Anatole Lécuyer
email: anatole.lecuyer@inria.fr

Playing with senses in VR: Alternate perceptions combining vision and touch

Virtual Reality is an immersive experience based on computer-generated stimulations perceived with multiple sensory channels. It is possible to manipulate these sensory stimulations independently and create conflicting situations in which, for instance, vision and touch are spatially and/or temporally inconsistent. In this article we show how to exploit these ambiguous sensorial situations in order to generate new kinds of percept but also plausible 3D interactions in virtual environments. We particularly insist on three results obtained by playing with visual and haptic senses in virtual reality: (i) pseudo-haptic effects, (ii) self-motion sensations, and (iii) bodyownership illusions.

Conflicts and coherence of senses in virtual reality

.

.

If "it works", and when this psychological conflict is solved in favor of the virtual situation, the resulting sensory experience and subsequent feeling of immersion can be very strong. This can be revealed by physiological or behavioral reactions of users. A good example is the vertigo sensation felt when positioned at the top of a virtual pit and looking down. People would often refuse to jump into the virtual void, even though they know they are safely standing on a flat floor in reality. Interestingly, when a real physical edge is added on the floor, for instance using a wooden plate, the physiological reaction is increasing significantly compare to the situation with no physical edge [START_REF] Meehan | Physiological measures of presence in stressful virtual environments[END_REF]. Thus, the vertigo sensation is stronger in presence of an additional and consistent tactile cue. The immersion feeling is higher in presence of a sensory redundancy in VR.

An immersive experience relies on realistic sensory stimulations, essentially visual, but sometimes audio or tactile (haptic). In practice, it is often impossible to perfectly reproduce a multi-sensory experience in VR with all the sensory stimulations involved. In a paradoxical manner, the VR experience is usually a situation of "sensory deprivation": a perceptual isolation with multiple senses being removed or cut off. Incidentally, in terms of interaction capabilities, the user is mostly in situation of handicap, being unable to achieve basic operations, provided with limited possibilities of perception and action.

Figure 1: What happens when vision and touch are not collocated ? In our experiment (Congedo, 2006), participants could watch and grasp a rotating virtual handle under two different conditions: VHc (Visual and Haptic information are spatially "Collocated" -as in HMD-based VR settings) and VHd (Vision and Haptics are "De-located", as in screen-based VR settings).

Sensory redundancy is an effective means for gaining a higher immersion. But the sensory stimulations are expected to remain consistent spatially and temporally. A spatial or temporal discrepancy between sensory sources is expected to decrease the plausibility of the virtual experience. In [START_REF] Congedo | The Influence of Spatial De-location on Perceptual Integration of Vision and Touch[END_REF], we could show that when visual and haptic information are not spatially colocated, the multi-sensory integration is negatively impacted, and the weight given to the haptic modality decreases strongly in favor of vision (see Figure 1). The spatial offset between the visual and haptic displays ends up with a masking of the haptic sensation. Interestingly, designers of VR systems can relate these findings to the relevance of the two sensory channels. For instance, if the contribution of touch is important for the task, great efforts should be undertaken to collocate as much as possible the visual and haptic percepts. On the other hand, in presence of a low-quality haptic feedback, the interest may be to contain the limitations of the haptic device by keeping distant the two displays.

Virtual Reality can be used to create experimental situations that can sometimes hardly be reproduced in a real setup such as artificial sensory conflicts. A sensory conflict implies that the information coming from one modality differs from the information coming from another one. Sensory conflicts can be a source of problems in VR. According to the sensory conflict theory, "cybersickness" is evoked as the result of an inconsistency between the visual and the vestibular or proprioceptive senses. But sensory conflicts can also help psychologists to better understand how humans perceive multisensory information, enabling for instance to compute the relative weights attributed to the various sensory channels [START_REF] Ernst | Humans Integrate Visual and Haptic Information in a Statistically Optimal Fashion[END_REF]. For instance, when spatial interaction tasks are concerned, visuohaptic perception was found to be characterized by a strong visual dominance [START_REF] Rock | Vision and touch: An experimentally created conflict between the two senses[END_REF]. In this context the concept of sensory coherence is also central when perceiving and representing the environment with multiple senses. In this perspective, sensory signals are not processed to directly estimate the relevant variables, but rather to estimate the difference between mental estimations and the relevant variables (Cornilleau-Pérès, 1993).

In this article, we will show how sensory conflicts can become a source of inspiration for VR designers and, more generally, how combinations of visual and haptic feedbacks can be exploited to generate alternate percepts and novel 3D interaction schemes. We will insist on three series of results obtained recently on this research topic: pseudo-haptic effects, self-motion sensations, and bodyownership illusions. All of them were obtained by playing with redundant or conflicting visual and haptic cues in virtual environments, bordering systematically on sensory illusions.

Haptic illusions and "pseudo-haptic feedback"

As a first example, "pseudo-haptic feedback" intends to produce a wide range of haptic sensations, such as friction, relief or stiffness, without using a haptic interface but playing with visual feedback.

We state four key assertions concerning pseudo-haptic feedback [START_REF] Lécuyer | Simulating Haptic Feedback using Vision: a Survey of Research and Applications of Pseudo-Haptic Feedback[END_REF]. First, pseudo-haptic feedback implies one or more sensory conflicts between visual and haptic cues. Second, pseudo-haptic feedback relies on the sensory dominance of vision over touch when perceiving spatial properties (distance, position, size, displacement, etc). Third, pseudo-haptic feedback corresponds to a new and coherent representation of the environment resulting from a combination of haptic and visual information. Fourth, pseudo-haptic feedback can create a haptic illusion, i.e., the perception of a haptic property different from the one present in the real environment.

Since our first article published on this topic in 2000 [START_REF] Lécuyer | Simulating Haptic Feedback using Vision: a Survey of Research and Applications of Pseudo-Haptic Feedback[END_REF], we have designed and studied numerous examples of pseudo-haptic effects. We provide hereafter a representative set of successful studies and setups.

The most famous technique based on pseudo-haptic feedback has been originally designed in a 2D context: the "pseudo-haptic textures" [START_REF] Lécuyer | Simulating Haptic Feedback using Vision: a Survey of Research and Applications of Pseudo-Haptic Feedback[END_REF]. This pseudo-haptic effect was meant to display the relief of 2D images using a simple computer mouse. When the user manipulates the mouse, the technique consists of altering the cursor's visual motion as it moves over the image, i.e. manipulating the Control/Display ratio. To create the impression that the cursor is climbing up a slope, it is slowed down. Inversely, to simulate the cursor sliding down a slope, it is speeded up. For example, to simulate the cursor moving over a bump (as illustrated in Figure 2A), the cursor is slowed down until it reaches the top of the bump. Once it is past the top, the cursor accelerates, until it reaches the foot of the bump. After that, it returns to its normal speed. This technique has been evaluated within an extensive series of experiments that demonstrated that participants were able to well recognize and precisely draw texture patterns simulated using pseudo-haptic textures. Later on, we proposed an extension of this technique called "Elastic Images" aiming at simulating the local elasticity of images [START_REF] Argelaguet | Elastic Images: Perceiving Local Elasticity of Images through a Novel Pseudo-Haptic Deformation Effect[END_REF]. The elasticity sensation is generated by a procedural image deformation algorithm that modifies the image according to its simulated physical properties and to the virtual pressure exerted by the user (see Figure 2C). The simulated pressure depends on the time the user keeps the mouse button pressed. A psychophysical experiment showed that users were able to recognize up to eight different elasticity configurations.

More recently pseudo-haptic feedback has been studied in the context of 3D interaction with virtual environments for improving the selection or manipulation of virtual objects. One example illustrated in Figure 2B improves the selection of items enclosed in a 3D carousel designed for virtual showcasing [START_REF] Gaucher | A novel 3D Carousel based on Pseudo-Haptic Feedback and Gestural Interaction for Virtual Showcasing[END_REF]. The carousel is a 3D ring menu rendered on a 3D display. The 3D interaction with the carousel is achieved by tracking user's gestures: in order to rotate the carousel, the user has to perform swipe gestures. Our pseudo-haptic effect is introduced to highlight relevant items, such as promotional products, by locally modifying the friction of the carousel. This effect is expected to "attract" the user towards these specific items when interacting with the carousel. When facing an item with a strong friction coefficient, the user must increase the amplitude of movement of the hand to move to the following or previous item. A second example is a unique interaction paradigm called the "Virtual Mitten" [START_REF] Achibet | The Virtual Mitten: A novel interaction paradigm for visuo-haptic manipulation of objects using grip force[END_REF]. It is meant to simulate 3D manipulation of objects using grip forces. It is based on the passive haptic feedback provided by a handheld elastic input device (an engineered hand-exerciser), and the visual metaphor of a mitten that enables to grasp and manipulate 3D objects (see Figure 2E). The grip force exerted on the device enables to grasp objects and achieve various manipulation tasks, such as opening a drawer or pulling a lever. The grasping performed by the virtual mitten is directly correlated with the grip force applied on the elastic device. A pseudohaptic effect is then introduced in order to generate the haptic perception of different levels of grasping effort. A psychophysical experiment could show that that participants were well able to perceive different levels of effort during several manipulation tasks thanks to this pseudo-haptic approach.

