics of the coefficients.

P-recursive sequences are also of interest in number theory, where there is a vast literature analyzing the modular congruences of famous sequences, e.g., for the binomial coefficients, or the Fibonacci, Catalan, Motzkin, Apéry numbers, see [START_REF] Deutsch | Congruences for Catalan and Motzkin numbers and related sequences[END_REF][START_REF] Gessel | Combinatorial proofs of congruences[END_REF][START_REF] Kauers | A method for determining the mod-2 k behaviour of recursive sequences, with applications to subgroup counting[END_REF][START_REF] Rowland | A case study in meta-automation: automatic generation of congruence automata for combinatorial sequences[END_REF][START_REF] Xin | Keywords: D-finite function, Modular properties, P-recursive sequence, supercongruence, Carmichael function, Wieferich primes. (Concerned with sequences A123692, A265165[END_REF]. For example, the Apéry numbers satisfy A(p e q) = A(p e-1 q) mod p 3e , in which the exponent 3e in the modulus grows faster than the exponent e in the function argument. This phenomenon is sometimes called "supercongruence", and find roots in seminal works by Kummer and Ramanujan (see [START_REF] Victor | A q-microscope for supercongruences[END_REF][START_REF] Osburn | Gaussian hypergeometric series and supercongruences[END_REF][START_REF] Straub | Supercongruences for polynomial analogs of the Apéry numbers[END_REF] for more recent advances on this topic). Accordingly, many articles consider sequences modulo m = 2 r , or m = 3 r , or variants of power of a prime number.

We now restate an important result which holds for any m (not necessarily the power of a prime number).

Theorem 1 (Congruences and periods for P-recursive sequences [START_REF] Banderier | Right-jumps and pattern avoiding permutations[END_REF]Theorem 7]). Consider any P-recurrence of order r:

P 0 (n)u n = r i=1 P i (n)u n-i ,
where the polynomials P 0 (n), . . . , P r (n) belong to Z[n], and where the polynomial P 0 (n) is invertible mod m. Then the sequence (u n mod m) n≥0 is eventually periodic2 . In particular, sequences such that P 0 (n) = 1 are periodic mod m. Additionally, the preperiod and the period p are bounded by m 2r+1 , therefore one can efficiently compute them via the Knuth-Floyd cycle-finding algorithm (the tortoise and the hare algorithm).

N.B.: It is not always the case that P-recursive sequences are periodic mod p. E.g., it was proven in [START_REF] Klazar | On integrality and periodicity of the Motzkin numbers[END_REF] that Motzkin numbers are not periodic mod m, and it seems that

(n + 3)(n + 2)u n = 8(n -1)(n -2)u(n -2) + (7n 2 + 7n -2)u(n -1) , u 0 = 0, u 1 = 1 ,
is also not periodic mod m, for any m > 2 (this P-recursive sequence counts a famous class of permutations, namely, the Baxter permutations). This is coherent with Theorem 1, as the leading term in the recurrence (the factor (n + 3)(n + 2)) is not invertible mod m, for infinitely many n.

For our sequence (b n ) n≥1 (defined by recurrence (1)), this theorem explains the periodic behavior of b n mod m. Thanks to the bounds mentioned in Theorem 1, we can get b n mod m, by brute-force computation, for any given m. For example b n mod 15 is periodic of period 12 (after a preperiod n * = 9): (b n mod 15) n≥9 = (10, 5, 10, 10, 0, 10, 5, 10, 5, 5, 0, 5) ∞ .

The period can be quite large, for example b n mod 3617 has period 26158144. More generally, for every positive integer m, the sequence (b n mod m) n≥1 is eventually periodic, for some period p depending on m, as defined in the footnote on the previous page. For each m, let T m be the smallest possible period p. In this paper, we study some properties of (T m ) m≥1 . This is sequence A306699 in the OEIS; here are its first few values T 2 , . . . , T 100 :

Periodicity mod m and links with number theory

Our main result is the following. b) We have T m = 1 if and only if m is the product of primes p ≡ 0, 1, 4 (mod 5).

c) For every prime p, we have

T p | 2p ord 5 (p). d) If T m > 1 then 2 | T m if m is even, and 4 | T m if m is odd.
e) For m ≥ 3, we have T m = 2 if and only if m is even and m 2 is the product of primes p ≡ 0, 1, 4 (mod 5). f) For every prime p, we have

T p k | 2p k (p -1).
The function T m thus shares some similarities with the Carmichael function introduced in [2, p. 39], and it is expected that its asymptotic behavior is also similar (following, e.g., the lines of [START_REF] Erdős | Carmichael's lambda function[END_REF]). In this article, we focus on the rich arithmetic properties of this function. Note that Theorem 2 allows computing T m in a much faster way than the brute-force algorithm mentioned in Section 1: the complexity goes from m 2r+1 via brute-force to ln(m) 3 via Shor's factorization algorithm [START_REF] Shor | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer[END_REF] (or to sub-exponential complexity in ln(m) with other efficient algorithms, if one does not want to rely on the use of quantum computers!).

