Saïda Bouakaz

Michel Vacher

Marc-Eric Bobillier-Chaumon

Frédéric Aman

Salima Bekkadja

François Portet

Erwan Guillou

Solange Rossato

Elodie Desserée

Pierre Traineau

M.-E Bobillier Chaumon

J.-P Vimont

T Chevalier

CIRDO: Smart companion for helping elderly to live at home for longer

Keywords: Ambient Assisted Living, Automatic recognition of aged voices, Scene analysis, Distress situation, User study

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Enabling seniors to stay in their own home as long as possible has become a major issue in developed countries for societal and economical motivations (improving well-being and autonomy, care and assistance). This concept, known as 'Ageing in place' has emerged due to the increase of life expectancy in all developed countries in the last decade. Indeed in the 2060s the number of people of 80 years old should be tripled [START_REF]Eurostat, Projections de population 2008-2060[END_REF]. In France, though the situation is less contrasted, 12 million people (out of 66 millions) received the Personalized Allocation of Autonomy (APA) 1 , where 61% of them were at home and 39% were in specialized institutions [START_REF] Bellamy | Bilan démographique 2012. la population croît, mais plus modérément[END_REF]. When a senior is loosing autonomy, the assistance from a carer(s) becomes essential. This carer role is often played by relatives (partner, children. . . ), who Email addresses: s❛✐❞❛✳❜♦✉❛❦❛③❅✉♥✐✈✲•②♦♥✶✳❢r (S. Bouakaz), ▼✐❝❤❡•✳❱❛❝❤❡r❅✐♠❛❣✳❢r (M. Vacher), ♠❛r❝✲❡r✐❝✳❜♦❜✐••✐❡r✲❝❤❛✉♠♦♥❅✉♥✐✈✲•②♦♥✷✳❢r (M.-E. Bobillier Chaumon) 1 This allowance from the state is given to people over 60 years old in loss of autonomy must ensure the care and often financial support. The challenge that our society must address is to enable seniors to age in place as autonomously and safely as possible while relieving some of the burden (stress, times, etc.) supported by the carers. This is the aim of the Ambient Assisted Living (AAL) research programs which intend to develop services and innovative products that improve the quality of life of older people, maintain their independence and quality of life in a normal living environment whereas seniors are more affected by physical or cognitive diseases. This is in this context that our project, called Cirdo, is developed.

The aim of the project is to develop an ubiquitous system in the home, involving audio and video processing, making it possible to automatically detect falls and distress situations to call for help if necessary. The expected impact of such project is the improvement of the safety feeling of the person in loss of autonomy. By contrast with current worn system, the person does not need to wear sensors 24h/day. Figure 1 illustrates the installation and the use of the system inside the home of a person. Moreover, if the person falls in the camera range, the system can react even if the person losses consciousness or is dizzy. If the person is out of the camera range, the person still has the option to call for help thanks to the audio sensors. Thus, the proposed system overcomes some of the limitation of the currently available solutions.

Several challenges must be addressed to achieve the project aims. Firstly, on a social aspect, though a large number of projects has been conducted in the domain, the fall process in the domestic setting is still difficult to model due to the variability of persons and home settings. Moreover, real acceptance of video/audio sensors is very variable according to the user's situation (does the intrusion worth the benefit ?). On the same line, not only the user's needs must be considered but also the feeling of their relatives and the professionals working in the home (house maiden, nurse, etc.). Secondly, on a computer science aspect, a number of issues must be addressed regarding video processing in varying light conditions, sound analysis with atypical users (aged voice) and noise, decision making with uncertain information (decision making from the inferences made by the video and audio processing), and Computer-Human interaction (does the system prompts the user and reacts properly?).

To tackle these issues, the project adopts a user-centred design [START_REF] Norman | User Centered System Design; New Perspectives on Human-Computer Interaction[END_REF] using a multi-disciplinary approach (partners includes computer scientists, social psychologists, field industrials, trainers of health professionals, members of a network of telehealth actors). The applied approach consisted in 1) modelling the fall process of aged people in their own home and to prospect the acceptability conditions of the Cirdo system by the users and their relatives 2) designing computational models on the basis of this study, 3) developing a complete monitoring system and 4) validating this system with the targeted users. This paper, presents the Cirdo approach and the main results of the studies conducted so far which led to the development of a monitoring system not only compatible with the specific needs of the users but also transferable in their own psychosocial life environment. It is organised as follow, after a short state of the art presented in Section 2, Section 3 details the results of the field study which led the specification of the Cirdo system. The technical aspects of this system are described in Section 4. To build inference models, an experiment has been conducted in realistic settings, involving targeted users and users simulating an aged body. This experiment is described in Section 5. Finally, the paper ends with a discussion of the results obtained so far and outlines the future work.

State of the art

The effect of ageing

The ageing process is characterised by a decay of the main bio-physiological functions, affecting the social role and the integration of the ageing person in the society [START_REF] Ogg | L'état des enquêtes sur l'entraide en europe[END_REF]. As a consequence, the person withdrawals from society to the home and her social and family role is weakened [START_REF] Rosow | Socialization to old age[END_REF], but, above all, she losses goals and identity (disengagement theory [START_REF] Cumming | Growing Old: The Process of Disengagement[END_REF]). However, a decrease of activities does not necessarily mean a disengagement in the remain ones [START_REF] Clément | Penser les vieillesses. Regards anthropologiques et sociologiques sur l'avancée de l'âge[END_REF]. Furthermore, through the ageing progression steps, these physiological and mental degradations are not irreversible. This progression can be normal, pathological, optimal or successful when the seniors succeed in adapting thereself to the changing situation. Thus, the emergence of disabilities is not originated only in the individual but also in the interaction with her environment. More precisely, it is the unfitness of the environment to the person entering into fragility that causes the incapacity situation [START_REF] Newel | User Sensitive inclusive Design[END_REF]. In that case, two approaches exist in the literature [START_REF] Ebersold | Le champ du handicap, ses enjeux et ses mutations : du désavantage à la participation sociale[END_REF]. An "integrative" approach which acts on the person (rehabilitation, devices. . . ) and a "participatory" approach in which the environment is adapted to the person in loss of autonomy. The Cirdo project by detecting falls and other risky situations without constraining the person to wear a specific device, is clearly situated in the latter approach.

