Sylvain Mahé

Raphaël Braud

Philippe Gaussier

Mathias Quoy

Alexandre Pitti

Exploiting the Gain-Modulation Mechanism in Parieto-Motor Neurons: Application to Visuomotor Transformations and Embodied Simulation

Keywords: post-parietal cortex, gain-field modulation, mirror neurons, spatial transformation, social cognition

The so-called self-other correspondence problem in imitation demands to find the transformation that maps the motor dynamics of one partner to our owns. This requires a general purpose sensorimotor mechanism that transforms an external fixation-point (partner's shoulder) reference frame to one's own body-centred reference frame. We propose that the mechanism of gain-modulation observed in parietal neurons may generally serve this types of transformations by binding the sensory signals across the modalities with radial basis functions (tensor products) on the one hand and by permitting the learning of contextual reference frames on the other hand. In a shoulder-elbow robotic experiment, gain-field neurons (GF) intertwine the visuo-motor variables so that their amplitude depends on them all. In situations of modification of the body-centred reference frame, the error detected in the visuo-motor mapping can serve then to learn the transformation between the robot's current sensorimotor space and the new one. These situations occur for instance when we turn the head on its axis (visual transformation), when we use a tool (body modification), or when we interact with a partner (embodied simulation). Our results defend the idea that the biologically-inspired mechanism of gain modulation found in parietal neurons can serve as a basic structure for achieving nonlinear mapping in spatial tasks as well as in cooperative and social functions.

Introduction

Over the last two decades, the studies of the postparietal cortex (PPC) have permitted to understand better the neural mechanisms involved in the spatial representation of oneself body. What we have discovered is that our body representation is far more labile as we previously thought and that the brain fully exploits the perceptual ambiguity yielded by the senses to represent spatially not only our body limbs but also the nearby objects and by extension the persons around us. The socalled mirror neurons found by Rizzolatti and his colleagues exemplify the most this discovery as they respond to action and to observation [START_REF] Rizzolatti | Premotor cortex and the recognition of motor actions[END_REF][START_REF] Rizzolatti | Neurophysiological mechanisms underlying the understanding and imitation of action[END_REF]. Mirror neurons appear to bear out the fundamental structure for achieving perceptual, cognitive and motor functions as well as cooperative and social functions [START_REF] Keysers | Demystifying social cognition: Hebbian perspective[END_REF][START_REF] Fogassi | Parietal lobe: from action organization to intention understanding[END_REF] although its mechanism is still poorly understood.

In this perspective, the studies by Iriki in the macaque monkeys are particularly interesting as they showed evidences of a dynamical readaptation of the body schema with respect to the ongoing situation [START_REF] Iriki | Coding of modified body schema during tool use by macaque postcentral neurones[END_REF]. By simply ma-nipulating the visual feedback on a TV set that a monkey scrutinizes to guide its arm motion, Iriki showed how the parietal neurons were readjusting continuously the body image (here the hand) in accordance to the new reference frame [START_REF] Okanoya | Tool-use training in a species of rodent: The emergence of an optimal motor strategy and functional understanding[END_REF]. The spatial transformations performed could be as complex and nonlinear as the combination of translation, rescaling and rotation. This result was also tested on tool-use where the spatial receptive fields of the parietal neurons associated to the hand extended to entail the tool [START_REF] Maravita | Tools for the body (schema)[END_REF][START_REF] Goldenberg | From sticks to coffee-maker: mastery of tools and technology by human and non-human primates[END_REF]. In terms of social cognition, this transformation mechanism is considered to take a central place in the process of understanding others as a mean to transform someone else visuomotor perception into our own thus simulating their actions [START_REF] Rizzolatti | Neurophysiological mechanisms underlying the understanding and imitation of action[END_REF][START_REF] Meltzoff | like me: a foundation for social cognition[END_REF][START_REF] Fogassi | Parietal lobe: from action organization to intention understanding[END_REF][START_REF] Lewkowicz | Reading motor intention through mental imagery[END_REF], see Fig. 1.

Thus, one may recognize that the neural mechanisms involved in spatial representations constitute a hard problem that requires at the same time non-linear transformations as well as rapid processing. On the one hand, the PPC is ideally placed for multimodal integration since it is one of the first cortical structure to receive the sensory signals coming from the different modalities [START_REF] Andersen | Multimodal integration for the representation of space in the posterior parietal cortex[END_REF][START_REF] Pouget | Spatial transformations in the parietal cortex using basis functions[END_REF]. On the other hand, its role to bind fastly the sensory signals is not trivial at all since each sensory signal is encoded differently and anchored to different body part or spatial reference frame; e.g., eye-, head-, shoulder-or hand-centered. One consequence is that patients with lesions of the parietal cortex present difficulty in spatial adjustment, coordination disorders and even spatial neglect [START_REF] Keysers | Demystifying social cognition: Hebbian perspective[END_REF]. Moreover, the spatial disorders also pervade in the social domain particularly in autism spectrum disorders with the importance of embodied self-rotation for visual and spatial perspectivetaking [START_REF] Surtees | The use of embodied selfrotation for visual and spatial perspective-taking[END_REF][START_REF] Pearson | A review of visual perspective taking in autism spectrum disorder[END_REF]. These studies have revealed a lack of multimodal integration and the disability to put in perspective the spatial location of objects and persons relative to our body.

Considering the mechanisms it may involve, the discovery of reach cells and postural cells for particular orientation of the hand associated to the current context or motor plan have permitted to discriminate further the functional organization at the network level [START_REF] Blohm | Fields of gain in the brain[END_REF][START_REF] Mcguire | Sensory transformations and the use of multiple reference frames for reach planning[END_REF][START_REF] Bremner | Coding of the reach vector in parietal area 5d[END_REF]. Andersen and colleagues [START_REF] Andersen | The influence of the angle of gaze upon the excitability of the light-sensitive neurons of the posterior parietal cortex[END_REF][START_REF] Andersen | Encoding of spatial location by posterior parietal neurons[END_REF] first discovered neurons firing for a specific eye saccade motor command, modulated by the position of the eye relative to the head. That is, this result demonstrates that (1) these neurons are bimodal neurons as they encode two information at once and that (2) their amplitude level is an informative quantity that can be modeled. Furthermore, the gain-modulation effect observed for this behavior does not correspond to a summing integration as it would be for integrate-and-fire neurons. Instead, a more correct mathematical model of the parietal neurons' response would be a multiplicative integration between the incoming sensory signals, which can be approximated as a nonlinear basis function [START_REF] Pouget | Spatial transformations in the parietal cortex using basis functions[END_REF].

The striking advantage of a gain-field representation of the signal is that a basis function representation may approximate any desired mapping, as it is the case for the Fourier series or the wavelet decomposition. Therefore, this kind of representation meets the requirements for a local-to-allocentric transformation because multiple reference frames can be derived from the same population of neurons allowing to use intrinsic as well as extrinsic reference frames [START_REF] Mcguire | Sensory transformations and the use of multiple reference frames for reach planning[END_REF][START_REF] Bremner | Coding of the reach vector in parietal area 5d[END_REF]. For instance, Shadmehr and Wise proposed that the gain-field neurons compute a fixation-centered frame by subtracting the vector between the gaze location and the hand position to derive the hand toward the target in eye-centred frame [START_REF] Shadmehr | The computational neurobiology of reaching and pointing : A foundation for motor learning[END_REF][START_REF] Bremner | Coding of the reach vector in parietal area 5d[END_REF]. Following this, different robotic experiments have been conceived using the linear combination of basis functions for sensorimotor transformations [START_REF] Chinellato | Implicit sensorimotor mapping of the peripersonal space by gazing and reaching[END_REF][START_REF] Hoffmann | Body schema in robotics: A review[END_REF][START_REF] Halgand | A robotics approach for interpreting the gaze-related modulation of the activity of premotor neurons during reaching[END_REF].

