V Arounassalame
email: vignesh.arounassalame@onera.fr

M Guénin

M Caes

I Ribet-Mohamed

P Christol
email: christol@ies.univ-montp2.fr

L Höglund

E Costard
email: eric.costard@ir-nova.se

TPC Non uniformity correction calculated with two reference temperatures

Keywords: focal plane array, infrared detector, residual fixed pattern noise, performance evaluation, robust statistic, temporal stability, type-II superlattice Parameter Acronym Feature Residual Fixed Pattern Noise RFPN Spatial

The temporal stability of the image quality of an infrared focal plane array (FPA) is one key parameter to consider for high performance imaging applications. It is generally evaluated through Residual Fixed Pattern Noise (RFPN) measurements realized after a two-point correction (TPC) and bad pixels exclusion. However, the effect of random telegraph signal (RTS) on temporal stability has never been quantified, the intermittent blinking of RTS pixels making measurements inherently complicated. In this paper, we propose a novel protocol and a novel data processing technique to fully characterize the temporal stability of an infrared FPA, thanks to simultaneous RFPN and RTS noise measurements. Using a robust statistics based on Median Absolute Deviation (MAD) leads to a more trustful RFPN evaluation, unaffected by the bad pixels exclusion step. This new protocol can be applied to any FPA technology, cooled or uncooled, operating in the [3-14µm] spectral domain. We chose to use a Type-II superlattice (T2SL) midwave infrared FPA integrated in a cryocooler at 80K. We show that the long-term stability of this 320x256 pixels FPA is excellent. The RFPN only slightly increases over 77 days, using a TPC that has been calculated on the first day of the measurements campaign. A simple offset update, compatible with operational contingencies, can further improve the temporal stability of this FPA. Only 0.3% of the pixels were affected by RTS noise at some moment. Those blinking pixels don't affect the RFPN evolution, which is a very encouraging result for T2SL detector technology.

I. INTRODUCTION

HERMAL infrared sensors are a wide field of study, thanks to a broad spectral range (from 3 µm to 14 µm), multiple applications (military, aerospace, industrial …), and different achievable technologies for the fabrication of one of the main parts of the sensor : the focal plane array (FPA). A growing demand for high performance sensors encouraged technological developments which aimed to propose detectors tailored for specific requirements given the intended application.

In the MWIR (3-5µm) and LWIR (8-14µm) domains, several well-established high performance cooled or uncooled technologies are available. For example, quantum detectors can be based either on II-VI materials or on III-V materials [START_REF] Rogalski | Infrared systems fundamentals[END_REF]. Antimony-based type-II superlattice (T2SL) detectors are a recently matured III-V technology [START_REF] Rogalski | InAs/GaSb type-II superlattice infrared detectors: three decades of development[END_REF] which could combine the advantages of several technologies: the cut-off wavelength can be tailored thanks to band gap engineering, the electro-optical performance (especially quantum efficiency) is close to that of HgCdTe [START_REF] Wu | High quantum efficiency mid-wavelength infrared type-II InAs/InAs1-xSbx superlattice photodiodes grown by metal-organic chemical vapor Fig.9. Comparison of the time evolution[END_REF], high operating temperatures are accessible [START_REF] Ting | Mid-wavelength high operating temperature barrier infrared detector and focal plane array[END_REF], and promising results in terms of spatial uniformity and operability have been reported [START_REF] Höglund | Manufacturability of type-II InAs/GaSb superlattice detectors for infrared imaging[END_REF]. This maturity enabled to study its potential for radiation thermometry for example [START_REF] Hobbs | InAs/GaSb Type-II Superlattice for Radiation Thermometry[END_REF].

The performance of infrared focal plane arrays is evaluated through commonly accepted figures of merit such as quantum efficiency or responsivity, completed by fixed pattern noise (FPN) and modulation transfer function (MTF). However, there is still a lack of appropriate and commonly accepted metrics to better assess the potential of a given technology for high performance imaging. For example, the temporal stability of the signal delivered by a focal plane array is of major importance. Indeed, it dictates how often the calibration of operational electro-optical systems has to be done, and thus reflects the availability of the system during an operational mission. In this way, long term stability could become a key merit function in technology comparison.

However, finding the appropriate figure of merit to quantify the temporal stability is not so easy. The commonly used residual fixed pattern noise (RFPN) is for example highly affected by the preliminary step consisting in excluding the bad pixels [START_REF] Ribet-Mohamed | Temporal Stability and correctability of a MWIR T2SL focal plane array[END_REF]. As a result, different criteria for bad pixels exclusion have been developed [START_REF] Breiter | State-of-the-art MCT IR-modules with enhanced long term and cycle stability[END_REF] [START_REF] Robo | QWIP focal plane arrays performances from MWIR up to VLWIR[END_REF][10] [START_REF] Rubaldo | State of the art HOT performances for Sofradir II-VI extrinsic technologies[END_REF] [START_REF] Martijn | QWIPs at IRnova, a status update[END_REF], making comparisons between detectors difficult. Moreover, this bad pixels exclusion step doesn't necessarily detect pixels with random telegraph signal (RTS) noise. The signal delivered by these pixels can oscillate between two (or more) levels at certain times, creating a particularly harmful blinking effect for image quality. To date, the possible link between RFPN degradation and the appearance of RTS pixels has not been reported.

This paper reports on the study of the temporal stability of a MWIR T2SL IDDCA (Infrared Detector with Dewar Cooler Assembly) from IRnova. A new experimental protocol was defined to realize simultaneously RTS and RFPN measurements, the latter being evaluated with a robust estimator. The new protocol and data processing enables to reach the true RFPN values even if there are some bad pixels or RTS pixels left in the pixels exploited. The method developed here can be applied for any infrared detector operating in the MWIR or LWIR spectral domains, cooled and uncooled.

II. EXPERIMENTAL PROCEDURE

The MWIR T2SL VGA/4 FPA with 320x256 pixels (30 µm pitch) used in this work is integrated in an IDDCA. TABLE 2 provides an overview of its specifications. The detector's signal output is coded on 16 bits, between 0.3 V and 2.7 V.

The experimental bench consists of the T2SL IDDCA detector facing a CI-system SR80-4D/A blackbody (Fig. 1), hence the detector receives a controlled irradiance. The integration time of the FPA is set to 4 ms with a frame time of 16 ms (60Hz frame rate). The blackbody's specifications are in TABLE 3.

