Anthony Tannoury
email: anthony.tannoury@ua.edu.lb

Rony Darazi

Christophe Guyeux

Abdallah Makhoul

Efficient and accurate monitoring of the depth information in a Wireless Multimedia Sensor Network based surveillance

Keywords: Wireless Multimedia Sensor Networks, WMSN, Disparity Map, Stereo Vision, 3D Scene Reconstruction, Surveillance, Event Detection

Wireless Multimedia Sensor Network (WMSN) is a promising technology capturing rich multimedia data like audio and video, which can be useful to monitor an environment under surveillance. However, many scenarios in real time monitoring requires 3D depth information. In this research work, we propose to use the disparity map that is computed from two or multiple images, in order to monitor the depth information in an object or event under surveillance using WMSN. Our system is based on distributed wireless sensors allowing us to notably reduce the computational time needed for 3D depth reconstruction, thus permitting the success of real time solutions. Each pair of sensors will capture images for a targeted place/object and will operate a Stereo Matching in order to create a Disparity Map. Disparity maps will give us the ability to decrease traffic on the bandwidth, because they are of low size. This will increase WMSN lifetime. Any event can be detected after computing the depth value for the target object in the scene, and also 3D scene reconstruction can be achieved with a disparity map and some reference(s) image(s) taken by the node(s).

I. INTRODUCTION

Monitoring and surveillance attract many researchers, specially with the development of the Internet of Things (IoTs). Within this discipline, Wireless Senor Networks (WSNs) are able to sense scalar data like temperature, light, and so on [START_REF] Sharif | Wireless multimedia sensor network technology: A survey[END_REF], while Wireless Multimedia Sensor Networks add the access to audio, images, or video data. They are built with low cost [START_REF] Boudargham | Investigating low level protocols for wireless body sensor networks[END_REF] Complementary Metal Oxide Semiconductor (CMOS) cameras and have a big range of applications in different fields like health, military, environmental, etc. The main challenges in WMSNs is to capture and process needed information with low energy consumption, fast computation time, and high Quality of Service (QoS).

The depth of an object or a monitored character gives us an essential clue for tracking or triggering an event. This depth will create a 3d representation of any object and it cannot be supplied by conventional gathered data. Most existing 3D reconstruction methods [START_REF] Ripolles | Smart video sensors for 3d scene reconstruction of large infrastructures[END_REF] use professional, complex, and expensive sensors with a specific network topology to obtain the required results. However, camera sensors in WMSNs have low energy resources because they are powered by limited batteries, so energy consumption is a primary constraint for WMSNs as we aim to prevent from decreasing uselessly the network lifetime. Our main objective is thus to compute 3Ddepth while maintaining low power consumption, high speed, and QoS in the specific context of wireless multimedia sensor networks.

This is why we propose the use of low-complexity disparity maps computed by a couple of low cost sensor nodes in a WMSN. This is done to efficiently provide, to the monitoring system, an important depth information of the object under surveillance. Indeed, having multiple images captured at low cost from different views, and applying Stereo Vision on different couples after regrouping each pair of sensors as left camera and right one, our system can calculate at low cost some disparity maps to recover the depth information.

This system is summarized in Fig. 1. As can be seen, data processing is made locally in each couple of sensors. This will reduce the transmission rate that has a direct and important effect on the network lifetime. Our system will thus not transmit high resolution images, but only gray-scale disparity maps, and on demand -depending on the triggered event or changes in the scene.

The remainder of this article is organized as follow. Section II introduces the WMSN specifications, challenges, and limitations. In Section III, two low cost disparity map extraction methods are presented, namely the Sum of Absolute Differences (SAD) and Sum of Squared Differences (SSD) ones. They are experimented in the next section, and a complexity estimation is provided. This research article ends by a conclusion section, in which the contribution is summarized and intended future work is outlined.

II. WIRELESS MULTIMEDIA SENSOR NETWORKS A. Composition of WMSNs

WMSN are primarily built with wireless sensors able to record images, audios, and videos. They are powered by batteries and they have the capability of sensing, processing, Fig. 1: Phases of our system and transmitting data [START_REF] Almalkawi | Wireless multimedia sensor networks: Current trends and future directions[END_REF]. Finally, the availability of low cost CMOS cameras and the development of low complexity signal processing technologies and algorithms gives to the WMSNs the ability to provide useful multimedia data at low cost.

B. Applications of WMSNs

WMSNs can be used in various applications for divergent fields like object tracking, agricultural monitoring, e-health, and so on [START_REF] Ang | Wireless Multimedia Sensor Networks on Reconfigurable Hardware: Information Reduction Techniques[END_REF]. For instance, it can be deployed in a scene to monitor visible and hidden objects from different angles. Ham et al. [START_REF] Ham | Visual monitoring of civil infrastructure systems via camera-equipped unmanned aerial vehicles (uavs): a review of related works[END_REF] and Golparvar-Fard et al. [START_REF] Golparvar-Fard | Monitoring changes of 3d building elements from unordered photo collections[END_REF] supervised buildings and constructions. Augmented reality can be mixed too with WMSNs to visualize real time some 3d representations using mobile software [START_REF] Goldsmith | Augmented reality environmental monitoring using wireless sensor networks[END_REF].

C. WMSN Challenges

The main constraint in WMSN is energy consumption, because sensors are powered by small batteries. So optimizing energy is very essential in our work independently from the type of application. Most of the energy is consumed by transmission. So decreasing the transmission rate and distance between nodes will increase network lifetime. As an illustration, 64KB data processed in a wireless sensor node leads to a consumption of 0.00195µJ for program execution (data processing), which has no comparison with the 377µJ of power required for data transmission (radio), as experimented in [START_REF] Zhao | Estimating the battery consumption of data processing in a wireless sensor node[END_REF].

WMSNs need higher data rate while using high resolution images. Capturing images in a short time can be achieved but video streaming requires continuous capturing and delivery. In some monitoring scenarios, the system should work on real-time. Hence, powerful hardware and software techniques are a must to deliver the QoS requested by the considered applications.

D. Problematic and Solution

Our main problematic is how to develop an efficient WMSN-based monitoring that runs for a long period of time, works in real-time, while guaranteeing a good QoS [START_REF] Boudargham | Investigating low level protocols for wireless body sensor networks[END_REF]? In the studied context, the transferred data will be operated later for 3D scene reconstruction, at sink level. Stereo matching is done on each pair of sensors in order to deliver low size disparity maps saving depth information. Our system uses distributed WMSN, so the disparity map processing is shared recursively on all sensor nodes during the transfer to the sink, in order to decrease network load and energy consumption.

The next section outlines two low cost disparity map calculation methods that can reasonably be applied on WMSN.

III. DISPARITY MAP CALCULATION ON WMSN

Disparity map shows the pixels difference or motion between two stereo images captured from two (left and right) sensors. Being an hot topic, new developments and techniques are introduced each year to improve the quality of the produced maps. We are however not focusing on the most upto-date methods of high quality, as they are complex, but we intend to choose the most appropriate disparity map method in the WMSN context. In other words, we do not target a disparity map of the best quality, but the best compromise between quality of this latter and complexity to obtain it.

Existing methods can be classified in two main categories, namely the local methods and the global ones.

In our previous work [START_REF] Tannoury | Introducing disparity map for 3d scene reconstruction in wireless multimedia sensor networks[END_REF], we have investigated all reputed disparity map calculation methods. And we have chosen the sole local methods as unique convenient solutions for our WMSNs context, in terms of low complexity, fast speed, and reasonable quality. It is indeed well known [START_REF] Yoon | Adaptive support-weight approach for correspondence search[END_REF] that global methods achieve good results, but are computationally expensive. Besides, local methods, especially the Sum of Absolute Differences (SAD) and Sum of Squared Differences (SSD) ones, bring off reasonable results while being low cost.

In SAD algorithm, the absolute difference between the intensity of each pixel in the reference block and the one of the corresponding pixel in the target block is computed with the following formula:

SAD(x, y, d) = (x,y)∈w |I l (x, y) -I r (x -d, y)| (1)
SAD makes the sum of differences over w, where w is the aggregated support window. The SSD algorithm, for its part, can be summarized as follows, see Equation 2:

SSD(x, y, d) = (x,y)∈w |I l (x, y) -I r (x -d, y)| 2 (2)
We now intend to extend the aforementioned study by a deep experimentation and complexity calculation, to choose a good trade-off between speed, computation cost, and disparity map quality. The experiment and complexity computations, provided in the next section, ensure how Sum of Absolute Differences (SAD) is better than Sum of Squared Differences (SSD) with respects to computation time and complexity. The Strecha et al. [START_REF] Strecha | On benchmarking camera calibration and multi-view stereo for high resolution imagery[END_REF] multi-view dataset has been used for comparison, while all simulations are computed using Matlab 2016. Fig. 2 shows different views of the same scene. Disparity maps using SAD and SSD are depicted in Fig. 3. Fig. 4a emphasizes that SAD is better than SSD when focusing on time performance, while the computation time decreases with the captured images resolutions. Note that the performance is independent from sensor coupling. This is obvious in Fig. 4b, where the processing time is approximately the same for different views. Finally, Fig. 4c shows how processing time increases linearly with the number of chosen couples.