Apart from the social aspect, ageing also has effects on the voice and movements of the person. Indeed, aged voice is characterized by some specific features such as imprecise production of consonants, tremors and slower articulation [START_REF] Ryan | Perceptual and acoustic correlates in the speech of males[END_REF]. From an anatomical point of view, some studies have shown age-related degeneration with atrophy of vocal cords, calcification of laryngeal cartilages, and changes in muscles of larynx [START_REF] Takeda | Aging effects on motor units in the human thyroarytenoid muscle[END_REF][START_REF] Mueller | Acoustic and morphologic study of the senescent voice, Ear, Nose[END_REF]. Thus, in the context of technology development aiming at helping aged people to live independently at home, the use of elderly voice can be an issue for performance of speech recognition systems. Some authors have reported that classical Automatic Speech Recognizers (ASR) systems exhibit poor performances with elderly voices because most acoustic models of ASR are acquired from non-aged voiced samples [START_REF] Baba | Acoustic models of the elderly for large-vocabulary continuous speech recognition[END_REF][START_REF] Vipperla | Ageing voices: The effect of changes in voice parameters on ASR performance[END_REF]. Apart from the physical change of voice induced by ageing many other challenges are to be tackled in a home such as noise caused by devices, emotive voice. For more information about these issues the reader is refereed to [START_REF] Vacher | Development of Audio Sensing Technology for Ambient Assisted Living: Applications and Challenges[END_REF].

The fall and its detection

In the literature, this domestic risky situation for an aged person is defined as: "inadvertently coming to rest on the ground, floor or other lower level, excluding intentional change in position to rest in furniture, wall or other objects" [START_REF]WHO global report on falls prevention in older age[END_REF]. Falls are the cause of 80% of daily life injuries and 9,300 deads per year in the French population above 65 years old. About a quarter of the persons between 65 and 75 years old declared to have felt in the year [START_REF] Inpes | Mieux prévenir les chutes chez les personnes âgées[END_REF]. It has been emphasized that the age, gender and health of the person can influence the gravity and the kind of injury [START_REF] Todd | A global report on falls prevention: reviews of socio-demographic factors related to falls and environmental interventions to prevent falls amongst older people living in the community[END_REF].

Given the serious consequences that can follow a fall, a number of projects have developed solutions to detect it as early as possible. In most cases, the authors propose the use of a worn sensor [START_REF] Bloch | Evaluation under real-life conditions of a stand-alone fall detector for the elderly subjects[END_REF]. A few projects have considered microphones to detect fall. It is often fused with other modalities like motion thanks to the use of an accelerometer [START_REF] Popescu | An acoustic fall detector system that uses sound height information to reduce the false alarm rate[END_REF][START_REF] Charalampos | Enabling human status awareness in assistive environments based on advanced sound and motion data classification[END_REF]. Doukas et al. [START_REF] Charalampos | Enabling human status awareness in assistive environments based on advanced sound and motion data classification[END_REF] proposed a patient awareness system to detect body fall and vocal stress in speech expression through analysis of motion data and acoustic, but in this case the person is constrained to wear sensors continuously. Another approach proposed a dialogue system in order to replace traditional emergency systems [START_REF] Hamill | Development of an automated speech recognition interface for personal emergency response systems[END_REF]. However, the vocabulary of the prototype was limited (yes or no) and the system was not tested with aged users. Another popular approach is the use of companion robots like in the CompanionAble project [START_REF] Milhorat | Companionable Hands-free speech-sound[END_REF]. The companion robot is able to move nearby the person so that the user is close to the sensor. The robot is able to recognize commands uttered in Dutch, no specific application to distress sentence recognition were reported by the authors. This system is limited by the capacity of the robot to move close to the person and we are not aware of studies reporting the transferability of such devices in the own home of the person. Recent studies have shown that audio sensor, though being intrusive, are accepted by the elderly population [24][25].

The originality of the Cirdo project lies in its combination of audio and video analyses together with a deep field study. Video processing is highly accurate to detect movements of the person but its action is limited to the camera range and the light condition while microphones can sense sounds outside the camera range and permits voice recognition but cannot detect person movement and is challenged by noise.

Field studies

As previously said, the fall process in the ageing people is often the conjunction of several "risky" factors of intrinsic type (related to fragility, people own apprehension. . . ) and extrinsic (environment) [START_REF] Ballinger | The construction of the risk of falling, among and by older people[END_REF] as well as sociocultural effects (personal or shared experience of the fall, life history. . . ) [START_REF] Bobillier-Chaumon | Can ict improve the quality of life of elderly adults living in residential home care units? from actual impacts to hidden artefacts[END_REF].

The objective of the field study is to analyse in depth the different fall cases in order to identify the key poses (behaviour, gestures and poses during the fall) and the key phrases (speech, sounds) characterising the risky situations that can be used to construct computational models. To do this, a number of questions guided the analysis: what are the daily living activities (ADL) performed at home by the seniors? What are the characteristics of the falls and the causing factors? On which descriptors the Cirdo system can be based to detect automatically these falls? What are the main priorities for the inclusion of the system both from a sensor point of view and a dissemination point of view (respect for private life vs. risky areas in the house)?

To do this, various methods have been deployed: semistructured interviews with 65 aged people, analysis of the activities of daily life conducted at seniors homes, analysis and script falls methods (recorded simulation and observations of falls and Personas method). From these, 12 "scripts" of typical falls have been constructed by describing on the one hand the fall context (person, activity been performed, location, moment, causes. . . ) and on the other hand the members of the body and their movements during the fall as well as the person's reaction once on the floor (try to get back on her feet, crawling. . . ) and the duration of the immobilisation. Figure 2 shows a script of a typical fall. Phrases uttered during the fall have also been identified "Aahhh, zuuut, qu'est-ce qui m'arrive? Oh merde, merde!". Using the Personas method, we defined scenarios which describe the context and the process of the falls to:

1. obtain a clear representation of the 28 identified falls so that a categorisation in 3 disjointed classes: falling, slipping and stumbling, was possible; 2. ask 20 participants to play a fall respecting these scenarios in order to record them in the controlled conditions of a living lab. Figure 3 describes an example of fall scenario obtained thanks to this method. The different components of the Cirdo system and their connection are described Figure 4. The video camera stream is processed by the Video module which extracts continuously the main positions and situations of the person as well as detect falls. The audio stream is processed by the Audio module which extracts continuously the speech and sound utterances in the home. These events are sent to the Fusion/Decision module which analyses the incoming events, detects risky situations and decides whether an emergency call must be emitted through the e-lio system (see http://www.technosens.fr/).

