Kui Chen
email: kui.chen@utbm.fr

Salah Laghrouche
email: salah.laghrouche@utbm.fr

Abdesslem Djerdir
email: abdesslem.djerdir@utbm.fr

Health state prognostic of fuel cell based on wavelet neural network and cuckoo search algorithm

Keywords: Proton exchange membrane fuel cell, remaining useful life, degradation prognosis, wavelet neural network, cuckoo search algorithm

This paper proposes a novel degradation prognosis of Proton Exchange Membrane Fuel Cell (PEMFC) based on Wavelet Neural Network (WNN) and Cuckoo Search Algorithm (CSA). The proposed method considering the main operating conditions of PEMFC can be applied to the health state prognostic of PEMFC under different conditions. First, the operating data of PEMFC are reconstructed by the locally weighted scatterplot smoothing method to filter noise. Then, the WNN that can analyze the degradation characteristics of PEMFC (global degradation trend and reversible phenomena) is adopted to build the degradation model of PEMFC. Finally, the structure and parameters of WNN are optimized by CSA to improve the accuracy for the degradation prognosis of PEMFC. The optimized degradation prognosis method is used to predict the remaining useful life of PEMFC. The proposed prognostic method is validated by 3 degradation tests of PEMFC under different conditions. The results show that CSA can greatly improve the degradation prognosis accuracy of PEMFC based on WNN. The proposed CSA-WNN can achieve higher precision than other traditional prognostic methods.

Introduction

The fuel cell system can directly convert the chemical energy of fuel into electrical energy [START_REF] Bougrine | Load estimator-based hybrid controller design for two-interleaved boost converter dedicated to renewable energy and automotive applications[END_REF], [START_REF] Wang | A robust data treatment approach for fuel cells system analysis[END_REF]. Fuel cells have a wide range of applications due to the advantages of high power generation efficiency, no noise, and no pollution [START_REF] Rubio | Robust feedback linearization for nonlinear processes control[END_REF][START_REF] Matraji | Robust control of the PEM fuel cell air-feed system via sub-optimal second order sliding mode[END_REF][START_REF] Laghrouche | Control of PEMFC air-feed system using Lyapunov-based robust and adaptive higher order sliding mode control[END_REF]. Among kinds of fuel cells, Proton Exchange Membrane Fuel Cell (PEMFC) is very suitable for usage in automobiles because of the advantages of low operating temperature (50 to 100 • C), atmospheric pressure operation and the absence of chemical hazards to humans [START_REF] Rana | A Novel dPdI feedback based control scheme using GWO tuned PID controller for efficient MPPT of PEM fuel cell[END_REF], [START_REF] Wang | Coordinating IMC-PID and adaptive SMC controllers for a PEMFC[END_REF]. The PEMFC is mainly composed of a proton exchange membrane, some platinum catalysts, two electrodes, two gas diffusion layers, and two bipolar plates. With the operation of PEMFC, these components face performance degradation, and the failure rate of PEMFC increases [START_REF] Li | A novel equivalent consumption minimization strategy for hybrid electric vehicle powered by fuel cell, battery and supercapacitor[END_REF]. The degradation of PEMFC will result in short Remaining Useful Life (RUL) and high maintenance costs. The service life of PEMFC in the commercial vehicle requires at least 5,000 h, while that of PEMFC in practice is less than 3,000 h [START_REF] Sulaiman | Optimization of energy management system for fuelcell hybrid electric vehicles: Issues and recommendations[END_REF]. The PEMFC is a complex dynamic system which contains multiple physics (electric field, velocity field, concentration field, pressure field, temperature field, etc.), multi-scale (component, cell, and stack), and multiphase (gas phase, liquid phase, and solid phase) [START_REF] Chen | Degradation model of proton exchange membrane fuel cell based on a novel hybrid method[END_REF]. For this complex system, the Prognostics and Health Management (PHM) can be considered as a very promising tool to accurately estimate the health state of PEMFC for improving its RUL and reducing its maintenance costs [START_REF] Javed | Prognostics of Proton Exchange Membrane Fuel Cells stack using an ensemble of constraints based connectionist networks[END_REF], [START_REF] Jha | Particle filter based hybrid prognostics of proton exchange membrane fuel cell in bond graph framework[END_REF].

PHM is a technology that uses sensor information and various algorithms to monitor, predict, and manage system health state [START_REF] Wu | Datadriven remaining useful life prediction via multiple sensor signals and deep long short-term memory neural network[END_REF]. PHM can provide a suitable maintenance plan based on system health state to avoid the problem of maintenance surplus and deficiency [START_REF] Mrugalska | A bounded-error approach to actuator fault diagnosis and remaining useful life prognosis of Takagi-Sugeno fuzzy systems[END_REF]. The concept of PHM first appeared in military equipment. Currently, PHM is widely used in different fields, such as aircraft, electronic systems, cars, ships, batteries, fuel cells, etc [START_REF] Li | An improved local mean decomposition method based on improved composite interpolation envelope and its application in bearing fault feature extraction[END_REF], [START_REF] Liu | ACCUGRAM: A novel approach based on classification to frequency band selection for rotating machinery fault diagnosis[END_REF].

In most literature, voltage is selected as the health indicator of PEMFC. The voltage is usually measured in order to control PEMFC operation and monitor PEMFC performance [START_REF] Chen | Degradation prediction of proton exchange membrane fuel cell based on grey neural network model and particle swarm optimization[END_REF]. Therefore, voltage is adopted as the health indicator of PEMFC to represent the degradation of PEMFC in this paper.

The health state prognostic methods can usually be divided into 3 categories: model-driven, data-driven, and hybrid methods. In model-driven methods, the health state of PEMFC is predicted by combining empirical or semi-empirical degradation models with extended Kalman filter [START_REF] Bressel | Extended Kalman Filter for prognostic of Proton Exchange Membrane Fuel Cell[END_REF], [START_REF] Bressel | Remaining useful life prediction and uncertainty quantification of proton exchange membrane fuel cell under variable load[END_REF], unscented Kalman filter [START_REF] Zhang | An unscented kalman filter based approach for the health-monitoring and prognostics of a polymer electrolyte membrane fuel cell[END_REF], [START_REF] Liu | Remaining useful life estimation for proton exchange membrane fuel cells using a hybrid method[END_REF], particle filter [START_REF] Jouin | Joint particle filters prognostics for proton exchange membrane fuel cell power prediction at constant current solicitation[END_REF], [START_REF] Jouin | Prognostics of PEM fuel cell in a particle filtering framework[END_REF], and unscented particle filter [START_REF] Chen | A novel health indicator for PEMFC state of health estimation and remaining useful life prediction[END_REF]. The model-driven methods take into account the degradation characteristics of PEMFC. However, it is difficult to establish an accurate degradation physical model because the PEMFC is a complex system.

For data-driven methods, the degradation trend is learned from recorded data, and the health state of PEMFC is predicted by the learned degradation trend. Common learning algorithms have echo state network [START_REF] Morando | Proton exchange membrane fuel cell ageing forecasting algorithm based on Echo State Network[END_REF], adaptive neuro-fuzzy inference system [START_REF] Silva | Proton exchange membrane fuel cell degradation prediction based on Adaptive Neuro-Fuzzy Inference Systems[END_REF], relevance vector machine [START_REF] Wu | A modified relevance vector machine for PEM fuel-cell stack aging prediction[END_REF], and group method of data handling [START_REF] Liu | Data-based short-term prognostics for proton exchange membrane fuel cells[END_REF]. The data-driven methods predict the health state of PEMFC without degradation model. However, it requires a lot of historical data.

