S M Popoff

G Lerosey

M Fink

A C Boccara

S Gigan

Controlling Light Through Optical Disordered Media : Transmission Matrix Approach

Keywords: numbers: 05.60.Cd, 71.23.-k, 78.67.-n

We experimentally measure the monochromatic transmission matrix (TM) of an optical multiple scattering medium using a spatial light modulator together with a phaseshifting interferometry measurement method. The TM contains all information needed to shape the scattered output field at will or to detect an image through the medium. We confront theory and experiment for these applications and we study the effect of noise on the reconstruction method. We also extracted from the TM informations about the statistical properties of the medium and the light transport whitin it. In particular, we are able to isolate the contributions of the Memory Effect (ME) and measure its attenuation length.

Introduction

For a long time, wave propagation was only studied in the simple cases of homogeneous media or simple scattering were the trajectories of waves can be exactly determined [START_REF] Goodman | Introduction to Fourier optics[END_REF]. Nevertheless, the effect of the multiple elastic scattering on propagation of waves is complex but deterministic, and information conveyed inside such media is shuffled but conserved [START_REF] Skipetrov | [END_REF]. Within the last decades, the study of complex and random media has been an interdisciplinary subject of interest, ranging from solid state physics to optics via acoustics, electromagnetism, telecommunication and seismology [START_REF] Ishimaru | Wave Propagation and Scattering in Random Media Academic[END_REF][START_REF] Sheng | Introduction to Wave Scattering, Localization and Mesoscopic Phenomena[END_REF][START_REF] Tourin | Waves in Random and Complex Media[END_REF][START_REF] Lee | [END_REF]7]. Recent works in optics have shown that it is possible to achieve wavefront shaping beyond a complex media despite multiple scattering [8,9] and even to take advantage of the randomness of such media to increase resolution [10] by reducing the size of the focal spots. These works amount to performing phase conjugation. Nevertheless, in acoustics [11], seismic [12], electromagnetics or telecommunication [START_REF] Gore | Introduction to Space-Time Wireless Communication[END_REF], other operations are done to transmit information through a random propagation systems. These techniques assume that the propagation between input and output is characterized in amplitude and phase, which is not trivial in optics. We recently introduced a method to experimentally measure the monochromatic transmission matrix (TM) of a complex media [START_REF] Popoff | [END_REF] and presented an application to image transmission through such media [15]. In the present paper, we will develop the TM model for optical systems, in particular for multiple scattering media and detail the experimental acquisition method. We will show that the TM can be exploited for different applications. The most straightforward ones are focusing and image detection using well-established operators used for inverse problems in various domains (telecommunication [START_REF] Gore | Introduction to Space-Time Wireless Communication[END_REF], seismology [12], optical tomography [16,17], acoustics [18,19] and electromagnetics [20]). We will confront theory and experiment for applications and study the statistical properties of the TM. We will in particular study the effect of the ballistic contributions on the statistics of the TM and its consequences on image transmission. We will finally study how the short range correlation of the Memory Effect [21] (ME) affects the TM and how to quantify its correlation length with the knowledge of the TM.

Matrix Model and Acquisition

Modeling Transmission Through a Linear Optical System

In free space, a plane wave propagates without beeing modified and its k-vector is conserved since plane waves are the eigenmodes of the free space propagation. In contrast, a plane wave illuminating a multiple scattering sample gives rise to a seemingly random output field. This output field is different for a different input k-vector. In order to use a scattering medium as an optical tool, one has to characterize the relation between input and output free modes. At a given wavelength, we will model such a system with a matrix linking the k-vector components of the output field to those of the input field. We will justify the validity and the interest of such a model.

For any linear propagation medium, the propagation of an optical wave is entirely characterized by its Green function [START_REF] Papas | Theory of electromagnetic wave propagation[END_REF]. If the sources are linearly polarized and the observation is made on the same polarization, one can use a scalar model. The scalar Green function G(r, r ′ , t, t ′ ) describes the influence of the optical field at position r ′ at time t ′ on the optical field at position r at time t. For a propagation in a medium stationary over time, the time dependence of the Green function is governed by t -t ′ . For a surface S src containing all the sources, the optical field on r at t reads :

E(r, t) = Ssrc ∞ -∞ G(r, r ′ , t -t ′ )E(r ′ , t ′ ) dt ′ d 2 r ′ (1)
This expression can be written in the spectral domain :

E(r, ω) = Ssrc G(r, r ′ , ω)E(r ′ , ω) d 2 r ′ (2)
Where E(r, ω) (resp. G(r, r ′ , ω)) is the temporal Fourier transform of E(r ′ , t) (resp. G(r, r ′ , t) ) at the pulsation ω. We want to characterize the relation between the optical field on the surface S src containing the sources and the optical field on the elements of an observation surface S obs . Experimentally, sources and receptors have a finite size. We note E out m = S out m E(r) dr the average optical field on the m th receptor of surface S out m ⊂ S obs . In a similar way, we note E in n = S in n E(r) dr the average optical field on the n th controlled area of surface S in n ⊂ S src . Therefore, we can define the mesoscopic TM of an optical system for a given wavelength as the matrix K of the complex coefficients k mn connecting the optical field (in amplitude and phase) at the m th output element to the one at the n th input element element. Thus, we have :

E out m = N n k mn E in n (3)
In essence, the TM gives the relationship between input and output pixels at a given frequency, notwithstanding the complexity of the propagation for a stationary medium. Along this paper, we will perform the Singular Value Decomposition (SVD) of TMs to study their transmission properties. The SVD consists in writting :

K = U ΣV † (4)
Where † denotes the transpose conjugate. The SVD decomposes the system in independent transmission channels characterized by their transmission coefficients and by their corresponding input and output modes. V is a unitary change of basis matrix linking input freemodes with transmission channel input modes of the system. Σ is a diagonal matrix containing real and positive values called singular values of K and noted λ m . These values are the square root of the energy transmission values of the transmission channels. U is the unitary change of basis matrix between transmission channels output modes and output freemodes. For practical purposes, we will study the normalized singular values defined by :

λ m = λ m k λ 2 k (5)
The SVD is a powerful tool that gives access to the statistical distribution of the incident energy injected thought the multiple transmission channels of the system. Within this matrix model, the number of channels of the observed system is min(N, M ). Nevertheless, this number does not quantify the information that could simultaneously be conveyed in the medium.

Quantifying information is of high interest and this work has been done in acoustics [START_REF] Derode | [END_REF]24] for Time Reversal focusing through a multiple scattering medium. In those theories, the number of spatial and temporal Degrees of Freedom (N DOF ) also called Number of Information Grains is introduced. These degrees of freedom represent as many independent information grains can be conveyed through the system. For a monochromatic experiment there are only spatial degrees of freedom and N DOF is given by the number of uncorrelated element used to recover information [START_REF] Skipetrov | [END_REF]. This number will be different on the type of experiment considered, focusing or image detection. A focusing experiment consists in finding the optimal input wavefront to send to maximize energy at a desired position (or a combination of positions) on the output of the sample. An image detection experiment consists in retrieving an image sent at the input of the system by measuring the output complex field. For focusing experiments N DOF is given by the number of independent input segments, which corresponds to the number of degrees of freedom the experimentalist can control. Conversely, for detection experiments, the experimentalist cannot access more than the number of independent output elements recorded. In the general case, we have N DOF ≤ N for focusing and N DOF ≤ M . We choose to fix the input (resp. output) pixel size at the input (resp. output) coherence length of the system. For this ideal sampling, N DOF is given by the input or output dimension of the TM according to the type of experiment : N DOF = N for focusing and N DOF = M for detection.

Experimental Setup

To measure the TM of an optical system, one needs a way to control the incident light and to record the complex optical field. The control is now possible thanks to the emergence of spatial light modulators (SLM) and the output complex field can be measured using an interferometric method detailed in section 1.3. We want to study an optical system governed by multiple scattering in which the propagation function can not be a priori calculated. The material chosen is Zinc-Oxyde, a white pigment commonly used for industrial paints. The sample is an opaque 80 ± 25 µm thick deposit of a ZnO powder (Sigma-Aldrich 96479) with a measured mean free path of 6 ± 2 µm on a standard microscope glass slide. The thickness of the sample being one order of magnitude higher than the mean free path, light transport inside the sample is in the multiple scattering regime. To generate the incident wavefront, the beam from a diode-pumped solid-state single longitudinal mode laser source at 532 nm (Laser Quantum Torus) is expanded and spatially modulated by a liquid crystal SLM (Holoeye LC-R 2500). Using a polarizer and an analyzer, we choose an appropriate combination of polarizations before and after the SLM to achieve up to a 2π phase modulation depth with less than 10% residual amplitude modulation. Only a part of the SLM is modulated and the rest of the surface is used to generate a static reference (details in 1.3). The surface of the SLM is then imaged on the pupil of a 20x objective with a numerical aperture (NA) of 0.5. Thus a pixel of the SLM matches a wave vector at the entrance of the scattering medium. The beam is focused at one side of the sample. We image the output pattern 0.3 mm from the opposite surface of the sample through a 40x objective (NA = 0.85) onto a 10-bit CCD camera (AVT Dolphin F-145B). We do not directly image the surface of the material to mitigate the effects of the possible presence of ballistic light. This point is detailed in section 3.4.