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Video analysis

Video capture in a private indoor space can benefit from some favorable operating conditions: not subject to vibration nor direct weather conditions. However, the daily living environments face specific difficulties which make the video analysis and the scene interpretation a hard task to accomplish. Among the most significant are: (i) changes in light levels between bright and dark areas variations caused by artificial light sources, (ii) sudden changes of lighting (turning on or off the indoor lights, a curtain that moves), (ii) foreground colors and patterns are close to background one (the problem of camouflage) as well as the lack of contrast between home furnishings, (iii) the variety of light sources, often artificial, which can generate shadows connected to objects and affect detection and identification of the subject of interest, (iv) the plethora of objects (furniture, ornaments, carpets, etc.) in the environment (living room, kitchen, bedroom, etc.) and some of which are likely to be moved [START_REF] Cucchiara | Computer vision system for in-house video surveillance[END_REF].

Cirdo is built in order to accomplish several tasks for monitoring the safety of elderly in home environments. The vision modules is responsible for both locating individuals present in the scene, tracking their movement and identifying some unusual events like falling, or a long period of inactivity. Given the application context, the cameras are static. The camera was then calibrated to determine its intrinsic parameters and extrinsic parameters with respect to the plane of the ground. OpenCV vision library provides efficient system calibration algorithms. Area inside the camera's field of view is as well estimated.

In order to comply with a strict code of ethics that fully protects user privacy and ensure an efficient detection, the video stream is acquired continuously and the process runs in background. The process chain is performed in automatic way. This means that the process runs without being connected to any server and doesn't require any operator intervention.

In this context we are interested in the detection of risky situations; with focus on detecting several types of fall. To this aim, as mentioned above we built scenarios simulating real life situations. Our video processing stage is structured as recognition and generating alert tag sent to the audio-video fusion process. Moreover, as with the audio processing, to respect the privacy of the person, the scenes that are not related to distress are removed. The first step performs background learning and subtraction for extracting the silhouette, which should be processed for identification or tracking. The difficult part of background subtraction is not the differencing itself, but the maintenance of a background model [START_REF] Toyama | Wallflower: principles and practice of background maintenance[END_REF] especially in case of moved objects, sudden or progressive changes of illumination or any other alterations of the appearance of the background. The model used is a variant of Stauffer and Grimson [START_REF] Stauffer | Adaptive background mixture models for real-time tracking[END_REF] method called Mixture Of Gaussian or MOG adapted for multi-level representation: pixel, region and image. On the first level, we use an enhanced Mixture of Gaussians, built on both chrominance-luminance and chrominance-only spaces, which handles global illumination changes [START_REF] Deeb | Real-time two-level foreground detection and person-silhouette extraction enhanced by body-parts tracking[END_REF].

On the second level, we improve segmentation results, in interesting areas, by using statistical foreground models updated by a high-level tracking of body parts. Each body part is represented with a set of template characterized by a feature vector built in an initialization phase. To improve tracking and identification of the posture, each body part is represented by a blob model. The approach starts with three body parts: head, upper part and lower part. Then analysis of upper and lower part adds new body parts when it is possible: arms and legs. Finally, high level tracking is done by finding blob-template correspondences via distance minimization in feature space. Correspondences are then used to update foreground models, and a graph cut algorithm [START_REF] Rother | grabcut": Interactive foreground extraction using iterated graph cuts[END_REF], which minimizes a Markov random field energy function containing these models, is used to refine segmentation and recognition [START_REF] Deeb | Extraction de silhouette et suivi des parties du corps dans un environnement non contrôlé[END_REF]. This step is reinforced by detecting and tracking the position of the user [START_REF] Deeb | Méthode robuste pour la détection de chute dans un environnement non contrôlé[END_REF]. To cope with the problem of real-time, the video processing module is improved with NVIDIA GPU cards with CUDA technology.

Audio module: automatic aged speech recognition

The audio module of the Cirdo project, CirdoX, is in charge of 1) detecting sound events; 2) discriminating between noise and speech and 3) performing aged speech recognition.

The module has been designed to operate as a background task with complete autonomy. It doesn't require any action from the elderly for initialization, settings and configuration. The system is continuously listening to the sound environment to recognize and detects any occurrence of speech that can be emitted during a distress situation. Anything that is not distress speech is rejected in order to preserve the privacy of the elderly. The system is operating in pseudo real time to trigger an alarm immediately after the recognition of a distress call. This service is intended for the elderly, it should therefore be adapted to the voice particularities of this part of the population. The system is designed for a user generic older voice model, without the need for an adaptation to speaker individually. Also, the stakes are that it is robust to the emotions when distress situations are occurring.

For enabling adaptation to aged speakers a large amount of aged speech is needed. However, there is no such corpus available in French. We thus performed our own data collection of French aged voice. However, this collection was insufficient to learn acoustic models from scratch. That is why we used acoustic models learned on typical speaker that we adapted using the corpus of aged voices. With this technique, the average Word Error Rate (WER) decreased to 14.52% compared with the 43.47% WER without adaptation. From an applicative point of view, this test shows that even though the size of the targeted user corpus is small, we can reach sufficiently good performance to use the audio module in Cirdo. For more details about the study, the reader is refered to [START_REF] Aman | Analysing the Performance of Automatic Speech Recognition for Ageing Voice: Does it Correlate with Dependency Level?[END_REF].

After a bibliographical study in collaboration with the GRePS laboratory and in the prolongation of previous studies [START_REF] Vacher | The SWEET-HOME Project: Audio Technology in Smart Homes to improve Well-being and Reliance[END_REF], we defined a list of home automation orders (intentional calls to the system) and of distress calls that the person could utter during a distress situation to request for assistance. This list was completed with casual sentences. At the beginning of the Cirdo project, a survey was conducted by the GRePS laboratory [START_REF] Bobillier-Chaumon | Démarche de développement de technologies ambiantes pour le maintien à domicile des personnes dépendantes : vers une triangulation des méthodes et des approches[END_REF] in order to determine what sort of sentences were truly uttered by aged people in distress case and during a fall and the collected sentences were added to the list. Ten samples of each kind (distress, home automation order, casual) are given in Table 1. To enable comparison of system performance between elderly and non-elderly, the corpus contains recordings made by these two groups of people. This corpus is called the AD80 corpus (Anodin-Détresse: anodin means casual and détresse means distress). The non-aged part of the corpus was previously recorded in our laboratory in 2004 and was complemented in 2013 with sentences based on [START_REF] Bobillier-Chaumon | Démarche de développement de technologies ambiantes pour le maintien à domicile des personnes dépendantes : vers une triangulation des méthodes et des approches[END_REF]. The collection of the aged part of the AD80 corpus was performed sporadically from 2009 to 2012 in collaboration with a rehabilitation centre, volunteers and a nursing home. Targeted speakers were persons aged of more than 60 years old. The recording was done with a single microphone positioned about 30 cm from the speaker's mouth. The recording was done using a computer and a home made software to prompt sentences to be read by the speaker and to record the utterances using voice activity detection.