In previous works, we demonstrated how the mechanism of gain-field modulation can be applied for integrating audio-visual signals and proprioceptive feedback in a head-neck-eyes robotic device as it is for some parietal neurons [START_REF] Pitti | Gainfield modulation mechanism in multimodal networks for spatial perception[END_REF]. In our studies, the gainfield neurons were successfully used to remap the location of one sound signal (in head-centered reference frame) into retina coordinates (in eye-centered reference frame). The gain-field based model enabled the system to increase the accuracy of a visual stimulus (i.e., the position of the mouth when a person speaks) by using the supplementary sound signal to estimate more precisely the spatial location of the mouth-voice stimulus.

In this paper, we consider to employ again the gain modulation mechanism but this time toward a fixationpoint reference frame, external to the body [START_REF] Shadmehr | The computational neurobiology of reaching and pointing : A foundation for motor learning[END_REF]. We propose that the same neural architecture may permit to derive cognitive functions involved in fixation-point reference frame (such as tool-use) as well as social functions involved in perspective-taking tasks such as joint attention and imitation. To this end, we first perform a rapid learning of the visuomotor associations in a gainfield architecture with a shoulder-elbow-like robotic arm and a camera. Once its body schema learned, a second gain-field module is added for situations of visuomotor mismatch and novelty. This second module is used to encode the new sensorimotor task set [START_REF] Pitti | Contingency perception and agency measure in visuo-motor spiking neural networks[END_REF][START_REF] Pitti | Neural model for learning-to-learn of novel task sets in the motor domain[END_REF] corresponding to the visuomotor distortions induced by novel fixation-centered tasks such as in front of a mirror, during tool-use or during the control of an avatar in a video game (self-observation in a TV set). In the social domain, the idea is then to retrieve the hidden visuomotor transformation responsible for the mismatch between the robot arm's own motion and what it currently sees (e.g., a person moving its hand aside). The hidden visuomotor transformation permits to reduce the resulting spatial error in the visual field and in the motor domain, which is associated to the so-called correspondence problem [START_REF] Heyes | Causes and consequences of imitation[END_REF][START_REF] Brass | Imitation: is cognitive neuroscience solving the correspondence problem?[END_REF] and to motor imaginary [START_REF] Kosslyn | Neural foundations of imagery[END_REF].

Methods

A Formal Model of Gain fields

Our architecture implements multiplicative neurons, called gain-field neurons, that multiply unit by unit the value of two or more incoming neural populations, see Fig. 2. Its organization is similar to radial basis function (RBF) as it transforms the incoming signals into a representation of basis functions, a functional space, that could be exploited to simultaneously represent stimuli in various reference frames [START_REF] Salinas | Gain modulation: A major computational principle of the central nervous system[END_REF][START_REF] Pouget | Spatial transformations in the parietal cortex using basis functions[END_REF]. The multiplication between afferent sensory signals from two different modalities (M 1 , M 2 ) is the element-wise multiplication between two probability distributions X m 1 and X m 2 , two vectors of dimensions {M 1 and M 2 } respectively, with {m 1 , m 2 ∈ M 1 , M 2 }. The featured matrix is the signal activity X m , m ∈ M 1 × M 2 that gain-field neurons learn the activity. The equation for n th gain-field neuron X GF n , n ∈ N, with synaptic weights w i , i ∈ M 1 , M 2 is:

X m = X m 1 × X m 2
(1)

X GF n = M 1 ×M 2 i=0 w i X i (2)
Hence, neurons X GF have the same equation as perceptrons without bias. The key idea here is that X GF map encodes a particular combination of the two values, the amplitude of the gain-field neurons encode one modality conditionally to another in a Bayesian fashion, in a lower dimension [START_REF] Braun | Motor task variation induces structural learning[END_REF][START_REF] Braun | Structure learning in action[END_REF]. We exploit this feature to model the parietal circuits from different sensory signals as a linear combination of gain-fields neurons f so that we have f (X GF n ) = N n=0 ω n X GF n , for n ∈ N gainfield neurons, and ω the pondering weights. In a sense, the linear combination of perceptron-like neurons corresponds to the neural architecture of RBFs.

We explain hereinafter how the gain-fields neurons learn the associations between various modalities, see Fig. 2. Once a first mapping is done -, say between variables X and Y in Fig. 2,-it is possible to chain the gain-field maps so that the activity of the laters (e.g., to encode the new modality θ 1 ) depends on the activity of the formers (i.e., the first modalities X and Y). As an example, we give the equations for the second map X GF2 n constructed from the activity of a first map f (X GF1 n ) and the new modality X m 3 , a vector of dimension M 3 :

X ′ m = X m 3 × f (X GF1 n ) ( 3
)
X GF2 n = M 3 ×N i=0 w i X ′ i ( 4
)
The new gain fields neurons X GF2 n depend on X m 3 and f (X GF1 n ) and encode then a relational map obtained from the activity of all the preceding modalities, X m 1 and X m 2 , with the new one, X m 3 . We make the point that X GF1 represent essentially the multiplication of the two input modality values, which do not uniquely represent one combination of the two values. However, from a biologically viewpoint, the observation of single gainfield neuron shows a similar behavior in parietal neurons [START_REF] Salinas | Gain modulation: A major computational principle of the central nervous system[END_REF]. While one X GF is not enough to disambiguate certain cases, the linear combination of multiple X GF1 , as the X GF2 n do, can do much for generalization, as it is for RBFs and perceptrons. Thus, the ROC neurons each represent a specific overall activity pattern of the X GF map which encodes a pair-wise combination of the two values. By doing so, the system retranscribes the relative information of the occurrence of one modality with respect to the others. This multimodal process is similar to the chaining of Bayesian conditional rules between multiple variables [START_REF] Deneve | Bayesian multisensory integration and cross-modal spatial links[END_REF]. As the number of dimension augments, a linear combination of GF neurons can permit to represent well the input data for an appropriate number of GF neurons, if the input space is sparse enough. In this case, we should not see a resolution loss after the learning stage.

The Rank-Order Coding Algorithm

In order to learn the sensorimotor mapping between input and output signals, we implement the hebbianlike learning algorithm proposed by Thorpe and colleagues [START_REF] Van Rullen | Face processing using one spike per neurone[END_REF][START_REF] Thorpe | Spike-based strategies for rapid processing[END_REF] called the Rank-Order Coding (ROC) algorithm that we have used already in previous researches [START_REF] Pitti | Gainfield modulation mechanism in multimodal networks for spatial perception[END_REF][START_REF] Pitti | Neural model for learning-to-learn of novel task sets in the motor domain[END_REF]. The ROC algorithm has been proposed as a discrete and faster model of the derivative integrate-and-fire neuron [START_REF] Van Rullen | Surfing a spike wave down the ventral stream[END_REF]. ROC neurons are sensitive to the sequential order of the incoming signals; that is, its rank code and the distance similarity to this code is transformed into an amplitude value. A scalar product between the input's rank code with the synaptic weights furnishes then a distance measure and the activity level of the neuron. More precisely, the ordinal rank code can be obtained by sorting the signals' vector relative to their amplitude levels or to their temporal order. Due to this feature to encode the rank, ROC neurons are more robust to amplitude noise than winner-takesall (WTA) [START_REF] Thorpe | Spike-based strategies for rapid processing[END_REF].