The experiment is made to perform both RFPN measurements and RTS pixels detection, in order to see if the degradation of the RFPN is correlated with the apparition of blinking pixels. RFPN measurements require varying the blackbody's temperature (to explore different well fill values of the integrating capacitor), while the study of RTS pixels requires long acquisitions at fixed blackbody temperature instead. The following protocol has therefore been established (see Fig. 2) : the blackbody temperature is varied from 10°C to 58°C with a 3°C step (17 cubes of images). These temperatures were chosen as they were easily accessible with an extended blackbody, integration time was then adjusted accordingly at 4 ms in order to reach the saturation for the highest blackbody's temperature.

For each temperature, a cube of 5000 images is acquired. The first 256 images are used to infer the RFPN (temporally averaged image is used for the calculation), whereas the whole cube is analysed to extract RTS pixels. The number of 5000 images is chosen as a good compromise between observation time and data processing complexity. In order to study the time evolution of the RFPN and RTS pixels, this measurement was repeated day after day, sometimes within the same cooling cycle and sometimes after letting the FPA warm up to room temperature and a new cooling cycle.

Data from two measurements campaigns have been used. Indeed, a pause of 45 days passed exactly between the two campaigns of measurements, in order to have a long time evaluation. A total of 74 RFPN measurements (one RFPN measurement equals to 17 cubes of images) were realized, representing a cumulated acquisition time of 28 hours. Between the first and last RFPN measurements, 77 days have passed.

III. DATA PROCESSING

The Fixed Pattern Noise corresponds to the spatial fluctuations of the signal delivered by the pixels composing an FPA when it receives a uniform irradiance. The fixed pattern noise can find its origin in two different parts of the FPA: the first one is the detection circuit made of T2SL technology (which converts photons into electrons) because of differences in cut-off wavelengths for example. The second one is the Read Out Integrated Circuit made of silicon (ROIC, which processes the pixel data to generate a video signal) for example if the output amplifiers have different offsets.

A. Two-point correction

A linear two-point correction (TPC) [START_REF] Schulz | Non uniformity correction and correctability of infrared focal plane arrays[END_REF] is commonly realized on the signal in order to reduce the Fixed Pattern Noise. This calibration is done using two extended blackbodies at different temperatures T1 and T2. The corrected signal for one pixel (i,j) 𝑆 𝑖,𝑗 ′ is calculated using the two coefficients of the linear correction 𝐺 𝑖,𝑗 and 𝑂 𝑖,𝑗 :

{ 𝑆 𝑖,𝑗 ′ (𝜙) = 𝐺 𝑖,𝑗 𝑆 𝑖,𝑗 (𝜙) + 𝑂 𝑖,𝑗 𝑆 𝑖,𝑗 ′ (𝜙 1 ) = 〈𝑆(𝜙 1 )〉 𝑆 𝑖,𝑗 ′ (𝜙 2 ) = 〈𝑆(𝜙 2 )〉 (1) …
where 𝑆 𝑖,𝑗 (𝜙) is the signal delivered by the pixel (i,j) for an incident power equal to 𝜙 (after removing the electrical offset of the pixel). 𝜙 1 and 𝜙 2 correspond to the incident powers of the two blackbodies. They are chosen such as the Well Fill reaches 33% and 66% respectively. They enable to have an efficient correction on the whole detector's dynamic. Other values can be chosen depending on the application. 〈𝑆(𝜙 1 )〉 (respectively 〈𝑆(𝜙 2 )〉) represents the spatially averaged signal at the incident power 𝜙 1 (respectively 𝜙 2 ). The two correction coefficients, gain and offset respectively, are calculated from (2) and (3) assuming that each pixel has a linear signal with the incident power:

𝐺 𝑖,𝑗 = 〈𝑆(𝜙 2 )〉 -〈𝑆(𝜙 1 )〉 𝑆 𝑖,𝑗 (𝜙 2 ) -𝑆 𝑖,𝑗 (𝜙 1 ) (2) …
𝑂 𝑖,𝑗 = 〈𝑆(𝜙 1 )〉 -𝐺 𝑖,𝑗 𝑆 𝑖,𝑗 (𝜙 1 )

This type of calibration is done only once, the correction coefficients are used for the RFPN measurements made afterwards to determine how long they remain valid.

B. Offset Update correction

Another type of correction is the offset update correction [START_REF] Orzanowski | Nonuniformity correction algorithm with efficient pixel offset estimation for infrared focal plane arrays[END_REF]. It is used in complementary of the TPC. Indeed, the TPC calculates 𝐺 𝑖,𝑗 and 𝑂 𝑖,𝑗 at the beginning, then before each RFPN measurement, the 𝑂 𝑖,𝑗 coefficient is refreshed. The following calculation is used :

𝑂 𝑖,𝑗 = 〈𝑆(𝜙 1 )〉 -𝐺 𝑖,𝑗 𝑡 0 𝑆 𝑖,𝑗 (𝜙 1 ) (4) …
with 𝐺 𝑖,𝑗 𝑡 0 the Gain from the first RFPN measurement with the TPC. Hence, only 𝑂 𝑖,𝑗 is updated and the Gain used for the TPC is from the first RFPN measurement. Reminder, 𝜙 1 corresponds to an incident power such as the Well Fill (WF) capacity of the detector reaches 33%. This type of complementary correction is often operationally accessible. Indeed, it can be performed if the operator uses the camera's shutter as a blackbody for the calibration before utilization. Therefore, we have chosen ambient temperature which corresponds to a WF of 33% in our case for the offset update.

C. Temporal stability definition

The two-point correction is very effective at the time it is implemented. However, the problem is to know if the correction used is appropriate for a long time. That is why, the Residual Fixed Pattern Noise (RFPN) which remains after the TPC is calculated. It indicates the effectiveness of the reduction of the Fixed Pattern Noise by its definition itself:

𝑅𝐹𝑃𝑁(𝜙) = √ 1 𝑁 ∑ ∑(𝑆′ 𝑖,𝑗 (𝜙) -〈𝑆′(𝜙)〉) 2 𝑗 𝑖 (5) …
where N is the total number of pixels of the FPA. The RFPN is thus a standard deviation representing the global behavior of the FPA and it is usually interpreted as the thickness of the Gaussian distribution expected for photon noise limited pixels.

The RFPN tends to increase with time. This may be due to instabilities or drifts in the detector as well as in the instrument in which it is integrated. Determining when the TPC needs to be calculated again is not easy because it depends on the application. One can use the ratio of the RFPN over the temporal noise (TN) [START_REF] Perry | Linear theory of nonuniformity correction in infrared staring sensors[END_REF], the latter being defined as the temporal fluctuations of the signal delivered by a pixel. Indeed, fixed pattern noise and temporal noise are in general quadratically summed [START_REF] Mooney | Responsivity non-uniformity limited performance of infrared staring cameras[END_REF]. Therefore, the criterion that the RFPN must be inferior to the temporal noise in order to get a good image quality is reasonable. This ratio defines the temporal stability of the FPA.