The Automatic Speech Recognition (ASR) toolkit implemented in the module was Sphinx3 [START_REF] Seymore | The 1997 CMU Sphinx-3 English broadcast news transcription system[END_REF]. This decoder used a context-dependent acoustic model with 3-state left-to-right HMM. The acoustic vectors are composed of 13 MFCC coefficients, the delta and the delta delta of each coefficient.

A general language model (LM) was estimated from French newswire collected in the Gigaword corpus. It was 1-gram with 11,018 words. Moreover, to reduce the linguistic variability, a 3-gram domain language model was learnt from the sentences used during the corpus collection, with 88 1-gram, 193 2-gram and 223 3-gram models. Finally, the language model was a 3gram-type which results from the combination of the general language model (with 10% weight) and the domain one (with 90% weight). This combination has been shown as leading to the best WER for domain specific application [START_REF] Lecouteux | Distant Speech Recognition in a Smart Home: Comparison of Several Multisource ASRs in Realistic Conditions[END_REF]. The interest of such combination is to bias the recognition towards the domain LM but when the speaker deviates from the domain, the general LM makes it possible to correctly recognise the utterances.

To discriminate distress sentences from other sentences (e.g., personal communications) each ASR hypotheses are compared with the list of distress sentences at a phonetical level (see Table 1) using the Levenshtein distance. Informally, the Levenshtein distance between two words is the minimum number of single-character edits (insertion, deletion, substitution) required to change one word into the other.

In order to assess the filtering, we realized a decoding with the 43 elderly speakers from the AD80 corpus, including 2796 distress/home automation sentences and 3006 casual sentences, for a total duration of 2 hours 12 minutes. The casual utterances were used as disrupter, with some sentences far from the distress ones: for instance Les patates sont cuites (Potatoes are cooked), or closer, for instance Le médecin a appelé (The doctor called). The average WER with the adapted acoustic model was 14.5% for the distress sentences, and was much higher for the casual sentences, 87.5%, due to the adapted-to-distress language model. At the point of equal error, we obtained the positive and negative test showed in table 2, and recall, precision and F-measure equal to 88.4, 86.9 and 87.2% .

Data fusion/Decision module

The data fusion module is in charge of sending an alarm when the video module detects a fall or when the last stage of the audio module identifies a distress call or an home automation order. This module acquires continuously the events sent by the video and audio modules and maintains the knowledge of the situation to provide context for the decision. Variables such as the number of persons in the camera range, the luminosity of the room and the audio energy in the room can enable the characterisation of the situation. For instance, when the luminosity in the room is low, the information coming from the video module will be less reliable so that decision would mostly be based on audio processing. Once a distress event is identified, a message is sent to the communication module that will handle the emergency call.

Communication module

The communication module is actually a commercial product named e-lio made by the Technosens company. In the house of the elderly person, this equipment, composed of video camera, a central unit and a 3 buttons remote control which is a phone as well, is connected to the user's televisions and gives her access to a lot of connected TV services such as: television, phone calls, video calls, radio, pictures sent by family, games. The system has also an emergency mode that handles the connection with a relative till it reaches a carer (in case the first contact did not answer the call).

Experiments for Corpus Collection

To develop the algorithms needed for the audio and video automatic detection of falls in the home, experiments were conducted in a smart room configured to look like a standard room (chairs, carpet, coffee table, TV. . . ). These experiments aimed at collecting significant number of samples of data to learn and test the algorithms by asking targeted users to replay the scenarios that were defined in the field study (cf. Section 3).

Experimental protocol

The protocol was as follows. Each participant was introduced to the context of the research and was invited to sign a consent form. The participants played five types of fall chosen from the 28 risky situations identified (1 slip, stumble 1, 2 falls in a stationary position and a position of hip blocked on the sofa). Figure 7a shows a excerpt of a participant simulating a fall. These situations were selected because they were representative falls in the home and because they could safely be played by the participants. Two other scenarios, called "truefalse", were added to challenge the automatic detection of falls.

The first "true-false" consisted in rapidly picking up magazines on the floor (close to a situation of fall) while the second was to try to catch a remote control on a coffee table when the person is sitting on the sofa (close to the situation in which the person has a blocked hip).

Before each scene to play, the experimenter explained the scene to the participant specifying the context (location, moment of the day. . . ), the activity (what the person was being to do and what she wanted to do when the incident occurred) and the gestures and words produced before, during and after the fall. Then, the person was asked to repeat several times the scene before starting the camera and microphone recording. In average, the experiment lasted 2h30 per person.

Materials

The experimental room and the equipment used are depicted Figure 8.

Regarding audio analysis, we conducted a multi-source capture with two wireless Sennheiser microphones. A eW-300-G2 type ME2 microphone was placed in the suspended ceiling of the experiment room, and a SKM-300-G2 microphone was placed on a furniture close to the subject. The recording was performed in the technical room by a computer connected to the microphones high-frequency receivers thanks to a National Instruments PCI-6220 8-channel card. A loudspeaker in the control room enabled experimenters to hear the progression of the scene in the smart room. The National Instruments PCI-6220 card was controlled by the StreamHIS software. This software enables data acquisition and was developed by the GETALP team.

Regarding video analysis, we conducted the video capture with two cameras (webcam Sony PSEye: 640x480 60Hz connected by a USB USB 2.0) fixed to the wall of the smart room in front of the sofa and pointing towards it, and a depth camera (Microsoft Kinect). Synchronization and recording of video streams were performed in the technical room, by a computer Intel i3 3.2GHz, 4GB RAM, NVidia GeForce GTS 450 512MB. The tools enabling video acquisition was developed within the team SAARA (based the OpenCV library). Video processing (background learning and subtraction, body parts segmentation) is performed on a PC cluster, taking advantage of algorithms parallelization on CPU and GPU. Motion tracking and identification of hazardous situations is done on a PC (Intel i7 3.5GHz, 4GB RAM, NVIDIA GTX 660 4GB GeFore).

Corpus

The targeted participants were elderly person that were still able to play the scenario defined in Section 3. However, the recruitment of such population is a long task. So, given that the aim was mostly to collect data that were realistic but not necessary real, people under 60 were recruited invited to wear an equipment (see Figure 7b) which hampered their mobility and reduced their vision and hearing to simulate aged physical conditions.

Overall, 17 participants were recruited (9 men and 8 women) with mean age of 40 years old (SD 19.5). Among them 13 people were under 60 and worn the simulator. The persons of the aged group were between 61 and 83 years old (mean 68.5). The details relative to the corpus composition are given Table 3. Scenarios are relative to fall (F1 to F4), blocked hip (B) and True Negative (TN). The necessary time for achieving one scenario was twice for the aged group.