Proof of Part a). The proof will use a little preliminary result. We call T m the "eventual period of the sequence mod m", or, for short, the "period", even if the sequence starts with some terms which does not satisfy the periodic pattern. The following lemma holds for all eventually periodic sequences of integers.

u a+d = u n+Aa+Bb ≡ u (n+Aa)+bB (mod m) ≡ u n+Aa (mod m) ≡ u n (mod m) so d < a is a period of (u n ) n≥0 modulo m, contradicting the minimality of a.
An immediate consequence is the following: Proof of Part b). We use the generating function [START_REF] Carmichael | The Theory of Numbers[END_REF], which tells us that

[x n ]B(x) = b n n! = (-1) n √ 5 α β n -β α n . (3)
Thus,

b n = (-1) n-1 √ 5 (βα(α -1) • • • (α -(n -1)) -αβ(β -1) • • • (β -(n -1))) . (4)
By Fermat's little theorem,

p-1 k=0 (X -k) = X p -X (mod p). (5)
Now, assume that p ≡ 1, 4 (mod 5). Then

p-1 k=0 (α -k) ≡ α p -α (mod p) ≡ 0 (mod p),
where for the last congruence we used the law of quadratic reciprocity: since p ≡ 1, 4 (mod 5), we have 5

p = p 5 = 1,
where • p is the Legendre symbol. Thus,

α p = 1 + √ 5 2 p ≡ 1 + √ 5 • 5 (p-1)/2 2 p (mod p) ≡ α (mod p), (6)
because 5 (p-1)/2 ≡ 5 p ≡ 1 (mod p) by Euler's criterion.

In the above and in what follows, for two algebraic integers δ, γ and an integer m we write δ ≡ γ (mod m) if the number (δ -γ)/m is an algebraic integer. This shows that

1 p p-1 k=0 (α -k)
is an algebraic integer. The same is true with α replaced by β. Now take r ≥ 1 be any integer and take n ≥ pr. Then, for each = 0, 1, . . . , r -1, we have that both

1 p p-1 k=0 (α -(p + k)) and 1 p p-1 k=0 (β -(p + k)) are algebraic integers. Thus, if n ≥ pr, then √ 5b n p r = (-1) n-1 β r-1 =0 p-1 k=0 (α -(p + k)) n-1 k=pr (α -k) -α r-1 =0 p-1 k=0 (β -(p + k)) n-1 k=pr (β -k)
is an algebraic integer. Thus, 5b 2 n /p 2r is an algebraic integer and a rational number, so an integer. Since p = 5, it follows that p 2r | b 2 n , so p r | b n for n ≥ pr. This shows that T p r = 1 for all such primes p and positive integers r. The same is true for p = 5. There we use that α -3 = √ 5β, so √ 5 | α -3. Thus, if n ≥ 10r, we have that

n k=1 (α -k) is a multiple of 2r-1 =0 (α -(3 + 5 )) in Z[(1 + √ 5)/2],
which in turn is a multiple of 5 (α -k) ≡ α p -α (mod p).

Since 5 (p-1)/2 ≡ -1 (mod p), the argument from [START_REF] Gessel | Combinatorial proofs of congruences[END_REF] shows that α p ≡ β (mod p). Thus

2p k=1 (α -k) = p k=1 (α -k) 2p k=p+1 (α -k) ≡ (β -α) 2 (mod p) ≡ 5 (mod p).
The same is true for α replaced by β. Thus, it follows that for n > 2p, we have

b n+2p = (-1) n+2p-1 √ 5 β n+2p-1 k=0 (α -k) -α n+2p-1 k=0 (β -k) ≡ (-1) n-1 √ 5 5 β n-1 k=0 (α -k) -α n-1 k=0 (β -k) (mod p) ≡ 5b n (mod p).
Applying this k times, we get

b n+2pk ≡ 5 k b n (mod p).
Taking k = p-1 and applying Fermat's little theorem 5 p-1 ≡ 1 (mod p), we get T p | 2p(p-1). We can optimize this idea by taking k = ord p (5), where ord p (5) is the order of 5 modulo p (the smallest k > 0 such that 5 k ≡ 1 (mod p)), this gives the stronger wanted claim: T p | 2p ord p [START_REF] Flajolet | Analytic Combinatorics[END_REF].

Proof of Part d).

By a), we know that T p | T pm . Taking p = 2, one gets 2 | T m . Now, if T m > 1, by b), there is at least a prime p = 2, 3 (mod 5) such that p | m. We then have T p | T m by a). We now prove by contradiction that T p is a multiple of 4.

Take a prime p ≥ 3 and assume ν 2 (T p ) < 2, where ν q (a) is the exponent of q in the factorization of a. That is, T p would either be odd or 2 times an odd number. Since T p | 2p(p -1), it would follow that if we write p -1 = 2 a k, where k is odd, then T p | 2pk. Thus, one would have

b n ≡ b n+2pk ≡ 5 k b n (mod p) (7)
for all n > n p . Since p = 2, 3 (mod 5), 5 is not a quadratic residue, and thus 5 k ≡ 1 (mod p) (since -1 ≡ 5 (p-1)/2 ≡ (5 k ) 2 a-1 (mod p)). So, the above congruence [START_REF] Gessel | Symmetric functions and P-recursiveness[END_REF]

b n = β n-1 k=0 (α -k) -α n-1 k=0 (β -k) ≡ 0 (mod p) , b n+1 = β n-1 k=0 (α -k) (α -n) -β n-1 k=0 (β -k) (β -n) ≡ 0 (mod p).
We treat this as a linear system in the two unknowns

(X, Y ) = β n-1 k=0 (α -k), α n-1 k=0 (β -k)
in the field with p 2 elements Z[α]/pZ[α]. This is homogeneous. None of X or Y is 0 since p cannot divide β n-1 k=0 (α -k). Thus, it must be that the determinant of the above matrix is 0 modulo p, but this is 1

-1 α -n -(β -n) = √ 5,
which is invertible modulo p. Thus, indeed, it is not possible that b n and b n+1 is a multiple of p for all large n, getting a contradiction. This shows that T p is a multiple of 4.

Proof of Part e).