D. Bad pixels detection

Before calculating the RFPN, some pixels have to be excluded because their behavior deviates too far from the whole FPA. Generally speaking, pixels whose mean level, response, noise and/or NETD (noise equivalent temperature difference) deviate too far from the median value are discarded (see TABLE 4), but the exact criteria vary from one team to another, which can have a strong influence on the RFPN value [START_REF] Ribet-Mohamed | Temporal Stability and correctability of a MWIR T2SL focal plane array[END_REF].

In this article, bad pixels are pixels which do not follow the main statistical distribution of the matrix. Among them can be found, hard defects pixels (dead pixels, short circuited pixels) and pixels which do not respond to the criteria used in TABLE 4. 256 images were acquired on day 1 as specified by the protocol detailed in the previous section. The total number of bad pixels detected is 49, which corresponds to an operability of 99.94%. This total is not equal to the sum of the number of pixels in each category, because one bad pixel can have multiple defects and fall simultaneously into several categories. Once the bad pixels have been set aside, the residual fixed pattern noise is calculated on the corrected images (after application of the two-point correction) thanks to equation ( 5).

E. RFPN evaluated with Median Absolute Deviation (MAD)

Even if the selection of bad pixels is careful, some pixels with atypical behavior may persist and have a significant influence on the RFPN. This is due to the fact that the RFPN evaluation relies on a standard deviation calculation, which is well known to encounter problems regarding outliers. To tackle this issue, the median absolute deviation (MAD) [START_REF] Huber | Scale estimates[END_REF] can be used as a robust estimator. Indeed, using the median value rather than the mean value as in equation ( 6) provides a robust measure of statistical dispersion. The MAD is very insensitive to the presence of outliers. One indicator of this insensitivity is the "breakdown point" [START_REF] Donoho | The notion of breakdown point[END_REF]. The estimator's breakdown point is the maximum proportion of data points that can be outliers without forcing the estimator to result in a false value. For the MAD, this value is 50%. It means that as long as less than half of the data points are outliers, the MAD will have the correct value. The MAD is defined as follows:

𝑀𝐴𝐷 = 𝑚𝑒𝑑𝑖𝑎𝑛(|𝑆 ′ -𝑚𝑒𝑑𝑖𝑎𝑛(𝑆 ′ )|) (6) …
With S' the corrected image. But MAD is only a robust measure of the variability of sample data, and does not represent the standard deviation. A scale factor 𝑘 [START_REF] Rousseeuw | Alternative to the median absolute deviation[END_REF] must be applied in order to define the corresponding RFPN :

𝑅𝐹𝑃𝑁 𝑀𝐴𝐷 = 𝑘 • 𝑀𝐴𝐷 (7) …
Where k is a constant scale factor, which depends on the distribution. Usually, k is taken equal to 1.4826, under the assumption of normality of the data, without taking into account the outliers.

The importance of a robust estimator can be observed on Fig. 3. Indeed, the robust RFPNMAD is clearly lower than the RFPN polluted by outliers. It can be noted that both the RFPNMAD and the RFPN polluted by outliers exhibit the usual "W" shape curve, making the effect of outliers difficult to detect.

F. Random Telegraph Signal (RTS) pixels

Recent attempts to increase the operating temperature of infrared focal plane arrays and to reduce their size, weight and power (SWaP) have shown that image quality could be degraded due to pixels affected by RTS (random telegraph signal) noise [START_REF] D'souza | Noise attributes of LWIR HDVIP HgCdTe detectors[END_REF][21] [START_REF] Pogány | Random Telegraph Signal noise instabilities in lattice-mismatched InGaAs/InP photodiodes[END_REF]. As can be seen in Fig. 4, those RTS pixels exhibit a signal oscillating between at least two levels, while the received power and operating conditions are unchanged. This results in a blinking effect as a function of time, which cannot be suppressed by TPC. Since the blinking is not permanent, detecting these pixels is a very difficult task and is the subject of intensive research [START_REF] Yuzhelevski | Random telegraph noise analysis in time domain[END_REF]- [START_REF] Goiffon | Multilevel RTS in Proton Irradiated CMOS Image Sensors Manufactured in a Deep Submicron Technology[END_REF]. The algorithm used here for the detection of RTS pixels is from [START_REF] Guénin | RTS noise detection and voltage effect on RTS in HgCdTe focal plane array[END_REF]. The main steps of the algorithm are:

• A list of candidate pixels is calculated using a criterion on temporal noise. • For each candidate pixel, the temporal evolution of the signal (also called continuous level) is analyzed by the Pruned Exact Linear Time (PELT) algorithm [START_REF] Killick | Optimal detection of changepoints with a linear computational cost[END_REF] in order to detect the jumps and level differences. The principle relies on changepoints detection of the continuous level's mean value. These changepoints are detected by using a cost function from the PELT algorithm. • Finally, a candidate pixel is confirmed as a RTS pixel if the difference of levels of the jumps is superior to the temporal noise.

IV. RTS PIXELS ANALYSIS RTS pixels are detected using the algorithm from [START_REF] Guénin | RTS noise detection and voltage effect on RTS in HgCdTe focal plane array[END_REF] and then classified. After exclusion of hard defects and continuous level defects, 280 pixels are identified as RTS pixels. However, RTS pixels' origins are not known in this FPA. Indeed, in an infrared FPA, RTS pixels can have different origins: either from the detection circuit which is composed of T2SL material or the ROIC which is in silicon. An experiment is made to distinguish if RTS pixels come from the ROIC or the detection circuit. In order to isolate RTS pixels coming only from the ROIC, measurements (cubes of 5000 images) are made at a near zero integration time (1µs). A first list of RTS pixels is computed, and then the definitive list is made from the common RTS pixels between the first list and the list of RTS pixels detected at 4ms. This comparison is made in order to know which RTS ROIC pixels were still observable under flux. The results show that only 22 RTS pixels under flux come from the ROIC, and 21 among them belong to hard defects or continuous levels defects. RTS ROIC pixels are now excluded, in order to focus the study on T2SL RTS pixels. 279 RTS pixels remain. These pixels are divided into three categories :

• Two-levels: RTS pixel with a continuous signal oscillating between two levels at least on one measurement (251 out of 279 RTS pixels, hence 90 %). • Multi-levels: RTS pixel with a continuous signal oscillating between multiple levels (more than two) at least on one measurement (0.4 %). • Noisy: RTS pixel whose continuous signal has too much noise to clearly identify different levels (10 %).