For hybrid methods, it combines the advantages of data-driven methods and model-driven methods. The physical degradation model combining with time delay neural network is proposed to predict the degradation of PEMFC [START_REF] Zhou | Online energy management strategy of fuel cell hybrid electric vehicles based on data fusion approach[END_REF]. However, compared with model-driven and data-driven methods, the hybrid method has the largest amount of calculation.

Considering the advantages and disadvantages of the above methods, the data-driven methods are adopted in this paper. Most of the current datadriven methods predict the degradation of PEMFC under constant load current. For constant load current, the operating conditions of PEMFC remain almost unchanged. Therefore, these data-driven methods do not take into account the effects of operating conditions on the performance of PEMFC. However, the operating conditions of PEMFC vary greatly in Fuel Cell Electric Vehicle (FCEV) under real conditions. More importantly, these operating conditions have a large impact on the performance of PEMFC [START_REF] Chen | An evaluation method of gas distribution quality in dynamic process of proton exchange membrane fuel cell[END_REF]. For example, flooding can cause gas starvation of PEMFC, and drying can cause the deformation and cracking of membrane [START_REF] Pei | Main factors affecting the lifetime of Proton Exchange Membrane fuel cells in vehicle applications: A review[END_REF]. In this paper, the main operating conditions which include current, temperature, hydrogen pressure, and relative humidity are considered in the proposed prognostic method.

The degradation characteristics of PEMFC include global degradation trend (irreversible degradation phenomena) and reversible phenomena [START_REF] Pei | The recovery mechanism of proton exchange membrane fuel cell in micro-current operation[END_REF], [START_REF] Gazdzick | Evaluation of reversible and irreversible degradation rates of polymer electrolyte membrane fuel cells tested in automotive conditions[END_REF]. With the operation of PEMFC, the PEMFC performance will occur irreversible loss, which is the global degradation trend [START_REF] Pei | The recovery mechanism of proton exchange membrane fuel cell in micro-current operation[END_REF]. The global degradation trend represents the degradation of catalyst, electrodes, gas diffusion layer, and membrane. The global degradation trend usually appears in the low frequency part. When a characteristic test of PEMFC is done or the operating conditions change greatly, the PEMFC performance will occur the recovery phenomenon, which is called reversible phenomena [START_REF] Gazdzick | Evaluation of reversible and irreversible degradation rates of polymer electrolyte membrane fuel cells tested in automotive conditions[END_REF]. The reversible phenomena are the transient process of PEMFC. The reversible phenomena usually appear in the high frequency part. Most of the previous prognostics methods only pay attention to the global degradation trend while both reversible phenomena and global degradation trend are considered by Wavelet Neural Network (WNN) in this paper.

WNN is an artificial neural network based on wavelet analysis. The wavelet analysis can overcome the dilemma of the time-domain resolution of the Fourier transform. The wavelet analysis can analyze the degradation characteristics of PEMFC at different frequency parts through the transformation of the wavelet basis function.

Compared with other neural networks, WNN has the following advantages. Firstly, multi-scale analysis of signals by wavelet analysis can effectively extract local features of signals [START_REF] Huang | Forecasting energy fluctuation model by wavelet decomposition and stochastic recurrent wavelet neural network[END_REF]. Secondly, the low correlation of neurons in WNN makes WNN have faster convergence speed. Thirdly, The wavelet basis function of hidden layer neurons brings two new variables (scale factor and translation factor), which makes WNN have stronger learning ability [START_REF] Adamowski | A wavelet neural network conjunction model for groundwater level forecasting[END_REF].

However, the application of WNN to the degradation prognosis of PEMFC has certain limitations. Firstly, the number of hidden layer neurons is difficult to determine for WNN [START_REF] Saengrung | Neural network model for a commercial PEM fuel cell system[END_REF]. When the number of hidden layer neurons is large, it is prone to over-learning, and the performance of degradation prognosis is always not satisfied. When the number of hidden layer neurons is small, it is difficult to learn the degradation trend of PEMFC. Secondly, the initialization of the parameters of WNN has a great influence on the learning and prediction performance [START_REF] Sun | A carbon price prediction model based on secondary decomposition algorithm and optimized back propagation neural network[END_REF]. If the initialization parameters of scale factor and translation factors are not appropriate, the learning process of WNN will not converge. In order to solve these problems, the Cuckoo Search Algorithm (CSA) is adopted in this paper.

CSA is a metaheuristic algorithm that combines the advantages of Particle Swarm Optimization (PSO), Differential Evolution (DE) and Simulated Annealing (SA) [START_REF] Ajenikoko | Cuckoo Search Algorithm Optimization Approaches for Solving Economic Load Dispatch: A Review[END_REF]. CSA has two search abilities: local search (local random walk, Eq. 14) and global search (global random walk, Eq. 15), which are controlled by handover/discovery probability P a . This makes it possible to explore the search space more efficiently, so that the global optimization can be obtained with higher probability [START_REF] Xu | Structural damage identification based on modified cuckoo search algorithm[END_REF]. In addition, it has been proven that CSA meets the requirements of global convergence and thus has guaranteed global convergence properties [START_REF] Wang | Markov model and convergence analysis based on cuckoo search algorithm[END_REF].

Another advantage of CSA is that its global search adopts Lévy flights instead of the standard random walk [START_REF] Yang | Nature-inspired metaheuristic algorithms[END_REF]. Since Lévy flights have infinite mean and variance, CSA can more effectively explore the search space than standard Gaussian algorithms. This advantage, combined with local search capabilities and guaranteed global convergence, allows CSA to find global optimization very efficiently [START_REF] Wang | Markov model and convergence analysis based on cuckoo search algorithm[END_REF]. Various applications have shown that CSA is very efficient [START_REF] Yang | Nature-inspired metaheuristic algorithms[END_REF][START_REF] Gandomi | Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems[END_REF][START_REF] Civicioglu | A conceptual comparison of the Cuckoosearch, particle swarm optimization, differential evolution and artificial bee colony algorithms[END_REF].

This paper proposes a novel degradation prognosis method based on WNN and CSA for PEMFC under different conditions. The proposed method considers the influence of the main operating conditions including current, temperature, hydrogen pressure, and relative humidity on the degradation of PEMFC. First, the locally weighted scatterplot smoothing is proposed to reconstruct the operating data of PEMFC. Then, the WNN is adopted to analyze the global degradation trend and reversible phenomena for the degradation prognosis of PEMFC. Finally, the parameters of WNN and the number of neurons in the hidden layer of WNN are optimized by CSA. The accuracy of the proposed method is validated by 3 degradation tests of PEMFC under different conditions. The main contributions are:

1. The proposed prognostic method considering the influence of the main operating conditions on the degradation of PEMFC can be applied to the health state prognostic of PEMFC under different conditions. 2. Different degradation phenomena (global degradation trend and reversible phenomena) can be analyzed by WNN, which can make an accurate degradation prediction of PEMFC. 3. The CSA optimizes the parameters of WNN and the number of neurons in the hidden layer of WNN, which greatly improves the accuracy of degradation prognosis for PEMFC. Compared with other traditional evolutionary algorithms, the CSA is more suitable for the health state prognostic of PEMFC. 4. Compared with other traditional prognostics methods, the proposed CSA-WNN has the highest accuracy of degradation prognosis for PEMFC. In addition, the proposed method has a very short calculating time and can be used in the online health state prognosis of PEMFC.