Table I contains structural similarity (SSIM) and Peak Signal-to-Noise Ratio (PSNR) functions applied on different pairs. SSIM and PSNR are used to measure the similarity between the two calculated disparity maps (using SAD or SSD). Getting 1.0 and infinity for different pairs ensure that the calculated disparity maps have approximately the same quality and are too similar. This will help us later to choose SAD as the best method -because it has the same quality as SSD, but with a lower complexity leading to a more efficient processing. Indeed, the complexity of the two methods is accessible theoretically, proving that SSD is more complex than SAD. In this first situation, WMSNs will need more time to process more complex algorithms, and then it will consume more energy [START_REF] Mao | Localization Algorithms and Strategies for Wireless Sensor Networks[END_REF]. Computing the complexity of the two approaches using Equations 1 and 2, we have obtained the following results. For the SAD, we have 2 loops, one for horizontal width and the second one for vertical height of the image, while the operation within the loop is a single subtraction. So the SAD complexity is O(n 2 ) in terms of elementary operations. The SSD, for its part, integrates a square within the same loop, increasing the complexity to O(n 3 ) elementary operations, where n is the number of lines (or columns) in the images. Such complexities are coherent with the simulations, leading to the choice of SAD as best compromise to achieve an acceptable depth evaluation in a WMSN-based video surveillance context.

V. CONCLUSION AND FUTURE WORKS

Disparity map is a main parameter to get the depth of a monitored scene and then reconstruct it. In this research article, we contributed by applying disparity map calculation on WMSN distributed nodes. Our approach is directed by the main WMSN limitations: energy consumption, processing capability, QoS, and communication. The experiments and studies we done help to choose the best disparity map calculation method for WMSN usage, namely the SAD approach.

In future work, we intend to investigate not the existing literature, but novel disparity map computation methods specifically designed for wireless multimedia sensor networks. They will reach the optimized compromise between quality of the maps and complexity to obtain them. We will further investigate the optimal way to transfer the map from the terminal couple of nodes until the sink, by updating the disparities at aggregator nodes when receiving maps from various close couples of sensors (that observe a similar scene). The proposal will finally be distributed in a real wireless sensor networks, in order to test in vivo the real performance of this optimized surveillance in operational context.

Fig. 2 :Fig. 3 :

 23 Fig. 2: Different views of the scene

  Processing Time with different coupling

Fig. 4 :

 4 Fig. 4: Processing time per resolution, views and number of pairs.

TABLE I :

 I

SSIM and PSNR

IV. EXPERIMENTS AND COMPLEXITY COMPUTATION

ACKNOWLEDGMENT

This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01), the France-Suisse Interreg RESponSE project, and the National Council for Scientific Research in Lebanon.


Francois Sanson
email: francois.sanson@inria.fr

Carolin Frueh

Noise estimation and probability of detection in non-resolved images: application to space object observation

Keywords: Space Situational Awareness, Signal Processing, Optical Astronomy, CCD

Charged Couple Device (CCD) technology is widely used in various scientific measurement contexts. CCD equipped cameras have revolutionized astronomy and space-related optical telescope measurements in recent years. They are also used in electroscopic measurements, e.g., in fields such as geology, biology, and medicine. The signal-to-noise ratio and the probability of detection are crucial to design experiments observation setups properly and to employ further mathematical methods for data exploitation such as, e.g. multi-target tracking methods. Previous attempts to correctly characterize the signal-to-noise ratio for star observations are revisited in this work and adapted for the application of near-Earth object observations and high precision measurements, leading to a modified CCD equation. Our formulation proposes a novel distribution of the signal noise that accurately accounts for the truncation noise and the presence of ambiguous pixels. These improvements are employed to derive the probability of detection and the SNR with significant improvements compared to existing formulations when ambiguous pixels are present.

Introduction

Since its introduction in the early seventies, Charged Coupled Device (CCD) technology revolutionized optical measurements in both consumer market and scientific imaging alike. In non-resolved imaging, details of the objects or structures of interest are not available. Such images are encountered in Astronomy or Medical imaging [START_REF] Smal | Quantitative comparison of spot detection methods in fluorescence microscopy[END_REF]. The object image is represented by one to a couple of hundred pixels that stand out from the background. In practice, working with non-resolved images raise several challenges: first, the object pixels have to be discriminated from the noisy background of the whole frame. The background is typically composed of clutter or spurious object images, sometimes with the same intensity as the signal of interest. In this case, the object signal does not clearly stand out from the background and may remain undetected.

Second, the sensing process itself is a stochastic process affected by noise that deteriorates our ability to single out object signal.

A CCD sensor is usually composed of a thin layer of photoactive semiconductors (typically silicons) and a transmitter region (see [START_REF] Janesick | Scientific charge-coupled devices[END_REF] for a complete review of CCD functioning). The photon bombardment leads to electron emissions according to a stochastic process that yields different results from one observation to another. The electrons are collected in a capacitor well at each so-called pixel. After the photo exposure, a control circuit performs the readout of the CCD, during which each capacitor transfers its charge to the neighboring pixel. The electrons are transferred to a voltage level. This readout process is subject to random errors that directly affect the resulting image. Hence the obtained signal image is the realization of a series of complex random processes which affect our ability to detect and track space objects [START_REF] Massey | CCD Data: The Good, The Bad, and The Ugly[END_REF]. Furthermore, hot and dead pixels can lead to false detections if they are not masked. Hot pixels constantly show a signal (charge) even when the camera shutter is closed while dead pixels do not measure any signal.

The community worked on deriving estimates of the Signal-to-Noise Ratio (SNR) and the probability of detection to quantify and predict the expected information obtained from an observation beforehand. The SNR is the expected intensity of a signal divided by the expected noise associated with the signal.

The probability of detection is the probability that an object is detected for a given scenario. Several approaches have been developed to estimate both quantities. One method consists in simulating a large number of CCD output realizations of the same observation to extract an average SNR and probability of detection for each observation scenario. Such Monte Carlo approaches are in particular proposed in [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF] where a computer simulator of a CCD is presented. The accuracy of the strategy depends on the number of samples that can be generated for each scenario. The major drawback of Monte Carlo methods is their computational cost: when a large number of observation scenarii has to be processed, they may become irrelevant because too computationally expensive. Alternatively, analytical approximations of the SRN have been proposed in the literature. The so-called 'CCD equation' is presented in its most classical form in the reference [START_REF] Howell | Handbook of Ccd Astronomy[END_REF]; [START_REF] Mortara | Evaluations of Charge-Coupled Device / CCD / Performance for Astronomical Use[END_REF]. The CCD equation has been subsequently improved in [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF] where the background subtraction noise is included. In [START_REF] Newberry | Signal-to-noise considerations for sky-subtracted CCD data[END_REF], the author proposes a CCD equation in the context of data reduction techniques (processing noise) and derives a model for the truncation noise.

Initially developed in the context of astronomical star observations, the expressions have certain shortcomings, especially when dealing with faint object images [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF]. Faint detections are frequent in the observation of near-Earth space objects and space debris objects in the field of Space Situational Awareness. Objects do not necessarily have a stable attitude, leading to time-varying detections. Nevertheless, precise and reliable estimations of the SNR and the probability are crucial inputs for object detection and tracking tasks. The SNR is an indication of the amount of information available in a signal.

It is directly related for instance to the variance in the estimated space object position [START_REF] Sanson | Quantifying uncertainties in signal position in non-resolved object images: Application to space object observation[END_REF]. Space Situational Awareness (SSA) heavily relies on ground observations to detect and track space objects. Automated procedures need to be developed to identify and keep tracking hundreds of thousand objects. In recent years, space object tracking techniques have been greatly improved by the use of multi-object filtering approaches. Filtering algorithms (Kalman, Extended Kalman, Unscented,Probability Hypothesis Density) are classical tools for multi-object observation and tracking (see [START_REF] Mahler | Statistical Multisource-Multitarget Information Fusion[END_REF]

Estimation of the Signal-to-Noise Ratio (SNR)

The image generation process can be described as follow (see Fig. 1a). All celestial light sources impinge on the sensor (object signal but also background stars, Zodiac light, etc. ). The sensor itself generates spurious electrons called dark noise. For the signal to be read out, the readout process collects the electrons from each pixel and the analog to numeric conversion takes place.