Let m be of shape different from the one required in Part b), i.e., m has now at least one prime p ≡ 2, 3 (mod 5) such that p | m. Then 4 | T p by what we have done above, and so 4 | T m by a). Thus, such m cannot participate in the situations described either at d) or e). Further, one has T 4 = 8 as (b n mod 4) n≥0 = (1, 0, 1, 2, 3, 0, 3, 2) ∞ . Thus, if 4 | m, then 8 | T m . Hence, if T m = 2, then the only possibility is that 2 | m and m/2 is a product of primes congruent to 0, 1, 4 modulo 5. Conversely, if m has such structure then T m = 2 by a) and the fact that T 2 = 2 and T p r = 1 for all odd prime power factors p r of m. This ends the proof of e).

Proof of Part f ). Finally, f) is based on a preliminary result: a slight generalization of (5), namely

p r -1 k=0 (X -k) ≡ (X p -X) p r-1 (mod p r ) (8)
valid for all odd primes p and r ≥ 1. Let us prove (8) by induction on r. We first prove it for r = 2. We return to (5) and write

p-1 k=0 (X -k) = X p -X + pH 1 (X),
where

H 1 (X) ∈ Z[X].
Changing X to X -p for = 0, 1, . . . , p -1, we get that

p-1 k=0 (X -(p + k)) = (X -p ) p -(X -p ) + pH(X -p ) ≡ (X p -X -pH(X)) -p (mod p 2 ).
In the above, we used the fact that H(X -p ) ≡ H(X) (mod p). Thus,

p 2 -1 k=0 (X -k) = p-1 =0 p-1 k=0 (X -(p + k)) ≡ p-1 k=0 ((X p -X -pH(X)) -p ) (mod p 2 ) ≡ (X p -X -pH(X)) p -(X p -X -pH(X)) p-1 p p-1 =0 (mod p 2 ) ≡ (X p -X) p -(X p -X -pH(X)) p-1 p p(p -1) 2 (mod p 2 ) ≡ (X p -X) p (mod p 2 ).
In the above, we used the fact that p is odd so p(p -1)/2 is a multiple of p. This proves (8) for r = 2. Now, assuming that (8) holds for p r , for some r ≥ 2, we get that for all ≥ 0, we have

p r -1 k=0 (X -(p r + k)) ≡ ((X -p r ) p -(X -p r )) p r-1 + p r H r (X -p r ) (mod p r+1 ) ≡ (X p -X) p r-1 + p r H r (X) (mod p r+1 ),
where H r (X) ∈ Z[X]. This allows concluding the induction step, and thus the generaliza-tion (8) that we wanted:

p r+1 -1 k=0 (X -k) = p =0 p r -1 k=0 (X -(p r + k))
≡ ((X p -X) p r-1 + p r H r (X)) p (mod p r+1 ) ≡ (X p -X) p r (mod p r+1 ).

Equipped with this preliminary result, letting p > 2 be congruent to 2, 3 (mod 5), evaluating the above identity in α, and using that α p ≡ β (mod p), we get that

p r -1 k=0 (α -k) ≡ (X p -X) p r-1 (mod p r ) ≡ (α p -α) p r-1 (mod p r ) ≡ (β -α) p r-1 (mod p r ). This shows that 2p r -1 k=0 (α -k) ≡ (β -α) 2p r-1 (mod p r ) ≡ 5 p r-1 (mod p r ).
The same is true for β; this leads to

b n+2p r ≡ (-1) n+2p r -1 √ 5 5 p r-1 β n-1 k=0 (α -k) -α n-1 k=0 (β -k) (mod p r ) ≡ 5 p r-1 b n (mod p r ).
Thus, applying this k times, we get

b n+2p r k ≡ 5 p r-1 k b n (mod p r ). (9)
By Euler's theorem a φ(n) ≡ 1 (mod n), one has 5 p r-1 (p-1) ≡ 1 (mod p r ). Thus, taking k = p -1 in (9), we get b n+2p r (p-1) ≡ b n (mod p r ). Therefore, T p r | 2p r (p -1). N.B.: As in the proof of c), we can optimize this idea; indeed ord 5 (p r ) = p r-1 ord 5 (p) and thus taking k = ord 5 (p), one gets T p r | 2p r ord 5 (p). Finally, it remains to prove f) for p = 2. Here, by inspection, we have

7 k=0 (X -k) ≡ (X 2 -X) 4 (mod 4).
By induction on r ≥ 2, one shows that

2 r+1 -1 k=0 (X -k) ≡ (X 2 -X) 2 r (mod 2 r ).
should be the zero element in the finite field Z[α]/pZ[α], with p 2 elements, it could be that the "probability" that this condition happens is 1/p 2 . By the same logic, the "probability" that p is base-5 Wieferich should be 1/p. Assuming these events to be independent, we could infer that the probability that both these conditions hold is 1/p 3 . Then, as the series p≡2,3 (mod 5)

1 p 3
is convergent, this heuristically suggests that there should be only finitely many primes p ≡ 2 or 3 (mod 5) such that T p 2 | 2p(p -1).

Finally, our results apply to other sequences as well. More precisely, let a, b be integers and let α, β be the roots of x 2 -ax -b. Let What are the periods mod m of such sequences?

B(x) = β β -α (1 -x) α + α α -β (1 -x) β -1 = n≥0 b n x n n! .
• In case α and β are rational (hence, integers), B(x) is a rational function, so b n = n!u n , where (u n ) n≥0 is binary recurrent with constant coefficients. It then follows that b n ≡ 0 (mod m) for all m provided n > n m is sufficiently large. Thus, T m = 1.

• In case α, β are irrational, then we get a result similar to Theorem 2 (where we had (a, b) = (1, 1)). Namely, b n ≡ 0 (mod m) for all n sufficiently large whenever m is the product of odd primes p for which the Legendre symbol ∆ p = 0, 1, where ∆ = a 2 + 4b is the discriminant of the quadratic x 2 -ax -b. In case p is odd and ∆ p = -1, we have that T p | 2p(p -1) and T p is a multiple of 4. Also, T p r | 2p r (p -1) for all r ≥ 1 in this case. The proofs are similar. In the case of the prime 2, one needs to distinguish cases according to the parities of a, b. For example, if a and b are odd, then ∆ ≡ 5 (mod 8), so 2 is not a quadratic residue modulo ∆, so T 2 r | 2 r+1 for all r ≥ 1, whereas if a is odd and b is even then T 2 = 1.