The duration of a typical measurement of the TM is of a few minutes, much less than the typical decorrelation time of the medium (more than an hour). From now on, we will ignore any time dependence of the medium. It is an important prerequisite for the measurement and exploitation of the TM.

Acquisition Method

Measuring a TM in optics raises several difficulties coming from the impossibility of having direct access to the phase of the optical field. Generally, one can have access to the complex optical field using interferences, for instance with a known wavefront and a full field phaseshifting interferometry method [25]. For any input vector, if the reference phase is shifted by a constant α, the intensity in the m th output mode is given by :

I α m = |E out m | 2 = |s m + N n e iα k mn E in n | 2 (6) = |s m | 2 + | N n e iα k mn E in n | 2 + 2ℜ e iα s m N n k mn E in n
where s m is the complex amplitude of the optical field used as reference in the m th output mode.

Thus, if we inject the n th input mode and measure I 0 m ,I m , respectively the intensities in the m th outgoing mode for α = 0, π/2, π and 3π/2 gives :

(I 0 m -I π m ) 4 + i I 3π 2 m -I π 2 m 4 = sm k mn (7)
Commercial SLMs cannot control independently phase and amplitude of an incident optical field. Our setup uses a phase-only modulator since the phase is the most important parameter to control in wavefront shaping [26]. A recent method [27] allows phase and amplitude modulation with similar commercial modulators, nevertheless, we use in this work a phase-only configuration to keep the setup as simple as possible.

Controlling only the phase forbids us to switch off or modulate in amplitude an incident mode. Thus we can not directly acquire the TM in the canonical basis. We want to use a basis that utilizes every pixel with identical modulus. The Hadamard basis in which all elements are either +1 or -1 in amplitude perfectly satisfies our constraints. Another advantage is that it maximizes the intensity of the received wavefront and consequently decreases the experimental sensibility to noise [START_REF] Larrat | Ferroelectrics and Frequency Control[END_REF]. For all Hadamard basis vectors, the intensity is measured on the basis of the pixels on the CCD camera. We obtain the TM in the input canonical basis by a unitary transformation. The observed transmission matrix K obs is directly related to the physical one K of formula 3 by

K obs = K × S ref (8)
where S ref is a diagonal matrix of elements where s ref mm = s m represents the static reference wavefront in amplitude and phase.

Ideally, the reference wavefront should be a plane wave to directly have access to the K matrix. In this case, all s m are identical and K obs is directly proportional to K. However this requires the addition of a reference arm to the setup, and requires an interferometric stability.

To have the simplest experimental setup and a higher stability, we modulate only 65 % of the wavefront going into the scattering sample (this correponds to the square inside the pupil of the microscope objective as seen in Figure 1), the speckle generated by the light coming from the 35 % static part being our reference. S ref results from the transmission of this static part and is now unknown and no longer constant along its diagonal. Nevertheless, since S ref is stationary over time, we can measure the response of all input vectors on the m th output pixel as long as the reference speckle is bright enough. We will show in the next subsection that we can go back to the amplitude elements of S ref and that the only missing information is the phase of the reference. We will quantify the effect of the reference speckle and show that neither does it impair our ability to focus or image using the TM, nor does it affect our ability to retrieve the statistical properties of the TM.

Multiple Scattering and Random Matrices

Under certain conditions, the rectangle N by M TM of a system dominated by multiple scattering [START_REF] Derode | Waves in Random and Complex Media[END_REF] amounts to a random matrix of independent identically distributed entries of Gaussian statistics. We will study the properties of such matrices thanks to the Random Matrix Theory (RMT) and investigate the hypothesis for which the TM of multiple-scattering samples are well described by this formalism.

We introduced previously that the SVD of a physical matrix is a powerful tool to study the energy repartition in the different channels. For matrices of independent elements of Gaussian distribution, RMT predicts that the statistical distribution ρ( λ) of the normalized singular values follows the Marcenko-Pastur Law [START_REF] Marčenko | [END_REF]. For γ = M/N , it reads :

ρ( λ) = γ 2π λ ( λ 2 -λ 2 min )( λ 2 max -λ 2 ) (9)
∀ λ ∈ [ λ min , λ max ]
With λ min = (1 -1/γ) the smallest singular value and λ max = (1 + 1/γ) the maximum singular value. In particular, for γ = 1, the Marcenko-Pastur law is known as the "quarter circle law" [31] and reads :

ρ( λ) = 1 π 4 -λ 2 (10)
Those predictions are valid for a TM of independent elements i.e. without any correlation between its elements. This hypothesis can be broken by adding short range correlation but also non trivial correlations such as ones appearing when introducing the flux conservation [32]. Since we only experimentally record and control a small part of the field on both side of the sample, we will not be affected by this conservation condition.

Another straightforward parameter that could have an influence on the hypothesis of independent elements is the size of the segments uses as sources (on the SLM) and receptors (on the CCD). If the segment size is less than the coherence length (i.e. the size of a speckle grain) two neighboring segments will be strongly correlated. The size of input (resp. output) segments is set to fit the input (resp. output) coherence length. This way, the TM K should respect the hypothesis of independent elements and follow the Marcenko-Pastur law.

Nevertheless the method of the acquisition introduces correlation in the TM recorded. For M = N , K should respect the Quarter Circle law but the methods detailed in section 1.3 gives only access to the observed TM matrix K obs = S ref .K. whose elements read :

k obs mn = N j s mj k jn = k mn s mm (11)
To study the singular value distribution of our TM, we have to be sure that we can go back to the precise statistics of K using the observed matrix K obs .

The effect of the reference S ref is static at the output. This means that its influence is the same for each input vectors. Calculating the standard deviation of an output elements over the input basis vectors we have :

|k obs mn | 2 n = | k mn | 2 n |s mm | (12) = |k mn | 2 mn |s mm | ∀m ∈ [1, M ]
We made the hypothesis that each input k-vector gives an output speckle of the same mean intensity, it reads |k mn | 2 n = |k mn | 2 mn . This is valid in the diffusive regime as long as the illumination is homogeneous. We select for the illumination only the center part of an expanded gaussian beam. We also neglected the correlation effects that are expected to be small regarding our system, such as Memory Effect [21] (see section 4) and long range correlations [33]. We can now define a normalized TM K f il whose elements k fil mn are normalized by

|k obs mn | 2 m . k fil mn = k obs mn |k| 2 mn |s mm | 2 (13) = k mn |k| 2 mn s mm |s mm | (14)
This matrix is not influenced by the amplitude of the reference and reads :

K fil = K |k| 2 mn × S φ (15)
Where S φ is a diagonal matrix of complex elements of norm 1 representing the phase contributions of the output reference speckle. In contrast with S ref , S φ is a unitary matrix since S φ S † φ = I where I is the identity matrix. Thus if K = U ΣV † then K fil = U ′ ΣV † where U ′ = S φ × U is a unitary matrix. Therefore, K fil and K have singular value decompositions with the same singular values.

We show in figure 2 that the statistics of the observed TM K obs do not follow the quarter circle (formula 10) law whereas the singular value distribution of the filtered matrix K fil is close to it. Remaining correlation effects are due to inter-element correlations between nearby pixels. Taking a sub-matrix of the TM with only one element out of two, its statistics finally follows correctly the expected quarter-circle law. Those results are comparable to the ones obtained for transmission matrices in acoustics [START_REF] Derode | Waves in Random and Complex Media[END_REF].

This clearly demonstrates that we can directly study the properties of K even if we only have access to the observed matrix K obs . We also want to point out that one has to be careful when interpreting deviation from a given prediction on SVD results. Once the matrix is measured and its singular values analyzed, one has to investigate the causes of the deviation from the Marcenko-Pastur law. Those causes could come from physical phenomenons involved in light propagation in the medium that break the hypothesis of the absence of correlation in the TM (like the apparition of ballistic contributions, see section 3.4) but could also be inherent to the experimental method. However, K obs brings all informations on singular values that describe the physics of the propagation in a given setup.

So far, we compared the experimental singular value distributions for a symmetric case (M = N ). But what would be the effect of breaking the input / output asymmetric raio M/N ? In the same setup, we experimentally acquired TMs with a fixed number of controlled pixels on the SLM (N = 1024) and changed the number of independent output pixels recorded by the CCD (M = γN ) to achieve various ratio values (γ ∈ {1; 1.5; 2; 3; 4; 5}). Normalized singular value distributions of experimental TM filtered are shown in figure 3 and follows qualitatively the RMT prediction of equation 10.

We see that with increasing γ, the range of λ narrows. This effect has been measured in experiment in the acoustic field [34]. Physically this means that when γ increases, all recorded channels tend to converge toward the mean global transmission value λ = 1. When M > N , we record more independant informations than the rank of the TM and averaging effects lead to a distribution that is peaked around a unique transmission value when M ≫ N . We notice that the experimental distributions deviate from the theoretical ones as γ increases. To have TMs with different values of γ, we record a large TM with γ = 11 and then create submatrices by taking lines selected randomly among the lines of the original TM. By increasing γ we increase the probability of having neighboring pixels in the TM and then are more sensitive to correlations between nearby pixels. This modifies the statistics and decreases the number of independent output segments. This explains the deviation from the theory. Other effects that could bring correlations in certain experimental conditions are discussed is section 3.4. Calculations on experimental TM are made using the filtered matrix K fil .