In the following, only two level RTS pixels are considered for the sake of simplicity. Long and numerous measurements (see II-Experimental Procedure) are realized in order to see if a fixed population of RTS pixels exists. If so, then excluding those RTS pixels could help to solve the problem of image quality deterioration. Indeed, if these pixels are identified, their signals can be replaced by their immediate neighbors thanks to image processing. That is the purpose of the 74 measurements, with each measurement containing 664s (8x83s) of cumulated observation time data. Only the 8 first blackbody's temperatures (from 10°C to 31°C) are exploited, because high blackbody's temperatures induce important shot noise which hides some RTS noises. Fig. 5 shows two curves: one is the number of RTS pixels detected for each measurement, the other is the cumulated number of RTS pixels, i.e. the number of pixels which blinked at least once since the beginning of measurements. The black rectangle between measurements 24 and 25 represents the 45 days of pause between both measurement campaigns. The number of pixels which were 2-level RTS pixels at some point is 251 (i.e 0.3 % of the FPA), in 13.7 hours of cumulated observation time. In average, there were 10 RTS pixels by measurement, which is very low and steady. The number of cumulated RTS pixels does not seem to reach a plateau, meaning that there is no fixed population of RTS pixels. Because of the numerous thermal cycles, it is difficult to see if there is a difference on the number of new RTS pixels appearing after each measurement. That is why, measurements are divided into two categories: measurements realized within a given cooldown and measurements realized after letting the FPA reach room temperature (new cooling cycle). Statistics in TABLE 5 show that the appearance of RTS pixels is correlated to thermal cycles. Indeed, there are five times more new RTS pixels appearing after letting the FPA reach room temperature than during the same cooldown.

V. RFPN ANALYSIS

The evolution of the RFPN/TN ratio over time (or number of RFPN measurements here) is studied in order to evaluate the temporal stability of the FPA. Fig. 6. represents the RFPN MAD /TN over the Well Fill at different times using the TPC from day 1. As it can be observed, RFPN MAD /TN corresponding to day 1 shows the best performance without surprise, since the TPC corresponds to the data set at that time. As expected, the RFPN MAD /TN cancels itself out at the reference points which correspond to 33% and 66% of the Well Fill. But as times passes, the TPC loses its effectiveness as it can be seen on the curves representing days 13 and 77. Fig. 7. represents the RFPN MAD //TN with the offset update. The improvement is important. Until 50% of the Well Fill, there is practically no difference with day 1. After 50% of the Well Fill, there is an increase of the RFPN MAD /TN, however, the increase is less important than without offset update. It can be interpreted that the main factor to the pixels' signal variation in time is only a shift, that is why the offset updates are very efficient.

The previous analyses were made for three measurements, for the sake of clarity. The whole data set is analyzed in the following discussions through the time evolution of the RFPN/TN (respectively RFPN MAD /TN). It is the evolution of the RFPN/TN (respectively RFPN MAD /TN) evaluated at 50% of the Well Fill. The black rectangle between measurements 24 and 25 represents the 45 days of pause between both measurements.

In Fig. 8 the time evolution of the RFPN/TN is represented. It is focused on determining the potential role of RTS pixels in degrading the temporal stability. For each evolution, the calibration is done on measurement one: the TPC correction coefficients are calculated and bad pixels are detected and excluded. Then, for each measurement after calibration, the correction coefficients calculated are applied. The diamond dots curve is the evolution of the ratio RFPN/TN without detection of RTS pixels for any measurement. As for the square dots curve, the same is applied but for each measurement, RTS pixels are detected and excluded from data exploitation. Before comparing the two curves, it can be noted firstly, that for the whole duration of the measurements, the RFPN/TN remains lower than one. Secondly, there is a sudden important increase of 60% after measurement 2, but the ratio quickly becomes relatively stable. Moreover, the RFPN/TN is not constantly increasing, it is also dropping sometimes from one measurement to another. Contrarily to what can be thought, the image quality is not growingly deteriorated. Sometimes it is improved compared to the previous measurement as it can be seen between measurements 20 and 21. These "relaxation phenomenon" of the RFPN/TN happen when the detector has been switched off for 15 hours.

At last, the comparison between the two curves shows that there is almost no difference between them. There are no sudden increases due to RTS pixels and there is not a constant shift throughout measurements between the two quantities Hence, RTS pixels for a detector temperature of 80K are not responsible for the deterioration of global image quality. Here it only has a slight effect on the RFPN/TN because there are only a few ones, their amplitude is weak and their frequency of apparition is low. However, it is known that increasing the temperature's detector increases the number of RTS pixels, and a deterioration of the global image quality can happen because of RTS pixels. That is why, as underlined in section III, the MAD criterion has to be utilized in order to ensure that the true image quality is evaluated. In order to properly evaluate RTS pixels impact when there are only a few ones, a way to evaluate the local image quality will have to be found. In Fig. 9, RFPN MAD /TN and RFPN/TN are compared. The RFPN MAD /TN is inferior to the RFPN/TN, as expected, thanks to its robustness. However, the two quantities are very close and excluding more pixels to reach the RFPN MAD /TN value could be counterproductive, since good pixels could be eliminated as well, for a small improvement. The comparison between RFPN MAD and RFPN MAD with offset update shows an improvement in favor for the last one. The offset update enables to have an almost constant RFPN throughout the whole measurement. However, it can be noted that the measurements 39 to 47 and 60 to 65 have a sudden increase compared to other measurements and it does not reappear in later measurements. In order to understand what phenomenon caused this increase, the temporal noise of the FPA against the blackbody's temperature was studied. And it was seen that the temporal noise for temperatures between 31 °C and 55°C is very important when the problem occurred, compared to other measurements. Indeed, it does not follow the evolution of the shot noise as it should be expected. Moreover, the observation of the temporal noise image shows that it is inhomogeneous. In view of the time constants characteristic of the phenomenon we concluded that convection occurred between the detector and the blackbody for these measurements. That is why this part should not be considered for the interpretation of temporal stability.

Finally, there is one last important accessible parameter that was left untouched: the number of images used for averaging. Indeed, this parameter is important because it determines if the detector can be used in operational condition or not. Moreover, the FPN is a very sensitive quantity, which can fluctuate because of the environment (room temperature, parasitic flux, hygrometry…). That is why it is hard to give a clear interpretation of fluctuations of the RFPN/TN ratio under one. Hence, the time evolution of the RFPN MAD /TN for 1 image instead of 256 is observed in Fig. 10.

The RFPN MAD /TN becomes superior to one, but it stays between values 1.1 and 1.3 for the whole time. As for the offset update method, the improvement is modest (8% of decrease compared to the RFPN MAD /TN without offset update). However, it is very stable and there is almost no variation, exception made of the measurements when the convection occurred. These results are encouraging for the temporal stability and uniformity of this detector.