This concludes our analysis of the periodicity of such P-recursive sequences mod m.

Theorem 2 .

 2 Let (b n ) n≥0 be the sequence defined by the recurrence of Formula 1. The period T m of this sequence b n mod m satisfies: a) If m = p e 1 1 • • • p e k k (where p 1 , . . . , p k are distinct primes), then 3 T m = lcm(T p e 1 1 , . . . , T p e k k).

Lemma 3 .

 3 T m divides all other periods of (u n ) n≥0 modulo m.Proof. Let T m = a and assume there is b (not a multiple of a) which is also a period modulo m. Thus, there are n a , n b such that u n+a ≡ u n (mod m) for all n > n a and u n ≡ u n+b (mod m) for all n > n b . Let d = gcd(a, b). By Bézout's identity, one has then d = Aa + Bb for some integers A, B. Let n a,b = max{n a , n b } + |A|a + |B|b and assume that n > n a,b . Then

Corollary 4 .

 4 We have T lcm(m 1 ,...,mr) = lcm(T m 1 , . . . , T mr ).Proof. First consider r = 2, and let a := m 1 , b := m 2 . Since lcm(T a , T b ) is a multiple of both T a and T b , it follows that it is a period of (u n ) n≥0 modulo both a and b, so modulo lcm(a, b). It remains to prove that it is the minimal one. To this aim, suppose that T lcm(a,b) < lcm(T a , T b ). Then either T a T lcm(a,b) or T b T lcm(a,b) . Since the two cases are similar, we only deal with the first one. In this case we would have that both T a and T lcm(a,b) would be periods modulo a. By the previous lemma, this would force gcd(T a , T lcm(a,b) ) < T a , which would obviously be a contradiction. Now, a trivial induction on the number r ≥ 2 gives that T lcm(m 1 ,...,mr) = lcm(T m 1 , . . . , T mr ) holds for all positive integers m 1 , . . . , m r . In particular Part a) of Theorem 2 holds: T m = lcm(T p e 1 1 , . . . , T p e k k ). Let us now tackle the proofs of Parts b)-f).

r = √ 5 5 2

 55 2r in Z[(1 + √ 5)/2]. Thus, if n ≥ 10r, then 5 r | b n . This shows that also T 5 r = 1 and in fact, m | b n for all n > n m if m is made up only of primes 0, 1, 4 (mod 5). This finishes the proof of b). Proof of Part c). The claim is satisfied for p = 2, as (b n mod 2) n≥0 = (1, 0) ∞ , thus T 2 = 2 | 4. Consider now p > 2. By Part b), it suffices to consider odd primes p ≡ 2, 3 (mod 5). Evaluating Formula (5) at α = 1+ √

  would imply that p | (5 k -1)b n but p 5 k -1, so b n ≡ 0 (mod p) for all large n. Take n and n + 1 and rewrite what we got, i.e., b n ≡ b n+1 ≡ 0 (mod p) in Z[α]/pZ[α] as

  Accordingly, the sequence (b n ) n≥0 satisfies b 0 = 1, b 1 = 0, and, for n ≥ 0 b n+2 = (2n -a + 1)b n+1 + (b + an -n 2 )b n .

  2, 12, 8, 1, 12, 84, 8, 36, 2, 1, 24, 104, 84, 12, 16, 544, 36, 1, 8, 84, 2, 1012, 24, 1, 104, 108, 168, 1, 12, 1, 32, 12, 544, 84, 72, 2664, 2, 312, 8, 1, 84, 3612, 8, 36, 1012, 4324, 48, 588, 2, 1632, 104, 5512, 108, 1, 168, 12, 2, 1,

24, 1, 2, 252, 64, 104, 12, 2948, 544, 3036, 84, 1, 72, 10512, 2664, 12, 8, 84, 312, 1, 16, 324, 2, 13612, 168, 544, 3612, 12, 8, 1, 36, 2184, 2024, 12, 4324, 1, 96, 18624, 588, 36, 8. Do you detect some hidden patterns in this sequence? This is what we tackle in the next section.

OEIS stands for the On-Line Encyclopedia of Integer Sequences, see https://oeis.org.

An eventually periodic sequence of period p is a sequence for which u n+p = u n for all n ≥ n * (n * is called the preperiod). Some authors use the terminology "ultimately periodic" instead. In the sequel, as the context is clear, we will often omit the word "eventually".

As usual, lcm stands for the least common multiple.

A prime p is a Wieferich prime in base b if b p-1 ≡ 1 (mod p 2 ). This notion was introduced (with b = 2) by Arthur Wieferich in 1909 in his work on Fermat's last theorem[START_REF] Wieferich | Zum letzten Fermatschen Theorem[END_REF].

Acknowledgments

We thank the editor and the referee for comments which improved the quality of our paper. This work has been initiated when the second author was invited professor at the Paris Nord University, in April 2018. In addition, Florian Luca was supported in part by grant CPRR160325161141 and an A-rated scientist award both from the NRF of South Africa.

Evaluating this in α, we get

The same holds for β, so

showing that T 2 r | 2 r+1 for all r ≥ 2.

Comments and generalizations

Along the proof of our main result we showed that if p ≡ 2 or 3 (mod 5), then

From here we deduced that T p | 2p(p -1) via the fact that 5 p-1 ≡ 1 (mod p). One may ask whether it can be the case that

, for some prime p?