Additional noise is generated and referred to as readout noise. An example of a CCD generated image is represented in Fig. 1b. The frame is cropped around the object image. Note that this is a high signal-to-noise ratio case. As the CCD image generation relies on photon absorption and electron emission, the resulting image is a stochastic process. Furthermore, to detect the object and find its centroid [START_REF] Hagen | Gaussian profile estimation in two dimensions[END_REF]; [START_REF] Sanson | Quantifying uncertainties in signal position in non-resolved object images: Application to space object observation[END_REF], the background has to be estimated and needs to be either locally or globally subtracted to the image [START_REF] Houtz | Streak detection and chatacterization in ground-based optical observations of space objects[END_REF]. Background estimation itself introduces statistical error [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF]. To illustrate the stochastic nature of the signal generated by the CCD after background subtraction, Fig. 2 shows three realizations of the same original light source, leading to three different realizations subject to the same stochastic processes. Additional, higher order effects such as gain variations due to the CCD fabrication process or the brighter-wider effects have been noticed by several studies [START_REF] Lage | Measurements and simulations of the brighter-fatter effect in CCD sensors[END_REF]; [START_REF] Beamer | A study of astrometric distortions due to "tree rings" in CCD sensors using LSST photon simulator[END_REF], but are neglected this paper.

Traditionally, the Signal-to-Noise Ratio (SNR) is referred to as the CCD equation. Signal-to-noise ratio is defined as the expectation value E(S) of a signal S over the standard deviation var(N ) of the noise N :

SNR := E(S) var(N ) , (1)
Three different versions of the CCD equation are compared here: the classical SNR classic as in Tiersch (1993) , Merline's derived in [START_REF] Newberry | Signal-to-noise considerations for sky-subtracted CCD data[END_REF]Mer-line and Howell (1995), SNR Merlin , and our improved derivation formulation, SNR impro , that is derived in this study. The assumptions of all three methods are discussed and compared.

Known Formulations of the CCD Equation

The signal of interest is the object signal S obj , that is the trace that the object on the detector. The signal is usually computed by adding the signal intensity in all pixels i containing the object image. Theoretically, the object image is spread over the entire frame. Realistically, only the pixels above the background n are used. The SNR is then defined as the ratio between the sum of the signal intensities and the sum of the noise in each pixel:

S obj = npix i S obj,i N = npix i N i . (2)
The noise N i in a given pixel comes from distinct sources. It comes from the object signal S obj,i , but also from the celestial and sky background sources S S,i , such as stars, and other light sources: the zodiac light airglow and others, that contribute to a non-zero photo background. Furthermore, the dark noise, N D,i , of the detector that results from atomic excitement in non-zero temperatures, and the readout noise N R,i , of the detector contribute to the noise of the CCD output. The temperature of the detector influences both sources. Moreover, due to CCD limited resolutions, not every single electron can be reported. Inevitably there is a truncation noise introduced N U,i . Then the total noise per pixel is:

N classic,i = S obj,i + S S,i + N D,i + N R,i + N U,i (3)
The classical derivation concludes at these noise terms. The formulation of [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF] adds a further term, stemming from the background estimation; it will be discussed below.

To determine mean and standard deviation for the SNR, the distributions of all those sources need to be determined. Here, there is a modeling choice given.

The basis of all three derivations is that the electron emittance after absorption of all incoming photons is modeled as a Poisson random variable (hypothesis 1).

Incoming photons per pixel are stemming from the object itself per pixel i, S obj,i

with mean and variance λ obj,i and from celestial and terrestrial background sources, S S,i with mean and variance λ S,i . Furthermore the dark noise is also modelled as a Poisson variable N D,i with mean and variance λ D,i . Thus:

E[S obj ] = var(S obj ) = npix i λ obj,i , E[S S ] = var(S S ) = npix i λ S,i , E[N D ] = var(N D ) = npix i λ D,i . (4)
When taking a summation over the single pixels, it is assumed that the pixels are independent (hypothesis 2). In the classical formulation of the CCD equation as well as in the formulation of Merline, the truncation noise is modeled by

an independent uniform random variable with support [-g 2 , g 2 ],
where g is the gain. Newberry (1991) (hypothesis 3), where U i are independent and identically distributed (iid) uniform random variables with support [-g 2 , g 2 ]. The readout error is chosen modeled by a centered Gaussian distribution with variance σ 2 R,i in the classical formulation and in the formulation by [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF] (hypothesis 4). Hence, the truncation and the readout noise variance is:

var(N U,i ) = g 2 24 var(N R,i ) = σ 2 R,i (5)
If it is assumed that all noise sources are independent (hypothesis 5), then the noise var(N classic ) can be deduced from Eq. (3) :

var(N classic ) = npix i var(S obj,i ) + npix i var(S S,i ) + npix i var(N D,i ) + npix i var(N R,i ) + npix i var(N U,i ) (6) var(N classic ) = npix i λ obj,i + npix i λ S,i + npix i λ D,i + npix i σ 2 R,i + npix i g 2 24 (7)
The classical formulation of the CCD equation hence results in the following expression, assuming a uniform background underneath the object image, allowing to multiply the background sources by n pix rather than performing a direct summation (hypothesis 6):

SNR classical = E[S obj ] var(S obj ) + n pix • (var(S S,i ) + var(N D,i ) + var(N R,i ) + var(N U,i )) = npix i λ obj,i npix i λ obj,i + n pix • (λ S,i + λ D,i + σ 2 R,i + g 2 24 ) (8)
In the classical CCD equation, the background subtraction is not included in the noise. It is equivalent to assuming that the background is perfectly determined (hypothesis 7). In the classical CCD equation and the derivation of Merline, it is furthermore assumed that the number n pix of object pixels is exactly known (hypothesis 8).

The CCD equation derived by [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF] differs from the classical derivation as it takes into account the background estimation process that generates additional uncertainty. In practice, the background subtraction leads to an additional term coming from the background estimation noise. In the case of a constant background, the background is estimated over the region in which the object image is located, background signal N B is:

N B = 1 n B nB i (S S,i + D i + R i + U i ) (9)
and n B is the number of background pixels used to estimate the background. A common way of estimating the background is the background pixel identification method explained in [START_REF] Schildknecht | CCD Algorithms for space debris detection[END_REF]. The CCD image is divided into groups of m cells. In every group, the cells are ranked according to their intensity.

Then the p lowest intensity cells and the p highest intensity cell are dropped.

The background is defined as the mean intensity of the remaining pixels. The size of the sub-frame should ideally be much larger than the signal. This leads to the following modification of the noise variance compared to the classical formulation, assuming the signal is independent of the background (hypothesis 9):

var(N Merline ) = var(N classical ) + n pix n B var(N B ) (10)
Based on the Central Limit Theorem, the distribution of the background estimation noise can be assumed to be Gaussian, denoting σ 2 B = var(N B ) Leading to the modified CCD equation of Merline:

SNR Merline = E[S obj ] var(S obj ) + +n pix 1 + 1 n b • (var(S S,i ) + var(N D,i ) + var(N R,i ) + var(N U,i )) = npix i λ obj,i npix i λ obj,i + n pix 1 + 1 n b (λ S,i + λ D,i + σ 2 R,i + g 2 24 )
(11)

Discussion of the Hypotheses of the Classical and Merline CCD Equation

In this section, we discuss the hypotheses used in the previous derivations of the CCD equations.

Hypothesis 1 -the number of electrons emitted after the absorption of photons is a Poisson random variable:. This assumption is plausible and is a classical model for electron emission. although non-Poisson distributed higher order effects exist [START_REF] Walter | The brighter-fatter and other sensor effects in CCD simulations for precision astronomy[END_REF].

Hypothesis 2 -the pixels are uncorrelated: As long as the Poisson parameter λ obj can be modeled as a deterministic value, the pixels can be safely viewed as independent. The light reflected upon the object can be modeled using geometric optic macroscopic laws under the assumption that the object and the illumination and observation geometry is known. However, atmospheric disturbance modeling could impose using to non-deterministic Poisson parameter, depending on the level of accuracy modeling. Furthermore, in a few particular cases with very high pixel intensity, bleeding effects can occur, and in this case, neighbor pixels may be correlated [START_REF] Barrett | Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions[END_REF] for example in the brighter-wider effect [START_REF] Lage | Measurements and simulations of the brighter-fatter effect in CCD sensors[END_REF]; [START_REF] Beamer | A study of astrometric distortions due to "tree rings" in CCD sensors using LSST photon simulator[END_REF].