In order to make the corpus exploitable, the team is currently annotating the distress situation in the data. This is made using the Advene software 2 developed at the LIRIS laboratory. Ad- vene allows to organize annotations elements -such as type of annotations, annotations, and relationships -under schemes defined by the user. In this work, a unique schema is used which contains the following annotation types: fall period and distress situation, and now we plan to extend it to include information about video and speech features.

Conclusion

This paper summarizes the research results achieved within Cirdo project which is focused on an innovative tools to facilitate safety of elderly living in domestic environments and caregivers. The project brought together several partners which are actually complementary, involving multidisciplinary studies, including sensors acquisition, audio processing, video analysis and humanities. In particular, the proposed solution have tackled both the algorithmic and architectural technologic solution and taking into account the human factors (acceptability, ethic). The aim of Cirdo project is to simplify access to equipment daily taking into account the needs and uses of elderly people. Studies conducted by the partners from humanities allow Cirdo to establish a link between the needs, people and technology, and proposes solutions by gathering knowledge from different domain.

The Cirdo project's key success is combining engineering sciences and the humanities to propose novel solution to help elderly people and persons with impaired. On the technical side, Cirdo proposes an original solution which relies on a fusion of audio and video data. In order to respect privacy and preserve personal data the processing is achieved without any external human intervention. The developed system will be integrated into the e-lio service platform.

To deal with real time constraint the video processing is done on GPU, allowing quick reaction to distress situation and in particular the fall cases. Regarding audio analysis, we focused initially on the recognition of the older voices, but affects also have an influence on the performance of the ASR [START_REF] Aman | How affects can perturbe the automatic speech recognition of domotic interactions[END_REF]. Further studies are still ongoing and we are currently recording a corpus adapted to this situation.

additional information

• Conflict of interest: none

Figure 1 :

 1 Figure 1: Global architecture of the Cirdo system.

Figure 2 :

 2 Figure 2: Different phases of a fall

Figure 3 :

 3 Figure 3: Fall scenario using the Personas method

Figure 4 :

 4 Figure 4: Global architecture of the Cirdo system.

Figure 5 .

 5 It based on the extraction of the silhouette vs. the separation between foreground and background. The extracted silhouette is used to obtain useful characteristics for detection of movements events "at risk". Tracking the movement of this figure allows us to identify a fall. The treatment consists of four steps: (1) learning the background, (2) extraction of the silhouette, (3) analysis of posture and motion tracking, (4) distress posture/fall

Figure 5 :

 5 Figure 5: Video analysis : the overall system

Figure 6 :

 6 Figure 6: distress situation when the person can not sit up (left sequence), Video analysis result: fall detection (left sequence) (a) initial record, (b) discrimination between the background and foreground, (c) extraction of silhouette (d) situation identification

Figure 7 :

 7 Figure 7: Snapshots of the experiment.

Figure 8 :

 8 Figure 8: Experimental room and equipment.

Table 2 :

 2 F-measure of the distress sentence detection task.

	Threshold = 0.7	Distress	Casual
	Distress	TP = 2472	FP = 374
	Casual	FN = 324 TN = 2632

Table 1 :

 1 Je saigne ! Je me suis blessé ! e-lio appelle les secours ! Ce livre est intéressant. Examples of sentences of the AD80 corpus

	Sample Distress Sentence	Home Automation Order	Casual Sentence
	1	Qu'est-ce qui m'arrive !	Appelle quelqu'un e-lio !	Bonjour madame !
	2	Oh là !	e-lio, appelle quelqu'un !	Ça va très bien.
	3 Oh là ! 4 Aïe ! J'ai mal !	e-lio appelle ma fille !	Il fait soleil.
	5	Je peux pas me relever !	e-lio, appelle du secours !	J'ai ouvert la porte.
	6	Aidez-moi !	e-lio appelle le SAMU !	Je dois prendre mon médicament !
	7	Au secours !	e-lio appelle les pompiers !	J'allume la lumière !
	8	Je me sens mal !	e-lio appelle mon fils !	Je me suis endormi tout de suite !
	9	Je suis tombé !	e-lio tu peux téléphoner au SAMU ?	Le café est brûlant !
	10	Du secours s'il vous plaît !	e-lio tu peux appeler une ambulance ? Où sont mes lunettes ?

Table 3 :

 3 Audio/Video Corpus for the two group of participants

	2 http://liris.cnrs.fr/advene/

✩ This study was supported by the French funding agencies ANR and CNSA through the project CIRDO -Industrial Research (ANR-2010-TECS-012). The authors would like to thanks the persons who agreed to participate in the survey or in the recordings.


Christophe Hazard
email: christophe.hazard@ensta-paristech.fr

On the absence of trapped modes in locally perturbed open waveguides

Keywords: Open waveguide, Helmholtz equation, generalized Fourier transform

This paper presents a new approach for proving that the presence of a bounded defect in a uniform open waveguide cannot produce trapped modes, contrary to the case of a closed waveguide. The originality of the proof lies in the fact that it relies on a modal decomposition. It shows in particular that the absence of trapped modes results from a strong connection between the various modal components of the field. The case of the three-dimensional scalar wave equation is considered.

Introduction 1.About trapped modes

In a non-dissipative propagative medium filling an unbounded part of the space, trapped modes, also referred to as resonance modes or bound states, designate localized free oscillations of the medium which do not radiate towards infinity. More precisely, these are time-harmonic solutions of the propagation equations which have a finite energy that remains confined in a bounded region despite the unboundedness of the propagative medium. In this paper, we wonder whether such trapped modes can occur in locally perturbed open waveguides.

A uniform cylindrical waveguide is a device whose physical features are invariant in one space direction, called here longitudinal, and which is able to propagate waves over long distances in this direction. Such a waveguide is called closed if it is bounded in the transverse directions by an impenetrable wall, and open if its cross-section can be considered as unbounded, as for instance optical fibers, immersed pipes, or other guiding devices embedded into a propagative matrix. Actually the term 'waveguide' is justified by the existence of guided waves, that is, time-harmonic waves that propagate without attenuation in the longitudinal direction and remain confined in the other transverse directions. For a closed waveguide, such a confinement is simply due to the boundedness of the cross-section. But for an open waveguide, this confinement results from a particular layout of the various materials which compose the waveguide. Numerous papers are devoted to the existence and the properties of guided modes for acoustic, elastic or electromagnetic open waveguides (see, e.g., [START_REF] Bonnet-Ben Dhia | Mathematical analysis of elastic surface waves in topographic waveguides[END_REF][START_REF] Bonnet-Ben Dhia | Mathematical analysis and numerical approximation of optical waveguides[END_REF] and references cited therein).