The neurons' output X GF is computed by multiplying, not directly the amplitude values of the sensory signal vector I, but the inverse of its rank order rank(I) by the synaptic weights w; w ∈ [0, 1]. For an input vector signal of dimension M and for a population of N GF neurons X, we replace eq. 2 by:

X GF n = m∈M 1 rank(I m ) w GF n,m (5)
The updating rule of the neurons' weights is similar to the WTA learning algorithm of Kohonen's selforganizing maps [START_REF] Kohonen | Self-organized formation of topologically correct feature maps[END_REF]. For the best neuron s ∈ N and for all afferent signals m ∈ M, we have:

∆w GF s,m = α ( 1 rank(I m ) -w GF s,m ), ( 6
)
where α is the learning rate; we set α = 0.01. We make the note that the synaptic weights follow a powerscale density distribution that makes the rank-order coding neurons similar to basis functions. This attribute permits to use them as receptive fields so that the more distant the input signal is to the receptive field, the lower is its activity level.

Learning Sensorimotor Transformation

Once the gain-field neurons have learned the visuomotor rules from their input space, they can be exploited for comparing any situation of visuomotor mismatches during physical perceptual changes as well as during social ambiguities. For example, a comparator model as plotted in Fig. 3 that computes the difference between a real signal value A with an estimated one  from the network can serve to retrieve back the possible transformations responsible for the sensorimotor errors.

In that sense, the GF neurons used in our model are similar to radial basis functions (RBF) used in image processing for morphing and registration problems where the linear combination of RBF can serve to deform the input space to a particular a-priori model. Let's consider the point A and the new point  estimated by the GF network with:

f (A) = n i=1 w i X GF i (A) f ( Â) = n i=1 w i X GF i ( Â) (7)
Let us express f ( Â) as a function of f (A):

f ( Â)= n i=1 w i (X GF i (A) + (X GF i ( Â)-X GF i (A))) f ( Â)= n i=1 w i X GF i (A) + n i=1 w i (X GF i ( Â)-X GF i (A)) f ( Â)=f(A) + n i=1 w i (X GF i ( Â)-X GF i (A)) (6)
This means that the closer  is to A, the closer f ( Â) will be to f (A), in other words, the closer a point  is from a point A used for the training of the GF neurons, the closer the associated transformation will be to the example transformation.

The second term corresponds to the transformation function that minimizes the distance between X GF i (A) and X GF i ( Â). One perceptron can estimate this function within the range around the point A by identifying the weights that minimize the correspondence problem, Although a population of perceptrons is more satisfactory since it can cover the whole space as well as different transformation functions with respect to the current situation.

Experimental Setup

In our experiments, we use the Kinova robotic arm with 7 degrees of freedom and a fixed camera to observe it, a graphic of the setup is shown in Fig. 4 a).

For the sake of simplicity, we limited the arm to two Degrees-of-Freedom (DoF) in the visual plan of the camera, but the architecture can be easily extended to more DoFs by chaining the GF maps. The two motor joints correspond respectively to the shoulder and elbow-like joints, θ 0 and θ 1 ; with θ 0 ∈ [0 • , 200 • ] and with θ 1 ∈ [0 • , 100 • ]. Each motor angle is translated into a discretized vector of 22 bins with a Gaussian curve centered on the current motor angle. A color-based vision system focusing on the red color provides the hands coordinate (x,y) on the retina reference frame. And similar with the motor angles, the visual coordinates are translated into two discretized vectors of 22 bins with a Gaussian curve centered on the current position. Using the gain-field architecture as in Fig. 2, the system is composed of three maps, where each ROC group is composed of 22 neurons in order to have a regular network. The last ROC group learns to associate then the motor angles (θ 0 , θ 1 ) with the coordinates of the hand in the (x,y) axis. The output of the gain-fields network will now give an RBF based representation of the learned (x, y) coordinates of the hand.

To take advantage of the gain-fields, an output layer of perceptron neurons will combine their amplitude level toward the estimation of the hands coordinates (x,y) in the visual field and of the motor angles; resp. ( x, y) and ( θ 0 , θ 1 ). This will correspond to the integration of the four inputs through three chained maps as in Fig. 2; see also section 2.1.

Then, in the second part of the experiment, we will use this output layer in order to learn the transforma-tions produced in situations of perceptual ambiguities; i.e., camera-shift, tool-use and social interaction, see section 2.3 and Fig. 4 b),c) and d). A third layer will be added that will learn the difference between the estimated values of the hand coordinates and the current ones (after the transformation is applied to the system); c.f., eq. 2.3. The output of the third layer of perceptron neurons provides the transformation for any given point that will cancel the transformation applied to the system and thus making the previously learned action still relevant.

Results

Generalization from Visuomotor Association

We first let the network learn the visuomotor coordination with the gain-field neurons and we check then how the learning operates toward the spatial location of the arm in the eye-field and in the peripersonal space.

At the initial stage, we perform a motor babbling in a randomized fashion so that the ROC neurons will map uniformly the visuomotor space. After the learning period, which corresponded to 12 minutes duration, 12, 000 samples with the exploration of a new arm configuration each 0.05 second, the ROC neurons selforganized themselves and reproduced the gain-field behavior of PPC neurons.

For instance, we plot in Fig. 5 the activity level of three different ROC neurons from the third map with respect to the variables pairs (θ 0 , θ 1 ) and (x, y). The color code indicates the activity value for each neuron and their respective receptive fields in the motor domain and in the visual domain. We make the note that the motor and visual receptive fields are not separated from each other, but the activity level of the gain-fields neurons is related altogether to the four variables: in Fig. 6 for instance, the amplitude level for one GF neuron varies with respect to the motor angle θ 0 for a given range of values on the X Axis; the color code indicates three different intervals of the motor angle θ 0 . We can see that its receptive field is not strictly centred on a particular value on the X axis but shifts with respect to the proprioceptive feedback; a phenomenon also observed in the biological PPC neurons [START_REF] Salinas | Gain modulation: A major computational principle of the central nervous system[END_REF].

One consequence for gain-fields neurons to simulate basis functions is that the linear combination can serve to estimate the current visuo-motor state in a specific reference frame. Gain-modulated neurons in PPC have been found to generate depth-dependent activity [START_REF] Blohm | Computations for geometrically accurate visually guided reaching in 3-d space[END_REF] as well as hand-centred maps [START_REF] Galeazzi | A self-organizing model of the visual development of hand-centred representations[END_REF] but because our setup does not have vergence and tactile information, we will not focus on these properties. Instead, we propose to use the gain-field neurons to estimate each incoming signal from the fusion of all, in their respective reference frame.

A population of perceptron neurons is used to adapt its weights correctly to the appropriate linear combination of gain-fields in order to estimate the (x,y) variables, ( x, y); see Fig. 7 a). The blue line corresponds to the real visual location on the Y axis whereas the green line and the red line are the linear combinations learned by the perceptron neurons respectively from the first ROC map ( y/ROC1) and from the third ROC map, ( y/ROC3); the first map is the one that receives information from the (x,y) position solely whereas the third map combines the information coming from the four variables. In this plot we observe a slight variance and bias in the estimation process for the two output networks, more pronounced for the second system, which integrates the motor signals to the visual ones. The histograms plotted in Fig. 7 b) display the normalized spatial error for the two neural populations. The two histograms correspond to two power-law curves but the variation in their shape exemplifies how the two neural networks generalize also differently. The larger error variance in the second perceptron map indicates that the motor variables (θ 0 , θ 1 ) have a defavorable influence in the estimation of the visual variables.