Pseudo-haptic feedback has then been introduced for simulating perception and interaction with a highly complex 3D object: the user's self-avatar. We indeed introduced the notion of "Pseudo-Haptic Avatars" (Gomez, 2014) and showed how the visual animation of a self-avatar could be artificially modified in real-time in order to generate different haptic perceptions. In our experimental setup participants could watch their self-avatar in a virtual environment in mirror mode (see Figure 2D).

They could map their gestures on the self-animated avatar in real-time using a Kinect. The experimental task consisted in a weight lifting with virtual dumbbells that participants could manipulate by means of a tangible stick. We tested three kinds of modification of the visual animation of the self-avatar: an amplification (or reduction) of the user motion, a change in the dynamic profile of the motion (temporal animation), or a change in the posture of the avatar (angle of inclination). Thus, to simulate the lifting of a "heavy" dumbbell, the avatar animation was distorted in real-time using: a decrease in user's visual motion, a slower dynamics, and a larger angle of inclination of the avatar. Experimental results showed that users were well able to discriminate weights using this pseudo-haptic feedback by relying only on the motion and posture. This technique could for instance be used in applications such as sport training, exercise games, or industrial training.

(A) (B) (C) (D) (E)

Figure 2: Pseudo-haptic effects. (A) Pseudo-haptic texture: simulating passing over a bump shape on screen by playing with the speed of the mouse cursor. (B) 3D carousel: a carousel-like ring menu is rotated using swipe gestures, and is augmented with friction pseudo-haptic effects that can repulse or attract the user towards pre-determined items. (C) Elastic image: simulating the local elasticity of a 2D image with a procedural deformation algorithm using the "pressure" corresponding to the time elapsed when clicking. (D) Pseudo-haptic avatar: the user can lift different virtual dumbbells using Kinect-based gestures recognition, and then perceive different virtual weights using pseudo-haptic effects applied on the self-avatar visual animation. (E) Virtual Mitten: an engineered hand-exerciser is used to grasp and manipulate virtual objects via a mitten metaphor augmented with pseudo-haptic effects enabling to feel different levels of grasping efforts.

All these examples illustrate how a spatiotemporal sensory conflict introduced and well controlled in the perception-action loop can produce a wide range of haptic sensations and improve 3D interaction. The visual motion is here distorted in a synchronized way with the user's physical motion or sensorymotor action. The resulting pseudo-haptic percept corresponds to the subjective reinterpretation of these stimuli, and to an optimal visuo-haptic perception of a world, which must remain coherent, depending on the interaction context. Interestingly a similar decrease in speed will be interpreted in one context as a texture effect and in another context as a change in mass. This suggests that many more pseudo-haptic effects could be invented, considering the great number of potential contexts and scenarios of VR applications.

Self-motion illusions with the "haptic motion"

A second series of results illustrate how introducing a haptic cue which is, this time, well consistent and synchronized with visual feedback can be used to improve self-motion sensations in VR.

The sensation of self-motion is intrinsically a multimodal perception involving many sensory channels: visual, tactile, proprioceptive, vestibular, and even auditory. A good way to generate selfmotion sensation in VR consists in watching a visual motion on a large screen or wearing an HMD with a large field of view. The peripheral visual stimulation can then be sufficient to induce a "vection" illusion. Vection is a well-known illusion of self-motion that most people have already experienced, for instance, when seated in a standing train while another train starts to move aside. In this situation, people would feel that they move in the opposite direction of the produced optic flow although they are steady.

We first showed in two experiments [START_REF] Ouarti | Haptic Motion: Improving Sensation of Self-Motion in Virtual Worlds with Force Feedback[END_REF] that the vection illusion induced by the visual feedback of a virtual environment could be strongly improved in presence of an additional haptic force cue applied in the hands of the user. This force must be proportional to the acceleration present in the optical flow (Figure 3A). We strapped the shoulders of participants to ensure that the effect was not related to vestibular or proprioceptive feedback. And we found that the presence of this redundant haptic information had a significant and positive effect on the occurrences of the vection illusion, but also on its duration, onset, and on the perceived intensity of the sensation of self-motion.

(A) (B) (C)

Figure 3: Haptic Motion. (A) Experimental apparatus used in (Ouarti, 2014), (B) HapSeat device, and (C) concept of driving simulator based on haptic motion.

One advantage of this technique, that we called "Haptic Motion", compare to the use of a classical motion platform, is the possibility to generate a sensation of acceleration during a very long time, and in any 3D direction or orientation. Thus, we can foresee a wide range of usages, notably in the entertainment industry, but also, more generally, in VR applications relying on motion or navigation in virtual worlds.

One application of this approach is the HapSeat device (see Figure 3B), which enables to produce motion sensations in a consumer living-space [START_REF] Danieau | HapSeat: Producing Motion Sensation with Multiple Force-feedback Devices Embedded in a Seat[END_REF]. Instead of moving the entire user body as it is traditionally done with motion platforms, we proposed to stimulate only several specific parts of the body. The approach consists in applying multiple force-feedback so to generate a complete (6 degrees-of-freedom) sensation of self-motion, while being seated and experiencing a passive VR navigation. In our prototype, we used three low-cost haptic actuators arranged around the seat, in the two armrests and the headrest. A user study was conducted and participants reported that using the HapSeat increased significantly the quality of experience when watching virtual navigations. A second application is the design of a new kind of driving simulator (see Figure 3C). In this case, the motion platform would be substituted by the use of an actuated driving wheel, able to exert forces at the level of the hands of the driver. Following our haptic motion approach, these forces could simulate the motion felt by the user inside the vehicle such as when braking or accelerating (front/back longitudinal forces) or when turning (lateral forces).

Body-ownership illusions using visuo-tactile stimulations

Another approach shows how to exploit a synchronized stimulation between vision and touch together with a spatial discrepancy (offset) between them, in order to generate powerful body-ownership illusions in virtual reality.

The "body-ownership sensation" is classically depicted as the feeling of owning one's body and experiencing that this body is the source of sensations. It is currently of great interest for the VR community to better understand how and to what extent it is possible to develop a sense of ownership in virtual environments towards more or less realistic self-avatars.

A famous example of body-ownership illusion is the Rubber-Hand Illusion (RHI) originally described by Botvinick and Cohen. In their setup, the participant was seated with the left arm resting upon a table . A standing screen was positioned to hide the arm from participant's view. A life-sized rubber model of hand was placed on the table in front of the participant who was explicitly asked to keep eyes fixed on the rubber hand. Two small paintbrushes were manipulated by an experimenter to stroke simultaneously both the real hand and the rubber hand, synchronizing the timing of the two brushings as closely as possible. After ten minutes of such visuo-haptic stimulation, it was found that the participant "feel as if the rubber hand had sensed the touch". In other words, an illusion of ownership is observed towards the artificial hand. Later on, follow-up studies showed that the RHI can be obtained in VR, and also stressed the importance of the synchronization between the visual and haptic brushing feedback. A temporal delay introduced between the two sensory stimulations is found to break the illusion.