Hypothesis 3 -the truncation noise is an independent additive uniform noise:

During the truncation process the signal is converted from electrons into ADU.

This conversion leads to losses in resolution: the CCD can only count a limited number of electrons at the time. This assumption is conceptually wrong and leads to inaccurate estimations of the truncation noise for faint signals (cf section 2.2 for more details), besides it implies that the signal remains a Poisson distribution after the round off error.

Hypothesis 4 -the readout noise is an independent additive Gaussian noise:

The readout error is a sum of independent random variables each accounting for a flaw in the electronics. The almost Gaussian distribution usually obtained [START_REF] Massey | CCD Data: The Good, The Bad, and The Ugly[END_REF] can be justified by Lindeberg-Feller theorem. Under mild assumptions on the U i such as finite second moment, we have Durrett (

∞ i U i is normally distributed. Hypothesis 5 -the background, signal and dark noise are independent: Independence is an accurate model since the electron emissions are emitted by different and independent sources, however an intense electric current increases temperature by Joule dissipation leading to an increase in the dark noise, which is usually not the case in a cooled sensor.

Hypothesis 6 -the background is constant over the signal: Some studies such as [START_REF] Schildknecht | CCD Algorithms for space debris detection[END_REF] propose more refined models of backgrounds. For instance, due to optical effects the background may be intense at the center of the image and celestial sources such as stars may vary from pixel to pixel, however for signal of reasonable size, the variation of the background are usually negligible or can normally be counterbalanced by estimating different background values for different parts of the image.

Hypothesis 7 -the background is perfectly determined: This assumption that is assumed in the classical CCD equation and has been improved upon by [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF], is wrong in general since only a limited number of pixels available to evaluate the estimated quantity of the background level.

Hypothesis 8 -the number of signal pixels is perfectly known: The diffraction pattern of the signal from the object of interest spreads the whole image frame, same as the background pixels. In general, the object image, the first maximum of the diffraction pattern, the so-called Airy disk, is denoted as signal pixels, discriminated against the background pixels, outside the Airy disk. Sometimes, they are also called the signal and background mask. As with the background estimation, the number of pixels that belong to the object is determined as the number of pixels above the background level. Especially for very faint signals,

this assumption is problematic. In this case, it may be impossible to determine for some pixels whether they should be included in the background mask or considered as part of the signal. This problem will be assessed in the derivation of the improved CCD equation in the following section.

Hypothesis 9 -the background is independent of the object signal: This appears to be legitimate as the signals stem from different sources. However, the true refraction pattern from the object signal is spreading across the whole frame, while the first order maximum, Airy disk, is usually associated with object signal, the secondary maxima are often included in the background determination (see hypothesis 8). A correlation between the background and the signal always exists. The correlation is especially significant when a sub-frame technique is utilized, using only a small number of background pixels around the trace of the object.

Derivation under more General Conditions: Improved CCD Equation

The derivation of the improved CCD equation is based upon the derivation of [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF]. The hypotheses 3 and 8 are relaxed: The modeling of the truncation error is improved upon Merline's formulation, and the uncertainty in the number of object pixels is taken into account. Improvements are most significant for faint object images with a low SNR, such as the one shown in Fig. 3. A summary of the hypotheses used in each derivation is given in Table 1.

First, we work on the truncation error term. To avoid the assumption of a uniform distribution, the signal after truncation has to be defined. Taking the Poisson processes, stemming from the actual photon impacts, we use the following notation for the independent Poisson signals S pois :

S pois,i = S obj,i + S S,i + N D,i (12)
We also define for later use, the Poisson signals excluded the object signal:

S SD,i = S S,i + N D,i (13)
The corresponding truncated signal is denoted by S pois,trunc,i . In contrast to the approximate representation in the previous section, S pois,trunc,i is no longer a Poisson distributed random variable. In the following, we derive the exact distribution of the signal after the truncation. For simplicity, we assume here that the gain g is even. We get for the probability of the truncated signal S pois,trunc,i based on the original signal in the interval subject to truncation, S pois,i , for any signal strength q: P (S pois,trunc,i = q) = P S pois,i ∈ g q -1 2 ; g q + 1 2

for any q > 0.

( This is equivalent to:

) Hypotheses classical Merline's CCD Improved CCD equation equation CCD equation Hypothesis 1 X X X Hypothesis 2 X X X Hypothesis 3 X X Hypothesis 4 X X X Hypothesis 5 X X X Hypothesis 6 X X X Hypothesis 7 X Hypothesis 8 X X Hypothesis 9 X X X 14
P (S pois,trunc,i = q) = g(q+1)-1 k=gq exp(-λ pois,i )λ k-1 2 g tot,i (k -g 2 )! = Γ(g(q + 1 2 ), λ pois,i ) Γ(g(q + 1 2 )) - Γ(g(q -1 2 ), λ pois,i ) Γ(g(q -1 2 )) , (15)
for any k > 0. Γ(q) = (q -1)! is the Gamma function and Γ(q, x) = ∞ x e -t t q-1 dt is the incomplete Gamma function. The noise variance of the Poisson sources, var(S pois,trunc,i ) can be derived using the previous analytical expression of P (S pois,i = q) in Eq.14: var(S pois,trunc,i ) = q>0 q 2 P (var(S pois,i ) = q) -q>0 qP (var(S pois,i ) = q) 2 (16)

In the next step, the background subtraction is more closely investigated. As previously discussed, there is indeed a correlation between the background and the object image. This correlation is exacerbated for very faint object traces relative to the (true) background, in which cases also the Airy disk's main maximum may merge with the surrounding background. Fig. 3 gives an example of a very faint signal where the pixels at the edge of the signal are extremely ambiguous. Ambiguous pixels are those pixels that could be part of the object image or part of the background. The following CCD equation derivation accounts for the ambiguous pixels that can be part of the signal, in terms of the first Airy disk or signal mask, and of the background, respectively background mask. We denote I amb the set of ambiguous pixels. The ambiguous pixels are included in the background mask and are considered as being part of the signal. In other words we have I amb ⊂ I obj andI amb ⊂ I B , where I obj is the set of signal pixels from the object image and I B is the set of background pixels used in the background determination. Note that in our model the membership of each pixel I amb , I obj or I B is deterministic and not subject to uncertainty.

The overall actual background B (analogous to Eq.( 9) ) can readily be defined as:

N B = 1 n B nB i S pois,i + N U,i + N R,i = 1 n B nB i S pois,trunc,i + N R,i = 1 n B nB i N total,i (17)
B and S pois are correlated since the some pixels are used to compute B and S.

As a result the background subtracted image noise is:

N impro = S pois -n pix B = i∈I obj N total,i - n pix n B i∈I B N total,i (18)
N impro = i∈I obj \I amb N total,i + (1 - n pix n B ) i∈I amb N total,i - n pix n B i∈IB\I amb N total,i (19)
where n pix are the number of object trace pixels (including the unambiguous and the ambiguous pixels) , n amb are the ambiguous trace pixels that could belong either to the object trace's first maximum or the background, n B are the number of pixels used in the background estimation.

) + n • σ 2 R,i (21)
Assuming that all the Poisson distribution parameters S pois are constant, the expression can be simplified to the following:

SNR impro,const = npix i λ obj,i
(n pix -

2n amb npix nB + n amb n 2 pix n 2 B )var(S pois,trunc ) +( npix nB ) 2 (n B -n amb )var(S SD,trunc ) + n • σ 2 R
Where var(S SD ) is the variance of the Poisson signals without light from the object trace S pois (see definition in Eq.( 13)). The truncated version of S SD is denoted S SD,trunc . Its variance is computed using Eq. ( 16).

Further simplifications can be proposed if the number of signal pixels is much smaller than the number of pixels used for the background determination. In that case, the ratio n amb n B is small and the background is well determined. In this case we can take n amb n B → 0 in Eq. ( 22):

SNR impro,const ≈ npix i λ obj,i npix nB var(S SD,trunc ) + n pix var(S pois,trunc ) + (1 + npix n B )σ 2 R . ( 22
)
The equation is similar to [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF] except for the modeling of the truncation noise. If the number of background pixel is small (a sub-frame technique has been used to reach a higher sampling rate) or the signal is faint, then Eq. ( 22) is more accurate.