VI. CONCLUSION

We have presented a new experimental protocol and the associated data processing technique to fully characterize the temporal stability of the image quality of infrared focal plane arrays. The experimental protocol has been defined so as to carry out simultaneous RFPN and RTS noise measurements. We thus combine measurements varying the blackbody temperature (to extract RFPN) and long acquisitions at fixed blackbody temperature (to study RTS pixels). Measurements are repeated, either within one cooling cycle or after letting the detector reach room temperature and a start new cooling cycle, in order to rule on a possible correlation between cooling cycles and RFPN / RTS pixels evolution. In order to overcome the limitations of the usual RFPN figure of merit, which is strongly affected by outliers and thus intrinsically depends on the bad pixels exclusion step, we have defined a robust estimator based on the median absolute deviation.

Using this new capacity, we have realized the first simultaneous RFPN and RTS noise measurements on an infrared focal plane array. The device under test was a commercial T2SL MWIR 320x256 pixels focal plane array integrated in a cooler assembly operating at 80K. Our results show that the temporal stability is excellent, with a remarkably stable RFPN up to 77 days, applying a TPC that was calculated on the first day. The number of RTS is very low, with 10 blinking pixels in average by measurement. From one measurement to the next, the RTS pixels are not necessarily the same, such that the number of pixels that have blinked at least once since the beginning of the measurement keeps increasing, leading to a total of 251 pixels (0.3% of the 320x256 pixels) in 13.7 hours of cumulated observation time. Our results allow us to conclude that RTS pixels do not affect the RFPN evolution of this FPA, which, to the best of our knowledge, has never been demonstrated before.

To finish with, we show that a simple offset update correction, which is generally available in operation, leads to an even better temporal stability. This focal plane array is also suitable for real time imaging, since RFPN remains close to one when it is evaluated on a single image. We believe the exciting new measurement capacity demonstrated here will allow detector manufacturers to better understand the physical origins of temporal instabilities in infrared focal plane arrays. It will also help system integrators in setting detector specifications for camera cores and delivery compliance testing. To finish with, it opens the path to a rigorous comparison of the temporal stability of image quality of infrared focal plane arrays from different technologies.

Fig. 1 .

 1 Fig.1. Representation of the experimental bench. On the right the blackbody whose temperature varies between 10°C and 58°C. On the left, the T2SL IDDCA. A cold shield defines the field of view (FOV) of the FPA, and the cryocooler cools down the FPA to 80K. A piece of pierced foam is placed between the blackbody and the detector to reduce thermal convection.

Fig. 2 .

 2 Fig. 2. Experimental protocol. Cubes of 5000 images are acquired for 17 values of the blackbody temperature. The first 256 images of each cube are used for RFPN measurements, while the 5000 images are used for RTS noise study.

TABLE 4 -Fig. 3 .

 43 Fig. 3. RFPN/TN and RFPN MAD/TN over the Well Fill calculated on a data set containing two outliers.

Fig. 4 .

 4 Fig.4. Raw signal (in digits) as a function of time, for a RTS pixel facing a blackbody whose temperature is stable over time. This RTS pixel is categorized as a 2-level RTS pixel. Level 1 refers to the low level and level 2 to the high level.

Fig. 5 .TABLE 5 -

 55 Fig.5. 2-levels RTS pixels detected for each measurement of RFPN. The purple curve is the absolute number of RTS pixels detected at each measurement. The green curve represents the cumulated number of RTS pixels. The black rectangle represents the 45 days of pause between both campaigns.TABLE 5 -Statistics of the number of new 2-level RTS pixels appearing, within a given cooldown, after letting the FPA reach room temperature, and between two consecutive measurements (regardless of thermal cycles).Number of new RTS pixelsbetween two consecutive measurements:

Fig. 6 .

 6 Fig.6. RFPN MAD/TN over the Well Fill at different times, using the TPC from the measurement day 1.

Fig. 7 .

 7 Fig.7. RFPN MAD/TN over the Well Fill at different times, using the TPC from the measurement day 1 and complementary offset update.

Fig. 8 .

 8 Fig.8. Comparison of the time evolution (in number of measurements) of RFPN with and without RTS pixels excluded. The black rectangle represents the 45 days of pause between both campaigns, it explains the discontinuity observed.

TABLE 2 .

 2 Specifications of the IRnova FPA under test.

	Format	320x256
	Pixel pitch	30 µm
	Spectral Range	3.5-5 µm
	F number	F/2
	Noise Equivalent Temperature Difference	12 mK
	(NETD)
	Operability	99.8%
	Operating Temperature	80 K
	Framerate	60 Hz

TABLE 3 .

 3 Specifications of the blackbody.

	Temperature range	[5°C -100°C]
	Uniformity	±0,01°𝐶
	Stability	±0,02°𝐶
	Emissivity	0.97 ± 0,02
	Dimensions (WxLxH) 161 mm x 155 mm x 165 mm
	Emitting surface	106 mm x 106 mm