Neural Extension to Visuomotor Transformation

To estimate the generalization properties of our system for visuomotor transformation, we first perform our experiments on three basic visual transformations as done by Iriki on a macaque in which the primate showed rapid adaptation to new visual position of its limbs with changes operated on the screen [START_REF] Iriki | Coding of modified body schema during tool use by macaque postcentral neurones[END_REF] or on the body [START_REF] Maravita | Tools for the body (schema)[END_REF][START_REF] Okanoya | Tool-use training in a species of rodent: The emergence of an optimal motor strategy and functional understanding[END_REF]. The first transformation consists on the small rotation of the camera in the plane of the robotic arm and the second transformation corresponds to a body extension of the robot hand with a tool (tool-use), the third transformation consists on a translational shift of the camera with respect to the arm plane of motion; see resp. Fig. 4 b) andc).

We present the results for the estimated transformations of three population outputs for each case presented above in Figure 8 a), b) andc) respectively for rotation, translation and tool-use. Each perceptron neuron of the output layer estimates the (∆x, ∆y) values learned from the current values received (x, y) and the expected ones (x ′ , y ′ ) from the GF maps.

In order to visualize the transformations that link the two sensorimotor spaces, we plot the arrows that link each point (x, y) with its associated ones (x + ∆x, y + ∆y) after transformation. In Figures 8, we plot in blue the Figure 6: Amplitude dynamics of one Gain-Field neuron with respect to the motor variable θ 0 on the X axis. It is noteworthy that the two variables are linked together with respect to the amplitude. The X centers slightly shift also with respect to the variable θ 0 as it is seen in biological neurons. trajectory of the hand before transformation and in red the trajectory of the hand after transformation. We make the note that the length of the arrows do not correspond to their actual size, they are reduced in order to be visualized better in the figure. The Figure 8 a) displays the transformation for the camera rotation. The blue line corresponds to the position of the hand before transformation and the red line corresponds to its position after transformation. As it can be seen, the arrows within the center, where the robot moves, show a correct approximation of the rotation as the generalization is made on the close neighborhood of the learned examples. However, past to a certain distance to the originally learned examples -, and outside the center area,-we can see some local discrepancies as the orientation of the arrows do not perfectly match the direction of the rotation anymore, although it is a global transformation.

The second transformation in Fig. 8 b) is the addition of a small stick at the end of the arm that effectively scales the arm up. During the second step of the experiment, the vision system would in this case recognize the position of the end of the stick instead of the position of the hand. Once again the transformation is successfully learned, and the generalization applies to a much larger part of the parameter space, which is probably due to the simplicity of the transformation. With respect to rotation, this transformation corresponds to a local transform, not applied on the whole image.

The last transformation performed is a simple vertical shift of the camera; see Fig. 8 c). Once again the transformation is successfully learned, and the general-a) b)

Figure 7: Neural estimation on the visual field. (a) The blue line corresponds to the location of the target to the Y axis whereas the red line and the green line correspond to the estimated location of the two output neural networks respectively calculated from the ROC neurons from the first map (visual inputs only) and from the ROC neurons from the third map (visual and motor inputs). (b) Histogram of the visual error for the two neural networks, the variance of the error distribution is larger for the network estimated from the visual and motor inputs (green) than from the sole visual inputs (red), although some big errors occur above 0.5: the motor inputs have an influence on the estimation of the spatial location of the visual targets.

ization is applying to almost the whole parameter space, which differs from the previous experiment.

What is interesting to notice on the three presented results in Fig. 8 is the capacity of the system to generalize at least to a certain extent; see Fig. 9 for a quantitative measure of the root mean square error of the estimated location of the visual target. For each transformation, all the learning input (the conditional signal of the perceptron neurons) were gain-field representations of a point on the red line, while the associated example (the unconditional link of the perceptron neurons) was on the blue line. This coverage of the parameter space is far from complete, however the generalization of the transformation to the rest of the parameter space is rather correct, at least in terms of direction. If we look closer, for any given point, the generalized transformation corresponds to a fraction of the closest learned example. if a point is in between two learned examples, then the transformation is an interpolation of these two example transformations.

Motor Imagery & Perspective-Taking in Social Context

The previous experiments were resolved with the estimation of the spatial error in the visual field. In comparison to body-centred sensorimotor transformation, the motor imaginary and perspective-taking tasks in the social domain require to observe the actions of one partner and to simulate his/her movements through our own motor system.

We investigate therefore how it is possible to estimate the motor configuration of someone else just from the visual input and to imitate one's arm posture with respect to what is seen. This situation corresponds to the so-called correspondence problem in social interaction for representing someone else's body posture [START_REF] Brass | Imitation: is cognitive neuroscience solving the correspondence problem?[END_REF]; see Figure 1 c) and Figure 4 d). Since there is no specific correspondence associations during role-taking, the new reference frame has to be computed from the estimated visual error from which the new motor action is calculated.

The experiment is as follows. The robot arm performs first a dynamic exploration of its arm visual location with a red toy in its hand and once it has learned its sensorimotor rules, a participant comes to interact with it by exploring his/her own peripersonal space with the same toy; see Figure 10 a) and b). We display in Figure 10 c) with red arrows the visual transform found by the output neurons, which corresponds to a combination of a translational shift in the X axis with a rotational effect centered on an external fixation point, roughly situated at the shoulder's location of the student. This situation combines the three experiments done in section 4.2 and Fig. 8.

Once the visual transform is defined, we analyze how the motor dynamics are estimated. We plot for this aim the robot personal space and the person's sensorimotor space in Fig. 11 a) respectively in blue and in red lines, where the arrows correspond to the visual transformation. We note that the two locations are different in space, therefore the neural architecture can differentiate its own robotic arm from one person's arm. Considering the motor estimation, the top chart of Fig. 11 b) displays the transition in the visual signal in the X and Y axis from the period when the robot waves its hand and stops (t < 500 iterations) to the one when the student starts to wave her hand (t > 500 iterations). The middle chart exposes the motor dynamics (θ 0 , θ 1 ) and the bottom chart displays the activity of the perceptron neurons involved in the motor transform for estimating the motor variables θ0 and θ1 . We can observe from the graph that the output variables θ0 and θ1 estimated by the neural system in the motor domain (bottom chart) are replicating the variations of the visual signal (top chart), which are mostly on the Y axis and for the motor variable θ 0 . The dynamics of the two variables are mostly in phase with the visual dynamics whereas the amplitude range is different. A plot of the renormalized estimated motor configuration is shown in Fig. 11 c) with respect to the visual stimulus. The neural system has mimicked the motor pattern of its partner and estimated its respective transform function.

Discussion

The Mirror Neurons System (MNS) shapes the sight information to the motor system when grasping an object. Its features extend to action recognition in the premotor cortex when it is the agent who is the observer so that the same cortical sites are activated during execution and observation. This system suggests that objectdirected actions, communication and body movement are intertwined.

What is so special in the mechanism called embodied simulation is that we reuse our mental state [START_REF] Jeannerod | Neural simulation of action: A unifying mechanism for motor cognition[END_REF][START_REF] Gallese | Neurophysiological mechanisms underlying the understanding and imitation of action[END_REF]. Embodied simulation should not be seen as a passive process by which mental states are "replayed", instead, it should be seen as an active process by which a transformation function is discovered online, mapping what it is seen to our own body dynamics. We propose that the processing involved in motor simulation in social cognition is a byproduct of the one involved in spatial transformation. We advance also that the gainmodulation effect in parietal neurons is part of the mechanisms responsible for it.