If the truncation error is modeled in the traditional way, assuming a uniform distribution, the simplified improved CCD equation Eq. ( 22) is identical to Merline's CCD equation Eq. ( 11).

Probability of Detection

The probability of detection is a probability measure of how likely one is to detect an object whose image is indeed in the field of view, that is, present in the declare detection, but it can be extended to detection strategies using multiple pixels.

In the following, we define the probability of detection via a single pixel only, instead of the whole object image. Naturally, the brightest pixel is selected.

The underlying idea is that the brightest pixel is more crucial for the detection than any part of the object image. It is, of course, noted that the theoretically brightest pixel in the center of the Airy disk might not be the brightest pixel in the realization of the stochastic process on the pixel grid. When spatial filters are applied, the brightest pixel after filter application is to be used.

Depending on the specific image processing method used, a particular threshold t between the background and the object image signal is set, above which detection is possible. Potent image processing algorithms can reach a threshold of 2.5 [START_REF] Houtz | Streak detection and chatacterization in ground-based optical observations of space objects[END_REF]. Hence, we define the probability of detection P d for a

given threshold:

P d = P (S obj,trunc,i bpix > t)
+P (S obj,trunc,i bpix < t at least one other j has S obj,j > t),

where S obj,trunc,i bpix is the signal of the expected brightest pixel of the object image. It is the measured signal and therefore includes the signal itself and the background noise, both truncated, plus the readout noise. In practice, if the brightest pixel is brighter than the other signal pixels, the second term is small.

In the following derivation, it is neglected. For the rest of the derivation, it is implicit that only the brightest pixel is considered. If we use the signal definition from Eq.( 17), we can express the probability of detection as:

P d = 1 -P (S pois,trunc,i bpix -N B,i bpix + N R,i bpix < t). (24)
Computing P (S pois,trunc,i bpix -N B,i bpix + N R,i bpix < t). can be complex and numerically expensive. Reasonable simplifications are proposed in order to explicate this expression. If the number of background pixels is large enough the central limit theorem allows us to approximate N B with a Gaussian random variable with mean µ B = E[S SD,trunc ] and variance var(S SD,trunc ) = σ 2 B . Moreover, -N B + N R is the sum of two independent Gaussian random variables which is also a Gaussian random variable with mean -µ B and variance σ 2 R + σ 2 B . Under those assumptions the probability of detection becomes, separating the Gaussian distributed parts with respect to the brightest pixel i bpix :

P d = 1 -P (S pois,trunc,i -< t| -B i + N R,i = )P (N B,i -N R,i = -)d for i = i bpix
Based on the definition of S pois,trunc,i Eq. 14 and from the fact thatS pois,i is assumed to be Poisson distributed, we obtain:

P (S pois,trunc,i -< t| -N B,i + N R,i = ) = P (S pois,trunc,i -≤ t + ) = P (S pois,i -≤ g t + - g 2 ) = Γ(g t + -g 2 , λ obj,i + λ S,i + λ D,i ) (g t + -g 2 )! . ( 25
)
x denotes the closest integer smaller or equal to x. Without any loss of generality, the gain g is assumed to be even. Using Eq. ( 25) the probability of detection can be written as:

P d =1 - ∞ -∞ Γ(g t + -g 2 , λ obj,i + λ S,i + λ D,i ) (g t + -g 2 )! • 1 2π(σ 2 B,i + σ 2 R,i ) exp - ( -µ B,i ) 2 2(σ 2 B,i + σ 2 R,i ) d .
for i = i bpix Eq. ( 26) can be simplified for any integer n as long as we have ∈ [n-t; n+1-t)

then t + = n:

P d =1 - 1 2 ∞ n=-∞ Γ(n -g 2 , λ obj,i + λ S,i + λ D,i ) n! • 1 2π(σ 2 B,i + σ 2 R,i ) ∈[n-t;n+1-t) exp - ( -µ B,i ) 2 2(σ 2 B,i + σ 2 R,i ) d (26)
P d =1 - 1 2 ∞ n=-∞ Γ(n -g 2 , λ obj,i + λ S,i + λ D,i ) n! •   erf   n + 1 -t -µ B,i 2g(σ 2 B,i + σ 2 R,i )   -erf   n -t -µ B,i 2g(σ 2 B,i + σ 2 R,i )     , for i = i bpix
where erf is the error function. In actual computation, the n summation is approximated by limiting it to a suitable value range.

Results

Performance of the Improved CCD Equation

The performance of the different expressions of the CCD equations, Eq. ( 8),

Eq. ( 11) and Eq. ( 22), is evaluated. To assess the performance of different CCD equations, we use Monte Carlo simulations as reference: a large number of CCD frames are simulated, and the average signal and noise are extracted to compute the SNR. A two-step validation is performed: in the first step, the effect of the mismodelling of the truncation error is investigated. In a second step, the different approximations of the true signal-to-noise ratio, represented in Eq. ( 8), Eq. ( 11) and Eq. ( 22) are compared against the Monte Carlo results, and their performance is quantified. In the Monte Carlo simulation of the CCD frames, a Gaussian noiseless signal is generated as the object image and constant background noise S S is added to it. Then, the photo-electron release by the external sources and internal dark noise is simulated. In the simulation of the readout process, the readout noise and the truncation process are also included.

Comparison of the Noise Models for the Truncation Error

In this section the truncation error modelling as uniform variable, Newberry (1991); [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF] is compared to the improved modelling, as

shown in section 2. The noise for a truncated signal has been computed and plotted in three different ways in Fig. 5 to optimally illustrate the difference between the two noise models. In this set of tests, the variances have been computed only considering the truncation noise and stochastic electron emission noise. The signal variance denoted by var MC , is obtained running 100,000,000 independent identically Poisson distributed samples of parameter λ, rounding off the signal and computing the variance of the signal. The second one is computed using the expression derived in the previous section (see Eq. ( 15)):

var impro = ∞ m=1 m 2 P (S trunc = m) - ∞ m=1 mP (S trunc = m) 2 with P (S trunc = m) = g(m+1)-1 k=gm exp(-λ)λ k-1 2 g (k -g 2 )! . ( 27
)
Finally the third expression, noted var classical , is computed as in the literature [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF]:

var classical = λ + g 2 -1 12 , (28)
where g 2 -1 12 is the variance of a uniformly distributed random variable. shows that the improved model of truncation noise performs better than the approximation used in literature, especially for faint signals with high gain.

Fig. 6 displays the relative error between var MC and var impro . It shows that when the signal is only three times bigger than the gain, the approximation used in the literature becomes inaccurate. Hence, the approximation made by [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF] and [START_REF] Newberry | Signal-to-noise considerations for sky-subtracted CCD data[END_REF] is acceptable for very bright star observations, where g λ is small. For faint signals, the ratio g λ may be much higher, and the improved expression of the noise for the truncation error should be used.

Comparison of the CCD Equations for Signals without Ambiguous Pixels

Now, the full expressions of the CCD equations, approximating the SNR of the image frame, Eq. ( 8), Eq. ( 11) and Eq. ( 22), are compared to Monte Carlo simulations. It is assumed that all object image pixels (Airy disc) are perfectly identified and discriminated against the background. Fig. 1 shows an illustrative example of a signal in which the object pixels can be easily identified, as the signal-to-noise ratio (SNR) in all object pixels is high. Fig. 7 shows the performance of the three approximations of the true SNR: the classical CCD equation, Merline's CCD equation as presented in [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF] and the improved CCD equation ( see Eq. ( 22) ) as a function of the normalized background level. The background is normalized by the expected intensity of the expected brightest pixel. All three are compared to a Monte Carlo estimation of the SNR obtained with 50 000 samples. The classical formulation is different.

The Monte Carlo simulation results are in good agreement with the improved and Merline CCD equation formulations while the classical formulation is biased.

The difference seems to be constant and is about 1 in the SNR. The constant offset is due to the background estimation noise that is neglected in the classical formulation.

Comparison of the Signal-to-noise ratio for Signals with Ambiguous Pixels

The situation is different when the presence of ambiguous pixels is taken into account. As the signal can be extremely faint, it may become extremely difficult to tell whether a pixel belongs to the background or the signal (see Fig. 3 and4b). Furthermore, in some cases, where very few pixels are available to compute the background, one is likely to include those ambiguous pixels in their estimation of the background level. In practice, the ambiguous pixels are pixels for which the intensity from the observed object is of the same order of magnitude as the background noise. 50 000 samples. Note that the ambiguous pixels are modeled as a deterministic quantity. Now, a clear distinction between all three formulations is visible. The improved formulation now lies in-between the SNR values estimated by the classical and the estimation from [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF]. The differences appear to be constant. The constant offsets are according to the constant offset introduced by neglecting the background estimation noise in the classical formulation and the constant amount of ambiguous pixels that are simulated. The Monte Carlo simulations align almost perfectly with the improved formulation. One particular result is that the introduction of ambiguous pixels does not necessarily lower the signal-to-noise ratio since the membership of the ambiguous pixels is deterministic. In practice, the difficulty behind differentiating the signal pixels from the background pixels makes the classical definition of the signal-to-noise ratio extremely ambiguous and subject to the observer subjectivity. It is hence more adequate to take the ambiguous pixels into account.