Well, first of all, it implies that 5 p-1 ≡ 1 (mod p 2 ). This makes p a base-5 Wieferich prime 4 . Despite the fact that it is conjectured that there are infinitely many such primes, only 7 base-5 Wieferich primes are currently known! (They are listed as A123692). Amongst them, only p = 2, 40487, 1645333507, and 6692367337 are additionally congruent to 2 (mod 5), and none is known to be congruent to 3 (mod 5). Note that the condition of p ≡ 2 or 3 (mod 5) being base-5 Wieferich is not sufficient to have the divisibility property [START_REF] Klazar | On integrality and periodicity of the Motzkin numbers[END_REF]. So, how many other primes could lead to T p 2 | 2p(p -1)? A close analysis of our arguments show that in addition to be a base-5 Wieferich prime, it should also hold that


est destinée au dépôt

La montée industrielle, l'agriculture intensive ainsi que l'évolution démographique font de la pollution de l'environnement un des enjeux majeurs de notre siècle. Les milieux impactés sont l'air, les sols et les eaux et différents procédés de décontamination voient régulièrement le jour. En ce sens, la décontamination des eaux à l'aide de supports solides est aujourd'hui un procédé qui tend à se développer fortement dans les années à venir. Les MOFs (Metal-Organic Frameworks) sont une nouvelle classe de matériaux poreux et apparaissent prometteurs pour ce mode de décontamination. Ces matériaux résultent de l'auto-assemblage d'un métal et d'un ligand organique. Ils sont étudiés et appréciés pour leur surface spécifique importante, leur taille de pores modulable, leurs propriétés contrôlables et leur stabilité chimique [1] . La littérature a montré que l'utilisation de MOFs pour l'extraction de polluants organiques de l'eau peut s'avérer efficace, comme c'est le cas des MOFs à base de zirconium UiO-66 qui ont révélé un effet adsorbant important sur les BTEX [2] . Les sources de MOF sont multiples. Ainsi pour des raisons environnementales, les matériaux utilisés dans cette étude proviennent du recyclage de batteries Li-ion. Cependant, la synthèse de MOF conduit généralement à l'obtention d'un matériau sous forme de poudre, ce qui s'avère peu adapté à l'application industrielle. Les recherches actuelles s'accentuent donc sur la mise en forme des MOFs sur supports solides et ainsi que sur leur assemblage sous forme monolithique. L'une des voies possible consiste réaliser un monolithe à partir d'émulsions de Pickering. Les émulsions de Pickering sont des émulsions stabilisées par des particules solides, des MOFs dans cette étude, qui s'adsorbant à l'interface phase aqueusephase organique. Dans le cas d'émulsions à haute phase interne (HIPE), ces dernières apparaissent sous l'aspect d'un gel et peuvent alors être utilisées en tant que « template » pour la synthèse de matériaux monolithiques macroporeux. La phase organique peut être éliminée grâce à des traitements spécifiques (CO 2 supercritique par exemple) et la taille des gouttelettes définit alors la porosité finale du monolithe. [3]

  Mise en oeuvre de MOF sous forme monolithique à partir d'émulsions de Pickering F. Lorignon, A. Gossard, M. Cognet, Michaël Carboni, D. Meyer To cite this version: F. Lorignon, A. Gossard, M. Cognet, Michaël Carboni, D. Meyer. Mise en oeuvre de MOF sous forme monolithique à partir d'émulsions de Pickering. Journees de Chimie de Coordination 2019, Jan 2019, Montpellier, France. hal-02411089

Figure 1 :

 1 Figure 1: Gel résultant d'une émulsion de Pickering-HIPE avant séchage au CO2 supercritique Statut : Doctorant 1 ière année

Journées de Chimie de Coordination de la SCF,

	31 janvier & 1 er février 2019, Montpellier	Oral/Poster

Mise en oeuvre de MOF sous forme monolithique à partir d'émulsions de Pickering

	Fabrice Lorignon a,b *, Marine Cognet a , Alban Gossard b , Michael Carboni a ,
	Daniel Meyer a
	a Institut de Chimie Séparative de Marcoule, UMR 5257, CEA/CNRS/ENSCM/UM,
	F-30207 Bagnols-sur-Cèze, France b CEA, DEN, DE2D, SEAD, Laboratoire des Procédés Supercritiques et de Décontamination,
	Marcoule, F-30207 Bagnols-sur-Cèze, France
	E-mail : fabrice.lorignon@cea.fr


Leonardo Iannucci
email: leonardo.iannucci@polito.it

Luca Lombardo

Marco Parvis

Régine Basséguy
email: regine.basseguy@ensiacet.fr

Pierangela Cristiani
email: pierangela.cristiani@rse-web.it

Emma Angelini

Sabrina Grassini
email: sabrina.grassini@polito.it

An imaging system for microbial corrosion analysis

Keywords: Microbial Corrosion, Image Processing, Implants, Scanning Electron Microscopy

This paper describes a viable and self-contained imaging system able to assess and quantify the effects of microbial corrosion on metals surface. The proposed image processing uses Scanning Electron Microscope micrographs to analyze bacteria attachment on sample surface and to estimate the degree of degradation of the material. After a preliminary brightness and contrast normalization, which refines the image taken by the operator, the software is able to identify dark spots on the clear metal surface. These are then attributed to singly attached bacteria or to larger clusters, which are the most dangerous ones, as they could overlay corrosion pits. After that, the degradation of the material is evaluated through the quantification of microbial attachment on the surface and through dimensional distribution of bacteria clusters.

I. INTRODUCTION

Prosthesis implantation is a surgical operation in which the choice of material is an issue of paramount importance. The interaction between a material (especially in case of metal alloys) and the body fluids can lead to corrosion phenomena that, in the most serious cases, can cause the failure of the implant. A particularly severe degradation is due to microbial corrosion, in which bacteria catalyze metal oxidation, significantly increasing the corrosion rate.