  Vignesh Arounassalame, received the Master degree in physics from the Institut d'Optique Graduate School, France, in 2018. He is currently pursuing the Ph.D. degree in optical engineering at ONERA, Palaiseau, France. His current research interests focus on electro-optical characterization of Type-II Superlattice photodetectors.Maxence Guénin, a Grenoble INP Phelma graduate with a Nanoscience Physics Master diploma, is a Ph.D student at ONERA, Palaiseau, France. His studies mainly focus on the physical causes of the RTS phenomenon in HgCdTe cooler infrared sensors.Marcel Caes, received his PhD degree in 1992, for the electro-optical characterisation of infrared focal plane arrays under low photon flux, for astronomy applications. He was then hired at ONERA to conduct research in the field of infrared detectors. He is now one of the mainstays of the "Infrared Detectors Characterisation Lab" at ONERA, and focuses on the development of tools to analyse the experimental data. Dr. Linda Höglund received a M.Sc. in Physics and Electrical Engineering 2001 and a Ph.D. in Material Science 2008 from Linköping University, Sweden. In 2001-2009, she worked at the Swedish research institute Acreo (now RISE), developing bolometers based on quantum structures and quantum dot infrared detectors. In 2009 she joined the Photonics group at NASA's Jet Propulsion Laboratory in Pasadena, California, USA. At JPL she was leading the optical characterization of type-II superlattice detector material. Since 2015 she has been working at IRnova, Sweden. At IRnova she is currently the R&D Team manager, leading the development of novel T2SL and QWIP detectors. Eric Costard, graduated as Doctor in Optics from the French University PARIS XI, joined Thales in 1985, where he has developed his background in semiconductor physics and processing. From the beginning of the 21 st century, he developed with his team innovation in infrared detection field: QWIP technology and InGaAs technology. With his team he has set up a pilot production line for QWIP FPAs for SWIR FPAs based on InGaAs material. Eric headed the IR Imaging Laboratory at III-VLab , he authored 15 patents in the field of QWIPs and received the Lamb Prize from the French Academy of Science in 2006. Subsequent to his work at Thales he also headed the III-V industrial transfer initiative at Sofradir in 2013. After many years working within the French IR industry, Eric moved to Sweden in the summer 2015 for joining IRnova. Eric is currently the CTO of IRnova where he continues to develop III-V detectors like QWIP and T2SL for gas detection and high end applicationsPhilippe Christol, is a professor of Electronic and Electrical Engineering at the Electronic Institute (IES) of Montpellier University, France, since 2005. He is specialist of infrared photodetection, in particular of antimonide-based photodetectors grown by Molecular Beam Epitaxy (MBE) on GaSb substrate. He is author/co-author of over 80 publications in refereed journals, a book chapter and contributed to over one hundred communications in international conferences including 15 invited talks. He has co-supervised 16 PhD students in the past ten years. His research interests currently focus on electrical and optical properties of Ga-free InAs/InAsSb and Ga-containing InAs/GaSb XBn and XBp type-II superlattice barrier photodetectors operating from MWIR to VLWIR spectral domain.Isabelle Ribet-Mohamed graduated in 1998 as an optical engineer of the Institut d'Optique (IOGS) in France. She received her PhD degree from Paris Sud university in 2001, for the development of non linear optical diagnostic techniques based on optical parametric oscillators. She was then hired at ONERA to conduct research in the field of infrared detection and electro-optical systems. She is now in charge of the orientation of emerging infrared technologies. She is author/co-author of over 40 publications in refereed journals, a book chapter and contributed to over 80 communications in international conferences. She has co-supervised 8 PhD students in the past 15 years.

	Linda Höglund,

This work was supported in part by the Agence Innovation Défense, from the French Ministry of Defence (MoD).


Sihem Tlili
email: sihem.tlili@gmail.com

Sami Mnasri
email: sami.mnasri@irit.fr

Thierry Val
email: val@irit.fr

A multi-objective Gray Wolf algorithm for routing in IoT Collection Networks with real experiments

Keywords: IoT routing, IoT Collection Networks, Multiobjective Grey Wolf, Prototyping, Optimization I

The routing is a real-world engineering problem in IoT Collection Networks. It has a direct impact on the network performance. The routing problem can be treated as a multi-objective optimization problem. Our objective is then to study the IoT routing problem using the Multi-Objective Grey Wolf Optimizer (MOGWO) [1], which consists in translating the routing process into a multi-objective optimization problem and implementing its execution with the MOGWO. In this context, the experimental study showed that the MOGWO provides better quality of routing (higher lifetime of the network, more efficient delivery delay and higher number of neighbors).

INTRODUCTION

Nowadays, several application areas are affected by the Internet of Things (IoT). Indeed, they take advantage of its benefits, such as efficiency, greater control, security, etc. In fact, the components of the IoT network are organized according to a topology the choice of which is a very important factor among those related to the well-functioning and efficiency of the network. Actually, several topologies are possible. The most common topologies are: point to point topology (Fig. 1(a)), mesh topology (Fig. 1(b)), star topology (Fig. 1(c)), hybrid topology (Fig. 1(d)) and tree topology (Fig. 1(e)).

In fact, the topology used to organize the network components is among the important factors that serve to specify the routing strategy to be applied. The transfer of data between the source and the destination takes place along a selected path. The choice of the appropriate path is made by applying a routing protocol the efficiency of which has a big effect on the growth of the IoT [START_REF] Agnihotri | A survey and comparative analysis of the various routing protocols of internet of things[END_REF]. Indeed, several network performance criteria, such as data rate, loss rate, network lifetime, delivery delay, etc., are influenced by the used routing protocol. In real applications, such as smart city, transport, industry, etc., since the IoT environments are more and more complex, the task of routing becomes more and more difficult.

Indeed, several challenges are related to the definition of the routing strategy the most important of which are the limited capacities of the objects, the heterogeneity of the network components, security, failure management, transfer reliability, support for dynamic topologies, etc. Therefore, to evaluate the performance of the applied routing protocol, several metrics including the consumed energy, the total path delay, the loss rate, etc., are defined. The definition of these evaluation metrics and the constraints to be satisfied makes it possible to translate the routing task into an optimization problem. The path to be used for the routing of the data from the source to the destination is defined through the resolution of the built optimization problem. In fact, the construction of an optimization problem consists in defining a number of decision variables, their value domains, a set of constraints and a number of objective functions. The possible values of the variables constitute a search space. The resolution of the problem serves to seek an optimal solution in this space. This solution must optimize the objective functions and satisfy the defined constraints. Besides, optimization brings better performance and success in several areas of the real world. It is therefore useful to improve the routing performance in IoT environments based on optimization. In other words, the routing task can be modeled and solved as an optimization problem to take advantage of this paradigm success.

As a matter of fact, the objective of this paper is to demonstrate the ability of optimization to remedy the complexity of routing and therefore improve the performance of IoT networks. For this reason, the Multi-Objective Gray Wolf Optimizer (MOGWO) [START_REF] Mirjalili | Multi-objective grey wolf optimizer: A novel algorithm for multi-criterion optimization[END_REF] is used to solve routing in an IoT environment. The remainder of this paper contains three sections. In section 2, we present recent works that propose routing strategies in IoT environments based on optimization algorithms. Section 3 describes our proposal to use the MOGWO algorithm to solve the routing problem. In section 4, the performance of the proposed algorithm is tested and compared to the Nondominated Sorting Genetic Algorithm III (NSGA-III) [START_REF] Deb | An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints[END_REF].

II. RELATED WORKS

There are some research studies that used optimization algorithms to solve the routing problem in the IoT networks.

In fact, the Multi-objective Imperialist Competitive Algorithm (MOICA) proposed in [START_REF] Ebrahimi Basabi | Secure routing in iot with evolutionary algorithm[END_REF] is based on the Imperialist Competitive Optimization Algorithm (ICOA) [START_REF] Atashpaz-Gargari | Imperialist competitive algorithm: An algorithm for optimization inspired by imperialistic competition[END_REF]. The routing task is modeled as an optimization problem constructed with the following parameters: integrity, confidentiality and availability, while the MOICA is a secure routing protocol the main objective of which is to reduce the probability of piracy. The contribution of the used optimization algorithm is reflected in the results of the carriedout tests although these tests are carried in small mesh network (20 nodes). In addition, the presented results do not contain comparisons with other protocols.