Phase Conjugation

Phase Conjugation and Degrees of Freedom

It was demonstrated [20,35] that one can take advantage of multiple scattering to focus on tight spots, thanks to the reversibility of the wave equation.

In such an experiment, the responses to short temporal signals emitted by an array of sources at a receiver at the output of a disordered medium are recorded. Those signals, linked to the Green's functions associated to the couples (source / receiver) are sent reversed in time. The waves generated converge naturally towards the targeted spot. The waves are focused in space and time. Phase Conjugation is the monochromatic equivalent to the previous concept named Time Reversal. Those techniques are robust methods to achieve focusing and imaging. In this chapter, we will experimentally and theoretically study the efficiency of phase conjugation on focusing.

When using Phase Conjugation for focusing, it corresponds to shape the input wavefront to put in phase the contributions of each input pixel at a desired output target. Denoting E target out the output target vector, the input vector that estimates the desired output pattern E target out using Phase Conjugation is given by :

E in = K † .E target out (16)
The effective output vector E out will be given by : (17) We will now demonstrate the dependence of the Phase Conjugation accuracy on N DOF for the simple case of a single focusing spot. We will compare the efficiency of perfect Phase Conjugation (with amplitude modulation) and of phase only Phase Conjugation. The intensity output field resulting from the Phase Conjugation focusing on the j th output pixel reads :

E out = K.E in = K.K † .E target out
|s out m | 2 = | N l k ml k * jl | 2 (18)
For m = j, the average intensity of sum of the the incoherent terms reads :

|s out m =j | 2 = N,N l,l ′ k ml k * ml ′ k * jl k jl ′ (19) = N l |k ml | 2 |k jl | 2 + N,N l =l ′ k ml k * ml ′ k * jl k jl ′ = N |k| 2 2 + 0
For m = j, the average intensity of the sum of the coherent terms reads :

|s out m=j | 2 = N l |k jl | 2 2 (20) = N l |k jl | 4 + N,N l =l ′ |k jl | 2 |k jl ′ | 2 = N (N -1) |k| 2 2 + N |k| 4 ≈ N 2 |k| 2 2 ∀N ≫ 1
We define the energy signal to noise ratio SNR for a single spot focusing as the intensity at the target point over the mean intensity elsewhere after phase conjugation. We directly have

SNR ≈ N = N DOF ∀N ≫ 1 (21)
Similar results are obtained considering detection instead of focusing. If E out is the output pattern corresponding to an unknown input target mask E target in to be detected, similarly to equation 17, the reconstruction E image of the target is given by :

E image = K † .E out = K † .K.E target in (22)
The reconstructed pattern for detecting a single target on the j th input pixel reads :

s img m 2 = M l k * lm k lj 2 (23)
And the signal to noise ratio for the reconstruction of a single pixel reads :

SNR ≈ M = N DOF ∀M ≫ 1 ( 24
)
Controlling Light Through Optical Disordered Media : Transmission Matrix Approach 12

Experimental Phase Conjugation

To perform such an ideal Phase Conjugation focusing, one has to control the amplitude and phase of the incident light. Our experimental setup uses a phase-only SLM to modulate the light. Thus the input l th component of the input field to display in order to focus on the j th output pixel reads k * jl /|k * jl |. The resulting output intensity pattern for a phase only experiment reads :

|s out m | = 1..N l k ml k * jl |k * jl | = 1..N l |k ml | (25)
Calculating the intensity of incoherent and coherent sums in the same way as equations 19 and 20 we have :

|s out m =j | 2 = N |k| 2 (26)
|s out m=j | 2 ≈ N 2 |k| 2 ∀N ≫ 1 (27)
We define SNR exp the expected signal to noise ratio that reads :

SNR exp ≈ N DOF |k| 2 |k| 2 (28)
Experimentally, for a setup and a random medium satisfying the hypothesis introduced in 1.4 (elements of the TM independent with a Gaussian statistic), we have |k| 2 |k| 2 = π 4 ≈ 0.78 [36]. Finally :

SNR exp ≈ π 4 N DOF ≈ 0.78N DOF (29)
To achieve such a SNR exp using the experimental setup presented in section 1.2 one has to remove, after measuring the TM, the reference pattern. If the reference part is still present, the intensity coming from the segments not controlled of the SLM will increase the background level. This effect (detailed in Appendix) modifies SNR exp . In our experimental conditions :

SNR exp ≈ 0.5N DOF (30)
We will now use equations 16 and 22 to experimentally achieve focusing and detection. We tested Phase Conjugation for simple targets focusing and for point objects detection through the medium. The results are shown in figure 4.

To experimentally study the effect of N DOF over the quality of focusing, we measure the energy signal to noise ratio for a single spot focusing. We represent in figure 5 this experimental ratio as a function of the number of controlled segments N on the SLM and compare it to the theoretical prediction SNR ≈ 0.5N DOF . Those experimental results correctly fit the theoretical predictions.

Comparison with Sequential Optimization

The first breakthrough using phase-only Spatial Light Modulator for focusing through a disordered medium were reported in [8]. The technique was a feedback optimization that leads to substantial enhancement of the output intensity at a desired target. We will present the principle of this approach and compare it to techniques requiring the measurement of the TM.

The experimental setup is roughly the same than the one presented in figure 1. The wavefront phase is controlled onto the SLM on N groups of pixels, and the full output pattern is recorded by the CCD camera. One point target is chosen and the optical field E out m on this pixel is the linear combination of the effects of the N segments on the modulator :

E out m = N n k mn E 0 e iφn (31)
where k mn are the elements of the TM as defined in equation 3 and φ n the phase imposed on the n th pixel of the SLM.

The input amplitude term E 0 is constant for a phase-only spatial light modulator with an homogeneous illumination. The idea is, for each pixel, to test a set of phase values and keep the one that gives the higher intensity on the target output point. The goal of this sequential algorithm is to converge toward a phase conjugation with an efficiency depending on the finite number of phase tested [8]. Different phase shifting algorithms were studied and their relative efficiencies in presence of time decorrelation are analyzed in [37], we will not discuss this problem in this paper.

In a single spot focusing experiment for a sufficient number of phase shifts tested, the sequential algorithm and the phase conjugation computed by the knowledge of the TM are equivalent. Nevertheless, both techniques have different advantages and drawbacks. To be able to change the position or the shape of focal pattern, sequential optimization needs a new optimization for each new pattern. This forbids real time applications. On the contrary, once the TM recorded, all information is known and permits to calculate the input wavefront for any output desired pattern, allowing for instance to change the focus location at a video rate. Nevertheless, sequential optimization presents two strong advantages. With a setup comparable to the one shown on figure 1, sequential algorithm is more robust to calibration errors than the TM measurement since, even if the exact values of the phase shifts tested are unknown, the algorithm automatically selects the best phase mask. Another asset of this technique is that no interferometry measurement is needed. Since only the intensity is recorded, it allows experiments using incoherent phenomenon such as focusing optical intensity on a fluorescence probe inside a disordered material [36].

Noise and Reconstruction Operator

Experimentally, one cannot perform a measurement without noise. In our setup, the main noise sources are the laser fluctuation, the CCD readout noise and the residual amplitude modulation. Because of the error made on the TM measurement, one has to find, for focusing or image detection, reconstructions operators very resilient to experimental noise. These operators only give an estimation of the solution. From now on, we will make the distinction between the two sources of perturbation on the reconstructed signal : the experimental noise inherent to the measurement errors, and the reconstruction noise, inherent to the reconstruction operator. We will discuss the influences of those two noises on the performance of the technique used. Due to spatial reciprocity, focusing and image detection are equivalent. We will thus concentrate in this section on the case of image detection.

Inverse problems are widely commons in various fields of physics. Optical image transmission through a random system is the exact analog of Multiple-Input Multiple-Output (MIMO) information transmission through an unknown complex environment. This has been broadly studied in the last decades since the emergence of wireless communications [START_REF] Gore | Introduction to Space-Time Wireless Communication[END_REF]. Similar problems have also been studied in optical tomography in incoherent [16] and coherent regime [17]. In a more general way, inverse problems has been widely study theoretically for more than fifty years. A. Tikhonov [38] proposed a solution for ill-posed system. Tikhonov regularization method was then broadly used in various domains to approximate the solutions of inverse problems in the presence of noise.