The degradation tests of PEMFC under different conditions are introduced in Section 2. Section 3 proposes data reconstruction for PEMFC, degradation prognosis of PEMFC based on WNN and CSA-WNN, and RUL prognosis of PEMFC. Section 4 presents the results and validation. Conclusions are presented in Section 5.

Degradation tests of PEMFC under different conditions

Degradation test of PEMFC in FCEV under real conditions

The European project MobyPost has designed and fabricated 10 FCEVs (Fig. 1) to carry out commercial postal delivery services on the real road. The PEMFC stack in FCEV consists of 40 cells with an activation surface of 100 cm 2 [START_REF] Chen | Fuel cell health prognosis using Unscented Kalman Filter: Postal fuel cell electric vehicles case study[END_REF]. The operating data of PEMFC in FCEV are measured and recorded at a frequency of 1Hz. The recorded voltage and current data of PEMFC in FCEV are shown in Fig. 2. The load current of PEMFC varies with the load of FCEV, and voltage is affected by operating conditions and degradation conditions.

Degradation test of PEMFC under dynamic load current

The degradation test of PEMFC under dynamic load current is made for 550 h in FCLAB, as shown in Fig. 3. The PEMFC stack under dynamic load current consists of 5 cells with an activation surface of 100 cm 2 [START_REF] Gouriveau | IEEE PHM 2014 data challenge: Outline, experiments, scoring of results[END_REF]. The operating data of PEMFC under dynamic load current are measured and recorded approximately every 30 seconds. In order to better characterize the PEMFC state, the polarization curve test and Electrochemical Impedance Spectroscopy (EIS) test are performed approximately weekly (0 h, 35 h, 182 h, 343 h, 515 h). The recorded voltage and current data of PEMFC under dynamic load current are shown in Fig. 4. From 0 h to 550 h, the overall voltage trend drops due to the global degradation phenomena of PEMFC. After each characteristic test, the reversible phenomena can be observed in PEMFC, and the voltage of PEMFC rises.

Degradation test of PEMFC under stationary load current

The degradation test of PEMFC under stationary load current is made for 982 h in FCLAB. The PEMFC stack under stationary load current contains 5 cells with an activation surface of 100 cm 2 [START_REF] Gouriveau | IEEE PHM 2014 data challenge: Outline, experiments, scoring of results[END_REF]. The operating data of PEMFC under stationary load current are also measured and recorded approximately every 30 seconds. The polarization curve test and EIS test are also performed approximately weekly (0 h, 48 h, 185 h, 348 h, 515 h, 658 h, 823 h). The recorded voltage and current data of PEMFC under stationary load current are shown in Fig. 5. From 0 h to 823 h, the overall voltage trend of PEMFC drops. After each characteristic test, the voltage of PEMFC rises due to the reversible phenomena of PEMFC.

Prognostic method of PEMFC

Data reconstruction for PEMFC

There are 1070370 raw operating data for PEMFC in FCEV under real conditions, 707508 raw operating data for PEMFC under stationary load current, and 393138 raw operating data for PEMFC under dynamic load current. The amount of recorded operating data of PEMFC is very large, which will cause a large amount of calculation. The raw operating data of PEMFC are measured and recorded at high frequencies, while the degradation is very slow. In addition, there are some disturbances in the recorded data due to noise. Therefore, the recorded raw data of PEMFC can be smoothed and filtered to improve calculation accuracy and reduce the amount of calculation.

The operating data of PEMFC are reconstructed by the Locally Weighted Scatterplot Smoothing (LOWESS). The main principle of LOWESS is that for a length of data, the weighted linear regression is performed at the center point with the weight function [START_REF] Liu | Remaining useful life prediction of PEMFC based on long short-term memory recurrent neural networks[END_REF]. The voltage reconstruction for PEMFC under different conditions is shown in Fig. 6.

The degradation prognosis of PEMFC based on WNN

WNN is a combination of wavelet analysis and BP neural network [START_REF] Lei | A comprehensive evaluation method for indoor air quality of buildings based on rough sets and a wavelet neural network[END_REF]. The wavelet analysis can analyze the global degradation trend and reversible phenomena of PEMFC through the transformation of wavelet basis function [START_REF] Majeed Alneamy | Utilizing hybrid functional fuzzy wavelet neural networks with a teaching learning-based optimization algorithm for medical disease diagnosis[END_REF]. The topology of WNN is shown in Fig. 7.

As shown in Fig. 7,

x =[x 1 ,• • • ,x n ]
that are the input variables of WNN, represent the operating conditions (time, current, temperature, hydrogen pressure, and relative humidity) of PEMFC. y=[y 1 ,• • • ,y m ] that are the output variables of WNN, represent the voltage of PEMFC. w ij is the input weight of WNN and w jk is the output weight of WNN. f is the activation function in the hidden layer of WNN. The Morlet wavelet basis function is chosen as the activation function in the hidden layer, and the mathematical formula is as follows.

f (x) = cos(1.75x)e -x 2 /2
(1)

The degradation prognosis of PEMFC based on WNN contains the following main steps:

1. WNN initialization: According to the recorded operation data of PEMFC, determine the WNN structure, and initialize the weight of WNN. 2. Calculate the output of hidden layer neurons: The input variables, input weights, and wavelet basis functions are adopted to calculate the output of hidden layer neurons as follows:

f (j) = f n i=1 w ij x i -b j a j j = 1, 2, • • • , l (2)
where f (j) is the output value of the jth neuron in the hidden layer, b is the translation factor of the wavelet basis function, and a is the scale factor of the wavelet basis function. 3. Calculate the predicted output of WNN. The output of hidden layer neurons and output weights are adopted to calculate the predicted output of WNN for PEMFC as follows:

y k = l j=1 f j w jk k = 1, 2, • • • , m (3)
where l is the number of neurons in the hidden layer, and m is the number of neurons in the output layer. 4. Calculation error: Calculate the error of WNN based on the predicted output and measured output as follows:

e = m k=1 y k -z k k = 1, 2, • • • , m (4)
where z is the measured output of PEMFC. 5. Update the weights of WNN and the parameters of wavelet basis function: the gradient learning algorithm is adopted to update the weights of WNN and the parameters of wavelet basis function as follows:

w n,k (i + 1) = w n,k (i) + ∆w n,k (i + 1) (5)
a k (i + 1) = a k (i) + ∆a k (i + 1) (6) b k (i + 1) = b k (i) + ∆b k (i + 1) (7)
where ∆w n,k (i + 1), ∆a k (i + 1), ∆b k (i + 1) are calculated based on the calculation error e.

∆w n,k (i + 1) = -β ∂e ∂w n,k (i) (8) ∆a k (i + 1) = -β ∂e ∂a k (i) (9) ∆b k (i + 1) = -β ∂e ∂b k (i) ( 10
)
where β is the learning rate. This paper adopts the method of additional momentum to improve the learning efficiency of network. Eq. ( 5)-( 7) can be rewritten as Eq. ( 11)- [START_REF] Wu | Datadriven remaining useful life prediction via multiple sensor signals and deep long short-term memory neural network[END_REF]. 11)

w n,k (i + 1) = w n,k (i) + ∆w n,k (i + 1) + η * (w n,k (i) -w n,k (i -1)) (
a k (i + 1) = a k (i) + ∆a k (i + 1) + η * (a k (i) -a k (i -1)) (12)
b k (i + 1) = b k (i) + ∆b k (i + 1) + η * (b k (i) -b k (i -1)) ( 13
)
where η is the momentum learning rate. 6. Determine if the WNN iteration ends. If it is not finished, run step 2.