In a recent study [START_REF] Hoyet | Wow! I Have Six Fingers!: Would You Accept Structural Changes of Your Hand in VR?[END_REF], we could show how to generate body-ownership illusions when being immersed with a virtual body that is structurally different from ours. We considered the peculiar case of having a virtual body with a 6-digit hand (see Figure 4A). We got inspired from the RHI conditioning method to foster the appropriation of the 6-finger hand and generate a "6-Finger Illusion" (6FI). Participants controlled the six-digit hand from a first-person view while wearing an HMD. They performed two tasks successively. In the first task, participants manipulated the virtual hand while mimicking finger movements presented in the virtual scene. The second task was directly inspired by the RHI experiment: an experimenter stroked the participants' real fingers with a brush while a synchronous virtual brushing was presented in the virtual scene (see Figure 4B). In the case when the virtual brush was stroking the sixth digit, the real ring finger was synchronously stroked so to provide a consistent tactile stimulation. We measured how participants behaved when asked to lift every virtual digit. Interestingly, when the additional (sixth) digit was pointed, participants lifted indeed a real finger (ring or pinky) in 96% of cases. Our questionnaires results support that participants responded very positively to the possibility of controlling the six-digit hand despite the structural difference, and accepted to some extent the virtual hand and all the individual digits as their own. Such conditioning protocol relying on synchronized visuo-tactile stimulations shows great potential for augmenting virtual embodiment in VR. Adding haptic cues synchronized with visual ones seems indeed to increase strongly the belief of participants, and could be encouraged when targeting applications in which the virtual body of the user plays an important role. In addition to videogames and immersive entertainment, we can think of numerous applications such as vocational or sport training, but also cybertherapy, or motor re-education.

Conclusion

When considering the combination of vision and touch in virtual environments we believe that "one plus one equals three". By playing carefully with two distinct sources of stimulation it becomes possible to create alternate and powerful perceptual experiences in VR. In our first example, introducing a spatiotemporal conflict and distorting visual motions can generate various haptic sensations and pseudo-haptic effects. In our second example, adding a force cue well synchronized with the visual stimulation can induce powerful self-motion sensations. In our third example, a tactile stimulation presented synchronously with visual feedback but with a spatial offset, enables to map the tactile sensations onto a fake virtual limb and generate body-ownership illusion.

Of course, several open questions remain related to the reconstructed percepts coming from these artificial situations. In particular, how do they differ from real-life situations? What do they reveal regarding human perception? Do they influence perception in reality, especially after long exposure in VR? Neuroimagery could help clarifying the cognitive processes involved, and eliciting the potential brain plasticity. In any case we need further works and fundamental studies to better understand and characterize multi-sensory integration in virtual environments.

The best is probably to come regarding the exploitation of vision and touch and, more generally, multi-sensory feedback in VR. We have addressed a subset of sensory combinations, and many more remain to be tested. We thus invite the VR community to further enter this playground and "play with senses" in virtual environments.

Figure 4 :

 4 Figure 4: The 6-Finger Illusion (Hoyet, 2016). (A) 3D model of the 6-digit hand used in our experiment. (B)Experimental procedure for visuo-tactile stimulation.

Acknowledgements

The author is deeply grateful to his colleagues and collaborators who have contributed to the results presented in this article: Merwan Achibet, Ferran Argelaguet, Alain Berthoz, Marc Christie, Marco Congedo, Fabien Danieau, Julien Fleureau, Pierre Gaucher, Edouard Gentaz, David Gomez, Philippe Guillotel, Ludovic Hoyet, Maud Marchal, Nicolas Mollet, Nizar Ouarti, and Jerome Royan.

Author

Anatole Lécuyer is senior researcher and head of Hybrid team at Inria, Rennes, France. Contact him at anatole.lecuyer@inria.fr.


Mark Campbell

Daniel Clark
email: daniel.clark@telecom-sudparis.eu

Time-Lapse Estimation for Optical Telescope Sequences Estimation par Intervalle de Temps pour les Séquences de Télescopes Optiques

Keywords: Sensor Calibration, Multi-Target Tracking Mots-clefs: Calibrage du Capteur, Suivi Multi-Cible

come

Introduction

The usage of low cost ground-based Complementary Metal Oxide Semiconductor (CMOS) optical sensors to track objects in Low Earth Orbit (LEO) and Geostationary Earth Orbit (GEO) has dramatically increased in popularity in the past decade, leading to a chance of integrating these sensors into the formal SSA environment. These Consumer O-The-Shelf (COTS) cameras and optics are inexpensive solutions and provide a relatively wide Field of View (FoV). A central challenge in the exploitation of these sensors is the extraction of multiple detections from a single image due to the wide FoV: some detections can stem from moving objects (e.g. satellites, debris), some others from statics objects (e.g. stars), some of them may be artefacts from the extraction process or false alarms. The association between the objects and the detections is unknown, as are the number of objects, both static and moving, in the FoV. In these conditions, exploiting the output of a sensor in order to estimate the number and trajectories of the detected objects becomes a multi-object estimation problem. The Probability Hypothesis Density (PHD) lter [START_REF] Mahler | Multitarget Bayes Filtering via First-Order Multitarget Moments[END_REF] was designed as an inexpensive ltering solution in the the context of multi-object ltering. Extensions to the PHD lter such as the Second-Order Probability Hypothesis Density (SO-PHD) [START_REF] Schlangen | A Second-Order PHD Filter with Mean and Variance in Target Number[END_REF] and the Cardinalized Probability Hypothesis Density (CPHD) [START_REF] Mahler | PHD Filters of Higher Order in Target Number[END_REF] provide additional information at the expense of computational complexity. Solving this multi-object scenario is crucial so that methods such as Initial Orbit Determination (IOD) and astrometry can take place. The multi-target tracking methods assume a constant time-lapse between observations e.g. a constant time between image captures. However this is often not the case when operating on real datasets. This could be due to a number of reasons: operator error, corrupted or missing metadata or even sensor failure. This paper explores the exploitation of multi-object tracking algorithms developed from the Finite Set Statistics (FISST) framework [START_REF] Mahler | Multitarget Bayes Filtering via First-Order Multitarget Moments[END_REF], known as the SC-PHD lter [START_REF] Lee | SLAM with Dynamic Targets via Single-Cluster PHD Filtering[END_REF][START_REF] Ristic | Calibration of Multi-Target Tracking Algorithms Using Non-Cooperative Targets, Selected Topics in Signal Processing[END_REF], in order to jointly estimate the time-lapse between the images and the objects' states in a sequence of images produced from an optical sensor. Variations of this method have been used to register microscopy [START_REF] Schlangen | Marker-Less Stage Drift Correction in Super-Resolution Microscopy Using the Single-Cluster PHD Filter, Selected Topics in Signal Processing[END_REF] and astronomy [START_REF] Campbell | Image Registration Using Single Cluster PHD Methods[END_REF] images. This paper is organized as follows. Sec. 2 presents the principle of joint multi-object ltering and sensor state estimation, and introduces the two multi-object lters exploited in the paper. Sec. 3 presents the details employed for the implementation of the algorithms. Sec. 4 describes the simulation tests and Sec. 5 discusses the results. Finally, the conclusions are provided in Sec. 6. The joint calibration and tracking method exploits a Simultaneous Localisation and Mapping (SLAM) based approach [START_REF] Lee | SLAM with Dynamic Targets via Single-Cluster PHD Filtering[END_REF] known as the Single Cluster Probability Hypothesis Density (SC-PHD) lter. There are two sources of uncertainty to be estimated, represented by two dierent processes:

• The sensor process Ψ estimates the sensor state: in this application, the time-lapse between subsequent images,

• The object process Φ estimates the number and states of moving objects: in this application, the orbiting objects moving through the image sequence.

Since the number and states of the objects is assumed unknown and possibly varying across the image sequence, the object process Φ is estimated through multi-object lters derived from the FISST framework: in the scope of this paper, the PHD lter [START_REF] Mahler | Multitarget Bayes Filtering via First-Order Multitarget Moments[END_REF] and the SO-PHD lter [START_REF] Schlangen | A Second-Order PHD Filter with Mean and Variance in Target Number[END_REF] [START_REF] Schlangen | Multi-object ltering with second-order moment statistics[END_REF]. These two solutions are explained in more details in Sec. 2.2.

Multi Object Filtering

This section focuses on the estimation of the object process Φ As seen in Sec. 2.1, the estimation of the object process is conditioned upon the sensor state (i.e. time lapse between observations). In practical terms, a multiobject lter estimating the object process is maintained for each possible sensor state y. In this section, y k denotes an arbitrary sensor state1 at time k (i.e., in the k-th image of the sequence). Each object is described by its state x in the (single) target state space X ⊆ R d , describing the physical characteristics of the object. An object may enter or leave the sensor FoV at any time during the image sequence, and thus the number of objects at any time is unknown and needs to be estimated. The multi-object state is represented by a Random Finite Set (RFS) Φ k , a random object whose size and elements are unknown, and whose realization is a set of target states X k = {x 1 , x 2 , ..., x n k } represents a specic multi-object conguration at time k [START_REF] Mahler | PHD Filters of Higher Order in Target Number[END_REF]. The evolution of the objects' state between time step k -1 and k is described by a Markov transition function t k|k-1 . The number and states of newborn objects is described by a RFS Φ b,k whose nature depends on the lter; its rst-order moment density or intensity is denoted by µ b,k (•). An observation collected from the sensor is described by a state z in the observation space Z ⊆ R d . The set of collected observation is denoted by Z k , which contains observations of all the targets in the image at time k.