Experimental Noise

Prospecting an optimal operator for image reconstruction, the first straightforward option is the use of inverse matrix K -1 obs for a symmetric problem (N = M ) since K -1 obs K obs = I where I is the identity matrix. Inversion can be extended when N = M with the pseudo-

inverse matrix K † obs .K obs -1 K † obs , noted K -1
obs . This operator provides a perfect focusing but is unfortunately very unstable in presence of noise. Indeed, the singular values of K -1 obs are the inverse ones of K obs . In other words, inverse operators tend to increase the energy sent through otherwise low throughput channels. This way, it normalizes the energy coming out from all channels. Singular values of K obs below noise level have the strongest contributions in K -1 obs but are aberrant. The reconstructed image is hence dominated by noise and bears no similarity with the input one. Another standard operator for image reconstruction or communication is Time Reversal. This technique has been studied in acoustics [18] and electromagnetic waves [39]. Being a matched filter [40], Time Reversal is known to be stable regarding noise level. Unlike inversion, it takes advantage of the strong singular values by forcing energy in the channels of maximum transmission. This way, singular values below noise level carry few energy and do not degrade reconstruction. Its monochromatic counterpart the Phase Conjugation uses the operator K † obs (the operator used in 2.2 for focusing and target detection). K † obs K obs has a strong diagonal but the rest of it is not null. We will see later that this implies that the fidelity of the reconstruction is not perfect and depends on the complexity of the image to transmit.

An intermediate approach is to use a Mean Square Optimized operator. This operator, that we will call MSO an denotes W is the Tikhonov regularization [38] for linear perturbated systems mentioned earlier. It takes the experimental noise into account minimizing transmission errors, estimated by the expected value E W.E out -E in W.E out -E in † . For an experimental noise of variance N o σ on the output pixels, W reads :

W = K † obs .K obs + N o σ .I -1 K † obs . ( 32
)
This operator stabilizes inversion through the addition of a constraint depending on the noise level. This operator is intermediate between the two previous one. If N o σ = 0, W reduces to the inverse matrix K -1 obs , which is optimal in this configuration, while for a very high noise level it becomes proportional to the transpose conjugate matrix K † obs , the phase conjugation operator. For an intermediate noise level, the channels of transmission much greater than the noise level are used 'like in inversion' operator and channels much below noise level are used 'like in phase conjugation'.

Reconstruction Noise

The reconstruction noise is the perturbation brought by the reconstruction operator, even without measurement noise. This noise is intrinsic to the techniques used. The only operator to provide a perfect reconstruction, i.e. without any reconstruction noise, is pseudo-inversion (or inversion for M = N ). Nevertheless, this operator is not usable in a general case since it is very sensitive to experimental noise. Phase conjugation is stable in presence of experimental noise but creates perturbations of the reconstructed signal. As seen in section 2.1, the signal to noise ratio for a single focal is proportional N DOF . The same goes for imaging a single spot, the target to detect generates a noise proportional to 1/N DOF on the rest of the image. For multiple targets, each target detection adds a perturbation for the reconstruction of the others, thus the intensity signal to noise ratio is approximately given by N f /N DOF where N f is the number of targets to detect. In other words, the fidelity of the reconstruction with Phase Conjugation is not perfect and depends on the complexity of the image to transmit. This limitation forbids the reconstruction of a complex image (with N f ≈ N ) by Phase Conjugation through a symmetric system (with N = M ) since the reconstruction noise is of the same order of magnitude as the useful signal.

For any reconstruction operator Op, if K is known exactly (not affected by the experimental noise), the mean normalized reconstruction noise N rec for a single target detection can be estimated by :

N rec = |Op × K| 2 mn m =n |Op × K| 2 mn m=n ( 33
)
N rec is define here by the ratio between the mean intensity |Op × K| 2 mn m =n brought by the reconstruction of a target outside its position and the mean intensity |Op × K| 2 mn m=n at the target position. The average is made over all input target positions. Graphically, N rec is the inverse of the signal to noise ratio defined by the mean of the intensity of the diagonal of Op × K over the mean intensity elsewhere. To study the reconstruction noise brought by MSO operator for different values of N o σ we show in figure 6 the simulation results for the reconstruction noise as a function of the N o σ . It is important to understand that, for a given N o σ , W is the ideal operator for an experimental energy noise level of N o σ . In that study, the simulated TM K is noise free. We see that for N o σ = 0 (i.e. for inversion operator), the reconstruction noise N o rec is null and for N o σ → ∞ (i.e. for Phase Conjugation operator) N o rec tends to N DOF (here N DOF = M = N ). This behavior is obvious when looking at the diagonal of |W × K| 2 compared to the others values.

Toward Optimal Reconstruction

In order to detect an input image with the best accuracy, one has to find the good balance between the sensitivity to experimental noise and the influence of the reconstruction noise on the recovered image. The mathematical optimum operator for those constrains is the MSO operator introduced in equation 32. To lower the reconstruction noise, one can only increase the parameter N DOF . We investigated two possibilities: averaging, and increasing the number of output modes M .

A possible way to virtually increase N DOF is to perform the transmission of the same image through different realizations of disorder. With several realizations, the reconstruction noise, depending on the TM, will be different and averaging will lower perturbations on the final image. It is the monochromatic equivalent of using broadband signals, which takes advantage of temporal degree of freedom [24] where uncorrelated frequencies are used to increase N DOF . Iterating with different physical realizations of disorder is restrictive since it requires to actively move or change the scattering sample. If we are only interesting in detecting an amplitude object, an other way to change the effective TM seen by the object is to illuminate it with different phase wavefronts. It is formally equivalent to transmitting the same image through different channels as if the image propagated through different realizations of disorder. For any amplitude object E obj with e obj m ∈ [0, 1], the effective incident field on the sample is S ′ φ .E obj where S ′ φ is a diagonal matrix containing only phase terms. The output complex field pattern for a given TM K reads :

E out = K.S ′ φ .E obj = K ′ .E obj (34)
With K ′ = K.S ′ φ . Physically, it is as if the same object E obj was transmitted through a different part of the medium. Another simple way to see it is that by changing S ′ φ (i.e. changing the 'virtual realization') we change the projection of E obj on the input singular modes. Thus the same image propagates through different channels. For a given phase illumination, the amplitude object is estimate by E img :

E img = |W.E out | = |W.K.S ′ φ .E obj | = |W.K ′ .E obj | (35)
With W the optimal MSO operator corresponding to the TM K for a given noise level. One now has to change the illumination wavefront S ′ φ and average the different E img obtained to lower the reconstruction noise. To illustrate this effect we show in figure 7 simulation results for the reconstruction of a simple amplitude object using two different phase illuminations and averaging over 10 different illuminations. The matrix is know exactly in the simulation so all perturbations come from reconstruction noise. We see that the two first realizations reproduce the amplitude object with different noises with amplitude of the same order of magnitude. Averaging over multiple illuminations significantly reduces the noise background. ) and the reconstruction averaged over over 10 'virtual realizations' (c.). Simulations are made for a 100 by 100 random matrix of independent elements of Gaussian statistics. The amplitude object consists of three points of amplitude one and the rest of the image to zero.

Since our SLM is a phase only modulator, in order to simulate amplitude objects, we generate what we call 'virtual objects'. We use different combinations of random phase masks to generate the same virtual amplitude object E obj by subtracting two phase objects. From any phase mask E

(1) phase one could generate a second mask E [START_REF] Skipetrov | [END_REF] phase where the phase of the m th pixel is shifted by s obj m .π. The optical complex amplitude of elements of the second mask are linked to those of the first mask by : e (2) m = e (1) m .e is obj m π

m the m th element of E (j)

phase . We have the relation between the two masks and the amplitude of the 'virtual object' :

|E (2) phase -E (1) phase | = 2. sin (E obj π/2) (37)
it is important to note that the norm of E

phase -E

phase is fixed and controlled whereas its phase is random, uncontrolled and differs from a 'virtual realization' to another. The term

E amp = |E (2) phase -E (1)
phase | can be estimated by :

E img = |W.(E (2) out -E (1) out )| (38)
where E

out (resp. E

out ) is the complex amplitude of the output speckle resulting from the display of the mask E

(1) phase (resp. E

(2) phase ). From a given set of random phase masks, we can define in the same way as in equation 34 a diagonal matrix S ′ φ containing only phase terms. This term differs from a 'virtual realization' to another. This way, the reconstruct image reads :

E img = |W.S ′ φ .E amp | (39)
This expression is formally equivalent to equation 35, thus generating 'virtual objects' using random phase masks is mathematically equivalent to illuminating a real object by random phase-only wavefront.

Optimal Reconstruction in Presence of Ballistic Light

We experimentally used this technique to virtually increase N DOF , and we average the results to lower the reconstruction noise. The complex amplitude image to detect is a grayscale 32 by 32 pixels flower image. The system is symmetric with N = M = 1024. The first step to reconstruct the input image with the best accuracy is to find the variance of the experimental noise N o σ in order to obtain the optimal MSO operator. To that end, we numerically iterate the reconstruction operation for different values of the constrain in the MSO operator (equation 32) and keep the one maximizing correlation between the reconstructed image and the image sent, hence obtaining an estimation of the experimental noise level. This step is made a priori knowing the image sent but could be performed by any technique leading to an accurate estimation of the noise level. We detailed in [15] the results in a purely diffuse system where ballistic contributions have no measurable effect. We study here the results in presence of quantitative ballistic contributions.