In previous works, we proposed a development scenario for the emergence of the MNS in two stages [START_REF] Pitti | A model of spatial development from parieto-hippocampal learning of bodyplace associations[END_REF][START_REF] Pitti | Neural models for social development in shared parieto-motor circuits, Book Chapter in 'Mirror Neurons: Theory and Possible Relationship to Language Disorders[END_REF]. We hypothesized that an automatic mimicry system is maturing at the fetal stage possibly in the superior colliculus (SC) forming the ground for the cortical circuits to organize the MNS [START_REF] Pitti | Modeling the minimal newborn's intersubjective mind: The visuotopicsomatotopic alignment hypothesis in the superior colliculus[END_REF]. At a first stage, we proposed that the topological alignment in SC across the modalities enables the automatic social responses of the newborns like facial preference and facial mimicry. This hypothesis has been also suggested by [START_REF] Nagy | Homo imitans or homo provocans? human imprinting model of neonatal imitation[END_REF][START_REF] Neil | Development of multisensory spatial integration and perception in humans[END_REF] who emphasized the central place that occupies the SC for fusioning the senses with respect to other brain regions not yet matured.

At a second stage, we propose then that a more complex spatial representation of the body in an allocentric metrics is emerging in the cortex as suggested in [START_REF] Bremner | Infants lost in (peripersonal) space?[END_REF][START_REF] Del Giudice | Programmed to learn? the ontogeny of mirror neurons[END_REF][START_REF] Pitti | A model of spatial development from parieto-hippocampal learning of bodyplace associations[END_REF]. This body mapping would be based on the GF mechanisms of sensorimotor transformation re-sponsible for body-centred reference frames [START_REF] Andersen | Multimodal integration for the representation of space in the posterior parietal cortex[END_REF][START_REF] Salinas | Gain modulation in the central nervous system: Where behavior, neurophysiology and computation meet[END_REF][START_REF] Blohm | Spatial transformations for eyehand coordination[END_REF] but also from the sensorimotor integration from the different modalities. As proposed by Shadmehr, such mechanism would permit to have far more plastic representations with fixation-point reference frames external to the body [START_REF] Shadmehr | The computational neurobiology of reaching and pointing : A foundation for motor learning[END_REF]. This system would permit to handle visuomotor changes such as during self-observation in a TV set or during tool-use, but also for solving the correspondence problem in inter-personal interactions.

Considering the construction of the inter-subjective mind, the PPC has been well-acknowledged for its contribution to agency [START_REF] Schwabe | Cognitive neuroscience of ownership and agency[END_REF][START_REF] Tsakiris | Having a body versus moving your body: How agency structures body-ownership[END_REF], intention [START_REF] Andersen | Intentional maps in posterior parietal cortex[END_REF][START_REF] Andersen | Intention, action planning, and decision making in parietal-frontal circuits[END_REF] and motor simulation [START_REF] Kosslyn | Neural foundations of imagery[END_REF][START_REF] Kosslyn | Mental imagery and the human brain[END_REF]. Blanke -, among other researchers,-proposes that the sense of agency is related to the level of anticipation across the sensorimotor signals. Hiraki and colleagues suggest further that the synchrony among the sensorimotor signals activates certain parietal regions while its disrupting activates others [START_REF] Miyazaki | Delayed intermodel contingency affects young childrens recognition of their current self[END_REF][START_REF] Shimada | The parietal role in the sense of self-ownership with temporal discrepancy between visual and proprioceptive feedbacks[END_REF][START_REF] Shimada | Infant's brain responses to live and televised action[END_REF]] (e.g., with time lags). More precisely, the right inferior parietal cortex is consistently activated in conditions involving imitation [START_REF] Ruby | Effect of subjective perspective taking during simulation of action: a pet investigation of agency[END_REF][START_REF] Decety | When the self represents the other: A new cognitive neuroscience view on psychological identification[END_REF][START_REF] Meltzoff | What imitation tells us about social cognition: a rapprochement between developmental psychology and cognitive neuroscience[END_REF]. We suggest that this region is associated to self-other simulation and encodes the transformations necessary for extending the body representation.

If we consider now this problem from a robotic perspective, Nagai pointed out that no computational models that aim to address imitation nor the architecture of MNS took self-other discrimination into account [START_REF] Nagai | Emergence of mirror neuron system: Immature vision leads to self-other correspondence[END_REF]. For instance, these works take advantage of self-observation and perceptual ambiguity for the emergence of imitation [START_REF] Andry | From visuo-motor development to low-level imitation[END_REF][START_REF] Kuniyoshi | From visuo-motor self learning to early imitation -a neural architecture for humanoid learning[END_REF][START_REF] Andry | Learning invariant sensorimotor behaviors: A developmental approach to imitation mechanisms[END_REF], but do not disambiguate the dynamics corresponding to oneself action and those of others [START_REF] De Rengervé | Emergent imitative behavior on a robotic arm based on visuo-motor associative memories[END_REF]. Among the few exceptions, the works done by [START_REF] Fuke | Acquisition of the head-centered peri-personal spatial representation found in vip neuron[END_REF][START_REF] Nagai | Emergence of mirror neuron system: Immature vision leads to self-other correspondence[END_REF] propose to discriminate the motor space with respect to observed actions (self ones or others'). We can cite notwithstanding the hebbian solution proposed first by Keysers [START_REF] Keysers | Demystifying social cognition: Hebbian perspective[END_REF] based on contingency detection at the neural level (i.e., the biologicallyinspired mechanism of spike timing-dependent plasticity) for categorizing the sensorimotor signals in situation of interaction with others [START_REF] Pitti | Contingency perception and agency measure in visuo-motor spiking neural networks[END_REF]. In comparison with these works, the present one emphasizes the learning of a transformation function that maps the robot visuomotor space (the egocentric reference frame) to the new one (the allocentric reference frame).We suggest that this mechanism of sensorimotor transformation based on gain-field neurons is general enough to serve for the mapping of spatial transformation during physical as well as social interactions.

Deneve and colleagues described in details the advantages and disadvantages of representing multimodalities using radial basis functions (tensor prod-ucts). The main advantages of basis functions is notably to reduce the computation of nonlinear functions into linear ones [START_REF] Deneve | Basis functions for object-centered representations[END_REF] especially for retinal images, which are highly nonlinear, requiring translation, scaling, and rotation of the image [START_REF] Olshausen | A multiscale dynamic routing circuit for forming size-and position-invariant object representations[END_REF]. Deneve acknowledges however that one of the main drawbacks is the curse of dimensionality as the number of basis functions required to approximate functions with high accuracy increases exponentially with the number of signals being combined: N 2 , N 3 , ... [START_REF] Deneve | Basis functions for object-centered representations[END_REF]. Deneve further argues in [START_REF] Deneve | Basis functions for object-centered representations[END_REF] that this weakness of the basis function approach is also one of its main strengths. Because it uses so many units, a basis function representation tends to be highly redundant. This redundancy can be exploited to filter out the noise optimally.