The observation process is plagued by observation noise, missed detections, and false alarms. The observation noise associated to some collected observation z is characterized by the likelihood function z,k (•|y), while the probability of detection for each individual object in the sensor FoV is denoted by p d,k (•|y). The number and states of false alarms is described by a RFS Φ , k whose nature depends on the lter; its intensity is denoted by µ , k (•|y), and its spatial distribution is denoted by s , k (•|y). For the rest of the paper, the notation µ(X ) will be used for various intensity functions µ to denote the integral X µ(x)dx.

Probability Hypothesis Density (PHD) Filter

The PHD lter [START_REF] Mahler | Multitarget Bayes Filtering via First-Order Multitarget Moments[END_REF] was designed as an inexpensive ltering solution in the the context of multi-object ltering, propagating only the intensity µ k of the object process Φ k . The key assumptions of the PHD lter is that both the predicted process Φ k|k-1 and false alarm process Φ , k are Poisson [START_REF] Mahler | Multitarget Bayes Filtering via First-Order Multitarget Moments[END_REF]. The prediction and update steps of the PHD lter are given by

µ k|k-1 (x|y) = µ b,k (x) + µ s,k|k-1 (x|y), (1)
µ k (x|y) = µ φ,k (x|y) + z∈Z k µ z,k (x|y) µ , k (z|y) + µ z,k (X |y) , (2)
where the survival µ s,k|k-1 (x|y), missed detection µ φ,k (x|y), and association µ z,k (x|y) terms are dened as

µ s,k|k-1 (x|y) = p s,k (x)t k|k-1 (x|x)µ k-1 (x|y)dx, (3)
µ φ,k (x|y) = (1 -p d,k (x|y)) µ k|k-1 (x|y), (4)
µ z,k (x|y) = p d,k (x|y) z,k (x|y)µ k|k-1 (x|y).
(5)

Second-Order Probability Hypothesis Density (SO-PHD) Filter

Recently, a second-order version of the PHD lter was introduced in [START_REF] Schlangen | A Second-Order PHD Filter with Mean and Variance in Target Number[END_REF] that combats some of the issues posed by the restrictive Poisson assumptions of the PHD lter. It propagating not only the intensity µ k of the object process Φ k , but also the variance var k (X ) of the number of objects in the whole target state space X .

The SO-PHD lter substitutes Panjer assumptions to Poisson assumptions in the PHD lter, thus providing more exibility in the description of the number of objects and false alarms. More specically, the Panjer distribution [START_REF] Schlangen | A Second-Order PHD Filter with Mean and Variance in Target Number[END_REF] describes the Poisson, binomial and negative binomial distributions in a unied formulation involving two parameters α and β which stand in one-to-one correspondence with the mean and variance of the distribution. Using this property, the SO-PHD lter is able to propagate the variances in the number of objects and number of false alarms via the Panjer parameters of the corresponding distributions. Before stating the recursion of the SO-PHD lter, some notations inspired by those introduced in [START_REF] Vo | Analytic Implementations of the Cardinalized Probability Hypothesis Density Filter[END_REF] for the CPHD lter will be states. The Pochhammer symbol or rising factorial (ζ) n for any ζ ∈ R and n ∈ N is dened with

(ζ) n := ζ(ζ + 1) • • • (ζ + n -1), (ζ) 0 := 1. (6)
Let α k|k-1 , β k|k-1 and α , k , β , k be the parameters of the predicted object process and clutter process at time k, respectively, and dene the expression

Y u [Z] := |Z| j=0 (α k|k-1 ) j+u (β k|k-1 ) j+u (α , k ) |Z|-j (β , k + 1) |Z|-j F -j-u d e j (Z) (7)
for any Z ⊆ Z k and for u = 1, 2, where F d is the scalar

F d := 1 + p d,k (x|y) β k|k-1 µ k|k-1 (x|y)dx, (8)
and the so-called elementary symmetric functions [START_REF] Schlangen | Single-cluster PHD lter Methods For Joint Multi-Object Filtering and Parameter Estimation[END_REF] e j are given by e j (Z) :=

Z ⊆Z |Z |=j z∈Z µ z,k (X |y) s , k (z|y) .
(

) 9
Where s c,k denotes the spatial clutter distribution at time k. With the help of Eq. ( 7), dene the corrective terms via

l u (φ|y) := Y u [Z k ] Y 0 [Z k ] and l u (z|y) := Y u [Z k \{z}] Y 0 [Z k ] (10)
for u = 1, 2, and

l 2 (z, z |y) :=    Y 2 [Z k \{z, z }] Y 0 [Z k ] if z = z , 0 otherwise. (11)
Assuming that p s,k (x) = p s,k is constant for all x ∈ X at time k, the prediction step of the SO-PHD lter is then given by

µ k|k-1 (x|y) = µ b,k (x) + µ s,k|k-1 (x|y), (12)
var k|k-1 (X |y) = var b,k (X ) + var s,k|k-1 (X |y),

where the survival term for the variance var s,k|k-1 is given by

var s,k|k-1 (X |y) = (p s,k ) 2 var k-1 (X |y) + p s,k [1 -p s,k ]µ k-1 (X |y). ( 14
)
The variance in the number of newborn objects var b,k is a parameter of the lter, and allows the operator to describe situations where the information on the number of objects entering the image frame.

The Panjer parameters of the predicted object process Φ k|k-1 are then given by [START_REF] Schlangen | A Second-Order PHD Filter with Mean and Variance in Target Number[END_REF] α

k|k-1 = µ k|k-1 (X |y) 2 var k|k-1 (X |y) -µ k|k-1 (X |y) , (15)
β k|k-1 = µ k|k-1 (X |y) var k|k-1 (X |y) -µ k|k-1 (X |y) . ( 16
)
With these, the corrective terms l u (10), [START_REF] Lee | SLAM with Single Cluster PHD Filters[END_REF] can be computed, and the update step of the SO-PHD follows:

µ k (x|y) = µ φ,k (x|y)l 1 (φ|y) + z∈Z k µ z,k (x|y) s , k (z|y) l 1 (z|y). ( 17
)
var k (X |y) = µ k (X |y) + µ φ,k (X |y) 2 l 2 (φ|y) -l 1 (φ|y) 2 + 2µ φ,k (X |y) z∈Z k µ z,k (X |y) s , k (z|y) l 2 (z|y) -l 1 (φ)l 1 (z|y) + z,z ∈Z k µ z,k (X |y) s , k (z|y) µ z ,k (X |y) s , k (z |y) l 2 (z, z |y) -l 1 (z|y)l 1 (z |y) . ( 18
)

Sensor State Estimation

This section focuses on the estimation of the sensor process Ψ. The state of the sensor is denoted by y ∈ Y, the sensor space Y describing the relative time dierence of an image frame with respect to the previous frame in the image sequence. Following the Sequential Monte Carlo (SMC) implementation in [START_REF] Lee | SLAM with Single Cluster PHD Filters[END_REF], the information regarding the state of the sensor at time k is described by a probability distribution p k approximated through a set of weighted particles {w i k , y i k } N i=1 , i.e.

p k (•) N i=1 w i k δ y i k (•). (19)
Note that the number of particles, N , is maintained constant throughout the scenario. Selecting an appropriate number of particles is a key consideration of the implementation of the lter. A large number of sensor particles provides a more robust estimator, but increases signicantly the computational cost of the overall algorithm: recall from the hierarchical structure (see Sec. 2.2) that a multi-object lter, either a PHD or SO-PHD lter is maintained for each possible sensor state, i.e., for each particle state y i k .

By construction, the initial states of y i 0 are uniformly distributed between some lower and upper parameters.