In previous experiments, the focal planes of the two objectives are not the same. If the two focal planes coincide, if we remove the sample, an incident k-vector matches an output k-vector. Thus a pixel of the SLM corresponds to a pixel of the CCD. In the same configuration with a scattering medium, if the part of the energy ballistically transmitted is important enough, this ballistic part can be isolated in the TM.

To observe those contributions and study the accuracy of the reconstruction methods in that particular case, we experimentally use a thinner and less homogeneous area of the medium with a symmetric setup (M = N ) where input and output segments are approximately the same size. We test the different reconstruction methods presented in 3.3 with one realization and averaging over 40 'virtual realizations' with random phase masks. Results are shown in figure 8. As expected, we see that the inversion operation does not allow image reconstruction, even with averaging, whereas optimal MSO allows a 93% correlation for 40 averaging (and a 60% correlation for one realization). Phase conjugation which is supposed to be stable in presence of noise gives a weak correlation of 31% with averaging.

Those results are similar to the ones obtained when we measure the TM of a system governed by multiple scattering [15] for inversion (11% correlation with averaging) and for MSO operator (94% correlation with averaging) but are very different for phase conjugation (76% correlation with averaging). We see in figure 8(e.) and (f.) that most of the energy of the reconstructed field is localized on very few points. In presence of strong enough ballistic components, the direct paths associated give rise to channels of high transmission linking one ) the spatial aspect of the input singular vectors corresponding to the first and the 100 th highest singular values, which should not be affected by balistic light. The 100 th singular vector is spatially randomly distributed in energy (as expected for random multiple scattering) whereas the first singular vector mostly contains energy in one k-vector. This confirms that, in this particular case, the channel of maximum transmission is mostly dominated by the ballistic contribution. Phase conjugation maximizes the energy transmission in channels of maximum transmission [40]. Therefore, ballistic high singular values contributions will be predominant in phase conjugation, regardless of the image E obj . Since the ballistic singular vectors are not homogeneously spatially distributed in energy, they will not efficiently contribute to an arbitrary image reconstruction. MSO is not affected since lower the weight of channels which do not efficiently contribute to the image reconstruction, including the ballistic ones.

Pseudo-Inversion

We showed in section 1.1 that when we set the output pixel size to fit the output coherence length, the number of degrees of freedom for detection is equal to the number M of segments read on the CCD. This way a second approach to increase N DOF and thus the quality of the image is to enlarge the output observation window. An important advantage is that the limiting time in our experiment is proportional to the number of input basis vectors N , thus, in contrast with a focusing experiment where N DOF is equal [8] to N , increasing the degrees of freedom for image reconstruction does not increase the measurement time. Another consequence of increasing the size of the image recorded, that is, of increasing the asymmetric ratio γ = M/N ≥ 1, is that the statistical properties of the matrix are deeply modified. We saw in section 1.4 that increasing γ the range of the normalized singular values decreases [START_REF] Marčenko | [END_REF]34]. A direct consequence is that the smallest non-zero singular value increases and reads λ 0 γ = (1 -1/γ). Controlling less information on the input than measured on the output results in an averaging effect. An interesting consequence is that for a reasonable experimental noise level, one can find a value of γ for which all singular values (and thus all channels energy transmission) are above this noise level. In such a situation, the TM recorded is barely sensitive to the experimental noise and since no singular values are drowned in the noise, pseudo-inverse operator can be efficiently used.

To highlight this effect, we experimentally measure the TM of our sample for different values of γ ≥ 1. For each TM we tested a single realization of the reconstruction of a complex image using optimal MSO operator and pseudo-inverse operator. The results are shown in figure 10(a.). For γ = 1 we are in agreement with previous results since optimal MSO gives a partially accurate image when results of inversion are totally drown in noise. Increasing γ and thus N DOF strongly improves the quality of the reconstructed image for both techniques and reaches a > 85% fidelity for the largest value of γ = 11, without any averaging. As expected, the curves of fidelity reconstruction as function of γ for both techniques become identical when the value of λ 0 γ reaches noise level (figure 10(b.)). Above this value, pseudo-inversion and optimal are totally equivalent. One notice that experimental λ 0 γ are always smaller than their theoretical predictions. We explain this deviation by the correlation induced in the matrix by the reference pattern |S ref | (see section 1.4). We saw in this part that increasing the degrees of freedom and using an appropriate operator that takes noise into account permits to drastically enhance the reconstruction accuracy of an image detection. Similar results can be obtain for focusing or for output wavefront shaping with the same techniques. Such experiments are more difficult to carry out since they require amplitude and phase modulation. Increasing γ = N/M and thus N in that case also presents the drawback to significantly increase the measurement time.

Memory Effect

To define the statistical properties of the TM of multiple scattering media in part 1.4 we did not take into account weak correlation effects that could break the hypothesis of independent elements of the TM. The strongest (first order) of those effects that can be recorded for system dominated by multiple-scattering (no ballistic contribution) in the diffusive regime is the so-called Memory Effect (ME). It brings correlation with characteristic length L me which depends, in transmission, on the geometry of the observation system [21]. If the speckle grain size L speckle is of the same order of magnitude or smaller than the ME range L me , ME brings additional correlations in the TM.

We define L speckle as the full width at half maximum of the autocorrelation function of a speckle intensity pattern and L me as the attenuation length of the ME. Both length are defined for a given observation plane.

The speckle size L speckle depends on the distance between the back of the sample and the observation plane D, the width of the illumination area W and the wavenumber k. It reads :

L speckle = 2πD kW (40)
In contrast, L me depends on k, D and the thickness e of the sample and reads :

L me = D ke (41)
Therefore, for W ≫ e we have L me ≫ L speckle . This kind of geometry can be convenient since it allows to move a focal spot along a distance greater that L speckle by just tilting the SLM [41]. We experimentally fixed D = 1.5 ± 0.2 mm, W = 0.5 ± 0.1 mm and the sample has a thickness e = 80 ± 25 µm. For those conditions, we expect L speckle ≈ L me ≈ 1.5 µm. The TM is measured. Each column of the TM is the complex amplitude response of a plane wave with a given incident angle. One can easily reconstruct the intensity patterns for different incident angles and calculate the correlation function between two patterns :

f (n-n ′ ),j = |k mn | 2 ⊗ m k (M -m)n 2 j m |k mn | 2 × m |k mn ′ | 2 (n-n ′ ) (42)
Where ⊗ m denotes the spatial convolution over the variable m and j is the variable of the correlation fonction corresponding to a displacement in pixel in the focal plane. We average results over all combinations of n, n ′ with n -n ′ = constant. Results are shown in figure 11. It is clear that the output speckle is coherently translated for distances greater than L speckle .

L speckle is deduced from the mean autocorrelation function of the speckle pattern f (n-n ′ )=0,j as response to an incident wave plane. We found L speckle = 1.3 ± 0.48 µm. We We found the expected correlation lengths with a good order of magnitude for autocorrelation (speckle grain size) and memory effect and we were able to qualitatively separate both effects.

Conclusion

We reinterpreted in this article previous works on optical phase conjugation in term of linear system defined by a monochromatic transmission matrix. Within this formalism, we show Controlling Light Through Optical Disordered Media : Transmission Matrix Approach 25 how to take advantage of multiple scattering to achieve image transmission and wavefront shaping. We also pointed out how to improve the quality of measurements in the presence of noise, taking benefit of Random Matrix Theory and information theory. We finally showed how to take benefit of the statistical properties of the TM to go back to physical phenomenons.

Figure 1 .

 1 Figure 1. (color online) Schematic of the apparatus. The 532 nm laser is expanded and reflected off a SLM. Polarization optics select a phase only modulation mode. The modulated beam is focused on the multiple-scattering sample and the output intensity speckle pattern is imaged by a CCD-camera. L, lens. P, polarizer. D, diaphragm.

Figure 2 .

 2 Figure 2. Singular values distributions of the experimental transmission matrices obtained by averaging over 16 realizations of disorder. The solid line is the quarter-circle law predicted by RMT. With + the observed matrix, with the matrix filtered to remove the reference amplitude contribution and with • the matrix obtained by filtering and removing neighboring elements to eliminate inter-element correlations.

Figure 3 .

 3 Figure 3. Normalized singular value distributions for different asymmetric ratios γ = 1, 1.5, 2, 3, 4, 5 and 6. (dash line) experimental results, (solid line) theoretical prediction from RMT. Calculations on experimental TM are made using the filtered matrix K fil .

Figure 4 .Figure 5 .

 45 Figure 4. Experimental results of focusing and detection using Phase Conjugation. Results are obtained for N = M = 256. We show typical output intensity pattern for a one spot target focusing a. and multiple target focusing b.. We present similar results for the detection of one point c. and two points d. before the medium. The point objects are synthesized using the SLM.

Figure 6 .

 6 Figure 6. Reconstruction noise as a function of N o σ . Results are obtained in simulation for a TM modelized by a 20 by 20 random matrix of Gaussian statistics. N o σ is normalized by the total transmission energy k λ 2 k . In inset we show the values of |W × K| 2 with a logarithmic scale.

Figure 7 .