One question that remains is how our framework can permit to describe an accurate geometrical information about the world. For instance, infants do seem to have an intuition of euclidean geometry that they retrieve from their senses [START_REF] Spelke | Beyond core knowledge: Natural geometry[END_REF]. Adults perform easily mental rotations by simulating how the object would move by applying affine-like transforms [START_REF] Kosslyn | Mental imagery and the human brain[END_REF]. One possible hypothesis that we can formulate could come from the findings of "cosine" gain-modulated neurons in the cortical motor areas and responding to certain visual direction and arm movements for performing 3D reaches [START_REF] Kakei | Sensorimotor transformations in cortical motor areas[END_REF][START_REF] Blohm | Spatial transformations for eyehand coordination[END_REF][START_REF] Blohm | Computations for geometrically accurate visually guided reaching in 3-d space[END_REF]. We can hypothesize for instance that such neurons may serve for extracting geometrical rules based on our actions [START_REF] Izard | geometry as a universal mental construction[END_REF]. We let nonetheless this unsolved question for future works.

Figure 1 :

 1 Figure 1: The problem of the frame of reference and of sensorimotor transformation. (a) Grasping is a complicated task as it requires to learn the visuomotor space in an ego-egocentric reference frame . (b) When we observed our own action in a TV set, we change reference frame, which requires to transform the spatial information of the visual coordinates with respect into the hand coordinates. (c) The same situation occurs when we observe someone else's actions and try to imitate them; this transformation is called the correspondence problem. The point A in visual space corresponds to the point A ′ in the new reference frame after transform and to the point A ′′ after a different transform.

Figure 2 :

 2 Figure 2: Neural architecture for gain-field neurons. Gain-Field neurons can bind the activity of one, two, three or four inputs by reusing gain-field neurons resulting from the multiplication of two modalities as one of the two modalities of another gain-field map. These neurons can serve as basis functions from which desired representations can be mapped. (ROC) stands for Rank-Order Coding neurons. The sensory signals I can be estimated then from the linear combination of ROC neurons as it is with perceptron neurons. The desired output, Î/ROC, means then the estimation of signal I by ROC.

Figure 3 :

 3 Figure3: Transformation mechanism with an error signal. We can compute an error signal ∆A between a variable A and its estimation  done by the neural network in Fig.2. This error can be learned by different maps with respect to the context it corresponds. The error map can serve then to learn the transformation to pass from the point A to the point Â.

Figure 4 :

 4 Figure 4: Experimental setups and sensorimotor transformations. (a) The first situation corresponds to the learning of the visuomotor correspondence between the motor variables (two degrees of freedom) and the (X,Y) position in the camera visual field. Although this task is nonlinear, a slight change of the reference frame does not require to relearn all the visuomotor links but instead, the transform function responsible for the global error. These changes can be due to a visual transformation as for a camera rotation (b) or to the body changes as during tool-use (c) or to the embodied simulation of one partner as in (d); the so-called correspondence problem.

Figure 5 :

 5 Figure 5: Activity of Gain Fields neurons. Neurons 1, 4 and 15 resp. a), b) and c) encode different sensorimotor locations in (θ 0 , θ 1 ) and (X, Y) as it would be for radial basis functions. Each variable is normalized within the interval [0, 1].

Figure 8 :

 8 Figure 8: Rotation, Tool-use and Shift Transformations. The arrows correspond to the estimated local transformations, the blue line corresponds to the original trajectory of the robot hand before any transformation and the red line corresponds to its trajectory after transformation. (a) Learned transformation after the rotation of the camera (b). Learned transformation after adding a stick to the robot hand. (c) Learned transformation after the camera shifting.

Figure 9 :

 9 Figure 9: Root mean square error computed from the position of the visual target and the estimated one for rotation (green), tool-use (red) and shift transformations (blue); resp. (a-c) in Fig. 8. The linear combination of gain-field neurons permit to reduce the error to a small range even for three different nonlinear transformations.

Figure 10 :

 10 Figure 10: Visual transformation and motor imagery during interaction with a partner. (a-b) Red toy handled by the robot and then handled by the student. This experiment reproduces the so-called correspondence problem posed in social interactions where one has to simulate the motor activity of one person into his/her own visuomotor system in order to imitate her or to recognize her intentions. (c) Within our framework, this simulation can be modeled as a visuomotor transformation by which the robot can learn the visual error.

Figure 11 :

 11 Figure 11: Visual transformation and motor imagery. (a) The arrows display the estimated visual transformation between the robot peripersonal space (blue) and the partner space (red). This function can serve for self-other discrimination purpose as well as the estimation of the motor activity of the partner. In b) and c), the plots of the (X,Y) visual signals with the estimated motor configuration ( θ0 , θ1 ) (normalized in c).

Acknowledgments

We would like to thank Jacqueline Fagard and Kevin O'Regan for our discussions on tool-use and observational learning in infants. We would like to thank also Antoine de Rengervé, Pierre Andry and Daniel Lewkowicz for discussions and comments as well as Olga Kamozina for her participation in the experiment in Figure 10. We acknowledge the grant from UCP-CNRS chair of Excellence and the ANR project IN-TERACT (ANR09-COORD014), program CONTINT. (http://interact.ensea.fr).


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Significant contribution of helicopter-borne resistivity surveys to the hydrogeological characterisation of volcanic islands: review of the recent researches

Benoit Vittecoq

To cite this version:

Benoit Vittecoq. Significant contribution of helicopter-borne resistivity surveys to the hydrogeological characterisation of volcanic islands: review of the recent researches. II Workshop "Estudio, aprovechamiento y gestión del agua en terrenos e islas volcánicas", Jan 2015, Las Palmas de Gran Canaria, Spain. hal-01074686

Significant contribution of helicopter-borne resistivity surveys to the hydrogeological characterisation of volcanic islands: review of the recent researches

Benoit Vittecoq 1 1 BRGM -Basse-Normandie, 4 avenue de Cambridge, 14200 Hérouville-Saint-Clair, France, b.vittecoq@brgm.fr

The demographic explosion of many volcanic islands, with associated increasing drinking water needs, requires detailed investigation of their subsurface in order to provide sustainable solutions and for an adapted management of groundwater resources. In this context, hydrogeological studies are challenging due to difficulties in accessing the study sites (steep slopes, dense vegetation and a limited access), and to the geological complexity of volcanic deposits (lateral and vertical lithological variability with superimposed intense weathering).

Furthermore, there are often few boreholes enabling detailed hydrogeological characterizations, which limit the detailed validation of the different approaches and methods.

The SkyTEM dual-moment Time Domain Electromagnetic system was used for the first time in Based on a literature review, the main contribution of this SkyTEM method to volcanic island hydrogeology will be presented. In this context, this method: (i) could help to highlight the main geological structures, playing key roles on the hydrogeological functioning of such islands even on a complex geologic context, and (ii) allows identification of seawater intrusions and depth of saltwater interface. For instance, the presence of a perched aquifer, potentially strategic for the island water supply, is suspected on Santa Cruz Island and confirmed on San Cristobal (Galapagos Archipelago). On these islands, seawater intrusions are typical of Hawaiian model basal aquifer. On Mayotte Island, influences of successive phases of volcanic construction and erosion, and consequently structural controls on groundwater flows have been highlighted. At a smaller scale a buried valley also clearly plays a key role on the hydrogeological functioning in the south of the island. Seawater intrusions here are clearly different from Galapagos Island and don't penetrate far inland. In Martinique islands, preliminary results are also promising.

a

  volcanic context in 2006 over the Galapagos Islands by UPMC-Sorbonne University and Aarhus University, then by BRGM in 2009 over Mayotte Island (with Aarhus University as partner), in 2013 over Martinique and Guadeloupe Islands (Lesser Antilles) and in 2014 over Reunion Island. The SkyTEM helicopter-borne TDEM system allows detailed mapping of the internal geological structure of the upper 200 m of these volcanic islands, through a quasi-3D resistivity model.