In this test, the number of ambiguous pixels considered is kept constant to facilitate the comparison between the three approaches. For lower SNR values, the number of ambiguous pixels is expected to increase, and therefore the difference between our estimation of the SNR and Merline's estimation is expected to grow. Note: When computing the probability of detection (see nex section), the difference between the formulation of Merline [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF] and the improved CCD equation can be limited even in cases of ambiguous pixels, since only the brightest pixel is taken into account.However, in practice, (in contrast to the use in probability of detection) the SNR of the whole object image is often of interest.

Validation of the Probability of Detection

To validate the expression for the probability of detection, derived in Eq. ( 27), the results are compared to 500 000 Monte Carlo simulations. As the choice of t is not unique but ultimately user-defined, the results are shown in Fig. 9 for different values of t. t is chosen as k × N , where N is the noise. The value of the threshold should be chosen such as it minimizes the risk of false detection and maximize the number of space objects detected. Fig. 9 shows the excellent agreement of the expression for the probability of detection with all Monte Carlo simulations, despite approximations that have been made to permit fast computationally efficient computations. The analytical expressions can be used in multi-target tracking scenarios. Note that the computation of the probability of detection does not require to model the ambiguous pixels as for the CCD equations. Moreover, it is possible to compute the probability of detection accurately, without using the CCD equation representation. Sometimes the SNR computation is not of interest once the probability of detection is accurately determined.

Note, that also the Monte Carlo validation also considers only the expected brightest pixel when estimating the probability of detection. This hypothesis is valid as t is chosen to minimize false positives.

Conclusions

In this work, the signal-to-noise ratio and the probability of detection of non-resolved object images on Charged Coupled Device (CCD) sensors are investigated. Non-resolved object images cover only a few pixels and do not feature any details of the object itself. The application cases in this paper are optical ground-based observations of human-made space objects. The results are, however, not specific to those observations.

Sensing is affected by external light sources entering the detector, but also internal detector noise. The detection process itself is stochastic. As a result, the signal-to-noise ratio can only be correctly computed using computationally intensive Monte Carlo simulations. Approximate expressions of the signal-to-noise ratio, so-called, CCD equation exist in literature. In this work, two common CCD equations and the modeling hypothesis they are based on are discussed. In a generalized approach, an improved CCD equation is derived. It features two modeling improvements. Firstly, it accurately models the truncation noise rather than using a uniform approximation. Secondly, the generalized formulation does not assume that the object image pixels above the background are perfectly known. Instead, the notion of ambiguous pixels is introduced. Avoiding the latter assumption proved to have a significant impact. The validation results

showed that if the object image is intensity is comparable to the background, and if some pixels can be classified as ambiguous pixels, belonging to the background or the object image, the improved CCD equation compares significantly better to the Monte Carlo ground truth. In case no ambiguous pixels are present, the performance is the same as the CCD equation first proposed by [START_REF] Merline | A realistic model for point-sources imaged on array detectors: The model and initial results[END_REF].

Based on the improved representation of the object signal and the background, an expression for the probability of detection has been derived. A computationally fast analytic approximation has been provided. The approximation is in excellent agreement with the Monte Carlo simulated ground truth. The expression for the probability of detection avoids the explicit computation of the CCD equation.

Both, the analytical expressions of the detection probability, and the improved CCD equation cut down computational cost avoiding expensive Monte Carlo simulations.

In this work, the expressions derived were compared to Monte Carlo simulations, future work could focus on further comparing our formulation to real observations, in particular using well-characterized stars.

Acknowledgements

We would like to acknowledge and thank for the support of this work via the AFSOR grant FA9550-14-1-0348 DEF.

  for reference) that have been adapted to SSA objectives Clark et al. (2007); McCabe et al. (2015); Delande et al. (2015); Clark et al. (2008); DeMars et al. (2015). All the mentioned algorithms strongly rely on the accurate and low-cost description of the SNR and the probability of detection. One other important aspect of SSA is the optimization of space object observation conditions to maximize the efficiency of a telescope or a network of telescopes. The algorithms proposed in Coder and Holzinger (2016); Frueh et al. (2017) directly use the SRN and probability of detection. Such optimization problems are typically non-convex and require the use of expensive genetic algorithms to obtain satisfying solutions. In this work, we propose a novel derivation of the probability of detection and the SNR that can directly be used in tracking algorithms or design of optical observations. In the first step, previous methods of noise estimation, here called classical CCD equation are revisited (Section 2.1). Then a new analytical approximation of the signal-to-noise ratio is developed featuring more accurate modeling of the CCD truncation noise and ambiguous pixels for faint signals. The results are compared to the Monte Carlo simulations and existing CCD equations in Section 4.1. Besides, expressions for the probability of detection for the use in multi-target tracking frameworks are derived. The accurate statistical determination of noise and the probability of detection with fully analytic expression is crucial for a fast determination of a closed-form approximation of those quantities solely based on the physical situation without actually simulating a full frame image. The results eliminate the need to perform expensive Monte-Carlo simulations Preliminary studies on the topic have been performed by the authors in[START_REF] Sanson | Noise Quantification in Optical Observations of Resident Space Objects for Probability of Detection and Likelihood[END_REF].

  Figure 1: a: Simplified signal generation processes on a CCD in ground-based astronomy observations. b: Space object image for an object with a relative movement to the observer and the signal obtained with a CCD, (source Purdue Optical Ground Station ).

Figure 2 :

 2 Figure 2: Three observation realizations of the same object (light source signal), background noise, sensor-detector, and observation geometry, generated numerically and projected on x-direction for clarity.

Figure 3 :

 3 Figure 3: Example of a faint signal with low signal-to-noise ratio (source Purdue Optical Ground Station ).

  fig/ (a) Schematic of non-ambiguous signal: the signal pixels clearly stand above the background level

  CCD frame. It is immediately clear that the probability of detection is directly related to the signal of the object image in comparison to the background of the frame. Several approaches for space object detection have been proposed using segmentation methods[START_REF] Virtanen | Streak detection and analysis pipeline for space-debris optical images[END_REF] where pixels above a given intensity are considered, Gaussian convolution[START_REF] Schildknecht | Streak detection algorithm for space debris detection on optical images[END_REF], suited space transforms (Radon or Hough)[START_REF] Ciurte | Automatic detection of meo satellite streaks from single long exposure astronomic images[END_REF];[START_REF] Zimmer | Gpu-accelerated faint streak detection for uncued surveillance of leo[END_REF]. Most detection algorithms rely on the entire object signal to detect the object (or even several images Yanagisawa et al. (2012)), and when computed, the probability of detection relies on a Gaussian noise assumption. Using several pixels to declare detection may be used in large faint streaks and permits to use a lower detection threshold. However, for faint and small signals, the probability of detection may rely on only one pixel. Faint and small signals are of particular interest because the detection probability is significantly lower than one. The derivation proposed in the following paragraph assumes only one pixel is used to

Figure 5 :Figure 6 :

 56 Figure 5: Comparison of the truncation noise for var MC , var impro and var classical . The variances are normalized by the variance of the Poisson signal λ.

Fig. 8

 8 Fig. 8 shows again, the three formulations of the SNR as given in the three formulations of the CCD equation and the Monte Carlo simulation obtained with

Figure 7 :

 7 Figure 7: Evolution to signal-to-noise ratio as a function if read out noise for g = 0.06λ obj , no ambiguous pixels are present.

Figure 8 :

 8 Figure 8: Evolution to signal-to-noise ratio as a function of normalized background level (g = 0.06λ obj ) with ambiguous pixels present.