The degradation prognosis of PEMFC based on CSA-WNN

Because the initial weight is randomly set, the WNN is easy to reach the local optimal value. At the same time, the number of neurons in the hidden layer of WNN has a great effect on the degradation prognosis of PEMFC. However, it is difficult to determine the optimal number of neurons in the hidden layer of WNN. In order to solve these problems, CSA is proposed to improve WNN. The weights, parameters of wavelet basis function, and the number of neurons in the hidden layer are optimized by CSA. The degradation prognosis of PEMFC based on CSA-WNN is shown in Fig. 8. CSA is an optimization algorithm by simulating the Cuckoo's brood parasitism. More details about CSA can be found in [START_REF] Yang | Cuckoo search: recent advances and applications[END_REF]. CSA explores new solutions through a local random walk and a global random walk. The local random walk can be expressed as:

x(t + 1) = x(t) + αs ⊗ H(P a -) ⊗ (x k (t) -x j (t)) ( 14
)
where s is the step size, P a is the probability of being discovered by the host bird, H(x) is the Heaviside function, x k (t) and x j (t) are two different solutions selected by random permutation, is a random number extracted from a uniform distribution, and ⊗ represents the product of two vectors.

The global random walk adopts Lévy flights [START_REF] Chiroma | Bio-inspired computation: Recent development on the modifications of the cuckoo search algorithm[END_REF]. Many works of literature have shown that many animals and insects exhibit typical characteristics of Lévy flights with power law [START_REF] Brown | Lévy flights in Dobe Ju/'hoansi foraging patterns[END_REF], [START_REF] Reynolds | Free-flight odor tracking in Drosophila is consistent with an optimal intermittent scale-free search[END_REF]. The Lévy flights can be expressed as follows:

x(t + 1) = x(t) + αL(s, λ)

L(s, λ) = λΓ(λ)sin(πλ/2) π

1 s 1+λ ( 16
)
where λ is the Lévy index, α is the step size scaling factor and is related to the scale of the problem of interest. In most cases, α = O(L/10) is often used, and L is the scale of the problem of interest. In this paper, α = O(L/100) is chosen, which can be more effective and prevent flying too far [START_REF] Yang | Cuckoo search: recent advances and applications[END_REF]. Eq. ( 15) is essentially a Markov chain whose next state only depends on the current state (the first term of Eq. ( 15)) and the transition probability (the second term of Eq. ( 15)). A large part of the new solution should be generated by far-field randomization, and their position should be far away from the current optimal solution, which will ensure that the system does not fall into local optimum [START_REF] Yang | Engineering Optimisation by Cuckoo Search[END_REF].

Generating a random number with Lévy flights should include two steps: the selection of the random direction and the generation of the step size obeying the Lévy distribution. Direction generation should be subject to the uniform distribution. The symmetric Lévy stable distribution is achieved by the Mantegna algorithm [START_REF] Mantegna | Fast, accurate algorithm for numerical simulation of Levy stable stochastic processes[END_REF]. In the Mantegna algorithm, the step size s can be calculated by the following random variables U and V , which obey the Gaussian distribution:

s = U |V | 1/λ ( 17
)
where

U ∼ N (0, σ 2 ), V ∼ N (0, 1) (18)
The variance can be calculated by the following formula:

σ 2 = Γ(1 + λ) λΓ((1 + λ)/2) sin(πλ/2) 2 (λ-1)/2 1/λ (19)
As shown in Fig. 8, the main steps for degradation prognosis of PEMFC based on CSA-WNN are as follows:

V f itness = N i=1 | ẑi -z i | ( 20
)
where N is the number of training data, ẑ is the predicted voltage, and z is the measured voltage. 4. Judging the end condition: If the individual reaches the fixed number of generations, jump to step 6; otherwise, carry out step 5. 5. Generate new host nests: The new host nests are generated by the local random walk and global random walk based on the current individual fitness value. Take the new host nests as the current host nests, jump to step 2. 6. Output optimal structure and parameters: Export the optimal input weights, output weights, hidden layer neurons, translation factors, and scale factors of WNN. 7. The degradation prognosis of PEMFC is got based on the CSA-WNN.

The RUL prognosis of PEMFC

With the long-term operation of PEMFC, the performance of PEMFC will decline. The end of life for PEMFC occurs when the performance of PEMFC decreases by 10% [START_REF] Sutharssan | A review on prognostics and health monitoring of proton exchange membrane fuel cell[END_REF]. In order to maintain the PEMFC in time, the RUL occurs when the performance of PEMFC decreases by 5%. The RUL prognosis of PEMFC is shown in Fig. 9. When the degradation prognosis of PEMFC decreases by 5%, the predicted time is RUL.

Results and validation

Setting of the prognostic method

According to the characteristics of recorded operation data of PEMFC, the input layer of WNN contains 5 neurons, which represent time, current, temperature, hydrogen pressure, and relative humidity. The output layer of WNN contains 1 neuron, which represents the voltage of PEMFC. The number of neurons in the hidden layer of WNN is determined by CSA. For CSA, the number of host nests is 20, the probability of host bird discovery is 25%, and the maximum number of iterations is 20.

The proposed method is implemented on a commercial computer with an i5-6300 Intel CPU (12 GB RAM and 2.3 GHz clock). In order to evaluate the computational complexity of different methods, the calculating time is recorded. In order to reduce the impact of randomness, 20 independent runs of each method are performed. The best prognostics result of each method is selected from the 20 runs for comparison. In order to assess the precision of different methods, the Absolute Percentage Error (APE) is adopted in this paper. Smaller APE means higher accuracy for the degradation prognosis of PEMFC. The APE is calculated as Eq. [START_REF] Liu | Remaining useful life estimation for proton exchange membrane fuel cells using a hybrid method[END_REF].

AP E = 100 * |ẑ -z| z ( 21
)
where ẑ is the voltage degradation prognosis, z is the measured voltage of PEMFC under different operation conditions.

The degradation prognosis for PEMFC based on CSA-WNN

The degradation prognosis for PEMFC in FCEV under real conditions is made by CSA-WNN and WNN with the different number of neurons in the hidden layer. The latter WNN is not optimized by CSA. 80% of recorded FCEV operation data are utilized to learn the degradation tendency of PEMFC, and remaining data are utilized to validate the degradation prognosis for PEMFC in FCEV. The degradation prognosis result for PEMFC in FCEV under real conditions is shown in Fig. 10. The accuracy of degradation prognosis and calculating time are shown in Table 1.