Yogesh Karpate

Olivier Commowick

Christian Barillot

Gilles Edan

Longitudinal Intensity Normalization in Multiple Sclerosis Patients

In recent years, there have been many Multiple Sclerosis (MS) studies using longitudinal MR images to study and characterize the MS lesion patterns. The intensity of similar anatomical tissues in MR images is often different because of the variability of the acquisition process and different scanners. This paper proposes a novel methodology for a longitudinal lesion analysis based on intensity standardization to minimize the inter-scan intensity difference. The intensity normalization maps parameters obtained using a robust Gaussian Mixture Model (GMM) estimation not affected by the presence of MS lesions. Experimental results demonstrate that our technique accurately performs the task of intensity standardization. We show consequently how the same technique can improve the results of longitudinal MS lesion detection.

Introduction

Multiple Sclerosis (MS) is an acquired inflammatory, demyelinating disease which causes disabilities in young adults and it is very common in the northern hemisphere. Quantitative analysis of longitudinal Magnetic Resonance Images (MRI) of subject taken at different time points provides a time varying analysis of the brain tissues which may lead to the discovery of new biomarkers of disease evolution. In MS, White Matter (WM) lesions are also present in addition to healthy brain tissues. Lesions can remain stationary, change volume, or disappear in later time points depending upon the state of MS. Due to protocol variations in the scanners, following the evolution of tissue intensities in a patient, e.g. changing appearance of lesions, makes quantitative evaluation of lesions difficult. In order to alleviate this problem, intensity normalization is necessary.

Histogram matching is a widely used technique in intensity standardization. In their seminal work, Nyul et al. [START_REF] Nyul | New variants of a method of MRI scale standardization[END_REF] proposed landmark based methods. It consists of matching the input image histogram landmarks onto standard histogram landmarks, obtained in a training phase, performing a linear interpolation of intensities between the positions. The technique in [START_REF] Nyul | New variants of a method of MRI scale standardization[END_REF] uses percentile landmarks, which is simple yet powerful. Jager et al. [START_REF] Jäger | Whole body MRI intensity standardization[END_REF] extended this principle to two or more jointly used MRI sequences (e.g., T1-w and T2-w), matching multidimensional joint histograms with nonlinear registration. With this method, no prior registration of the reference and normalized MR images is required. An algorithm proposed by Wang et al. [START_REF] Wang | Correction for variations in MRI scanner sensitivity in brain studies with histogram matching[END_REF] expands or shrinks a windowed part of the input image histogram with a multiplicative factor, found by minimizing the bin-count difference between the source and moving images histograms. The window is used to include only voxels of interest and exclude the background. This makes the technique linear in the intensity range of interest. Other techniques use parametric models, such as the technique proposed by Hellier [START_REF] Hellier | Consistent intensity correction of MR images[END_REF]. It models histogram of a reference image and of the standardized image with two GMMs and aligns their means through a polynomial correction function. Weisenfeld et.al. [START_REF] Weisenfeld | Normalization of joint image-intensity statistics in MRI using the Kullback-Leibler divergence[END_REF] have proposed to estimate a multiplicative correction field that alters the intensity statistics of an image or set of images to best match those of a model. In that paper, the Kullback-Leibler divergence between the source and moving images is minimized iteratively to estimate the parameters of a model, thus histograms are equalized. All these methods may be affected by the presence of white matter lesions.

We propose a longitudinal intensity normalization algorithm for multichannel MRI in the presence of MS lesions, which provides consistent and reliable longitudinal detections. The tissue intensities from multichannel MRI are modeled with parametric transform using a robust GMM estimation based on γ divergence, thereby keeping the lesions unaffected. The proposed technique is built on ideas similar to Hellier [START_REF] Hellier | Consistent intensity correction of MR images[END_REF] but taking into account the presence of pathological tissues in the intensity transformation function. It provides a technique that (1) uses tissue-specific intensity information by modeling them using a robust GMM; (2) provides a consistent intensity normalization between longitudinal images. Subsequently, we demonstrate its crucial role for further lesion analysis.

This paper is organized as follows. The modeling and parameter estimation of multi-sequence MRI with γ divergence followed by intensity normalization are reviewed in Section 2. The details of experiments and their results on longitudinal MS patients are discussed in section 3.

Methodology

Given two MR images of a single MS patient at time instant t 1 and t 2 , we seek to estimate a correction factor such that corresponding anatomical tissues adopt the same intensity profile. We model the image intensities of a healthy brain with a 3-class GMM, where each Gaussian represents one of the brain tissues White Matter (WM), Gray Matter(GM) and Cerebrospinal fluid (CSF). We consider the m MR sequences as a multidimensional image with n voxels. Each voxel i is represented as

x i = [x i1 ...x im ]
. The probability of intensity x i is calculated as follows:

f (x i |θ) = 3 k=1 π k N (µ k , Σ k ) (1)
where the mean µ k and covariance Σ k define the parameters N (µ k , Σ k ) of each Gaussian of the model along with their mixing proportions π k merged into parameter θ. If the proportions were known, θ could be estimated through the Maximum Likelihood Estimator (MLE):

θ = argmax θ L(θ) = argmax θ n i=1 f (x i |θ) (2)
Where x i are considered as i.i.d. samples. However, as π k are unknown, an Expectation Maximization (EM) algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF] is used to estimate the parameters.

γ-loss Function for the Normal Distribution

The parameter estimation with classic MLE for GMM can deviate from its true estimation in presence of outliers. In MS patients, such outliers may be of crucial importance as they may denote appearing or disappearing lesions. Notsu et al. [START_REF] Notsu | Spontaneous clustering via minimum gammadivergence[END_REF] proposed a modification of the MLE in order to make it more robust to outliers. The basic idea is to maximize (2) in the form of γ divergence. We consider the γ-loss function for the Normal distribution with mean vector µ and covariance matrix Σ.

L γ (µ, Σ) = Σ -γ 2(1+γ) n i=1 exp - γ 2 (x i -µ) T Σ -1 (x i -µ) (3)
Where |.| indicates the determinant. The bounded influence function of an estimator is an indicator of robustness to outliers. The influence function for GMM with γ loss function is bounded whereas the one for regular GMM is unbounded. As γ grows larger, bounds become tighter. For a sufficiently large γ, (γ ≥ 0.1), the estimating equation has little impact from outliers contaminated in the data set. Equation (3) can be casted to yield an EM style algorithm as follows.