Figure 9 :

 9 Figure 9: Detection probability as a function of the signal-to-noise ratio (SNR).

Table 1 :

 1 Summary of the hypotheses used in the derivation of the different CCD equations

presented in this work. The 'X' means the hypothesis is used to derive the SNR. Hence the fewer 'X's, the better. The hypotheses are presented and discussed in Section 2.1.1

  \I amb var(S pois,trunc,i

	respectively.
	SNR impro =		npix i i∈I obj \I amb var(S pois,trunc,i ) + (1 -λ obj,i npix n B ) 2	i∈I amb var(S pois,trunc,i )
	+		npix n B	2	i∈I B
								Then the noise variance
	can be computed as :
	var(N impro ) =	i∈I obj \I amb	var(S pois,trunc,i ) + (1 -	n pix n B	) 2	i∈I amb	var(S pois,trunc,i )
	with n = (n pix -2	n amb npix n B	+	n 2 pix n B ). Finally this leads to the improved CCD
	equation:
	SNR impro =	E[S obj ] var(N impro )
	=							E[S obj ]	npix n B ) 2	i∈I amb var(S pois,trunc,i )
	+		npix n B	2	i∈I B \I amb var(S pois,trunc,i ) + n • var(N R,i )	(20)
	denoting as before with λ obj,i the corresponding Poisson parameter of the object
	signal and with σ 2 R,i the Gaussian variance of the readout noise in the ith pixel,

i∈I obj \I amb var(S pois,trunc,i ) + (1 -


G Rimpault
email: gerald.rimpault@cea.fr

P Dufay
email: paul.dufay@cea.fr

P Leconte
email: pierre.leconte@cea.fr

G Truchet

I Kodeli
email: ivan.kodeli@ijs.si

Assessing the Uncertainty of Reactivity Worth Scale Measurements

The characteristics of any nuclear power plant should be determined according to welldefined calculation conditions and accuracies. The neutronic tools should go under a series of verification, validation and uncertainty quantification processes.

The proof of the best estimate values plus uncertainties of the neutronic tools should be achieved using measurements in critical facilities.

Concerning the sodium void reactivity worths in sodium fast reactor cores (SFR), measurements can take the usual form of successive substitutions of different materials (voided rodlets replacing Na rodlets for instance) and different sizes in order to vary the relative importance of central and leakage components.

The reactivity variations induced by these changes are compensated by a change in the position of the shim rod if the reactivity variation is smaller than half a  eff and if larger by the addition or the removal of peripheral sub-assemblies.

The calibration of the shim rod is performed by measuring a fission chamber response when dropping a control rod and solving the Nordheim equation. This gives a S curve where the position of a control rod is associated to a reactivity expressed in $. In this study, the uncertainties associated to this calibration are revisited, not only due to  eff value (~2%) but also to decay time constants associated to each family (~6%), these being quite different from one nuclear data evaluation to another.

INTRODUCTION

In critical facilities, reactivity worths are usually measured by moving the position of the shim rod if the reactivity variation is smaller than half a  eff and if larger by the addition or the removal of peripheral sub-assemblies.

The calibration of the shim rod is performed by measuring a fission chamber response when dropping a control rod and solving the Nordheim equation. This gives a S curve where the position of a control rod is associated to a reactivity expressed in $. The purpose of this study is to revisit the uncertainties associated to this calibration, not only of the  eff value but also of the decay time constants associated to each families, these being quite different from one nuclear data evaluation to another.

The work has been done on the ZONA2 core of the BERENICE experimental programme whose analyses are summarised in chapter 2. Incidentally, this core is also part of the CIRANO programme which had the purpose of studying Plutonium burning cores.

Using different sets of delayed neutron constants, KEEPIN, ENDF/B-VII.1 and JEFF3.1.1 the work conducted at first was to calculate kinetic values for the same ZONA2 core. This is presented in chapter 3.

With the use of results presented in chapters 2 and 3, it was possible to solve the Nordheim equation and derive reactivity as a function of time. The use of the different kinetic sets led to a significant change in the reactivity scale.

BERENICE EXPERIMENTAL ANALYSES

The effective delayed neutron fraction (β eff ) is an important neutronic characteristic which deserves attention. The BERENICE measurements campaign which took place in the experimental facility MASURCA at CEA Cadarache was devoted to the experimental validation of the βeff with the two cores R2 reference and R2 experimental using enriched uranium fuel and one core ZONA2 using MOX fuel [START_REF]Evaluation of βeff measurements from BERENICE programme with TRIPOLI4® and uncertainties quantification[END_REF]. The progresses in neutronic codes and nuclear data have enable to revisit these experiments with modern tools such as the Iterated Fission Probability method [START_REF] Truchet | Computing adjoint weighted kinetics parameters in TRIPOLI-4 by the Iterated Fission Probability method[END_REF] implemented in the Monte Carlo code TRIPOLI4® [START_REF] Brun | TRIPOLI-4®, CEA, EDF and AREVA reference Monte Carlo code[END_REF]. This code gives credit to deterministic codes such as ERANOS [START_REF] Rimpault | The ERANOS code and data system for fast reactor neutronic analyses[END_REF] for calculating β eff . However, the asset of TRIPOLI4® is the possibility to get a better representation of experimental cores. It is also important for calculating parameters entering in the determination of the experimental values.

For JEFF3.1.1 [START_REF] Santamarina | The JEFF-3.1.1 Nuclear Data Library. Validation results from JEF-2.2 to[END_REF] , the revised C/E discrepancies are of 1.2% ± 3.6% for the ZONA2 core and -1.2% ± 3.7 % for the R2 experimental core when using the Noise measurement technique [START_REF]Evaluation of βeff measurements from BERENICE programme with TRIPOLI4® and uncertainties quantification[END_REF].

The nuclear data uncertainty propagation [START_REF] Archier | COMAC: Nuclear data covariance matrices library for reactor applications[END_REF][START_REF] Tarride | Mathematical Aspects of Nuclear Data Covariance Matrix Preparation -An Example of Delayed Neutron Data[END_REF] has been leading to a 2.6% uncertainty for U-Pu core and 2.8% for enriched uranium cores with main contributors being the delayed neutron fission yield and the fission cross section of U238 values consistent with the Noise Technique re-analyses [START_REF] Kodeli | Uncertainty Analysis of Kinetic Parameters for Design, Operation and Safety Analysis of SFRs[END_REF][START_REF] Kodeli | Study of Uncertainty in Kinetic Parameters in the Scope of the UAM-SFR Project of OECD[END_REF].

ZONA2 DELAYED NEUTRON CONSTANTS

The effective delayed neutron fraction (β eff ) has been calculated with the ERANOS code for the ZONA2 core using the same nuclear data, i.e. JEFF3.1.1 with the exception of delayed neutron constants which can be KEEPIN [START_REF] Keepin | Delayed Neutrons from Fissionable Isotopes of Uranium, Plutonium and Thorium[END_REF], ENDF/B-VII.1 [START_REF] Chadwick | ENDF/B-VII.0: Next Generation Evaluated Nuclear Data Library for Nuclear Science and Technology[END_REF] and JEFF3.1.1 [START_REF]Delayed neutron data for the major actinides[END_REF]. The values for β eff differ significantly with KEEPIN giving 325 pcm, ENDF/B-VII.1 339 pcm and JEFF3.1.1 357 pcm. Compared to BERENICE measurements, the discrepancies are respectively of -7.8%, -3.9% and 1.2% to which a 3.6% experimental uncertainty should be associated. KEEPIN constant which have been often used at the time looks highly discrepant.

The delayed neutron constants are for KEEPIN and ENDF/B-VII.1 given in 6 families while JEFF3.1.1 is using 8 group. The same set of eight-group half-lives for all fissioning systems in the JEFF3.1.1 data set, with the half-lives adopted for the three longest-lived groups corresponding to the three dominant long-lived precursors: 87 Br, 137 I and 88 Br. Two main reasons for adopting this new delayed neutron group data structure can be briefly mentioned here [START_REF]Development of a Reference Database for Beta-Delayed Neutron Emission[END_REF]:

1. the need for a more consistent description of the delayed neutron emission from the longest-lived precursors to avoid distortions in the reactivity measurement analysis (today it is recognised that the half-lives used in Keepin's six-group structure do not accurately reproduce the asymptotic die-away time constants associated with the three longest-lived dominant precursors);

2. the advantage of using a single set of precursor half-lives (for all fissile isotopes and incident neutron energies) in calculations of reactor kinetics.

Values of delayed neutron constants calculated with ERANOS using KEEPIN, ENDF/B-VII.1 and JEFF3.1.1 for the ZONA2 core of the BERENICE experimental programme are presented in tables 1, 2a and 3 respectively. Values for ENDF/B-VII.1 have been also calculated with SUSD3D (table 2b) and confirm those of ERANOS (Table 2a). With these values, one can calculate sum (beta i /lambda i ) for ZONA2 using different nuclear data constants (Table 4).