Since corrosion is an electrochemical reaction, electrochemical techniques are generally the most effective in the study of material-environment interaction and of corrosion mechanisms [START_REF] Iannucci | iHomeX: An Internet-Enabled Laboratory for Long-Term Experiment Management[END_REF]. In the field of biomaterials, Electrochemical Impedance Spectroscopy, monitoring of open circuit potential, polarization curves and other traditional electrochemical measurements are routinely employed to assess the corrosion behavior of different metals and alloys when in contact with electrolytic solutions that simulate the aggressiveness of body fluids. However, most of them can't be considered as stand-alone techniques, as they must be coupled to each other or to other chemico-physical characterization techniques such as optical and electron microscopy, in order to investigate the mechanism and the morphology of the corrosion reactions occurring on the metallic surface. From this considerations, the necessity to develop easy-to-use solutions able to quantitative evaluate the extent of the degradation phenomenon occurred on the material surface arises.

Many are the possible approaches related to imaging analysis when microbial corrosion (MIC) is concerned. In particular, these methods can be classified taking in account the technique used for image collection or the main purpose of the investigation (corrosion assessment or biofilm growth). When surface morphology is the main concern, often stereomicroscope, reflected light microscopy and SEM (Scanning Electron Microscopy) are used [START_REF] Caines | Analysis of pitting corrosion on steel under insulation in marine environments[END_REF]- [START_REF] Pereira | Morphology Characterization and Kinetics Evaluation of Pitting Corrosion of Commercially Pure Aluminium by Digital Image Analysis[END_REF]. In order to have a more accurate 3D reconstruction of the surface, white light interferometry or Fourier Transform Profilometry can be employed, so as to detect pit formation [START_REF] Holme | Characterisation of pitting corrosion by white light interferometry[END_REF], [START_REF] Jang | Evaluation of aluminium pit corrosion in Oak Ridge research reactor pool by quantitative imaging and thermodynamic modeling[END_REF]. Main advantages of these techniques are related to the easy way in which corrosion pits can be detected and quantified (e.g. through parameters such as the pit density); drawbacks are related to the detection of biofilm and attached bacteria, as magnification is too low. On the other hand, to have the best morphological characterization of the biofilm, CLSM (Confocal Laser Scanning Microscope) is generally used [START_REF] Li | Microbiologically Influenced Corrosion of 2707 Hyper-Duplex Stainless Steel by Marine Pseudomonas aeruginosa Biofilm[END_REF]- [START_REF] Hannig | Visualization of adherent micro-organisms using different techniques[END_REF]. This allows to have quantitative information about biofilm coverage and also its 3D structure, providing important information to researchers in biology. Main limitation is that sample preparation requires staining techniques, that are not always available in material science laboratories.

Aim of this research work is to develop a simple and fast imaging system able to quantitatively estimate the microbial corrosion effects and the biofilm growth on metallic specimens subjected to MIC laboratory tests. Imaging based systems have been employed in several fields [START_REF] Delsanto | Characterization of a Completely User-Independent Algorithm for Carotid Artery Segmentation in 2-D Ultrasound Images[END_REF]- [START_REF] Lu | A digital imaging based multifunctional flame monitoring system[END_REF] mainly thanks to their limited cost and non invasiveness [START_REF] Corbellini | Exposure-Tolerant Imaging Solution for Cultural Heritage Monitoring[END_REF]- [START_REF] Grassini | Frequency domain image analysis for the characterization of porous products[END_REF].

The proposed approach is based on the processing of surface micrographs collected by means of a Scanning Electron Microscope (SEM), which provides a sufficient resolution at high magnification to examine bacteria attachment, even in case of low biofilm coverage. Electron microscopy is a traditional characterization technique routinely used in all material science laboratories, so it does not require additional equipment or skills. The system is based on the open-source openCV library, which can count many applications in the identification of specific objects present in images [START_REF] Lombardo | Optical solution for particulate distribution estimation[END_REF], [START_REF] Li | Automatic Human Spermatozoa Detection in Microscopic Video Streams Based On OpenCV[END_REF].

After a preliminary processing of the image, which is described in section II, the software is able to identify bacteria aggregates that are present on the sample (section III). Obtained results are finally presented and discussed in section IV.

II. IMAGE PROCESSING A. Image acquisition

To assess the microbial corrosion effects and the bacteria growth onto the metal, surface images are collected by means of a Field Emission Scanning Electron Microscope (FESEM -Supra 40 Zeiss). Fig. 1 shows, as an example, a FESEM image of a metallic specimen after a microbial corrosion test carried out in an electrolytic solution added by an inoculum containing a wide variety of bacteria; after 200 h of exposure to the MIC test a flat surface on which some bacteria agglomerates, highlighted as dark areas are present, can be observed. The image magnification is selected by the operator and automatically computed by the microscope; magnification can range from few nanometers per pixel to several tens of micrometers per pixel, while the image dimension can be 2000 × 1500 pixels. Since the expected microorganisms have dimensions in the range of 2 µm to 15 µm, a resolution of the order of 0.2 µm/pixel can be conveniently used; therefore, the analyzed area is of the order of 400 µm × 300 µm. This value has been used in all the images processed in this paper.

B. Image brightness and contrast normalization

In addition to magnification and pixel size, the operator selects also the images brightness and contrast in order to obtain the best visualization of bacteria aggregates and localized corrosion attacks (pits). Setting these parameters is a critical operation, which makes any automatic image processing difficult to be applied.