At each time step, the prediction step of the sensor state follows a Markov transition model h k|k-1 denoting the operator's knowledge in the nature of the sensor movement (i.e. linear drift). Each particle state is thus resampled according to the transition model, i.e.

y i k ∼ h k|k-1 (•|y i k-1 ). ( 20
)
The update step of the sensor state works as follows. Since the likelihood function z,k (x|y) associating a collected observation z to an object with state x is dependent on the sensor state therefore the estimation of the object process Φ is also conditioned on the sensor state y and is exploited to update the sensor state distribution as follows:

w i k ∝ L(Z k |Φ, y i k )w i k-1 N i=1 w i k-1 , (21)
where the multi-object likelihood L(Z k |Φ, y i k ) quanties the match between the set of collected observations Z k and the estimation of the object process Φ conditioned on y i k . These multi-object likelihoods can be seen below in Sec.2.4.1 and 2.4.2 After the updated weights w i k are calculated, the highest weighted particle is found. The resampling step then occurs by uniformly distributing between some minimum and maximum parameters using the highest weighted particle.

PHD Multi-Object Likelihood

If the object process is estimated through a PHD lter (see Sec.2.2.1), the multi-object likelihood is given by [START_REF] Swain | Group and Extended Target Tracking with the Probability Hypothesis Density Filter[END_REF]:

L(Z k |Φ, y i k ) = z∈Z k µ , k (z|y i k ) + p d,k (x|y i k ) k (z|x, y i k )µ k|k-1 (x|y i k )dx exp µ , k (z|y i k )dz + p d,k (x|y i k )µ k|k-1 (x|y i k )dx . ( 22
)

SO-PHD Multi-Object Likelihood

If the object process is estimated through a SO-PHD lter (see Sec.2.3), the multi-object likelihood is given by

L(Z k |Φ, y i k ) = |Z k | j=0 (α k|k-1 ) j (β k|k-1 ) j (α c,k ) |Z|-j (β c,k + 1) |Z|-j F -α-j d F -α c,k -|Z|-j c Z ⊆Z k |Z |=j z∈Z µ z,k (X |y i k ) z ∈Z k \Z µ c,k (z|y i k ), ( 23
)
where

Fd = 1 - 1 β k|k-1 p d,k (x|y i k )s k|k-1 (x|y i k )dx, (24)
F c = 1 + 1 β c , (25)
where s k|k-1

(x|y i k ) = µ k|k-1 (x|y i k ) µ k|k-1 (x|y i k )dx
is the spatial distribution associated to the intensity µ k|k-1 .

All of the multi-object lters used in the experiments are implemented using a Gaussian Mixture (GM) approach following [START_REF] Vo | The Gaussian Mixture Probability Hypothesis Density Filter, Signal Processing[END_REF], [START_REF] Schlangen | A Second-Order PHD Filter with Mean and Variance in Target Number[END_REF] and [START_REF] Vo | Analytic Implementations of the Cardinalized Probability Hypothesis Density Filter[END_REF]. Measurement driven birth is also used. Since the parent process is implemented using a SMC particle lter approach, a preset number of particles shall be used.

The motion of the sensor state , herein referred to as the time lapse ∆ k , (20) shall be modelled using two motion models, Brownian motion and static motion, each accounting for a dierent drift observed in common scenarios. The Brownian motion model, or random walk, describes a scenario where the images are perhaps taken manually by the operator at random intervals. The static model describes a xed oset with some zero mean Gaussian noise possibly due to a control software error. The time lapse state space is characterised by a simple one dimensional space x = ∆ k . When satellites and debris are observed using telescope, a streak is left behind in the image due to the long exposure times needed. The length of this streak can provide an estimate of the target's velocity v, given by:

v = l T (26)
Where l and T are the measured streak length and camera exposure time respectively. The inclination (or heading) θ of the streak can also be measured.

This motion using the Nearly Constant Heading (NCH) motion model [START_REF] Kountouriotis | Maneuvering Target Tracking Using an Unbiased Nearly Constant Heading Model[END_REF]. It provides a better representation of the target dynamics than a regular Nearly Constant Velocity (NCV) motion model since the targets move along a fairly xed line. Dynamic targets are described via their x and y position, the speed v and the inclination θ using a four-dimensional state space X ⊆ R 4 , where specic states at time k are of the form

x k = x k , y k , v k , θ k T (27)
The motion model for the dynamic objects is non-linear so an Extended Kalman Filter (EKF) [START_REF] Anderson | Optimal Filtering[END_REF] is used to propagate through the time steps. It is given as follows:

t k (x k |x k-1 ) = N (x k ; xk , Q k ), (28)
where the intermediate state xk is obtained with

xk = x k-1 + ∆ k v k-1 cos(θ k-1 ), sin(θ k-1 ), 0, 0 T , (29)
and where Q k is a covariance matrix of the form:

Q k = ∆ k     0 0 0 0 0 0 0 0 0 0 σ 2 v,k 0 0 0 0 σ 2 θ,k     , (30)
where σ v,k and σ θ,k are the standard deviations of the velocity and inclination, respectively.

Simulations

In order to test the algorithm, several scenarios shall be simulated, each using a dierent time lapse motion model. This simulated data will be generated as follows. Firstly a number of initial targets are generated N t , the targets' states are then initialised following the NCH model stated above. The generated x k , y k positions are limited to within the sensor's FoV, the velocities v k are drawn from a uniform distribution between some minimum v min k and maximum v max k velocity parameters and the inclinations are drawn from a uniform distribution such that 0 ≤ θ k ≤ 360. As stated before the target transition model follows the NCH model Eq.28. Spontaneous target birth may occur at each time step k and is modelled using a Poisson process with rate λ b .

Target death and detection rates are other factors and they are modelled using Bernoulli processes with probabilities of survival p s and detection p d . False alarms may also occur at each time step k, in these simulations a Poisson process with rate λ f a is used to represent these clutter process. The clutter process is also assumed to be uniformly distributed over the entire sensor FoV. At each time step measurements z k are obtained from the simulated targets' states x k using the observation model:

z k = H k x k + N (0, R k ), (31)
Where H k is given by

H k =   1 0 0 0 0 1 0 0 0 0 0 1   , (32)
And R k is given by

R k =   σ 2 x,k 0 0 0 σ 2 y,k 0 0 0 σ 2 θ,k   , (33)
For ease of simulation, the exposure time of the camera T is xed to 1 second. The measured velocity is only used for initialising new targets. The simulation of the time lapse between successive images ∆ k is dependent upon the scenario. It shall be done in one of two ways:

• Static ∆ k : A value of ∆ k is generated at the start of the scenario. This remains constant throughout.

• Random ∆ k : A value of ∆ k is generated at each time step k, drawn from a uniform distribution between some minimum ∆ min k and maximum ∆ max k parameters

The two simulated scenarios share the same parameters with the exception of how the time lapse ∆ k evolves.

These parameters are shown below in The accuracy of the time lapse estimation shall be measured using the Root Mean Square Error (RMSE) between the estimate obtained from the particle lter and the ground truth. Also the presented results are the averaged results over the MC runs. The estimates used in the results are obtained using a Maximum A Posteriori (MAP) estimate of the posterior likelihood distribution. The output from the highest weighted particle's PHD or SO-PHD lter, at each time step, shall be compared to the simulated ground truths using the Optimal SubPattern Assignment (OSPA) metric [START_REF] Schuhmacher | A Consistent Metric for Performance Evaluation of Multi-object lters[END_REF]. The OSPA metric introduces the concept of a miss distance for multi-target lters, which jointly weights the accuracy of a lter's spatial estimates and its estimated target cardinality to produce a single value. For the OSPA results shown, a cut-o parameter of c = 100m and order parameter p = 2 is used. The estimated number of targets alongside the ground truth shall also be shown.

Results

The execution times of the methods can be seen in Table .2, as expected the SO-PHD implementation is more computationally expensive by a factor of approximately 1.5. Fig. 1 shows the acrmse of the ∆k estimates. These show that for both scenarios the Single Cluster Probability Hypothesis Density (SC-PHD) method performs incredibly well (0.086 peak RMSE), regardless of the multi-object lter used. Note that the ∆k estimate is highly dependent on the number of particles as more particles will allow a more accurate estiamte to occur. The OSPA (Fig. 2) and estimated cardinality (Fig. 3) results show that the estimation of the multi-object state is also accurate, with the lters often being less than 50 for both scenarios. As expected the SO-PHD lter performs better than the PHD lter. Note that the variance estimated by the SO-PHD lter is not shown here. From the results shown above, it is clear that the Single Cluster Probability Hypothesis Density (SC-PHD) lter can accurately estimate the time lapse that occurs between images whilst also simultaneously tracking the dynamic objects in the image sequence. It has also been shown to work on real data although that is outside the scope of this paper. This method is not just limited to this application, it can be extended to almost any sensor calibration problem.