Expectation

Step. In the case of a GMM, the latent variables are the point-tocluster assignments k i , i = 1, ..., n, one for each of n data points. The auxiliary distribution q(k i |x i ) = q ik is a matrix with n×K entries. Each row of q i can be thought of as a vector of soft assignments of the data points x i to each of the Gaussian modes.

q ik = π k exp -γ 2 (x i -µ k ) T Σ -1 k (x i -µ k ) K l=1 π l exp -γ 2 (x i -µ l ) T Σ -1 l (x i -µ l ) (4)

Maximization

Step. The maximization step estimates the parameters of the Gaussian mixture components and the mixing proportions π k , given the auxiliary distribution on the point-to-cluster assignments computed in the expectation step. The mean µ k of a Gaussian mode is obtained as the mean of the data points assigned to it (accounting for the strength of the soft assignments). The other quantities are obtained in a similar manner, yielding to:

µ k = n i=1 q ik x i n i=1 q ik (5)
Σ k = (1 + γ) n i=1 q ik (x i -µ k )(x i -µ k ) T n i=1 q ik (6)
π k = n i=1 q ik n i=1 K l=1 q il (7)

Selection of Parameter γ

The estimation of power index γ plays a critical role in our approach, since γ affects the estimated parameters in presence of outliers. Notsu et al. [START_REF] Notsu | Spontaneous clustering via minimum gammadivergence[END_REF] suggested the selection of γ as a model selection problem based on Akaike information criterion (AIC). Let K be the number of clusters, p be the total numbers of parameters of a model and (µ k , Σ k ), k = 1, .., K be the means and the covariance matrices of the clusters respectively. From (1), the AIC is defined as follows:

AIC γ = -2 n i=1 log f γ (x i |θ) + 2 K p(p + 3) 2 + K -1 (8)
The value of γ which minimizes AIC is used as the optimal γ. For various values of γ, equation ( 8) is evaluated in cross validation manner and the γ which results in minimum value is chosen for the experiment.

Intensity Correction

We obtain the means and covariances of tissues for the source and target images using the procedure mentioned above. We chose a linear correction function such that g(x) = Σ i β i x i . The coefficients β i are estimated to minimize the following cost function: Σ l=n l=1 (g(µ source,k )µ target,k ) 2 . This function can be solved by linear regression. Using the results of the linear regression, the intensity profiles of the two images are normalized by mapping the intensity of the source image to the target image. The resulting correction function is smooth and interpolates the intensity correction.

Experiments and Results

Dataset and Preprocessing

Whole-brain MR images were acquired on 18 MS patients. T1-w MPRAGE, T2w and FLAIR modalities were chosen for the experiment. Expert annotations of lesions were carried out by an expert radiologist on all MS patients. The volume size for T1-w MPRAGE and FLAIR is 256×256×160 and voxel size is 1×1×1 mm 3 . For T2-w, the volume size is 256×256×44 and voxel size is 1×1×3 mm 3 . All imaging experiments for this study were performed on a 3T Siemens Verio (VB17) scanner with a 32-channel head coil. MR images from each patient are de-noised [START_REF] Coupé | An optimized blockwise nonlocal means denoising filter for 3-D magnetic resonance images[END_REF], bias field corrected [START_REF] Tustison | N4ITK: improved N3 bias correction[END_REF] and registered with respect to T1-MPRAGE volume [START_REF] Ourselin | Block matching: A general framework to improve robustness of rigid registration of medical images[END_REF][START_REF] Commowick | Block-matching strategies for rigid registration of multimodal medical images[END_REF]. All the images are processed to extract intra-cranial region using BET (Brain Extraction Tool) [START_REF] Smith | Fast robust automated brain extraction[END_REF].

We show the effect of longitudinal intensity normalization followed by detection on both normal tissues and lesions for 18 MS subjects, having 4 time-points each, approximately separated by a period of three to six months. The first time point is considered as the reference point to which the subsequent time points (moving ones) are aligned using intensity normalization. First, the parameters of reference and moving images are estimated using γ likelihood estimator as described in section 2.1. Secondly, voxels of moving image are aligned with respect to reference image using the procedure in section 2.3. Each patient and each time point t = 2, ..., t n , are rigidly registered to the T1-w MPRAGE of first time instance. The obtained difference image is processed further to obtain a soft detection by using heuristic thresholding iteratively (1) by Otsu's threshold [START_REF] Otsu | A threshold selection method from gray-level histograms[END_REF];

(2) erosion of image by one voxel. The detections from this difference image are compared with difference image of ground truth at corresponding time points.

Intensity Correction Evaluation

To evaluate the quality of intensity normalization, we compare the histograms of reference, moving and intensity normalized moving image using chi-squared distance given by χ 2

x,y = 1 2 (xi-yi) 2

xi+yi . Lower values of this distance indicate better alignment of intensities. Table 1 reports the chi-squared distance for various imaging sequences. Different methods are compared against the proposed one. We report the mean χ 2 distance for our method as 0.18(±0.045), 0.28(±0.037) 0.32(±0.038) for T1-w MPRAGE, T2-w and FLAIR respectively, outperforming other state of the art methods.

Longitudinal Lesion Detection

To show the quantitative improvement for identification of lesions, we report in Table 2 the precision (Positive Predicted Value) and recall (Sensitivity) of lesion detection averaged across the 18 patients for various overlap thresholds. The lesion is said to be detected if Rc∩R GT R GT ≥ ϕ where R c , R GT and ϕ are respectively the candidate region in the image, the ground truth and a threshold. Table 2 reports values of precision and recall for various thresholds. As from the figures, our approach outperforms other methods. We have a very high recall of 0.90 at ϕ = 0.2 and 0.82 even at ϕ = 0.4.

Figure 2 depicts the detected lesions for a representative image. The green label shows new lesions at t 3 , orange shows stationary lesions which are also a We proposed a new intensity normalization technique based on a robust GMM estimation with γ divergence. The efficacy of our method was evaluated through histogram matching distance method and longitudinal lesion detection. Compared to Nyul and Hellier method, our methodology is more suitable for longitudinal MS lesion analysis because of its ability to preserve the intensity variations caused by pathological changes. Our system relies heavily on a robust parametric modeling of tissue intensities based on γ divergence. The resulting system is both efficient and accurate, outperforming the state of the art methods. This performance suggests that it can provide valuable assistance in detecting the longitudinal MS lesions in clinical routine with high reliability. Our models are already capable of detecting highly variable lesion patterns, but we would like to move towards richer models. The framework described here allows for exploration of additional MR sequences with or without contrast agents. For example, one can consider infusing T1-w Gadolinium and DTI.

Figure 1 FLAIRFig. 1 .

 11 Figure1shows the intensity correction results for T1-w MPRAGE, T2-w and FLAIR images. Three time points and their corresponding MR modalities of a subject are shown before and after normalization. Each row represents the imaging modality and each column depicts the first time point, second time point, the absolute difference image without and with intensity normalization respectively. This figure demonstrates visually the ability of our approach to

Fig. 2 .

 2 Fig. 2. Lesion detection examples. For top and bottom, from left to right: Slice of FLAIR for t0, t3, |t0t 3(N ormalized) |, ground truth and lesions detected by our algorithm.

Table 1 .

 1 Chi-squared distance analysis for histogram matching

		Before Normalization	After Normalization
	Modality	Proposed	Hellier	Nyul
	T1-w	0.56 (±0.03) 0.18 (±0.045) 0.35 (±0.029) 0.3 (±0.019)
	T2-w	0.62 (±0.029) 0.28 (±0.037) 0.414 (±0.03) 0.315 (±0.042)
	FLAIR	0.56 (±0.027) 0.32 (±0.038) 0.45 (±0.051) 0.39 (±0.045)

Table 2 .

 2 Performance analysis for lesion detection. 73±0.04 0.90±0.05 0.68±0.03 0.85±0.04 0.63±0.03 0.82±0.01 Hellier 0.65±0.02 0.74±0.03 0.64±0.06 0.68±0.04 0.62±0.03 0.59±0.05 part of t 1 ; blue shows false positive detections. We are able to accurately detect appearing and disappearing lesions thanks to the proposed method.

	ϕ = 0.2	ϕ = 0.3	ϕ = 0.4
	Method Precision	Recall	Precision	Recall	Precision	Recall
	Nyul 0.63±0.01 0.60±0.02 0.61±0.04 0.67±0.02 0.58±0.03 0.64±0.03
	Proposed 0.