As shown in Fig. 10.a, the degradation tendency of PEMFC in FCEV under real conditions can be learned and predicted by CSA-WNN and WNN with the different number of neurons in the hidden layer. The number of neurons in the hidden layer optimized by CSA is 7 for the degradation prognosis of PEMFC in FCEV. Compared with the WNNs without optimization, the voltage degradation prognosis of CSA-WNN is the closest to the actual voltage degradation of PEMFC in FCEV most of the time. As shown in Fig. 10.b, the maximum APE for degradation prognosis of PEMFC is less than 2%. As shown in Table 1, the mean APE of CSA-WNN is the lowest. The reason why the mean APE of CSA-WNN is the lowest is that the CSA optimizes the weights, parameters of wavelet basis function, and the number of neurons in the hidden layer for WNN. In addition, the CSA leads to an increase in calculating time. However, compared with the degradation time of PEMFC, the calculating time of CSA-WNN is very short. Therefore, the proposed CSA-WNN is the most appropriate degradation prognosis method for PEMFC in FCEV under real conditions.

In order to illustrate the optimization performance of different evolutionary algorithms, the degradation prognosis for PEMFC in FCEV under real conditions is made by SA-WNN, DE-WNN, GA-WNN, PSO-WNN, and CSA-WNN. The main parameters of SA, DE, GA, and PSO are shown in Table 2. The degradation prognosis for PEMFC in FCEV based on different evolutionary algorithms and WNN are shown in Table 3.

As shown in table 3, the mean APE of CSA-WNN in learning and prognosis is the lowest among the four evolutionary algorithms. The calculating time of CSA-WNN is also the least. For the health state prognostic of PEMFC, the Lévy flights of CSA can be more conducive to the search for globally optimal solutions. Compared with SA, DE, GA, and PSO, CSA is more suitable for optimizing the degradation prognosis model of PEMFC based on WNN.

Comparison between CSA-WNN and other traditional methods

The degradation prognosis for PEMFC under dynamic load current is made by CSA-WNN, Elman [START_REF] Guo | Optimization of critical parameters of PEM fuel cell using TLBO-DE based on Elman neural network[END_REF], Back Propagation Neural Network (BP) [START_REF] Saengrung | Neural network model for a commercial PEM fuel cell system[END_REF] Extreme Learning Machine (ELM) [START_REF] Javed | Prognostics of Proton Exchange Membrane Fuel Cells stack using an ensemble of constraints based connectionist networks[END_REF], Relevance Vector Machine (RVM) [START_REF] Wu | A modified relevance vector machine for PEM fuel-cell stack aging prediction[END_REF], and Support Vector Machine (SVM) [START_REF] Kheirandish | Modeling of commercial proton exchange membrane fuel cell using support vector machine[END_REF]. 80% of recorded operation data are utilized to learn the degradation tendency of PEMFC under dynamic load current, and remaining data are utilized to validate the degradation prognosis for PEMFC. The main parameters of Elman, BP, ELM, SVM and RVM are shown in Table 4. The degradation prognosis result for PEMFC under dynamic load current is shown in Fig. 11. The mean APE of degradation prognosis and calculating time are shown in Table 5.

As shown in Fig. 11.a, the voltage degradation prognosis of six methods can follow the voltage degradation tendency of PEMFC under dynamic load current. As shown in Fig. 11.b, the maximum APE of six methods for degradation prognosis of PEMFC is less than 2.3%. Compared with Elman, BP, ELM, RVM, and SVM, the voltage degradation prognosis of CSA-WNN is the closest to the actual voltage degradation of PEMFC under dynamic load current most of the time. As shown in Table 5, the mean APE of CSA-WNN in learning and prognosis is the lowest among the six prognosis methods. It shows that the CSA-WNN has higher precision than other traditional prognosis methods. The reasons why the CSA-WNN has higher precision are that the CSA optimizes the WNN structure, and the WNN analyzes the global degradation trends and reversible phenomena during the degradation process of PEMFC. The only cost is the increase in calculating time. The calculating time of proposed CSA-WNN is less than 2 s, while the degradation time of PEMFC is more than 1000 h. Therefore, the calculating time of the proposed CSA-WNN is very short compared with the degradation time of PEMFC. The proposed CSA-WNN can be applied to the online degradation prognosis for PEMFC.

The RUL prognosis of PEMFC

After obtaining the RUL of PEMFC, some effective measures can be adopted to prolong its RUL and improve its performance. The RUL prognosis of PEMFC under stationary load current is made by CSA-WNN. The αperformance metrics are applied in the evaluation for the RUL prognosis of PEMFC. If the RUL belongs to the time area [(1-α)t w , (1+α)t w ], the RUL prognosis can be regarded as a good prediction. 10% is chosen as the value of α in this paper. The RUL prognosis of PEMFC under stationary load current is shown in Fig. 12.

As shown in Fig. 12, the RUL prognosis of PEMFC under stationary load current is 220 h at 600 h learning data. As the learning data increases, the RUL prognosis of PEMFC is closer to the real RUL of PEMFC. 220 h RUL prognosis (more than one week) is enough for engineers to maintain PEMFC. If some effective measures can be adopted based on RUL prognosis, the performance and RUL of PEMFC will be enhanced.

Conclusions

As the core of PHM, prognostics provide the system health state to maintain the system and improve its RUL. This paper proposes a novel degradation prognosis method of PEMFC under different conditions based on WNN and CSA. The influence of current, temperature, hydrogen pressure and relative humidity on the performance of PEMFC is considered by the proposed prognostic method. Firstly, the LOWESS is adopted to reconstruct the operating data of PEMFC for filtering noise and reducing the calculation amount. Secondly, WNN that can analyze the global degradation trend and reversible phenomena is applied to the degradation prognosis of PEMFC. Then, the weights, parameters of wavelet basis function, and the number of neurons in the hidden layer of WNN are optimized by CSA. Finally, the RUL of PEMFC is predicted by CSA-WNN. The accuracy of the proposed method is validated by three degradation tests. The main conclusions are as follows:

1. The operating conditions and different degradation phenomena have been considered in WNN, which benefits for establishing an accurate degradation prognosis model of PEMFC under different conditions. 2. The CSA solves the problem about how to find the optimal number of neurons in the hidden layer and optimize the parameters of WNN. The CSA can greatly improve the accuracy of the degradation prognosis for PEMFC. Compared with SA, DE, GA, and PSO, the CSA has the best optimization effect for the degradation prognosis of PEMFC. 3. Compared with Elman, BP, ELM, RVM, and SVM, the CSA-WNN has the highest accuracy of degradation prognosis in the learning and prognosis stage. Moreover, the proposed CSA-WNN that has a very short calculating time can be applied to the online prognosis. 4. The RUL prognosis of PEMFC under stationary load current is 220 h at 600 h learning data, which is long enough for the engineer to develop a maintenance plan.