 7 Figure 7. Comparisons of reconstruction simulations of a simple object for two different 'virtual realizations' (a. and b.) and the reconstruction averaged over over 10 'virtual realizations' (c.). Simulations are made for a 100 by 100 random matrix of independent elements of Gaussian statistics. The amplitude object consists of three points of amplitude one and the rest of the image to zero.

Figure 8 .

 8 Figure 8. Comparisons of the reconstruction methods in presence of ballistic contributions. a. initial gray scale object and b. a typical output speckle figure after the opaque medium. c., e. and g. are experimental images obtained with one realization using respectively inversion, phase conjugation and MSO operator, d., f. and h. are experimental images averaging over 40 'virtual realizations' using respectively inversion, phase conjugation and MSO operator. Values in insets are the correlation with the object a..

Figure 9 .

 9 Figure 9. (a.) normalized singular values λ sorted by decreasing order for a thin non homogeneous region of the sample with experimental conditions sensitive to ballistic waves (dash line) and for a thicker region of the medium with the two focal planes objectives separated by 0.3 mm (solid line). With • we show the normalized singular values of a simulated matrix of independent Gaussian elements following the 'quarter circle law'.(b.) amplitude of the singular vector corresponding to the maximum singular value (left) and to the 100 th singular value (right). Both share the same amplitude scale.

Figure 10 .

 10 Figure 10. Influence of the number of output detection modes. a. correlation coefficient between E img and E obj as a function of the asymmetric ratio γ = M/N for MSO (dashed line) and for pseudo-inversion (solid line), without any averaging. Error bars correspond to the dispersion of the results over 10 realizations. b. Experimental (solid line) and Marcenko-Pastur [30] predictions (dashed line) for the minimum normalized singular value as a function of γ. The horizontal line shows the experimental noise level N o opt σ .

Figure 11 .Figure 12 .

 1112 Figure 11. Spatial correlation functions of output speckle for a variation of the input incident angle. The correlations functions are drawn as function of the spatial coordinate corresponding to the displacement of the speckle pattern in the observation plane. Arrows show the maximum of correlation functions for the highest displacements. Those peaks are barely observable due to an exponential decay of the maximum correlation value.

Acknowledgments

This work was made possible by financial support from "Direction Générale de l'Armement" (DGA), Agence Nationale de la Recherche via grant ANR 2010 JCJC ROCOCO, Programme Emergence 2010 from the City of Paris and BQR from Université Pierre et Marie Curie and ESPCI.

Appendix

To take into account the effect of the reference beam in the effective SNR, we define Γ < 1 the ratio of the intensity coming from the controlled part over the total intensity. This way, Γ is also the ratio of the area of the controlled part of the SLM over the total illuminate area. One way to understand this situation is that N tot independent segments of the SLM are illuminated and we control only N = Γ.N tot of them. We injected the ratio Γ in equation 19 and 20. SNR now reads :

For the setup shown in figure 1, the illuminated area on the SLM represents the circumcircle of the square part that we control. The segments illuminated but not controlled are used to produce the reference pattern. For this geometry, with an illumination disk of radius R we have Γ =

π . The effective SN R corrected to take into account the reference beam is therefore :


Romain Bénard

Cyril Bossard
email: bossard@enib.fr

Pierre De Loor
email: deloor@enib.fr

Romain Benard
email: benard@enib.fr

Context's modeling for participative problem solving

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

This article introduces a context modeling for mking up a Virtual Environment for Training (VET). This VET will be used to simulate dynamic situation. For the learner, the aim is to recognize this type of situations and manage them.

Simulation in virtual reality allow the training of a learner to solve dynamic problem in a constructivist way (?). He will learn by doing. VET use Intelligent Tutoring System (ITS) (?). Those tutors implement learning strategies. Those one allow to have a pedagogical support which consists in interventions to guide the learner in his task. There is different way to help him, for example by asking about his mistake or by showing him the relevant information that he has missed.

Our work is centred on a more complex problem. We are interesting in the dynamic and collaborative situations. The learner will not follow a well defined procedure, so his mistakes are not predictable. Moreover, he has to be immersed in a virtual environment and he has to collaborate with autonomous agent to solve the problem. That is called participation. Participation is the integration of the human in the loop (Dautenhahn 1998;Schuler & Namioka 1993), which fully takes part in the resolution. He makes, imposes or accepts some decisions. He communicates with other actors of the resolution. This introduces a new problem: it is necessary for the various actors of the system to understand each other and to communicate.

This article deals with a contextual approach of this problem. We put forward advantages of context modeling for decison-making of autonmous agents. One can find a lot of definition of this context. We decide to keep the definition given by P. Brzillon and J-CH Pomerol [START_REF] Pomerol | About some relationships between knowledge and context[END_REF]: A context is a collection of significant conditions and surrounding influences that make a situation unique and comprehensible. We have also based our work on the studies of R. M.Turner (Turner 1993) which considers that the context-sensitivity is fundamental to create an intelligent behaviour.

A contextual approach will allow a better explanation (Brezillon & Pomerol 1999) of mistakes done by the learner. In Virtual Reality people are represented by an avatar. This one will have a context representation of the situation. In this way, significant information when a mistake is done can easily be shown to the learner. Explanations could be too more precise about the mistake.

As we have said, we are interesting in the learner particpation in the simulation, we need to have some credible action selection for the other agents in the simulation. Like (Brzillon 1999), we think that the context integration is a way to refine this decision-making. We are using context for the agent reasoning. Like (), we argue that case-based reasoning (Aamodt & Plaza 1994) is a good solution for reasoning about context. This CBR will take in entry the context and it will return the most appropriate solution according to old solved cases of the base.

The last point, we need in a participative simulation, is the substitution. This concept consists in allowing the learner, during an exercice, to take control of another protagonist. The situation can be played again but the learner should be aware of the situation with the new point of view. He has to know the past experience of the agent.(R. Mendes de Araujo 2005) proposes, that the context allows to better take into account interactions between the human and the machine. We argue that it can be a relly good way to share knowledge between human and agent. At last, we will see that it contributes to a simpler use of the concept of substitution.

This article is structured as follows, the part explains what we means by dynamic and collaborative situation, the part introduces our approach of context modeling. This part begins by a definition of context, continues with our formalism and ends with the advantages of our definition. The part deals with the actions selection. At last we show an example of the decision-making of our agent apllied to a football simulation.

Situation dynamique collaborative

D apres Tiberghien G. ( 2002), une situation comprend :

• des donnees interpretables en fonction des objectifs des participants.

• des agents capables de modifier la situation selon ces objectifs (inclure aussi les changements spontanes de la situation, depourvus d'objectifs).

• des agents capables d'adopter des points de vue epistemiques sur la situation en fonction de leur role (ces points de vue font aussi partie de la situation).

Le point commun de ces situations est qu'elles sont intuitivement complexes (J-M. Hoc & J-M. Cellier, 2001). Differents facteurs de complexite permettent de decrire ces situations dynamiques collaboratives.

Les facteurs de complexite issus des caracteristiques de la tache a accomplir ont une place centrale dans la litterature. Cette complexite a l'avantage d'etre objective, mesurable et reproductible. Synthetiquement, Woods (1988) propose deux types de facteurs de complexite: Les caracteristiques du systeme: la dynamique, le risque, le caractere incertain ou vague des informations, en bref les exigences de la tache. Les caracteristiques des agents: le nombre d'agents associes au fonctionnement du systeme, le caractere dynamique des roles de ces agents.

Les facteurs de complexite issus de l'expertise ou internes, c'est a dire propre aux sujets (Amalberti, 1996).Une complexite lie a la representation du systeme, des buts a atteindre et egalement des propres connaissances et savoirfaire de l'agent (de quoi suis-je capable?). Une complexite liee au partage de la representation entre agents impliques dans la meme situation, particulierement importante dans les activites collectives.

Les situations dynamiques collaboratives sont des situations evoluant de facon autonome, c'est a dire pour partie independamment des effets des actions et, au moins 2 personnes communiquent sur les actions existantes. Ces situations se caracterisent par leur complexite, les incertitudes qu'elles produisent, les risques associes, des contraintes temporelles importantes et, l'autonomie des entites integrees dans le collectif. Les interactions entre les differentes entites permettent de repondre a des objectifs communs identifies, de planifier et de gerer le temps, de partager le travail collectif et l'experience, de capitaliser des informations et de developper des competences individuelles et/ou collectives.

Ces situations se rencontrent dans de nombreux domaines lies au travail en equipe en temps contraint (urgence, securite) [START_REF] Hoc | Towards a cognitive approach to human-machine cooperation in dynamic situations[END_REF]; ?) ou dans les sports collectifs (G.K. 2003). La prise de decision de chacun des intervenants est faite sous pression temporelle. Il n'est donc pas possible de mettre en oeuvre des mecanismes de negociation complexes. Ceci n'exclut pas toute communication. Celle-ci est simplement breve et souvent non verbale

Les simulations en realite virtuelle permettent d'immerger un apprenant dans une situation dynamique collaborative. Dans ces environnements virtuels de formation, l'apprenant est represente par un avatar. Pour obtenir un comportement collectif, nous utilisons une approche a base d'agents simulant des prises de decision autonomes. Nous associons un agent par intervenant intgr la situation dynamique collaborative.