Here we are rather interested in trapped modes, which remain confined in both transverse and longitudinal directions. Trapped modes generally do not occur in cylindrical uniform waveguides, simply because there is nowhere to confine energy. More exactly, because of the longitudinal invariance, if a trapped mode existed, any translation of this mode in the longitudinal direction would remain trapped, which would yield an infinite-dimensional space of trapped modes. This may occur in very exceptional situations which are not considered here (for instance at the interface between a dielectric and a metamaterial [START_REF] Gralak | Negative index materials and time-harmonic electromagnetic field[END_REF]). In fact, a natural way to allow the energy confinement is to break the longitudinal invariance 2 by introducing a local perturbation, that is, a bounded defect, in a uniform cylindrical waveguide. For closed waveguides, it is now well understood that such local perturbations may produce trapped modes (see [START_REF] Linton | Embedded trapped modes in water waves and acoustics[END_REF] for a review). In a similar vein, there is an extensive literature about the occurrence of trapped modes in curved closed waveguides (see, e.g., [START_REF] Duclos | Curvature-induced bound states in quantum waveguides in two and three dimensions[END_REF][START_REF] Dauge | Plane waveguides with corners in the small angle limit[END_REF]). Those who are familiar with closed waveguides could be surprised to discover that the case of open waveguides differs singularly. Actually the pioneer work of Weder [START_REF] Weder | Absence of eigenvalues of the acoustic propagator in deformed wave guides[END_REF][START_REF] Weder | Spectral and Scattering Theory for Wave Propagation in Perturbed Stratified Media[END_REF] shows that no trapped mode can occur in locally perturbed stratified media. This result was then extended to more general situations in [START_REF] Debièvre | Spectral analysis for optical fibres and stratified fluids II: Absence of eigenvalues[END_REF], using an abstract argument which became very popular during the last decades in the mathematical physics community: the Mourre's commutator theory.

The purpose of the present paper is to explore an alternative proof for the absence of trapped modes in a locally perturbed open waveguide. The final result is not new but the proof is. Its originality lies in the tools we use, which are natural and usual tools for physicists, contrary to the abstract technique proposed in [START_REF] Debièvre | Spectral analysis for optical fibres and stratified fluids II: Absence of eigenvalues[END_REF]. Actually the proof proposed here relies on a modal decomposition which consists in representing any wave as a superposition of modes. It shows in particular that the absence of trapped modes results from a strong connection between the various modal components.

The mathematical model

We consider acoustic waveguides filling the whole space R 3 equipped with an orthonormal coordinate system (x, y 1 , y 2 ), where x ∈ R represents the longitudinal direction, that is, the privileged direction of propagation, and y := (y 1 , y 2 ) ∈ R 2 , the transverse directions. We are interested in the solutions u = u(x, y) to the time-harmonic acoustic wave equation

-∆u -k 2 u = 0 in R 3 , (1)
where k = k(x, y) is a real wavenumber function assumed such that

0 < k m ≤ k(x, y) ≤ k M < +∞ ∀(x, y) ∈ R 3 , (2)
for two positive constants k m and k M .

In the particular case of a uniform cylindrical waveguide, this wavenumber function, denoted by k uni , depends only on the transverse variable and we suppose moreover that k uni is constant outside a bounded region, i.e.,

k uni = k uni (y) with k uni (y) = k ∞ > 0 if |y| := (y 2 1 + y 2 2 ) 1/2 > d (3)
for some d > 0. The inhomogeneous region represents the core of the waveguide. The above assumption means that the surrounding cladding is homogeneous. We are interested in the case where k is a localized perturbation of such a function k uni in the sense that k -k uni is compactly supported.

Such a function k describes a uniform waveguide perturbed locally by the presence of a bounded penetrable defect (see Figure 1). Trapped modes are defined as non-zero finite-energy solutions to [START_REF] Aubin | Applied Functional Analysis[END_REF], which amounts here to assuming that u ∈ L 2 (R 3 ), where L 2 denotes the usual space of square integrable functions. Our aim is to prove the following theorem, which asserts the absence of such modes.

Theorem 1. Let k and k uni satisfy assumptions (2), ( 3) and (4). Then the only solution u ∈ L 2 (R 3 ) to the Helmholtz equation (1) (in the distribution sense) is u ≡ 0.

Guiding ideas of the proof

The scheme of the proof follows and extends the approach proposed in [START_REF] Bonnet-Ben Dhia | Diffraction by a defect in an open waveguide : a mathematical analysis based on a modal radiation condition[END_REF] for studying the scattering by a defect in a two-dimensional step-index waveguide. The keystone of this approach lies in the use of a modal decomposition of the acoustic field in the transverse directions. Such a decomposition is often used for closed waveguides. In this case, the acoustic field appears as the sum of a finite superposition Figure 1: A locally perturbed uniform waveguide (possibly empty) of guided modes and an infinite superposition of evanescent modes (see, e.g., [START_REF] Vassalo | Optical Waveguide Concepts[END_REF]). On the other hand, for open waveguides, such a modal decomposition still exists but it involves a continuum. The acoustic field appears as the sum of a finite superposition of guided modes, which are evanescent in the transverse directions, and a continuous superposition of radiation modes, which are so called because they are oscillating in the transverse directions. This continuum actually consists of two parts according to the longitudinal behavior of the modes, which can be either oscillating or evanescent. Those which are oscillating can be classified within a category of propagative modes which also contains the finite set of guided modes, since both kinds propagate without attenuation in the longitudinal direction.

When looking at the modal decomposition of a trapped mode, a simple energy argument tells us that outside any transverse section which contains the defect of our waveguide, all the modal components associated with these propagative modes must vanish. In fact, if some of these components were not zero, the energy of the trapped mode would be infinite, since propagative modes carry energy towards infinity. Then a more subtle argument enters the scene: because of the boundedness of the defect, the modal components extend to an analytic function of the modal parameter (which is related to the longitudinal wavenumber). It seems difficult to propose a physical interpretation of this analyticity property which provides us the key to the riddle. Indeed, as we already know that the modal components vanish on a continuous interval, analyticity implies that they must vanish everywhere: the modal components associated with evanescent modes are also zero. We conclude that the only finite-energy solution to the propagation equation is zero: trapped modes do not exist.

It is worth noticing that this approach cannot be used for a closed waveguide. In this case, the energy argument shows as above that the modal components associated with propagative modes must vanish. But these components compose a finite set, so the analyticity argument cannot apply. Actually, nothing can be said about the components associated with evanescent modes. This is precisely why trapped modes can occur in closed waveguides. To a certain extent, in an open waveguide, the above analyticity property means that the propagative and evanescent components are connected in a subtle but strong way, whereas they are independent in a closed waveguide.