In addition, the image brightness can change over the scanned surface because of the different topography and roughness and as a consequences of charging effects occurring when the electron beam hits non conductive species (bacteria aggregates). The low voltages used by the field emission microscope and the efficiency of its detector greatly help to reduce the charging effects making them almost negligible. However, brightness can still significantly change due to the different surface morphology in the different areas (i. e. different positions) of the scanned image [START_REF] Grassini | Frequency domain image analysis for the characterization of porous products[END_REF].

Therefore, an image normalization has to be performed in order to obtain FESEM images whose contrast and brightness are reasonably independent from the operator; this processing involves three steps:

1) Estimation of brightness changes due to the different morphology of the surface area; 2) Correction of brightness changes in the different positions; 3) Normalization of the image brightness to eliminate the operator setting effect.

The first step requires identifying a brightness plane whose value is described by the following equation:

B = B 0 + α BX x + α BY y ( 1
)
where B is the brightness of the metallic surface area not interested by microbial corrosion, B 0 is the average brightness value, α BX and α BY are the brightness changes with position coefficients, and x and y are the position coordinates. Since the samples have an average brightness which is almost constant in the absence of corrosion, the identification of the brightness average value could be easily performed if one is able to find on the image the areas not affected by corrosion, i.e. in the case of the described images, areas not affected by bacteria growth. Such a manual selection can be avoided in most cases since the area covered by the bacteria is usually limited with respect to the total scanned metallic surface, so that a the image brightness as a function of the coordinates can be easily estimated by solving in the Least Square sense the matrix equation:

B = U • λ (2)
where B is the column vector of all pixel brightness, U is nP ix × 3 matrix in which is row is [1, x, y] accounts for the position of each line in the B vector and λ = [B 0 , α BX , α BY ] ′ is the vector containing the sensitivities whose estimation is referred to λ.

Of course the actual estimated brightness values depend on the way the images are acquired, i.e. on the FESEM operator; however, tests performed on several images with average brightness in the range of 140 to 220 out of a maximum value of 255 led to position coefficients whose maximum value is of the order of 9 • 10 -3 on x axis and of 3 • 10 -2 on the y axis. This leads to maximum brightness changes of the order of 40 units out of 255 units.

The second pre-processing step, once λ has been estimated, can be simply obtained by computing the corrected brightness B c as:

B c = B 1 + α BX x + α BY y (3)
obtaining therefore an image whose brightness is no longer affected by brightness changes in the different surface areas.

The third step can be obtained simply by changing image contrast by modifying the brightness values, so that they covers all the available gray levels. In order to avoid a normalization affected by the presence of few abnormal white and black spots, the image brightness B min and B max can be obtained by:

B min = pixelnumber P xy : B p < B min ; kP tot (4)
B max = pixelnumber P xy : B p > B max ; kP tot (5)
where P xy is the generic pixel, P tot is the number of pixels in the image and k is the coefficient usually in the range of 5% to 10%, which is used to define the number of pixes shown as black and white. Once B min and B max have been estimated, the image brightness can be normalized as:

B s =      0, if B ≤ B min (B -B min ) M Bmax-Bmin , if B min < B < B max M axB, if B ≥ B max (6
) where M axB is the maximum brightness value of the image. Fig. 2 shows four different examples of normalized images collected on metallic specimens which have undergone to the MIC test in different solutions. The normalized images appear to have similar brightness and contrast even though the bacteria proliferation and coverage appears to be quite different.

III. BACTERIA IDENTIFICATION

On FESEM images collected on metallic samples exposed to MIC test, bacteria appear as dark areas whose dimension ranges from few squared micrometers for aggregate of few bacteria to several hundreds of micrometers for large bacteria aggregate.

The identification of the bacteria aggregates has been performed by using the OpenCV opensource environment [START_REF] Zelinsky | Learning OpenCV-Computer Vision with the OpenCV Library[END_REF], In the identified images, as an example, a green contour has been used to flag aggregates with size of less than 200 µm 2 while a red contour has been used to flag bigger aggregates with size above 200 µm 2 .

IV. RESULTS

After a microbial corrosion test, FESEM characterization provides morphological information about biofilm growth and corrosion attacks onto the specimen surface.

Depending on the material and on the electrolytic solution employed in the MIC test, a different amount of microorganisms singly attached or agglomerated in big clusters can be found. So important information can be gathered by means of an imaging analysis able to quantitatively estimate the amount of bacteria grown on the metallic surface as a function of the aggressiveness of the MIC test; therefore the imaging approach could be very useful to compare the behaviors of different metals and alloys exposed to MIC and to put in evidence also the different aggressiveness of bacteria inocula.

In order to validate the proposed image processing, two standard microbial corrosion tests have been carried out, by immersing stainless steel specimens (AISI304, composition in table I) in two different electrolytic solutions. AISI 304 Stainless Steel is an alloy basically employed in biomedical applications, such as orthopedic implants. The specimens have been immersed in two electrolytic solutions, the Stankey medium and artificial seawater (solutions composition are reported in table I). The former is a classic growth medium for bacteria, containing mainly sulphates and chlorides ions; the latter has a more complex composition simulating marine environment (also a favorable solution for microorganisms proliferation) and has been chosen in order to obtain a different corrosion morphology affecting the metallic surface after the test and compare it through the image analysis. Salt marsh sediments have been used as inoculum for the two solutions (it was added in the concentration of 10 vol%). Results from dimensional distribution of microorganism are shown in fig. 4 and5 as histograms for the samples immersed respectively in artificial seawater and Stankey medium. Data have been divided into two plots, one collecting single bacteria or small aggregates (having area up to 20 µm 2 ) and the other one grouping larger ones (until 3000 µm 2 ). Different information can be gathered arranging data in this way, as smaller objects provide indications about bacteria attachment, while bigger ones can be considered the most dangerous as far as microbial corrosion is concerned. Actually, they could generate preferential sites for pitting formation, as they create areas of differential aeration or where corrosive compounds (such as chlorides ions) can accumulate or pH changes can occurs; so identifying and quantifying them is an important point to assess the microbial corrosion behavior of the metallic specimen.