The proposed prognostic method that considers the operating conditions and degradation characteristics of the system can be used in different prognostic applications, such as bearing, bridge, battery, transformer, electronic system, and motor. The health state prognosis of the proposed method can be combined with the control theory and energy management strategy to improve its RUL and system economy. The adaptive population extremal optimization inspired by the Bak-Sneppen model of self-organized criticality [START_REF] Zeng | Adaptive population extremal optimization-based PID neural network for multivariable nonlinear control systems[END_REF][START_REF] Chen | Constrained multi-objective population extremal optimization based economic-emission dispatch incorporating renewable energy resources[END_REF][START_REF] Chen | A many-objective population extremal optimization algorithm with an adaptive hybrid mutation operation[END_REF], will be considered for use in the degradation prognosis of PEMFC in future work. The proposed prognostic method combined with the energy management strategy to improve the durability and economy of FCEV will be considered in future work. MAPE in learning (%) MAPE in prognosis (%) Time (s) Elman [START_REF] Guo | Optimization of critical parameters of PEM fuel cell using TLBO-DE based on Elman neural network[END_REF] 0.3641 0.0874 0.9224 BP [START_REF] Saengrung | Neural network model for a commercial PEM fuel cell system[END_REF] 0.5624 0.1500 0.1960 ELM [START_REF] Javed | Prognostics of Proton Exchange Membrane Fuel Cells stack using an ensemble of constraints based connectionist networks[END_REF] 0.3812 0.0770 0.0118 RVM [START_REF] Wu | A modified relevance vector machine for PEM fuel-cell stack aging prediction[END_REF] 0.4878 0.1253 0.2351 SVM [START_REF] Kheirandish | Modeling of commercial proton exchange membrane fuel cell using support vector machine[END_REF] 0.4767 0.0866 0.0182 CSA-WNN 0.2384 0.0518 1.9843

Figure 1 :

 1 Figure 1: MobyPost FCEV [44]

Figure 2 :Figure 3 :

 23 Figure 2: Voltage and current data of PEMFC in FCEV

Figure 4 :Figure 5 :Figure 6 :Figure 8 :Figure 9 :

 45689 Figure 4: Voltage and current data of PEMFC under dynamic load current

Figure 10 :Figure 11 :Figure 12 :

 101112 Figure 10: The degradation prognosis results for PEMFC in FCEV under real conditions

  1. Establish the degradation model of PEMFC based on WNN: Considering the characteristics of PEMFC working conditions, the PEMFC degradation model structure based on WNN is determined. The PEMFC degradation model structure contains the number of input layer neurons and the number of output layer neurons. 2. Initialize host nests: The input weights, output weights, hidden layer neurons, translation factors, and scale factors of WNN are randomly initialized to generate the host nests. 3. Calculation fitness: The predicted voltage of PEMFC is calculated through WNN based training data. The individual fitness is calculated by Eq. 20.

Table 1 :

 1 The accuracy and calculating time for PEMFC in FCEV

		MAPE in learning (%) MAPE in prognosis (%) Time (s)
	WNN-6	0.2750	0.0875	0.9440
	WNN-7	0.2350	0.0782	0.9998
	WNN-8	0.2423	0.0856	5.4491
	CSA-WNN	0.2273	0.0543	18.4846

Table 4 :

 4 The main parameters of Elman, BP, ELM, SVM and RVM

	Algorithm	Parameter	Value
		Input weight	Random setting
	Elman	Output weight Context layer weight	Random setting Random setting
		Hidden layer threshold	Random setting
		Input weight	Random setting
	BP	Output weight	Random setting
		Hidden layer threshold	Random setting
		Input weight	Random setting
	ELM	Output weight	Random setting
		Hidden layer threshold	Random setting
		Kernel function	Radial basis function
	SVM	Kernel's parameter	100
		Penalty factor	100
	RVM	Kernel function Kernel's parameter	Radial basis function 100

Table 5 :

 5 The accuracy and calculating time for PEMFC under dynamic load current


Anthony Berdeu

Thomas Olivier

Fabien Momey

Loïc Denis

Frédéric Pinston

Nicolas Faure

Corinne Fournier

Joint reconstruction of an in-focus image and of the background signal in in-line holographic microscopy

Keywords:

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

Introduction

In-line holographic setups [START_REF] Gabor | A new microscopic principle[END_REF] are much simpler and less sensitive to vibrations than off-axis setups. As illustrated in Fig. 1(a), in-line holography simply consists in recording the intensity of the patterns diffracted by all the elements located in the optical path of a coherent light beam. In holographic microscopy, an objective and other optical elements (not represented in this simplified figure) are placed between the object and the sensor to obtain a magnified image of the sample. After numerical reconstruction, this technique enables access to the absorption and phase shift of the objects, making it particularly suitable for studying absorbing as well as transparent objects in many domains including biology [START_REF] Garcia-Sucerquia | Digital in-line holographic microscopy[END_REF], fluid mechanics [START_REF] Katz | Applications of holography in fluid mechanics and particle dynamics[END_REF] and for particle characterization [START_REF] Lee | Characterizing and tracking single colloidal particles with video holographic microscopy[END_REF].

By applying the diffraction equations [START_REF] Goodman | Introduction to Fourier optics[END_REF], the diffracted wave can be back propagated to the object plane quite simply. However, the reconstructed wave is often distorted due to the diffraction of unwanted out-of-focus objects, such as dust on the optical elements, giving rise to a "background signal" (cf. Fig. 1). Background correction in in-line holography is problematic because the sensor is sensitive to the intensity of the interferences between the wavefront scattered by the sample of interest and this unwanted background. Standard reconstruction techniques assume that these interferences are negligible, so that the diffraction patterns due to the objects and those that form the background can be incoherently summed. Standard background corrections are performed by subtracting a background hologram from the sample hologram. These corrections are used to compensate for illumination inhomogeneities [START_REF] Xu | Digital in-line holography of microspheres[END_REF][START_REF] Seifi | Accurate 3d tracking and size measurement of evaporating droplets using in-line digital holography and "inverse problems" reconstruction approach[END_REF] or to remove [START_REF] Xu | Digital in-line holography of microspheres[END_REF][START_REF] Carter | Reconstruction of inhomogeneous scattering objects from holograms[END_REF]. Solution (i) requires that a part of the glass slide be free of objects of interest to only capture the contribution of the out-of-focus objects that remain static whatever the glass slide position (cf. Fig. 1). Solution (ii) requires sufficiently long hologram video and sufficient motion of the objects of interest to ensure each point in the field of view to be most of the time free of objects. Solution (iii) is analog to high-pass filtering the hologram before reconstruction, which leads to a loss of information and cannot suppress the high frequency fringes of unwanted objects in the optical path. In addition, none of these solutions account for the interferences between the background wavefront and the wavefront scattered by the objects of interest, by assuming a simple summation of their respective intensity.

In the proposed method, we apply a different strategy: joint reconstruction of the background and of the sample of interest, using an inverse problems approach. In order to separate the objects and the background, we use two holograms acquired at two different locations of the sample (either longitudinal or lateral shifts or both). In these two holograms, the background structures, fixed in the optical path of the setup, remain at the same location. The inverse problems framework is based on an image formation model that accounts for the interference between the object wave and the background, to reduce twin-image artifacts, and to improve the reconstruction of the absorption and phase images.

Proposed method

Inverse problems methods reconstruct an image by considering both an optical model (the forward model) of the hologram formation and some prior knowledge about the sample. These methods iteratively reduce the discrepancies between the modeled hologram and the data [START_REF] Denis | Inline hologram reconstruction with sparsity constraints[END_REF][START_REF] Jolivet | Regularized reconstruction of absorbing and phase objects from a single in-line hologram, application to fluid mechanics and micro-biology[END_REF][START_REF] Berdeu | Reconstruction of in-line holograms: combining model-based and regularized inversion[END_REF]. We use this general framework to jointly retrieve the sample and the background signal using two holograms. As illustrated in Fig. 1, between the two acquisitions, the objects of interest (green) are shifted by , , . The background (red) is assumed to remain unchanged, which is the case of patterns created by dust adhering to the setup interfaces such as lenses or filters. The background and the objects are unmixed thanks to the reconstruction process.