This very powerful argument of analyticity seems to have been used for the first time by Littman [START_REF] Littman | Spectral properties of the Laplacian in the complement of a deformed cylinder[END_REF] for proving the absence of eigenvalues in the continuous spectrum of the Laplacian outside a deformed cylinder. Shortly after, it was used independently by Weder [START_REF] Weder | Absence of eigenvalues of the acoustic propagator in deformed wave guides[END_REF] for proving the absence of trapped modes for a perturbed stratified medium. The approach proposed by Weder is based on the usual Fourier transform in the direction of the layers. In [START_REF] Bonnet-Ben Dhia | Diffraction by a defect in an open waveguide : a mathematical analysis based on a modal radiation condition[END_REF], we adopted a similar but orthogonal point of view. Indeed the above mentioned modal decomposition amounts to using a generalized Fourier transform in the transverse direction, which simplifies considerably the proof and allows to deal with more complex situations such as the junction of two open waveguides, as shown in [START_REF] Bonnet-Ben Dhia | Mathematical analysis of the junction of two acoustic open waveguides[END_REF]. In the latter papers, we investigated the time-harmonic scattering problem, not only the question of absence of trapped modes (which appears as a by-product of the solvability of the scattering problem). But for a technical reason which is related to the possibility of extending the generalized Fourier transform to non-L 2 fields [START_REF] Hazard | A distribution framework for the generalized Fourier transform associated with a Sturm-Liouville operator[END_REF], we restricted ourselves to very simple two-dimensional step-index open waveguides. In the present paper, the fact that we are interested only in trapped modes allows us to deal with more general waveguides, for which the question of the solvability of the scattering problem remains open.

The paper is organized as follows. In section 2, we introduce the main ingredients which provide us the modal decomposition of the acoustic field. The proof of Theorem 1 is the subject of section 3. We finally present in section 4 some possible extensions of Theorem 1.

In the whole paper, √ • denotes the following determination of the complex square root:

ζ := |ζ| 1/2 e i(arg ζ)/2 for ζ ∈ C with -3π/2 < arg ζ ≤ π/2. ( 5
)
2 Modal analysis of a uniform waveguide

Spectrum of the transverse Helmholtz operator

As mentioned in the introduction, our approach is mainly based on a modal analysis of a uniform open waveguide which allows us to represent a time-harmonic wave as the sum of a finite superposition of guided modes and a continuous superposition of radiation modes. These modes are simply obtained by the method of separation of variables applied to the Helmholtz equation ( 1) with k replaced by k uni .

Setting u(x, y) = ϕ(y) exp(γx) for some complex number γ, we are led to find λ := γ 2 ∈ C and a non-zero function ϕ defined in R 2 such that

-∆ y ϕ -k 2 uni ϕ = λ ϕ in R 2 ,
where ∆ y denotes the two-dimensional Laplace operator in the y-plane. In other words, we search for the eigenelements of the unbounded selfadjoint operator A defined in L 2 (R 2 ) by

Aϕ := -∆ y ϕ -k 2 uni ϕ ∀ϕ ∈ D(A) := H 2 (R 2 ), (6)
which represents the transverse part of the Helmholtz operator. Here, H 2 denotes the usual Sobolev space of order 2. It is well-known (see, e.g., [START_REF] Bonnet-Ben Dhia | High-frequency asymptotics of guided modes in optical fibers[END_REF]) that the spectrum Λ of A is composed of two parts: a continuous spectrum Λ c := [-k 2 ∞ , +∞) and a finite point spectrum Λ p which is non-empty if and only if

k * := sup y∈R 2 k uni (y) > k ∞ , (7)
which means that somewhere in the core, the wave speed is smaller than in the cladding. In this case, we have

Λ p ⊂ (-k 2 * , -k 2 ∞
) and every eigenvalue λ ∈ Λ p has a finite multiplicity.

Remark 2. Inequality (7) is a necessary and sufficient condition to ensure the existence of guided modes, which legitimate the use of the word 'waveguide' throughout this paper. However this assumption is not crucial in our analysis. Our results hold true for non-guiding devices.

What about eigenfunctions? On the one hand, if λ ∈ Λ p , the associated eigenfunctions ϕ belong by definition to H 2 (R 2 ) : they actually decay exponentially as |y| → ∞. The corresponding modes ϕ(y) exp(± √ λ x) are called guided modes for they propagate without attenuation in the longitudinal direction x (since √ λ is purely imaginary) whereas in the transverse directions, they remain confined near the core. On the other hand, if λ ∈ Λ c , the associated solutions ϕ are usually called generalized eigenfunctions for they no longer belong to L 2 (R 2 ) (otherwise, λ would be an eigenvalue). Actually, these functions are oscillating as |y| → ∞. Because of this transverse behavior, the corresponding modes ϕ(y) exp(± √ λ x) are referred to as radiation modes. They are propagative in the longitudinal direction if λ ∈ (-k 2 ∞ , 0) but exponentially increasing or decreasing if λ > 0. It seems natural to wonder if the knowledge of the above mentioned eigenfunctions and generalized eigenfunctions allows us to diagonalize our operator A. More precisely, can one build a generalized spectral basis associated with A, that is, a complete family of eigenfunctions and generalized eigenfunctions in which A becomes diagonal. Although the signification of both words 'complete' and 'diagonal' is somewhat imprecise at this stage, one can assert that such a construction is always possible for any selfadjoint operator in a Hilbert space. This result can be seen as an advanced version of the spectral theorem [START_REF] Berezansky | [END_REF]. However the practical construction of a generalized spectral basis highly depends on the operator and its rigorous justification is in general a difficult task. In order to understand such a construction in our situation, we begin with the case of a homogeneous medium which easily derives from the usual Fourier transform.