On the other hand, the IoT-MANET networks contain IoT components that are connected to Mobile Ad-hoc Network (MANET). In [START_REF] Sharma | Remit accretion in iot networks encircling ingenious firefly algorithm correlating water drop algorithm[END_REF], the service quality of these networks is improved based on the Intelligent Water Drop Optimization Algorithm [START_REF] Alijla | A modified intelligent water drops algorithm and its application to optimization problems[END_REF] and the Firefly Optimization Algorithm [START_REF] Yang | Firefly algorithms for multimodal optimization[END_REF]. Moreover, the conducted simulations gave results that proved the contribution of these two algorithms. The evaluated metrics are the data rate, the transmission delay, the percentage of successfully transmitted packets. The obtained results showed the improvement of these metrics. On the other hand, this proposal does not address the energy consumption, which is a very important challenge for the IoT. Besides, it does not support i) the self-stabilization (act automatically in case of problems) and ii) the identification of malicious nodes that are capable of disrupting the routing. These two challenges, which are cited in [START_REF] Karlsson | Secure routing ¨ for manet connected internet of things systems[END_REF], are considered as the most important ones in the IoT-MANET networks.

For their part, the authors in [START_REF] Jedari | Risa: routing scheme for internet of things using shuffled frog leaping optimization algorithm[END_REF] used the Shuffled Frog Leaping Optimization Algorithm (SFLA) to solve the routing of multimedia data in an IoT network. This algorithm consists of content-based, multi-hop routing. As for the simulations, they were carried out to evaluate the data rate, the consumed energy and the percentage of successfully transmitted packets. Although the proposal does not support the data redundancy, this challenge has a great effect on the performance of a multimedia network [START_REF] Shen | Routing in wireless multimedia sensor networks: A survey and challenges ahead[END_REF] since data redundancy can consume the resources available in the network.

On the other hand, the Heterogeneous Modified Grey Wolf Optimizer (HMGWO) [START_REF] Zhao | Routing protocol for heterogeneous wireless sensor networks based on a modified grey wolf optimizer[END_REF] is a routing protocol that uses the Gray Wolf Optimizer (GWO) [START_REF] Mirjalili | Grey wolf optimizer[END_REF]. HMGWO is applied in a network composed of fixed and heterogeneous nodes since they have different energy capacities. Then, the carried out simulations showed improvements in terms of data rate and consumed energy. Although the optimizing energy consumption improves the lifetime of network components, its performance is not validated by real experiments. On the other hand, energy consumption is the only parameter taken into consideration by the HMGWO [START_REF] Zhao | Routing protocol for heterogeneous wireless sensor networks based on a modified grey wolf optimizer[END_REF]. In addition, this protocol does not support the management of mobile entities.

Furthermore, the HMGWO algorithm [START_REF] Zhao | Routing protocol for heterogeneous wireless sensor networks based on a modified grey wolf optimizer[END_REF] is not interested in data exchanges between communicating objects and between star heads. The studied environment contains only communication between a base station and the star heads.

III. ROUTING RESOLUTION WITH THE MULTI-OBJECTIVE GREY WOLF OPTIMIZER ALGORITHM

A. The Multi-Objective Grey Wolf Optimizer algorithm

The authors of [START_REF] Mirjalili | Grey wolf optimizer[END_REF] propose an optimization algorithm called the Gray Wolf Optimizer (GWO). In fact, the proposed algorithm is inspired by the behavior of gray wolves when hunting. Therefore, in the GWO algorithm, the optimization problem is the hunting process, where:

• The optimum of the problem is the prey.

• {Xi} (i = 1, 2, ..., n) is a search space, which consists of the wolves that represent the search agents of the problem. Each wolf is represented by its coordinates in the studied environment (a vector 𝑋𝑖 ⃗⃗⃗⃗ ).

the optimal solutions are wolves called α, β and δ.

the best solution is the α wolf.

the ω wolves are the other search agents.

• ω wolves are dominated by α, β and δ wolves that guide the hunt. This dominance comes from the fact that α, β and δ wolves have a better knowledge of the prey's position. Indeed, the wolves move in order to encircle it where ω wolves follow α, β and δ wolves by moving in order to get closer to the prey. Moving a wolf is changing its coordinates. Its new position is then calculated based on its previous position. This calculation is done using three vectors 𝑎 , 𝐴 and 𝐶 .

The GWO algorithm presents the method used to choose and calculate the components of these vectors.

In fact, the purpose of this method is to get closer and closer to the prey.

MOGWO approach [START_REF] Mirjalili | Multi-objective grey wolf optimizer: A novel algorithm for multi-criterion optimization[END_REF] is a multi-objective version of the GWO algorithm [START_REF] Mirjalili | Grey wolf optimizer[END_REF], which adds two concepts:

• Non-dominated solutions are stored in an archive.

• A comparison of the solutions, which is performed according to a proposed mechanism, is used to select the optimal solutions Xα, Xβ and Xδ for the purpose of orienting the research towards unexplored research agents.

B. The Multi-Objective Grey Wolf Optimizer for the Internet of Things routing 1)

The network model: In this proposal, we consider a network with an organization carried out according to a hybrid topology and divided into stars, each of which has a head. The star heads communicate with one another according to a mesh organization. In other words, if an object wants to send a message, the routing begins by sending the message to the head of the star to which it belongs then, the message is passed from one head to another using a multi-hop routing. Then, if the destination does not belong to the IoT network, multi-hop routing ends when the message reaches a gateway to the outside, otherwise it ends when it arrives at the star that contains the final destination. The possible communication in this network model is exchanged between objects and data processing centers, inter-objects and with an external destination.

2) Routing resolution by the Multi-Objective Grey Wolf Optimizer algorithm: Suppose that :

• S1 and S2 are two stars in the studied network.

• H1 and H2 are the heads of S1 and S2, respectively.

• O1 and O2 are two communicating objects, O1 ∈ S1 and O2 ∈ S2.

• B is a gateway to the outside.

• O1 wants to send a message m to O2 (or to B if the final destination is not in the network).

Therefore, we propose to solve the routing with the MOGWO algorithm where the routing task is the optimization problem and the optimum of the problem is the destination. The objective functions must take into consideration the routing challenges while the search agents are the heads of the stars. Indeed, the routing to be carried out is multi-hops between the star heads where each iteration of the algorithm selects the destination of the next hop. Therefore, the proposed routing resolution is presented as follows:

• The message m is sent from O1 to H1.