Method

The main limitation of this implementation is that it requires some measurable information about the targets' velocity or that the targets all move at roughly the same speed. A possible extension to this method can be seen in [START_REF] Lee | SLAM with Dynamic Targets via Single-Cluster PHD Filtering[END_REF], where a joint update step for the PHD lter is used to discriminate better between the dierent target populations. the method could be extended to the physical spherical plane (right ascension α and declination δ). This would allow a more meaningful physical interpretation of the results to occur and an easier integration with common orbital determination methods.

Figure 1 Figure 2 Figure 3

 123 Figure 1 RMSE results of the ∆k estimates

  Table.1

	Number of Monte Carlo (MC) Runs	40	Number of MC Particles	50
	Number of Time Steps	30	Initial Number of Targets M	10
	Rate of Target Birth λ b	0.2 births / frame	Minimum Target Velocity v min k	5 pixels/image
	Maximum Target Velocity v max k	15 pixels/image	Survival Probability ps	0.95
	Detection Probability p d	0.9	False Alarm Rate λ f a	5 / frame
	State Space Dimensions (X × Y )	1000 pixels × 1000 pixels	Minimum Time Lapse ∆ min k	0.1 seconds
	Maximum Time Lapse ∆ max k	4 seconds
		Table 1 Simulation parameters

Table 2

 2 Execution time results in seconds per time step

The estimator associated to the sensor state will be presented later in Sec.

2.4


Keywords: Morphosyntactic mismatches, mixed categories, Participial Stem, underspecification, Distributed Morphology

1 Morphosyntactic Mismatches revisited: the case of Romanian Supine * Elena Soare Université de Paris 8 -CNRS UMR 7023 elena@linguist.jussieu.fr

Preliminaries

The boundary between inflectional and derivational processes is not always clear.

Moreover, sometimes word-formation and inflection are using the same material and the same operations. In some cases, this results in different sorts of mismatches: inflectional (cf.

the fact that a form like a participle can project a full VP like in he kept [ VP-prp putting PRP the book behind the couch]) and categorial mismatches (cf. the double categorization of the English gerund in the structure called "POSS-ing").

This presentation is concerned with morphosyntactic mismatches, i.e. mismatches between features that count for the external syntax (distribution) and features that count for the internal syntax. Such cases are the English "Poss-ing" Gerund or the Romanian Supine, which will be our main concern.

This last form is an interesting case into that picture. Among other syntactic uses, it has one which is a nominalized Participle (by a process like "conversion"). This form is clearly used in derivational as well as in inflectional processes, and in this respect, it represents evidence in favor of approaches that try to unify inflectional and categorial mismatches.

So, we will try to take a look at the dual/mixed nature of the Supine; determine its relationships with other parts of the system (inflectional and derivational aspects), and find the appropriate framework to fit the facts.

1.

What is Romanian Supine

The core facts about Romanian Supine

There are two major syntactic uses of Perfect Participle crosslinguistically: a) the perfect verb formation (+HAVE): b) the passive verb formation (+BE).

NB: Nothing special will be said about the mapping from "participle" to "adjective" (a case of automatic "inflectional mismatch". We could assume following [START_REF] Baker | Lexical Categories. Verbs, Nouns, Adjectives[END_REF] that the Participle is in fact "the verbal adjective".

Romanian has a third use of the Perfect Participle: the "Supine". The form of past participle in Romanian is used in three types of syntactic environments: nominal, adjectival and verbal. There is on the one hand a past participle, with verbal and adjectival uses:

(1) a am cules căpşuni 2)-( 4)). In [START_REF] Yoon | Dual Lexical Categories and Inflectional Morphology[END_REF], the participial form appears in a nominal structure, in combination with a definite determiner, and the complement is assigned Genitive case. In (3), the supine follows an aspectual auxiliary, expressing the end of an activity; in this case, it does not take an article, and requires the presence of a marker, a functional "preposition", de. In (4), the supine is preceded by a subcategorized preposition in the complement position of a verb of motion (expressing the Goal).

A. nominal supine: D° + participle:

( NB. There is no (morphological) reason to separate the "Supine" from the "Participle", as

shown by the fact that the irregularities are exactly the same for the two cases, cf. the "sigmatic" participles mers, cules.

1

The examples illustrate the supine as complement to aspectual verbs. The same form appears in copular structures, reduced relatives, Tough constructions, i.e. in predicative contexts involving (small) clauses in other languages.

(5) cânta -cântat -de cântat/ cântatul sing -sung -to sing -the singing In the generative literature, some verbal nouns have been analyzed as "mixed categories" (+N, +V), for instance the Arabic masdar (Fassi Fehri 1991), the Welsh verbal noun [START_REF] Rouveret | Principes généraux et typologie[END_REF] as well as English gerund. Is the supine also a verbal noun?

2 Mixed or underspecified?

A criterion for the mixed character is to find properties of two kinds of projections in the same projection, at the same time. For instance, the distribution would be nominal, as for the English gerund, which can appear in contexts excluded by non-nominal projections:

(6) a we were concerned about Pat's watching television b *we were concerned about that Pat was watching television

The same type of projection is characterized by internal properties specific for verbs, i.e.

Accusative case assignment (7a) and adverbial modification (7c):

( to pick strowberries (is a pleasant activity)

'picking strawberries is a pleasant activity'

This observation leads to the idea that the supine cannot assign itself a case to its complements. In the case of the nominal projection, this is done by the presence of nominal determination. Within the verbal projection, the supine combines with auxiliaries, forming a verbal complex, which, as a whole, is responsible for case-licensing of the complements. The result is not a "mixed" projection because the different properties of the supine do not manifest at the same time, in the same projection, but in different projections.

Another possibility would be to say that the auxiliary assigns an "aspectual role" to this object, and contributes to fully identify it, and that the supine only assigns a weak case to this object. This way of analysis was proposed for the Scottish Gaelic verbal noun by [START_REF] Ramchand | Aspect and Predication. The semantics of Argument Structure[END_REF], who considers that this characteristic is due to the nominal status of this verbal noun.

We therefore retain as a conclusion the following idea:

The supine changes its projection type as it changes its morphological properties; it does not match the mixed-category criterion.

There is an apparent exception to this generalization, represented by the supine inside a PP projection, in which the Preposition is selected by the main verb or has an autonomous lexical meaning (such as Goal), in the case of adjuncts. In traditional grammars, this prepositional supine is considered to be verbal (Accusative-Case Assigner). It is the essential argument of traditional grammars in saying that the supine keeps its verbal properties in this kind of contexts.

(12) am plecat la cules căpşuni have gone at picking strawberries 'I'm going to pick strawberries'

On the basis of this type of examples, traditional grammars take the view that the supine is a case-assigner when it is introduced by a Preposition. We have to note that these cases are restricted to supine expressing Goal. Also, when the supine is preceded by lexical subcategorized Prepositions, the prepositional accusative becomes impossible for the object of that supine. The strong structural Accusative Case is not allowed:

( Therefore, it seems that a direct object in such supine constructions manifests a special behavior, to be distinguished from the regular behavior of a direct object in an ordinary verbal construction. The only "objects" allowed by the supine in the absence of an auxiliary are bare, predicative NPs (not DPs), which do not accept strong accusative case.

What they bear is a weak case, that may appear with defective forms (quasi-nominal, or uncategorized forms; see also in this sense [START_REF] Ramchand | Aspect and Predication. The semantics of Argument Structure[END_REF] proposal for the Scottish Gaelic VN). Another explanation that could be adopted is that this behavior is due to the fact that the structure is frozen, and relies probably on a composition of the type N-N, generated by lexical rules (as compounds). If this is true, structures like ( 12) and the same, with PPs containing supines, involve nominal supine compound structures, in which the "light" object enters in a lexical compounding relation to the supine. An argument in this sense is given by the existence of structures of the type in ( 16), appearing in enumerations or other particular contexts (titles, labels), where the model of composition is possible with participles:

(16) n-am uitat nimic: cumpărat bilete, făcut bagaje… not have forget anything: buying tickets, packing bags….

'I didn't forget anything: buying tickets, packing bags…' Moreover, present-day Romanian tends to favor the construction with de-insertion instead of the direct one with the presumed Accusative object. Therefore, the perspective offered by productive supine structures is that of an element whose verbal-nominal nature is not "double" or "mixed", but clearly context-dependent, distinguishing this form from "true" verbal nouns.