Nous pensons qu'une modelisation fine du contexte des situations dynamiques collaboratives permet de repondre a ces besoins. En effet, le contexte est un moyen de reifier les informations d'une situation dynamique collaborative. Nous prsentons cette modlisation dans la partie suivante.

Architecture for Training Collaborative Dynamic Situation

The comportemental architecture of agents involved in our virtual environment is shown on the figure 1. We underline that the work originality is not in this architecture. A similar one can be found in (?). We find interesting to introduce it in order to facilitate the comprehension of our approach of using context. Each agent has a comportemental cycle, that begins with perceptions. This perception is compared to the old one that are stored in a casebase. The most similar case is extracted and allow to define the comportement for this type of situation. The comportement consists in doing or not an action. In the framework of the developpement of a virtual environment for training, we have identified three axes of context using.

Simulation CBR Context

1. La construction de la base de cas : La definition en amont d'une telle base est tres difficile (?). Elle est le fruit d'une analyse fine des situations. Cette analyse repose sur des elements pertinents (affordances ?) qui sont une abstraction de la perception, associe a une abstraction du passe du systeme. Si les regles comportementales sont difficiles a definir, les elements pertinents peuvent etre identifies par un expert du domaine. Ils constituent alors le contexte d'une situation necessaire a la prise de decision. According to (Shoham 1991), context makes it possible to simplify the representation and the comprehension of the problem by focusing the actors attention.

2. Builbing the casebase is a hard task, the crucial point is the definition of the significant information to describe a case. C'est la simulation qui revele des cas pour lesquels aucun comportement ne semble defini.

Context modeling for dynamic and collaborative situation

In the framework of the collective decision-making we consider, as (R. Mendes de Araujo 2005) proposes, that the context allows to better take into account interactions between human and machine. This point seems really important to us because we want to create a participative simulation in wich human agent have to be immerged. We will introduce our approach in this section, we first detail our distinction between differents contexts, and then we present the formalism of our contexts.

Context delineation

If we can work out a contextual representation of the problem allowing to evaluate its distance to typical schemas, we will be able to simulate fine decision-making closer to the human one. Another aspect which justifies the use of the context is underlined in (Turner 1998). R. M.Turner argues that context allows agents to better react to unexpected events by enabling them to detect event, to evaluate it and to answer to it. This point seems important to us because our system must be open as humans or agents can enter or leave it. Moreover, if one can foresee the agents behavior in the system, one can not claim to know all the possible actions of human during the resolution. That is the reason why we have chosen to formalize our context as (McCarthy 1993), in order to be able to compare them. Before introducing our context representation we will propose a delineation of contexts. The sum of those contexts will form the global context of the agent. We decide to follow the psychologist's point of view as it is underlined in (KOKINOV 1997) and we divide the global context in two parts. The external one refers to the physical and social environment, the second one refers to the internal state of the agent, such as skills, physiological, strategical.. The external context is made up of two more specific contexts that we will detail:

• Environmental context : refers to the physical elements of the simulation. This can be an actor, an object, or everything that the agent can see or hear.

• Communicative context : is the reunion of all the communication for the agent. We choose this name instead of social because, for us, social context means the representation of the group. But it is evident that the social context is totally dependant of the communicative one.

The internal context is the sum of five distinctives contexts that are:

• Skill context : represents the agent skill in the domain of the simulation. This context is usefull is the action selection when two or more actions seems to be good. The agent will choose the most appropriated one according to his skill context.

• Physiological context It reflects the psychological and physiological state of an agent. This agent can be disturbed, exhausted, inclined to collaboration or in peak form. It influences mainly the strategy of collaboration. It depends on the historical context

• Social context : It allows the agent to have an idea of the general mood of the group. This context strongly depends on the number of communications between actors, and especially on the type of conversations, such as for example number of refusals or number of requests for assistance...

• Strategical context : refers either the strategy that the agent should execute, or the future situation according to the agent point of view. It depends of the historical, social and environmental contexts.

• Historical context : It allows the agent to keep a trace of his past experience. The casebase is a way to keep information but they are not ordered. This is the role of the historical context. In practice, it is a graph containing the case used by the agent.

The last one is the collaborative context. We decide to place it in the center of the figure to show its dependance on the other contexts.

This delineation of contexts is usefull for our aim because:

• It simplifies the definition of the pertinent context to describe a situation. The expert can tell what context is important for this situation. This is important for building the casebase in the simplest way.

• This delineation allows a better explanation to the learner when he make a mistake. We can easily underline contexts that were relevant in this situation and so the learner can be concentrated on the interesting point of the explanation.

• The last point is our selection action method. The differents contexts do not have the same role in the action selection. Some of them like environmental, social and physiological are importants. As we illustrate before, the skill context will be usefull in the selection action, when two or more actions are chosen. Else it is used to specify action parameter (dexterity for an action).

Representing context

However, comprehension is not a deliberative and enumerative simple mechanism. It is often based on analogies and gatherings, what psychologists call typical schemas (Rosch 1975). This concept was reused by R. M.Turner (Turner 1993) as contextual schemas allowing the agents to reason with their representation of the context. We use schemas to represent the different contexts These schemas have a general form and are all, at least, built up from the following attributes:

Each context can be describe with this prototype:

• attribute: represents either a physical object or an actor involved in the current context.

• feature : is a part of the attribute and represents a particular point which is interesting for this attribute.

• action: is the action associated to this context. In literatur, a case in a CBR system is a problem and its solution. In our approach, a case is a context and the appropriate action. We choose to use XML to represent a case. CBML(Case-Based Markup Language) (Coyle, Hayes, & Cunningham 2002) is a good example of using XML to represent cases.

A case is divided in attributes. An attribute has been mentionned by a domain expert as relevant for this situation. For example, in the environmental context, it will be a physical object

The expert gives a weight to every attribute of a case. This weight is given in fonction of the interest of the attribute for the case.

The type of an attribute can be a value in the list:

• fuzzy allows to represent a concept by using words to describe a feature of the concept. An example can be closer,close,far ...

• string represents a string value

• compound an attribute can be more complex so we have to separate its features.(cf

• exact represents an exact value.

• ontology allows to calculate the similarity between two concepts stored in an ontology.

We have described our vision of the context for our agents, we will see how it is used in a case based reasoning system for actions selection.

Action selection

As we have said before we use Case-based reasoning in association with context to simulate a credible decison-making for our agents.

We will describe a CBR system and we will explain the CBR cycle. We will not explain in details our selection action like the similarity measurement, retrieving methods or learning algorithm... We just want to give an overview of the CBR that we use with context retrieval to select agent's actions. A CBR system reuses old solved problems to solve a new one. In order to be able to do that the system is made up of a base containing solved cases and the new case we want to solve.

According to (Aamodt & Plaza 1994), reasoning by reusing past cases is a powerful and frequently applied way to solve problems for humans. We can take, for example a little mathematical problem: Somebody asks you to solve 11*11 and find 121 which the good result and we ask you to find 11*12. You will not try to calculate 12*11 but adapt the result of 11*11, by doing 11*11+11, and find 132. This simple example illustrates how we can use experience to solve new problems by reusing previous one. This example and figure 3 illustrate the principle of the CBR. The first step consists in elaborating the new case. In our approach, this step is the context retrieval. Since the agent knows the context, it will compare it to those it has in its casebase. This step is called Retrieve. After that, the case is adapted in order to be more appropriated, this is called Reuse. If the case is not contained in the casebase, the expert decides if the new case should be stored in the base.

Application example

An example to illustrate our proposition is the simulation of a football match. We are currently working with psychologists to create a virtual environment for training, called CoPeFoot for Collective Perception in Football. The aim is to allow users to recognize collective situation in a football match. We are not interesting in the technical aspect of football, but in the strategical one.

The example we give here is a situation where a player of the team a is face to a player of the team b and have to eliminate him. The figure illustrates the current situation.

We have chosen three contexts of the player that we will describe, those are the environmental, social and personnal. The environmental one looks like:

Conclusion

We have introduced in this article our approach for creating a virtual environment for training. After having developped what we mean by dynamic and collaborative situation, we have described the way we use context and CBR to simulate the agent comportement in this type of situations. At last, we illustrate the action selection by an example extracted of our simulation.

We argue that using context is a good way to simulate credible actions selection for autonous agents. We have the certitude that context is a good manner to enhance explanations in a non procedural training. The association with a CBR will enhance the two points we have developped just before.

We argue that the context contributes to the two previous problems: based on the ideas of Araujo (R. Mendes de Araujo 2005), according to whom a group context allows a newcomer to more quickly take into account the current situation. We believe by extension that this assertion is also true when using the context for exchanging information between a human actor and his substitute.