Forward model

A 2D complex transmittance plane ( , ) = exp ( , ) is completely described by its complex phase = r + i , where complex values are underlined for the sake of clarity. With this convention, the real and imaginary parts of are directly linked to the complex refractive index of the transmittance plane whose real part accounts for the phase delay it introduces and whose imaginary part accounts for its absorption.

As shown in Fig. 1, under the first Born approximation [START_REF] Goodman | Introduction to Fourier optics[END_REF] (single scattering hypothesis), the forward model accounts for the interferences between three waves: the incident (unperturbed) wave inc (blue), the wave diffracted by background objects bg dif (red), and the wave diffracted by the sample s dif (green). The interference of the incident and background waves is modeled on the sensor plane by a single term exp( bg ) which corresponds to a non uniform wave with a wavefront distorted by propagation through the background objects. The objects of interest, located in the sample plane, induce a phase shift noted s . The model consequently depends on the complex phases Φ = ( bg , s ) defined respectively in the sensor and sample planes as well as on a set of geometrical parameters = , , , . These geometrical parameters are the distance between the sample and the sensor during the first acquisition and the 3D shift performed before the second acquisition ( , , ).

As under first Born approximation the complex transmit-tance of the object plane is considered to be illuminated by an unperturbed plane incident wave, the location of the background transmittance plane can be set arbitrarily. Therefore, it can be considered before or after the object plane without changing the imaging model. For the sake of simplicity, we place this plane at the sensor plane in the following. Considering a monochromatic plane wave illumination inc ( ) = exp (-) of wavelength , wavenumber = 2 ∕ , its propagation can be described by the first Rayleigh-Sommerfeld solution [5, Section 3.5.1]. The corresponding Rayleigh-Sommerfeld propagator [START_REF] Berdeu | Reconstruction of in-line holograms: combining model-based and regularized inversion[END_REF] at propagation distance is:

ℎ ( , ) = 1 1 - 1 with 2 = 2 + 2 + 2 , ( 1
)
and the complex amplitude on the sensor plane can be modeled, for the first acquisition = 1, by:

1 (Φ, ) = bg ⏟ ⏟ ⏟ inc (0)+ bg dif + inc ( ) s -1 * ℎ ( , ) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ s dif . (2)
The term s -1 models only the light diffracted by the sample plane (first Born approximation). The term bg takes into account both the plane wave illumination contribution and the background contribution by modeling their interference.

For the second acquisition = 2, the shift is to be taken into account:

2 (Φ, ) = bg + inc ( ) s -1 * ℎ + + , + . (3)
Finally, the forward model of the intensity measured by the sensor for acquisition = 1 or 2 is:

̃ (Φ, ) = | | | (Φ, ) | | | 2 . ( 4
)

Inverse problems formulation

Estimating parameters and reconstructing the images Φ can be stated as a regularized minimization problem involving the residuals between the data intensity =1..2 and its model ̃ (Φ, ) =1..2 :

Φ, ̂ = argmin Φ∈D, C data 1 , ̃ 1 (Φ, ) + C data 2 , ̃ 2 (Φ, ) + sp,s C sp s + sm,s C sm s + sm,bg C sm bg . ( 5
)
The term

C data = min ∑ , , -̃ , 2
[ ] , is the data fidelity term, where ( , ) stands for the sensor pixels locations and is a real-valued coefficient to adjust the range of the model to the data acquisition dynamic, and is a map of weighting coefficients accounting for either the unmeasured pixels or the limited size of the sensor [START_REF] Berdeu | Reconstruction of in-line holograms: combining model-based and regularized inversion[END_REF][START_REF] Soulez | Inverse-problem approach for particle digital holography: accurate location based on local optimization[END_REF].

Regularization terms C sp and C sm applied on bg and s are weighted by hyperparameters sp,s , sm,s and sm,bg .

C sp = ∑
, the weighted sum of the 1 -norms of r and i , ensures the sparsity of the reconstructed sample image s , which prevents twinimage artifacts [START_REF] Denis | Inline hologram reconstruction with sparsity constraints[END_REF] by favoring objects with compact supports, i.e., such that s is zero at many pixels of the sample plane where no absorption or phase shift occur. Since this sparsity prior can bias the reconstruction [START_REF] Deledalle | On Debiasing Restoration Algorithms: Applications to Total-Variation and Nonlocal-Means[END_REF], it is debiased using the method proposed in [START_REF] Mário | Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems[END_REF].

C sm = ∑ , Δ 2 , r + 2 Δ 2
, i smooths the reconstructed phases by favoring low-energy spatial gradients, which prevents noise amplification [START_REF] Berdeu | Reconstruction of in-line holograms: combining model-based and regularized inversion[END_REF].

Δ 2 , [ • ] = [ • ] +1, -[ • ] , 2 + [ • ] , +1 -[ • ] ,
2 corresponds to a finite difference operator. Hyper-parameters 1 and 2 are introduced to account for the different ranges of the real and imaginary parts r and i , they have a limited influence and can be set once for all.

Finally, Φ ∈ D are the constraints on the domain of the set Φ. In our case, Φ ∈ D ⇔ ℑ [ s ] , ≥ 0, i.e. we impose the positivity of the objects' absorbance. This constraint is justified by the fact that objects cannot emit light in the wave propagation process. Conversely, no hard constraint is enforced on the domain of the background that is thus allowed to fluctuate positively and negatively to account for any illumination inhomogeneity.

Alternating minimization algorithm and initialization

Solving (5) provides the complex phases Φ jointly with the parameters . In practice, we alternate a minimization step with respect to Φ (with fixed to its previous estimation) and a second with respect to (with Φ fixed to its previous estimation). The reconstructions of Φ with fixed are performed by a limited-memory quasi-Newton method with bound constraints, VMLM-B [START_REF] Berdeu | Reconstruction of in-line holograms: combining model-based and regularized inversion[END_REF][START_REF] Thiebaut | Optimization issues in blind deconvolution algorithms[END_REF]. The regularization hyperparameters sp,s , sm,s and sm,bg are tuned empirically to obtain the best visual reconstruction quality. The optimizations of with Φ fixed are done via a simplex search method [START_REF] Lagarias | Convergence properties of the nelder-mead simplex method in low dimensions[END_REF]. The method is implemented in the Global-BioIm framework [START_REF] Unser | GlobalBioIm: A Unifying Computational Framework for Solving Inverse Problems[END_REF][START_REF] Soubies | Pocket guide to solve inverse problems with globalbioim[END_REF].

Parameters are initialized based on the simple backpropagated transmittances 1 and 2 of the holograms. The initial lateral shifts , are first estimated by rigid registration of 1 and 2 . The initial focus distances and + are obtained by maximizing the sharpness of

| | | -1 | | |
[20], a focusing criterion that works for both absorbing and dephas- ing objects.

Results

The proposed method was tested on Gram-stained bacteria, fixed on a slide, using a standard procedure (bioMérieux PreviColor Gram system): Gram-positive cocci Staphylococcus epidermidis (S. epidermidis) and Gram-negative rods Escherichia coli (E. coli), see Fig. 2.

The data were acquired via a microscope (Olympus BX-61, oil-immersion 60x/NA1.4 objective, 10x tube lens, Basler monochromatic camera with a pixel pitch 3.45 µm) [START_REF] Jolivet | Regularized reconstruction of absorbing and phase objects from a single in-line hologram, application to fluid mechanics and micro-biology[END_REF]. The illumination source was a fiber-coupled red LED (617 nm, Mightex FCS-0617-000), filtered by a 5 nm band-pass filter centered at 610 nm. The tip of the fiber was placed approximately 5 centimeters from the sample. A ( , , ) motorized stage was used to shift the sample.