Capturing mismatches

There is a recognized difference between lexicalist and non-lexicalist approaches with respect to the treatment of inflectional mismatches versus category mismatches. Lexicalist approaches treat them as completely different, whereas non-lexicalist approaches treat them alike, via Head-Movement -type analyzes.

A. Lexicalist approaches

Feature-percolation systems (Head feature principle) account for the transmission of morphological features, i.e. "participle" from the head to the maximal phrase:

(17) [+V -N prp] [+V -N] [prp] put- -ing
In the case of mixed categories, this system would result in contradictory feature specifications:

(18) [+V -N -V +N] [+V -N] [-V +N] put -ing
A lexicalist solution for mixed-type of mismatch (English Gerund) would propose:

a "split-head" analysis of the type of [START_REF] Lapointe | Dual Lexical Categories and the Syntax of Mixed Category Phrases[END_REF][START_REF] Lapointe | Dual Lexical Categories vs. Phrasal Conversion in the Analysis of Gerund Phrases[END_REF]'s Dual Lexical Categories:

(19) X/Y external synt internal synt a "mixed-head" analysis of the type of Malouf' (1998)'s Hierarchical Lexicon.

(20) Ger

Nominal outside Verbal inside

The main idea of this type of framework is that the items at study are doubly categorized items and that there is feature percolation from the base categories to the resulting category. Roots bear POS specifications. The inflectional and categorial mismatches are not alike (more precisely, there is no inflectional mismatch).

There have been also attempts to unify inflectional and category mismatches, based on the argument that features contributed by inflectional affixes are exclusively relevant for external syntax [START_REF] Yoon | Dual Lexical Categories and Inflectional Morphology[END_REF]). This was also the sense of Haspelmath (1996)'s proposal that inflection should cover also word-class changing relations, i.e. transpositional inflection:

word-class changing inflection are special cases of inflection in his view. Another attempt was to say that features contributed by inflectional affixes are exclusively relevant for external syntax [START_REF] Yoon | Dual Lexical Categories and Inflectional Morphology[END_REF]). The result is that inflectional and category mismatches are treated alike.

B. Non-lexicalist approaches

A syntactic view would more or less say that the head affix would move and incorporate giving rise to categories accumulating features all the way up, like in current generative models since the late '80s (from [START_REF] Abney | English NP in its Sentential Aspect[END_REF] to [START_REF] Alexiadou | On the Syntax of Nominalization and Possession[END_REF]). This is the option adopted by Distributed Morphology [START_REF] Marantz | No Escape from Syntax: Don't try morphological analysis in the privacy of your own Lexicon[END_REF], [START_REF] Harley | Mixed nominalizations, short verb movement and object shift in English[END_REF]). In this theoretical framework, the items listed in the vocabulary have no category, categorization being contributed by the syntactic component. The insertion of an element in the appropriate syntactic context makes it a nominal, or verbal, or adjectival… element.

In the vision of syntactic approaches then, inflectional and derivational mismatches are assigned a unified treatment; categories are built up by operations of head-movement and incorporation of affixes; there is no POS specification on the Roots.

Word formation involves ROOT derivation:

V N v √DESTROY n √DESTROY
A Root = an l-morpheme A Noun = a Root whose nearest c-commanding element is a D° A Verb = a Root whose nearest licenser is Aspect or Tense

In the appropriate context, a Root becomes a Noun, a Verb… via the combination with an f-morpheme: n, v…

There is a clear connection between the syntactic behavior and categorization.

Lexical categories, heads of syntactic projections, determine the internal structure of the projection (selection, projection, complement licensing), and the type of position in which the corresponding phrases will be inserted, as well.

For the case of the supine, there are empiric facts that contribute to an "underspecification"-type analysis, as we tried to show in section 2.

What kind of mismatch in the Supine?

Is the type of mismatch manifested in the Romanian Supine purely inflectional? The answer is NO: we do not find a morphological process that is "relevant to syntax"; the -at form does not determine the global behavior and the functional nodes, inasmuch it can feed a nominal OR a verbal structure. Then, the process of making a Participle from a Verb is not clearly category-neutral. There are researchers who proposed that the participle morpheme is a derivational affix (Schütze (2003) for instance).

Is it purely derivational then? The answer is again NO: we do not clearly obtain different "lexemes" by the process of Supine formation.

However, if we see inflection and derivation as a continuum, we may think about the corresponding mismatches as being alike: neutral l-morphemes combined with f-morphemes which can compete for different values. In the case of Gerund for instance [START_REF] Harley | Mixed nominalizations, short verb movement and object shift in English[END_REF] the -ing suffix can be analyzed as a default nominalizer; it would then be a n or a v head according to what we find in higher structure (Asp or Det).

Distributed Morphology-style treatment of the supine/participle

If our view is correct, an analysis that seems to impose itself is the "underspecification" analysis, proposed in the Distributed Morphology framework. The Participle has no categorial features, being categorized by the syntactic context.

We can then propose the following supine-formation system (DM-style):

(21) N V A n √PTP v √PTP a √PTP
According to the view of Distributed Morphology, there are no categorial features at the "Vocabulary" level. Participle would then be a single uncategorized item competing for several syntactic contexts.

At this point, we could ask the following question: should we talk only about uncategorized roots or about uncategorized stems? If PTP is a root, then what is Infinitive?

What is less clear indeed, it is which label to put on the participial form itself. In DM, there are category-neutral "Roots" (l-morphemes), and there are affixes with features competing for a specific value to express (f-morphemes). The analysis we would like to propose is that Participle, here above PTP, is itself category-less. Or, it is not really a Root, being composed of a verbal root and the participial affix. We shall try now to find a solution to this puzzle.

3

Emergence of a latin-type "third stem" in Romanian?

Many modern theoretical views are coherent with the existence of a un-categorial level of grammatical representation. [START_REF] Baker | Lexical Categories. Verbs, Nouns, Adjectives[END_REF], for instance, considers that category is given by syntax. The categorial identification is done by the syntax in the following way:

A Noun -has a referential index A Verb -has a specifier An Adjective -by default: it is -N, -V

In his system, however, the Participle is not really discussed; it is considered a 'verbal adjective' without further investigation.

Schütze (2003) takes the participial suffix as being category-changing, creating

Participles from Verbs. This makes them derivational affixes. As for the Participle, it is considered as not being (really) a Verb; it does not carry Voice, but only (lexical?) aspect. So, the Participle is a de-categorized Verb. This, however, takes in fact the Participle to be a distinct category.

Aronoff (1994) discusses the problem of (English) Participle, which, according to him, illustrates the "morphomic" level; being purely morphological, this element is appropriate for the very different syntactic constructions of past and passive.

Another argument for the "morphomic" level discussed by Aronoff (1994) is the existence of the Latin "third stem", realized in participle, Supine, and future active participle.

In this case, a single stem, also a "morpheme", is used in various syntactic environments.

The supine was a verbal noun; derived from a participial stem (from a synchronic point of view), it was an item that allowed nominal inflection (Case marking) and appeared as Goal adjunct with verbs or adjectives: that the one of the Participle, even if the values (aspect, voice) of the categories derived from this stem are different (the Supine does not admit the passive interpretation in Latin). The same stem is used to derive a number of deverbal nouns in Latin, such as pictura, derived from pingo, pict-. Aronoff (1994) builds in this way an argument for the view that the morphologic level should be kept distinct from syntax, semantic or phonology; morphemes do not encode (grammatical) meaning since, in the cases illustrated, they do not always have the same value.

Indeed, the same Thematic element appears in very different formations, like the active future participle, meaning "those which will V" and in the supine, denoting the activity without further specifications, and in the past participle, denoting a state, a result.

One question to ask is whether Romanian morphology can be considered to be based on stems, and if the Participle is a stem. Such a view could be supported by the fact that the -AT formation is also used in derivation. -AT can also attach to non-verbal roots Root +V V-at (categorially neutral element) (25) Mîncat 'eating', cîntat 'singing', citit 'reading', mers 'walking'

Root +N [Adj] (state of someone who has…) (26) Sprîncenat 'eye-browed', migdalat 'almond-ed' -AT could be considered also as participating in the derivation of Agent Nouns. The idea that it would be the realization of the same stem is however contradicted by the variation illustrated in (23)d-e. However, the correspondence between Participle and the stem of Agent Nouns is stable for the "regular" classes of verbs.