As can be seen in fig. 3, the software is able to correctly identify almost all microorganisms present on the sample. Main limitations are due to agglomerates which lie near the edges of the image, as their contour can't be determined by the findContour algorithm. In virtually all other cases, bacteria are identified and labeled according to their dimension. Thus, analyzing data from the histograms, it is possible to recognize the different morphologies of the two samples. Immersion in artificial seawater lead to a condition in which bacteria are singularly and randomly attached on the surface with only few bigger agglomerates, which can be recognized in the histogram ranging until 3000 µm 2 . On the other hand, immersion in Stankey solution induced the formation of a higher number of bigger aggregates (area above 20 µm 2 ), which can also be quantified through their density:

aggregates density = number of aggregates surf ace area

This parameter has been found equal to 41.7 mm -2 for sample immersed in artificial seawater and equal to 133.3 mm -2 for the sample immersed in Stankey solution, providing an important additional clue for the material characterization. Such values have been calculated dividing the total number of bigger aggregates (reported in table II) by the micrograph area (expressed as mm -2 ) Finally, another useful parameter that can be derived from the image processing is the biofilm coverage, defined as: biof ilm coverage = pixel covered by bacteria total number of pixel • 100 (8)

Its value has been calculated for the two samples, resulting 10.0 % for the test in artificial seawater and 18.8 % for the test in Stankey medium. Table II summarizes further data that can be gained from the processing, such as: number of aggregates (distinguished according to the size), their total surface, the average surface of aggregates and the percentage of covered surface (from which the biofilm coverage can be calculated).

As a further analysis, analogous corrosion tests have been carried out in sterile conditions (using the same electrolytic solutions and material). In this case of course no microorganisms were present, but information about pitting corrosion could be gained through the image processing. Same magnification was used to collect micrographs and results from contour identification are presented in fig. 3. In the case of Stankey solution, the sample revealed initial hints of degradation, properly detected and classified by the software. On the other hand, for the artificial seawater sample no signs of corrosion could be found, as clearly visible from the FESEM micrograph and then from the histogram.

Obtained results have been summarized in table II.

V. CONCLUSIONS A novel imaging system has been proposed for bacteria identification after microbial corrosion test. The processing, based on OpenCV environment, is able to quantify microbial attachment on specimen surface, providing to researchers important information about material degradation, such as biofilm coverage, aggregates density and bacteria dimensional distribution. Results obtained from the software have been presented, showing the effectiveness in bacteria identification and the main information that can be derived from the processing.

Fig. 1 .

 1 Fig. 1. FESEM image of a metallic surface affected by microbial corrosion.

Fig. 2 .

 2 Fig. 2. Example of results of the normalization process on images with different brightness, contrast, and bacteria coverage.

[ 23 ]

 23 by using the findContour function set to identify dark spots. This function returns a list of contours which can be easily shown on the FESEM image as it possible to observe in fig. 3. The figure shows normalized and identified images collected on four metallic specimens immersed in different solutions. Two of the images (Artificial seawater and Stankey solution) refer to samples immersed in solutions containing the inoculum and therefore rich of bacteria, the other two (Abiotic artificial seawater and Abiotic Stankey solution) refer to samples immersed in solutions having the same chemical composition but without the presence of bacteria.

Fig. 3 .

 3 Fig. 3. Example of results of the identification process where in green are shown bacteria clusters with size below 200 µm 2 and in red cluster of bacteria above such value. Four examples are shown: in two cases samples were immersed in solution containing bacteria, and in the other two without bacteria. The corrosion difference is immediately visible

Fig. 5 .

 5 Fig. 5. Histograms showing dimensional distribution (expressed as agglomerate area) as a function of the covered surface percentage for the samples immersed in the Stankey solution.

TABLE I MATERIAL

 I AND ELECTROLYTIC SOLUTION COMPOSITION HP O 4 0.50, N H 4 Cl 1.00, N a 2 SO 4 1.00, CaCl 2 2H 2 O 0.100, M gSO 4 7H 2 O 2.00, N aCl 24.53, CH 3 COON a 3.00 Artificial Seawater N aCl 24.53, M gCl 2 5.20, N a 2 SO 4 4.09, CaCl 2 1.16, KCl 0.695, N aHCO 3 0.201, KBr 0.101, H 3 BO 3 0.027, SrCl 2 0.025, N aF 0.003, CH 3 COON a 3.00

	Material	Chemical Composition [wt%]
	AISI 304	Cr 18, Ni 10, Mn 2, Si 1, C 0.7
	Electrolytic Solution	Concentration [g/L]
	Stankey Medium	K 2

TABLE II RESULTS

 II OBTAINED FOR SAMPLES IMMERSED IN ARTIFICIAL SEAWATER, IN STANKEY SOLUTION AND IN ABIOTIC STANKEY SOLUTION.

	Electrolytic	Number of	Total	Surface	Average	Percentage	Number of	Total	Surface	Average	Percentage
	Solution	Small aggl.	of small aggl.	surface	of	covered	Large aggl.	of large aggl.	surface	of	covered
			(µm) 2		(µm) 2	surface		(µm) 2		(µm) 2	surface
	Artificial	6660	11000		1.6	8.7		5	1640		380	1.3
	seawater
	Stankey	16400	15700		0.9	12.5		16	8000		500	6.3
	Abiotic	20300	3800		4.5	2.9		-	-		-	-
	Stankey
	Fig. 4. Histograms showing dimensional distribution (expressed as agglom-
	erate area) as a function of the covered surface percentage for the samples
	immersed in artificial seawater.