Figure

  Figure 1: contexts division

  Figure 2: contexts division

  Figure 3: The CBR cycle

  L'expert est alors interroge sur ce cas, qui lui est presente apres filtrage contextuel. Cette presentation utilise les mots cles de son domaine. Elle lui est donc explicite et lui permet d'indiquer la decision a prendre. 3. Mise en place de support pedagogiques : De nombreuses strategies d'apprentissages sont definies par les pedagogues. Par exemple, il est interessant de montrer les consequences d'une erreur ou de mettre en evidence les elements de l'environnement qu'il doit considerer. Il est aussi important d'expliquer ou de poser des questions pour aider l'apprenant a comprendre le probleme. La encore, toutes ces aides pedagogiques peuvent faire l'objet

d'une contextualisation. Cette fois le contexte correspond a une fusion entre le contexte du domaine, de l'historique et des notions pedagogiques plus abstraites (pourquoi, comment ? a traiter ulterieurement) 4. La subsitution : Lorsque la machine prend le controle d'un avatar, elle doit adopter le comportement le plus credible possible avec la situation courante. Si le contexte utilise pour la definition du comportement des agents est egalement calcule durant l'execution du comportement de l'avatar, cela facilitera grandement cette prise de controle. En effet, les variables contextuelles necessaires a la prise de decision, representant une abstraction de l'historique de la simulation, seront toutes renseignees au moment de la prise de contrle. De plus, dans le cas inverse, ou c'est l'apprenant qui prend le controle d'un agent, s'il recupere son contexte. Tous les mecanismes support d'explications ou d'autres interactions entre l'apprenant et l'environnement de formation seront immediatement utilisables.


Romain Bénard
email: benard@enib.fr

Pierre De Loor
email: deloor@enib.fr

Jacques Tisseau
email: tisseau@enib.fr

UNDERSTANDING DYNAMIC SITUATIONS THROUGH CONTEXT EXPLANATION

Keywords: context, explanation, contextual graph, simulation, contextual help

This article presents advantages of using context to set up a pedagogical assistance for recognition of collectives situations in Virtual Environment for Training (VET). We are focusing on generation of explanations to the learner. Two assistances types have been envisaged thanks to context using, the first one consists in guiding the learner before action and the second can be used during action. Those assistances have been set up thanks to contextual graph and consists of animations in the virtual environment.

Introduction

This article deals with the interest of using contextual graphs to set up pedagogical tool in a VET dedicated to dynamic and collaborative situation (DCS) learning. For the learner, the aim is to manage this type of situations. It can be guided thanks to typical intervention such as the highlighting of relevant elements to enhance awareness of a particular concept. In order to do that, some pedagogical techniques such as enrichment, degradation, simplification or the heightening of simulations are used [START_REF] Fuchs | A theoretical approach of the design and evaluation of a virtual reality device[END_REF].

In the case of a dynamic and collaborative situations, relevant elements extraction is complex because of their dependency to context which is dynamic. That is the reason why we decide to show here the interest of using contextual graph to represent strategical and historical context for explanations. This article is structured as follows, it begins with a definition of the constructivism learning of dynamic and collaborative situations and help types that can be set up in order to recognize them. The section 3 illustrates the interest of using contextual graphs to make up a visual explanation. An example is given before the conclusion of this article.

Constructivist learning of DCS

Virtual environment for training allows a constructivist perspective of learning: Learner will enhance his knowledge by being face to new situations. He will build his own representations [START_REF] Piaget | The Psychology of the Child[END_REF]. DCS implie temporal or organisational constraints that make an automatic explanatory system hard to define. We will now introduce those situations

Dynamic and collaborative situations

Dynamic and collaborative situations can be found in various domains of teamwork with time pressure (rescue, security) [START_REF] Rognin | Cooperation, interactions and socio-technical reliability: the case of air-traffic control. comparing french and irish settings[END_REF] or in team sport [START_REF] Argilaga | Detection of realtime patterns in sports: Interactions in soccer[END_REF].

This implies various protagonists that interact in a common environment and have to solve a problem. A collaboration between protagonists can be needed to solve the problem. Situation data can be interpreted according to the protagonists point of view and roles. Consequently, elements taken into account by a protagonist to take a decision are not easily predictable [START_REF] Woods | Coping with complexity: the psychology of human behaviour in complex systems. Tasks, errors, and mental models[END_REF][START_REF] Cellier | La gestion d'environnements dynamiques[END_REF].

We can exemplify those situations through an example of collective action in soccer. This action is illustrated by the figure 1. A player A1 of the team A has the ball. He is face to a player B1 of the team B. There is a player A2 of the team A to the left of B1. A collaboration can be possible between A1 and A2 to keep the ball and eliminate B1.

A1 B1 A2

Figure 1. A simple dynamic and collaborative situation

Possible types of assistance

For this purpose, we have identified two help types that can be set up. They are called on-line help and can be divided in two guidances:

• A guidance thanks to actions: that means a strong guidance. One shows a priori linkings that allows the sequence realisation. In order to do that, we introduce graphical elements which will represent actions that must be done before they appear. In the example given below, it is possible to add a ball going from the learner to his partner, or we can represent its trajectory with an arrow as it is shown on the figure 2. This paper is focused on this help type.

• A guidance thanks to affordances: That means the learner will execute his action and if he makes a mistake, we can highlight some relevant elements he has missed. Those elements or affordances [START_REF] Gibson | The Ecological Approach to Visual Perception[END_REF] must allow him to understand why he was mistaken. In our example, if he kept the ball, we could make his opponent bigger according to his distance. The closest he is, the biggest the opponent will be. Another solution consists in making his partner blinking. The crucial point is that the help is done only if the learner is mistaken.

Those two types can be made up thanks to contextual graphs. Now, we will explain what are those graphs that we are using to model our strategical contexts.

Contextual graphs as learning support

Our agent decision-making is based on the notion of context [START_REF] Turner | Context-sensitive reasoning for autonomous agents and cooperative distributed problem solving[END_REF]. More precisely, we are using context-based reasoning (CxBR) that has been introduced in [START_REF] Avelino | Contextbased representation of intelligent behavior in training simulations[END_REF]. The definition and modelling of our context is based on psychologists point of view [START_REF] Kokinov | A dynamic theory of implicit context[END_REF]. The modelling of our context have been described in [START_REF] Bénard | Context's modelling for participative simulation[END_REF]. This paper deals with contexts used for explanation, that are modelling thanks to contextual graphs [START_REF] Brézillon | Context dynamic and explanation in contextual graphs[END_REF] that we will introduce now:

Context for explanation

In our approach, a contextual graph represents actions sequence and relevant states to lead a collaboration until its end. Relevant states are characterized by objects or by particular attributes of objects. Those can be likened to the notion of affordances. Every state of the graph represents a context at a given instant of the collaboration and is called contextual node. The transition between two nodes can be done thanks to an action. The figure 3 shows a contextual graph, with contextual nodes and actions that we will describe now:

Actions

Transitions in the graph represent actions that can be done according to the current context. They allow to highlight the link between two contextual nodes. They can be written thanks to a triplet that consists of an actor, the action and an object or a protagonist on which the action is performed. An action in a contextual graph looks like: A = < Actor, Action, Object > where

• Actor: represents the player which performs the action • Action: is the action, technically it is the name of the method which is called. Applied to soccer, an action can be pass, dribble or shakeOff...

• Object is an object or a protagonist on which the action is performed.

For example, < P 1, pass, P 2 > means that the player P1 passes the ball to the player P2. In the soccer example, an action is performed by the player which we are describing the context.

We will see later how this formalism of actions and contextual nodes allows to easily highlight relevant elements to set up pedagogical tool to help recognition of dynamic and collaborative situations. A triplet is used to represent an affordance too, as we will see in the next example.

Example of contextual assistance

We will now illustrate why and how our model allows to set up the pedagogical assistance. The trainer has first defined strategical context representing the situation that learner has to recognize. Every relevant actions and affordances are stored in the contextual graph. Each action and affordance can be linked to a graphical primitive for explanations. The strategical context looks like the graph of figure 3. Where b is the ball, j the player, p a partner, o an opponent, dist(X) represents the distance to X , position(X) the

Guidance before action

A graphical primitive is associated with every action of the contextual graph. Thanks to this primitive, relevant action can be shown to the learner. We have already defined various graphical primitives like a dotted arrow used to represent a move. The distance between the learner and an object can be represented with a double arrow, that can illustrate if the object should be nearer or further.

The example given at the beginning of this article on the figure 2 shows an assistance that have been set up to highlight the relevant action. The current context is C1 of the strategical context of A1. A curved arrow is drawn to represent the pass that the learner should do. To put in place, this graphical assistance, agent uses the triplet describing the action. Thanks to this it can determine the orientation and the position of the arrow.

conclusion

The aim of this article is to highlight the interest of using context in order to bring back relevant information to users (trainer or learner) in situations that can not be totally described by advance. We have described here the interest and the representation of our strategical context.

Using contextual graphs allows us to highlight relevant elements that correspond to the current situation according to the objectives. We have put in place two types of assistances, the guidance before action which is a strong guidance. The second type of guidance during action allows to bring to the fore elements that has not been taken into account.

We have focused our purpose on the guidance before action, but we could have done the same with the guidance during action. We are currently working on a third help type that allows to replay a situation.

Figure 2 .

 2 Figure 2. contextual help

Figure 3 .

 3 Figure 3. Strategical context of A1