• If O2 ∈ S1, m is sent from H1 to O2.
• Otherwise, multi-hop routing will be applied between the heads of the stars. H1 is the Xcurrent, which determines the head to reach in the first hop (the next Xcurrent). f. Xα is therefore the Xcurrent.

g. While (Xcurrent ≠ H2 (or Xcurrent ≠ B)) do :

i.

The coordinates of the elements of {Xi} are changed using the coordinates of the three best solutions Xα, Xβ and Xδ. MOGWO provides the equations for these changes.

ii.

For each element of {Xi}: replace its coordinates with those of the nearest star head.

iii.

For each element of {Xi}: calculate its objective functions' values.

iv. Determine the elements of {Xi} that represent non-dominated solutions.

v.

The archive is updated with the determined non-dominated solutions.

vi.

The new solutions Xα, Xβ and Xδ are selected, the MOGWO algorithm provides the method to search for these solutions in the archive.

Xα is the destination of the next hop.

vii.

The message m, Xβ, Xδ and the elements of {Xi} are sent to Xα.

viii.

Xα is the Xcurrent.

h. The last found Xα represents H2 or B. The message m is sent from H2 to O2, or to the outside by B.

IV. NUMERICAL AND EXPERIMENTAL EVALUATION

In this section, the performance of the introduced MOGWO algorithm is assessed by evaluating its Hypervolume values compared to a recent multi-objective algorithm, the NSGA-III [START_REF] Deb | An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints[END_REF]. Then, the MOGWO performance is evaluated by prototyping tests on a real testbed. Table 1 shows the used parameters for NSGA-III and MOGWO. Table II illustrates the parameters of the experiments.

Since the IoT routing is real-world engineering problem, the Hypervolume (HV) is used rather than the IGD to evaluate the behaviors of the optimizers [START_REF] Mnasri | A new multi-agent particle swarm algorithm based on birds accents for the 3D indoor deployment problem[END_REF]. Table III Table III highlights the influence of the number of considered objectives on the quality of the solution. Indeed, when the number of objectives increases, the values of the Hypervolume decreases (becomes worst). Regarding the number of generations, higher Hypervolume values (better results) are obtained for higher number of generations. It is also noticed that MOGWO outperforms NSGA-III in terms of Hypervolume values in most cases. Objective Grey Wolf Optimizer. In fact, we took advantage of the GWO algorithm, which has proven its effectiveness in large combinatorial problems to develop the proposed routing algorithm. We also carried out real experiments to compare the proposed routing algorithm to a recent multi-objective algorithm, the NSGA-III. The experimental results have shown that our proposal is more efficient than NSGA-III. The MOGWO algorithm outperforms it in terms of Hypervolume. In addition, the proposed routing algorithm is better than NSGA-III in terms of lifetime of the network (energy consumption), delivery delay and number of neighbors).

As continuation of this work, we are thinking of extending the objective functions of the constructed optimization problem in order to take into consideration more challenges for the IoT routing.

Fig. 1 .

 1 Fig. 1. The most common topologies for IoT networks

a.

  Search agents are initialized by neighboring star heads to construct {Xi}. b. The values of the objective functions are calculated for each Xi. c. The elements of {Xi}, which represent nondominated solutions, are determined to initialize an archive. d. The Xα, Xβ and Xδ which represent the best solutions in the archive are selected.e. The message m is sent to Xα. Xα also receives the coordinates of Xβ, Xδ and the elements of {Xi}.

Fig. 2

 2 Fig. 2 compares the behavior of the introduced MOGWO with the NSGA-III according to a set of network metrics indicating the quality of the proposed routing. The number of neighbors in the routing table is a metric indicating the quality of routing. The lifetime can be computed by the time in which the first IoT object shuts off. The lifetime is also an indicator showing the energy efficiency of the routing protocol. The delivery time indicate the time in which the transmitted message reaches its destination. The routing protocol and the number of hops considerably influences the delivery time. The RSSI represents the quality of the emitted signal. RSSI is better when the number of hops proposed by the routing protocol is less.It can be deduced from Fig.2(a) that MOGWO outperforms NSGA-III in terms of average number of neighbors which indicated a better quality of routing with MOGWO. When the number of iterations exceeds 400, the improvement of the number of neighbors becomes little for both algorithms which indicate that 400 iteration is the convergence value, especially for MOGWO.

Fig. 2 (

 2 Fig. 2(b) highlights the better performance of MOGWO compared with NSGA-III regarding the overall average lifetime of the network. This indicates that routing with MOGWO is more energy efficient than NSGA-III.

Fig. 2 (

 2 Fig.2(c) computes the average delivery time of the network messages. The NSGA-III seems to be better than NSGA-III regarding the needed average delivery time, especially for a high number of iterations. Here the convergence of the algorithms regarding the metric of delivery time is not reached even after 650 iterations.

Fig. 2 (Fig. 2 .

 22 Fig. 2(d) represents the average values of RSSI obtained after the routing using the NSGA-III and the MOGWO. The performance of the two algorithms considering the RSSI is comparable.

  presents the best, average and worst values of the Hypervolume on 30 execution of the MOGWO and the NSGA-III for different number of iterations and objectives. Bolded values in Table III are the best (highest) HV values in each line.

TABLE I .

 I SETTINGS OF NSGA-III AND MOGWO

		Parameter		Value
	Size of population [Nbr of grey wolfs]	300
	Number of runs		30, using distinct
				initial populations
	Number of objectives		3 to 5
	Number of generations		10 to 650
		Archive size		50
	MOGWO	alpha		0.1
		beta		4
		gamma		2
		Mutation	Operator	Swap Mutation
			distribution index 45
	NSGA-III		probability	0.0025
		Cross-over	Operator	sequential
				constructive (SCX)
			distribution index 25
			probability	0.9
	TABLE II. PARAMETERS OF THE PROTOTYPING EXPERIMENTS
		Nodes number		22 stationary nodes, 1
				mobile node
		Nodes repartition	300 * 300 m²
	Received Signal Strength Indicator	Variable (initiated to 120
		(RSSI)		dBm)
	Frame Error Rate(FER)	Variable (initiated to
				0.01)
	Average number of runs	25
		Transmission range	27 m
		Sensing range		27 m
		Transmission range	27 m
		Transmission power	110 mW

TABLE III .

 III HYPERVOLUME VALUES (BEST, AVERAGE AND WORST) FOR MOGWO AND NSGA-III ON THE ROUTING PROBLEM

	Nbr of	Nbr of	NSGA-III	MOGWO
	objectives	generations
			0.921185	0.921594
	1	350	0.916874	0.914165
			0.881268	0.901168
			0.981134	0.982056
	2	450	0.981023	0.980005
			0.978096	0.979087
			0.972858	0.973267
	3	650	0.972434	0.972601
			0.972232	0.972503