Two reconstruction experiments were performed from two acquired datasets: one for which the second hologram has been laterally shifted relatively to the first hologram, and one for which the two holograms have been acquired at two different defocus distances (i.e. axial shift). Figure 3 and Figure 4 present the hologram data used in our experiments and estimation results in the data space. Figure 5 illustrates reconstruction results in the object space (information of interest). Each figure is detailed hereafter. A compilation of all the presented images (data, background estimation, sample estimation in the data and object spaces, residuals), for each experiment, are proposed in two respective supplementaries that are referred as Visualization 1 and Visualization 2. bacteria can hardly be seen in the phase image, as zoomed in Fig. 5(j).

In the proposed method, the spatial coherence of the illumination source and other resolution-limiting factors such as the numerical aperture are included in the forward model [START_REF] Vandewiele | Low coherence digital holography microscopy based on the lorenz-mie scattering model[END_REF]. Given that the refractive index of the bacteria is larger than that of the surrounding medium, a positivity constraint is also applied on the real part of s , enforcing a positive phase shift.

Figures 5(c)-5(d) show the sample retrieved using a standard inverse problems approach [START_REF] Jolivet | Regularized reconstruction of absorbing and phase objects from a single in-line hologram, application to fluid mechanics and micro-biology[END_REF], that aims at applying the proposed approach without estimating the background contribution: Φ ∈ D ⇒ bg = 0, all other things being equal. The result suggests that the darker region, framed in green, strongly pollutes the retrieved modulus and phase, even when two shifted holograms are used in the reconstruc-tion. This shows that when the background signal is strong, standard reconstruction methods in digital holography microscopy such as, among others, modified Gerchberg-Saxton algorithms [START_REF] Zhang | Reconstruction of in-line digital holograms from two intensity measurements[END_REF][START_REF] Allier | Imaging of dense cell cultures by multiwavelength lens-free video microscopy[END_REF], regularized approaches with priors [START_REF] Rivenson | Sparsity-based multi-height phase recovery in holographic microscopy[END_REF], inverse problem approaches from multiple holograms [START_REF] Herve | Multispectral total-variation reconstruction applied to lens-free microscopy[END_REF] or even single hologram [START_REF] Jolivet | Regularized reconstruction of absorbing and phase objects from a single in-line hologram, application to fluid mechanics and micro-biology[END_REF], bias the reconstruction of the object of interest, as they do not account for the background contribution.

Finally, Figs. Table 1 shows that, in contrast to the lateral shift initial- ization that is very precise with corrections of only 10 nm, the initial auto-focus distance (estimated as described in Sec. 2.3) is overestimated: the correction after several iterations of our alternating minimization algorithm exceeds 140 nm. Some tests (not shown here) show that the refinement of the focus distance is beneficial to the reconstructions.

Discussion and conclusion

In this paper, we focused on the reconstruction of objects of interest that are shifted relative to a fixed background in digital holography microscopy. We have shown that an inverse problems approach jointly estimating this background signal and the signal of interest can effectively unmix these two components, using only two shifted in-line holograms as input. This separation is achieved by inverting the forward model that accounts for the interferences between the two signals. This also makes possible a refinement of the focus distance. The transparent objects are correctly reconstructed in the sample phase.

This procedure was applied on two 2D laterally or axially shifted (differently defocused) holograms. It can be naturally extended to a higher number of acquisitions or to objects moving in front of a static background (such as microscopy of in-flow objects).

The remaining high frequency fringes in the residues indicate that there is still some room for improvements. They might be reduced by further refining the focus or the coherence length estimation or by considering a more complex modeling of hologram formation.

Finally, the different parameters such as sp,s , sm,s and sm,bg are currently manually set. Methods to automatically tune these hyperparemeters could be investigated.

⋆

  Funding. French National Research Agency (ANR) (LABEX PRIMES of Université de Lyon ANR-11-LABX-0063, Investissements d'Avenir ANR-11-IDEX-0007); Region Auvergne-Rhônes-Alpes (project DIAGHOLO). * Corresponding author fabien.momey@univ-st-etienne.fr (F. Momey) ORCID(s): out-of-focus objects perturbations [2, 8, 9]. The background hologram can be either: (i) captured by acquiring a hologram after shifting the objects out of the field of view, (ii) computed by averaging a hologram sequence of moving objects [2, 7, 9], (iii) computed by low-pass filtering of a single hologram

Figure 1 :

 1 Figure 1: Hologram formation principle in the presence of disturbing objects (black circles): background patterns originate from dust (top, red), the 2D sample diffracts light (middle, green), in the sensor plane a hologram is captured (bottom). Optical elements are not shown here, and a fixed background plane accounts for all the disturbances. Between acquisitions 1 and 2, the sample is shifted.

Figure 2 :

 2 Figure 2: (a) Bright-field image of Gram-stained bacteria. (b) Zoomed-in view. (c) Under red illumination, absorbing, translucent and weakly absorbing bacteria are respectively identified with black, red and purple arrows.

Figure 3 (Figure 3 :

 33 Figure 3: (a) An in-line hologram of the bacteria shown in Fig. 2(a). The green frame emphasizes a darker region in the perturbed background. The colored arrows point at the diffraction patterns produced by the objects highlighted in Fig. 2. A pair of holograms recorded with a lateral shift of the sample is reconstructed. The contribution of the retrieved object is shown in (b) and the background signal in (c) with the same color bar as in (a). (d) gives the residuals 1 -̃ 1 between the first hologram and the predicted intensity.

  5(e)-5(f) and Figs. 5(g)-5(h) present the sample reconstructed with the proposed method, respectively for laterally shifted acquisitions and for acquisitions with different defocus. Compared to Figs. 5(a)-5(b), the sample absorption, given in Fig. 5(e)-5(g), and the sample phase, given in Fig. 5(f)-5(h), are almost free from background perturbations or twin-image artifacts, as is particularly visible in the region framed in green. The transparent objects appear to be well segmented in the phase image.

Figure 4 :

 4 Figure 4: (a) A pair of in-line holograms of the bacteria shown in Fig. 2(a), acquired at 2 different defocus distance. (a,b,d) We visualize both holograms respectively on the upper and lower half parts of the field of view (separated by a red dashed line). This pair of holograms is reconstructed. The contribution of the retrieved object at the 2 corresponding defocus are shown in (b) and the background signal in (c) with the same color bar as in (a). (d) gives the residuals 1 -̃ 1 and 2 -̃ 2 between the holograms and the predicted intensity.

Figure 5 :

 5 Figure 5: (a,b) Modulus and phase (radian) obtained with a simple back-propagation of Fig. 3(a). (c,d) Modulus and phase (radian) of the retrieved sample s without the estimation of the background bg via a state-of-the-art inverse problem approach.

Table 1

 1 Estimates of the different parameters at the beginning and at convergence of our alternating minimization algorithm.

		Initial (µm) Final (µm) Difference (nm)
	∕	4.48 ∕ 0.11	4.47 ∕ 0.12	-10 ∕ 10
	∕ +	7.80 ∕ 7.72	7.62 ∕ 7.58	-180 ∕ -140