Root + t +-or Agent Nouns There are, we think, reasons to treat the Participle form as one of the stems of the verb in Romanian. It would be a rather particular use of this notion, which should be used for "what remains when all affixes are set apart". In our case, contrary to Latin, there are no affixes going with the nominal vs. verbal or adjectival status2 . But we could assign the status of a stem to the part which is common to all these syntactic uses mentioned above, and it would correspond to "what remains when the categorial features are set apart".

4

Importing stems into [START_REF] Marantz | No Escape from Syntax: Don't try morphological analysis in the privacy of your own Lexicon[END_REF] system?

Let us sketch now a way of making sense of all the intuitions above. We will assume that the most qualified model accounting for the facts outlined in this paper is Distributed Morphology, as depicted in [START_REF] Marantz | No Escape from Syntax: Don't try morphological analysis in the privacy of your own Lexicon[END_REF]. The only inconvenient would be that our Participles seem to be in the same time basic and constructed.

In fact, the relationship between destruct and destroy, for instance, in this framework, is ensured by "readjustment rules". In our case, there seem to be not different instantiations / spell-outs of a same abstract ROOT, but indeed two roots -or two stems, if it is possible to talk about stems that are neutral from the point of view of syntactic category.

I assume that in Romanian there are perfective and imperfective neutral stems, i.e.

participial and infinitive stems.

In the system proposed here, stems as neutral elements are combined with nominal, adjectival, verbal heads. In our case, that of Romanian Participle, these heads do not have phonological content. Put differently, we assume that the difference between Romanian Participle / Supine formation and English Gerund formation is that -ing is a nominal head, whereas -at is not. We illustrate hereafter the different type of word-formation corresponding to Gerund and to Supine; all of them take place in the syntactic component, according to the Distributed Morphology framework.

(28) a) v b) n DESTROY v DESTROY n to destroy destruction c) n d) n/a/v n v n/a/v CITIT DESTROY V
destroying Is -AT (simply) an empty morpheme?

What -at is, then? The answer, in Aronoff's terms, would be that -at is an empty thematic morpheme, and we already seen the arguments (section 2). We may have some arguments for taking this morphological piece as the expression of (lexical) Aspect. Participial stems, as we will argue below, encode Aspect. If this is correct, the view of Aronoff (1994) about the complete absence of semantic-grammatical value for the participial stem could be at least for Romanian Participle. The thematic affix (AT or Thematic Vowel + T) seems to keep a certain value in Romanian, which we take to be an aspectual one.

We take the basic aspectual value of the participial stem to be a perfective one. This is indeed the default value of the past participle constructions, as well known.

This aspectual value of the participial stem will be changed by the contribution of different markers. The first one is the nominalizer element, i.e. the determiner. Event supinebased nominalizations, as shown by [START_REF] Cornilescu | Aspect and Nominalizations. The Case of Romanian[END_REF], take most of the time an atelic reading. This is a consequence of the nominal status of the supine and of the weak status of the object. The supine nominal can take implicit arguments, and in particular implicit objects, hence its atelic value:

(29) cîntatul este un dar singing-the is a gift 'singing is a gift' However, those are contexts with a generic reading, the supine denotes a generic event, and the aspectual value is shifted to the iterative-habitual reading.

The aspectual value attached to the participial stem that we supposed was present in agent -tor derived Nouns is not clear, but in any case they seem to be atelic. In those examples too, we could find an aspectual component which is habitual (someone has to perform an activity regularly in order to be, for instance, a dancer or a smoker).

In some periphrases, the participial stem is attached to the expression of completion, a value that has also to do with perfectivity. See for example (28), where the action of reading has to be completed, or the movement to reach its goal: The partition that we suggested seems to be supported by the fact that the imperfective stem is used in some derived nouns but also in the future form. In Old Romanian, the so-called "long infinitive" which is nowadays a noun was used as an infinitive.

In the same way, the participle is used in nominalizations and in complex (perfective) tenses.

( We may assume, then, that the Supine/Participle stem encodes Aspect; its basic value could be considered to be perfectivity. Some contexts, however, may involve shifting to an atelic iterative-habitual reading.

As for the Voice value of this stem, there is a discussion in the Romanian literature, about its voice ambiguity, which may go in the sense of Aronoff (1994): supine has been considered as ambiguous between active and passive reading. But in fact, supine's properties lead to think that it is rather a non-active form (maybe a middle). Active reading is not possible unless the supine has nominal properties, i.e. in the prepositional context discussed above. In the other cases, if it does not have a clear passive reading (which holds for the supine reduced relatives), the active reading is associated with an arbitrary reading of the subject.

A suggestion that could be made with respect to the aspectual value of the perfective stem is that it may shift its aspectual nature according to the way of realizing its object (i.e., in the basic position inside or outside VP). For instance, in the past participle constructions, the object will be realized in its basic position and assigned strong structural Accusative case, the whole construction Aux + Participle being able to case-mark it. In other constructions with the traditional supine, the object would only be assigned weak case, having a predicatemodifier status. The two options are associated with a telic and atelic aspectual value respectively.

To summarize

We argued in this paper that the participle, ROOT+AT by itself is not [+N], [+V].

Participles need syntactic supporters -functional elements, i.e. auxiliaries or determiners, in order to receive a category. (Stative and Resultative) Participle, verbal "Supine", "Supine"

Event Nouns in Romanian are all syntactic realizations of the "3 rd stem". The AT-Stem (or PART, or the 3 rd stem) combines with different functional layers attributing categories: a, n or v. AT itself is not a, n or v.

Stems are categorially neutral and accessible to inflection and to derivation. This leads to the triple use of the Romanian Participle known as the Past Participle / Supine parallelism in Romanian. The view of the grammar which could fit the facts discussed in this paper is the one outlined in Distributed Morphology, where "mixed categories" are not needed. In such a framework, it is possible to have an analysis in which a single morphological piece corresponds to three linguistic units. The only device we would have to add is that stems can also be categorially neutral and represent starting points in word formation.

  hand, traditional grammars take some of the contexts of the Romanian participle (preceded by prepositions) to involve a distinct form of the verbal paradigm, called supine, considered to have a mixed [+N, +V] nature (cf. (

  these examples is the fact that the arguments are not assigned case, which in Romanian corresponds to the morphological case, or to the insertion of a "preposition".The problem disappears when the internal argument has the form of a NP in the Genitive or that of a PP adjunct: licensing of the object seems not to be a property of the supine form in itself. The possibility to assign strong accusative case appears only if there is more verbal structure around. This is the case for 11a, in which the supine combines with an

  favor of a verbal noun analysis comes from the possibility of case marking on the Direct Object by the Supine, and the co-occurrence of the nominal inflection on it. According toAronoff (1994), the Supine should be treated as the manifestation of the same Stem (in the strict morphological sense, at the "morphomic" level)

(

  27) a cînta 'sing'-cîntăt-or 'singer' b măguli 'flatter'-măgulit-or 'flatterer' c hotărî 'decide'-hotărît-or 'decisive'

  to go to the market place' There are also event nominalizations in which a value of completion is encoded, in cases where the object is assigned strong genitive case. (31) a cititul ziarului de dimineaţă reading-the newspaper-GEN of morning 'the reading of the newspaper in the morning b ?cititul de dimineaţă reading-the of morning 'reading in the morning'

There are, of course, marks of agreement in gender and number when the Participle is in an adjectival position, but no marks of declension for the Supine use.

We are not able to propose, at this moment, an analysis for both Latin and

Romanian, but we may simply retain Aronoff's suggestion for Romanian, that there is a single stem at work in the two cases. This stem can be used as a base as well for verb as for noun formation. In sum, none of these forms is basic; they are all derived from a single sound form -a stem.

Therefore, we would like to apply the same view to Romanian Participle and Supine, which, as we saw, are perfectly homophonous. The advantage would be that we would unify two categories of the non-finite verbal system of Romanian that seem to have all in common. But in this case, we would rather like to say that a single morphological element, categorially neutral, is used to build a Noun (the Supine), a Verb (in combination with an Auxiliary, see before), or an Adjective. This is the analysis proposed above, and it goes somehow in the same direction as Aronoff's discussion.

Abstract

This article proposes a general survey of a supposed mixed category in Romanian, namely the Supine. It is shown that what seems to be a mixed verbal-nominal nature is a contextdependent behaviour, determined by syntactic categories that may select a supine form. The core argument is that the two behaviours are associated with different projections. These