Table 4. Average lifetimes of delayed neutrons  for ZONA2 using different nuclear data constants

One can notice a significant difference (10% between the extreme values) between the 3 sets.

SOLUTIONS OF THE NORDHEIM EQUATION

With the use of results presented in chapters 2 and 3, it was possible to solve the Nordheim equation and derive reactivity as a function of time.

Variations of the neutron population in the core are described by the point kinetics equations:

After the cancellation of transients, the doubling time can be measured and the reactivity can be derived by the following kinetics relationship:

For instance, for a 100 pcm reactivity insertion at the initial of the transient, we obtain the following curve presented in Figure 1. The use of different kinetic sets leads to a significant change in the doubling time. In the current work, the reactivity is set to 100 pcm and the doubling time is derived. The difference on the doubling times using different kinetic sets is significant. This has also been shown for thermal systems [START_REF] Santamarina | Validation of LWR Reactivity Versus Reactor Period Feedback on the Delayed Neutron Data (i, i)[END_REF] but is also true for SFR. Now if we use the same measured doubling time (for instance the one of JEFF3.1.1) to set up the reactivity scale, we get a deviation of -5.1% for KEEPIN and -8.4% for ENDF/B-VII.1. This means that a significant deviation is existing when analysing past experiments and they should be revisited with the most recent nuclear data.

CONCLUSIONS

The calibration of the shim rod is performed by measuring a fission chamber response when dropping a control rod and solving the Nordheim equation. This gives a S curve where the position of a control rod is associated to a reactivity expressed in $.

Revisiting the uncertainties associated to this calibration, we use the experimental results of the BERENICE programme and we found a discrepancy of 1.2% with an experimental uncertainty of 3.6%. Uncertainty due to nuclear data on  eff value is of 2.6%. However, the decay time constants associated to each family are quite different from one nuclear data evaluation to another. Solving the Nordheim equation with different sets lead to a deviation on reactivity of -5.1% for KEEPIN and -8.4% for ENDF/B-VII.1 when using JEFF3.1.1 as a reference. The reactivity scale can be seriously impacted by such deviations. Analysing past experiments require hence to change the scale with which reactivity worths were measured. The uncertainty on the reactivity due to delayed neutron data amounts to 6% mainly linked to the neutron data average lifetime .

Figure 1 .

 1 Figure 1. Doubling time T d [s] as a function of time [s] for  = 100 pcm

Table 1 . Values of delayed neutron constants calculated with ERANOS & KEEPIN Table 2a. Values of delayed neutron constants calculated with ERANOS & ENDF/B-VII.1 Table 2b. Values of delayed neutron constants calculated with SUSD3D & ENDF/B-VII.1

 1

	Family number	1	2	3	4	5	6
	beta i	8.4	69.3	60.6	116.8	52.7	17.4
	lambda i	0.0127	0.0317	0.116	0.311	1.4	3.87
	Family number	1	2	3	4	5	6
	beta i	8.6	60.5	52.4	120.9	70.7	25.9
	lambda i	0.0133	0.0309	0.113	0.293	0.857	2.73
	Family number	1	2	3	4	5	6
	beta i	8.4	60.7	52.1	122.7	72.2	26.5

Table 3 . Values of delayed neutron constants calculated with ERANOS & JEFF3.1.1

 3

Proceedings of the International Conference Nuclear Energy for New Europe, Portorož, Slovenia, September 10 -13, 2018

ACKNOWLEDGMENTS

The authors acknowledge the financial support from the Slovenian and French Research Agencies under the bilateral JSI-CEA project BI-FR/CEA/16-18-005 (2016-2018).


domain: stat math info

text:
Nassim Sahki

Anne Gégout-Petit

Sophie Wantz-Mézières

Change

Change-point detection method for the prediction of dreaded events during online monitoring of lung transplant patients

Change-point detection method for the prediction of dreaded events during online monitoring of lung transplant patients Nassim Sahki 1 -Anne Gégout-Petit 1 -Sophie Wantz-Mézières 1

1 Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France

Context

• Survival after lung transplantation is about 80% at 1 year and 50% at 6 years.

• The two main complications responsible for deaths in lung transplant patients are infection and/or rejection.

Main objective

• Test the monitoring of lung transplant patients by connected sensors ;

• Propose a methodology for real-time prediction of a serious event (infection and/ or rejection) via the change-point detection in the evolution of the multivariate signals collected by these connected sensors.

Clinical test & Health data

• AP-HP (Assistance Publique-Hôpitaux de Paris) launches the EOLE-VAL Test (duration= 2 years, observation= 6 months, patients number 25) at Bichat Hospital.

• Health data come from the real-time medical surveillance of some respiratory health parameters (physiological and spirometry) of lung transplant patients by connected objects.

Online change-point detection

• The application context places us in the sequential framework where the series {x t } t=1,..,n = {x 1 , ..., x n } is sequentially observed until time n, not fixed.

• The challenge here is to minimize the average detection delay "ADD" while maintaining a given probability of false alarm "α".

• Statistically, the problem of change-point detection is to sequentially test for each new observation x n , the hypotheses :

                         H 0,n : v > n, X t ∼ f 0 (•) ∀ t = 1, ..., n H 1,n : ∃ v ≤ n, X t ∼ f 0 (•) ∀ t = 1, ..., (v -1) X t ∼ f 1 (•) ∀ t = v, ..., n (1)
• Change-point detection here is based on the choice of a recursive statistic and the threshold it must reach to signal a detection.

⇒ CUSUM statistics of Page based on the score S t : W t (δ, q) = max{0, W t-1 (δ, q) + S t (δ, q)}, t ≥ 1, W 0 (δ, q) = 0 -The score function S t (δ, q; X 1 , ..., X t ) of Tartakovsky & al. ( 2012) is calculated according to the observations and the detection objective : δ = (µ 1 -µ 0 )/σ 0 , q = σ 0 /σ 1 respectively the minimum change on the mean and on the variance that we want to detect. µ 0 , σ 2 0 and µ 1 , σ 2 1 the mean, the variance of the pre-change and the post-change regimes.

⇒ The traditional method suggested for setting a constant threshold is based on Wald inequality, after fixing the tolerated false alarm rate "α", while respecting : h α ≤ -ln(α).

Contribution

⇒ We propose new detection thresholds : the empirical constant, the empirical instantaneous and the empirical instantaneous dynamic ;

⇒ The thresholds are built by an empirical method which consists in performing simulations of the statistic W t (δ, q) under the pre-change regime and constructing the threshold by the empirical quantile of the law of statistics, as following :

1. Empirical constant threshold is the quantile of the maximum values of the simulated statistics obtained along the trajectory.

2. Empirical instantaneous threshold is the quantile of the values of the simulated statistics obtained at each time of trajectory.

3. Empirical instantaneous dynamic threshold consists to use the previous instantaneous threshold and adapt it to the behavior of the statistics (data-driven).

It moves in time when the statistic returns to its initial value (zero).

⇒ The thresholds depend on the chosen objective detection. • The results show that the empirical thresholds are faster than that of Wald.

Thresholds performance

• The best threshold is the conditional instantaneous because it makes a compromise between the detection delay and the false alarm level. It gives the best average detection delay while respecting the tolerated false alarm rate.

Perspectives

-Estimation of signal parameters (mean and variance) of the pre-change regime.

-Adaptation of the change-point detection methodology to the multivariate case.

-Application of proposed methodology to respiratory health data collected from lung transplant patients.

-

  Physiological : • Skin temperature • Pulse oximeter oxygen saturation (Sp0 2 ) • Heart rate • Respiratory rate • Physical activity • Sleep quality -Spirometry : • Forced Expiratory Volume in 1 second (FEV1).

Figure 1 :

 1 Figure 1: Connected objects

Figure 2 :

 2 Figure 2: Sample Tukey Patch Test -Skin Temperature

⇒

  [START_REF] Thomas M Margavio | Alarm rates for quality control charts[END_REF] suggest a conditional instantaneous threshold by controlling the false conditional alarm rate at each instant of the trajectory.

Figure 3 :

 3 Figure 3: Comparison of the different empirical thresholds and that of Wald, built for α = 0.02 and according to different detection objectives δ ∈ {0.5, 1, 2}, q = 1, σ 0 = 1.

Figure 4 :

 4 Figure 4: Simulation results under the pre-change regime (estimation of α) and under the post-change regime (estimation of ADD) obtained by the different detection thresholds and according to three detection objectives on the mean δ ∈ {0.5, 1, 2}, q = 1. We have the results for three different values of the tolerated false alarm rate α. The real change-point is of a level of δ R = 1.