Diagonalization: the homogeneous case

Consider the operator A ∞ defined as in [START_REF] Bonnet-Ben Dhia | Mathematical analysis of the junction of two acoustic open waveguides[END_REF]

with k uni replaced by k ∞ , that is, A ∞ := -∆ y -k 2 ∞ .
Of course its spectrum is purely continuous and equal to Λ c = [-k 2 ∞ , +∞). Let F denote the usual Fourier transform in the y-plane defined by

Fϕ(ξ) := 1 2π R 2 ϕ(y) e -iξ•y dy ∀ξ ∈ R 2 , (8)
which is unitary from L 2 (R 2 y ) to L 2 (R 2 ξ ). We clearly have

(FA ∞ ϕ)(ξ) = (|ξ| 2 -k 2 ∞ ) Fϕ(ξ) ∀ϕ ∈ D(A ∞ ) := H 2 (R 2 ). (9)
It is then natural to introduce the change of variables

R 2 ∋ ξ → (λ, κ) := |ξ| 2 -k 2 ∞ , ξ |ξ| ∈ Λ c × S 1 ,
where S 1 denotes the unit circle in R 2 . Noticing that

R 2 |ψ(ξ)| 2 dξ = 1 2 +∞ -k 2 ∞ S 1 ψ κ k 2 ∞ + λ 2 dλ ds κ ∀ψ ∈ L 2 (R 2 ξ ),
we infer that the transformation C defined by

ψ(ξ) → Cψ(λ, κ) := 1 √ 2 ψ κ k 2 ∞ + λ (10) is unitary from L 2 (R 2 ξ ) to L 2 (Λ c × S 1 ). Hence, U ∞ := C F is unitary from L 2 (R 2 y ) to L 2 (Λ c × S 1
). Moreover, we deduce from (9) that

A ∞ ϕ = U -1 ∞ λ U ∞ ϕ ∀ϕ ∈ D(A ∞ ),
where λ stands here for the operator of multiplication by λ in L 2 (Λ c × S 1 ). The latter formula means that U ∞ diagonalizes A ∞ in the spectral space L 2 (Λ c × S 1 ). From ( 8) and [START_REF] Debièvre | Spectral analysis for optical fibres and stratified fluids II: Absence of eigenvalues[END_REF], this unitary operator can be rewritten in the convenient form

U ∞ ϕ(λ, κ) = R 2 ϕ(y) Φ ∞ λ,κ (y) dy where (11) Φ ∞ λ,κ (y) := 1 √ 8π exp i k 2 ∞ + λ κ • y ∀(λ, κ) ∈ Λ c × S 1 . (12)
Function Φ ∞ λ,κ represents a plane wave of direction κ. It is a generalized eigenfunction of A ∞ associated with the spectral value λ in the sense that it satisfies

(-∆ y -k 2 ∞ ) Φ ∞ λ,κ = λ Φ ∞ λ,κ in R 2 , (13)
but it does not belong to L 2 (R 2 ). And U ∞ appears as the operator of 'decomposition' on the family {Φ ∞ λ,κ ; (λ, κ) ∈ Λ c × S 1 } which can be seen as a generalized spectral basis for A ∞ .

Diagonalization: the perturbed case

We come back now to the transverse operator A defined in [START_REF] Bonnet-Ben Dhia | Mathematical analysis of the junction of two acoustic open waveguides[END_REF]. The formal construction of a generalized spectral basis for A simply consists in a perturbation of our generalized spectral basis for A ∞ , where each Φ ∞ λ,κ defined in ( 12) is interpreted as an incident plane wave which is scattered by the core. Hence we search for generalized eigenfunctions of A in the form

Φ λ,κ := Φ ∞ λ,κ + Φ sc λ,κ for (λ, κ) ∈ Λ c × S 1 , (14)
where Φ sc λ,κ stands for the associated scattered wave, which is the solution to

-∆ y Φ sc λ,κ -(k 2 uni + λ) Φ sc λ,κ = (k 2 uni -k 2 ∞ ) Φ ∞ λ,κ in R 2 , ( 15
) ∂Φ sc λ,κ ∂|y| -i k 2 ∞ + λ Φ sc λ,κ = O(|y| -3/2 ) as |y| → ∞. (16)
Recall that the outgoing radiation condition ( 16) ensures the uniqueness of the solution Φ sc λ,κ to (15) (see, e.g., [START_REF] Colton | Inverse Acoustic and Electromagnetic Scattering Theory[END_REF]). Note that instead of this condition, one could choose the incoming condition obtained by replacing -i by +i : the results given below hold true. Each Φ λ,κ is a generalized eigenfunction of A associated with λ for it satisfies by construction

(-∆ y -k 2 uni ) Φ λ,κ = λ Φ λ,κ in R 2 , but it does not belong to L 2 (R 2 ). We actually have Φ λ,κ ∈ H 2 loc (R 2 ), i.e., Φ λ,κ | O ∈ H 2 (O) for all bounded O ⊂ R 2 .
When the point spectrum Λ p is not empty (see Remark 2), this continuous family of generalized eigenfunctions must be completed by a finite family of eigenfunctions which simply consists of the union of orthonormal bases of the various eigenspaces. For the sake of conciseness, we use the same notation Φ λ,κ for both families: if λ is an eigenvalue of multiplicity m λ , we denote by Φ λ,κ ∈ H 2 (R 2 ) for κ = 1 to m λ , an orthonormal basis of the associated eigenspace. This leads us to consider the completed family defined by

{Φ λ,κ ; λ ∈ Λ, κ ∈ K λ } where K λ := {1, . . . , m λ } ⊂ N if λ ∈ Λ p , S 1 if λ ∈ Λ c .
The fact that this family constitutes a generalized spectral basis can be formulated by considering the analogue of operator U ∞ defined in [START_REF] Duclos | Curvature-induced bound states in quantum waveguides in two and three dimensions[END_REF], that is, the operator of 'decomposition' on this family, given by

U ϕ(λ, κ) := R 2 ϕ(y) Φ λ,κ (y) dy ∀λ ∈ Λ, ∀κ ∈ K λ . (17)
If λ ∈ Λ p , the above integral is simply the L 2 inner product of ϕ by the eigenfunction Φ λ,κ ∈ L 2 (R 2 ). On the other hand, if λ ∈ Λ c , this definition does not make sense for all ϕ ∈ L 2 (R 2 ) since Φ λ,κ no longer belongs to L 2 (R 2 ). It applies for instance to compactly supported ϕ. As for the usual Fourier transform, a density argument shows that this definition extends to all functions ϕ ∈ L 2 (R 2 ). This result is part of the following proposition, which collects the main properties of U . Its proof dates back to the early seventies [START_REF] Shenk | Eigenfunction expansions and scattering theory for perturbations of -∆[END_REF] and can be generalized for numerous scattering problems [START_REF] Hazard | Generalized eigenfunction expansions for conservative scattering problems with an application to water waves[END_REF].

Proposition 3. (i)

The operator U defined by [START_REF] Littman | Spectral properties of the Laplacian in the complement of a deformed cylinder[END_REF] for compactly supported ϕ ∈ L 2 (R 2 ) extends (by density) to a unitary transformation from L 2 (R 2 ) to the spectral space

H := H p ⊕ H c where H p := λ∈Λp C m λ and H c := L 2 (Λ c × S 1
).

(ii) The inverse transform U -1 appears as the operator of 'recomposition' on the family {Φ λ,κ ; λ ∈ Λ, κ ∈ K λ } : for all ϕ ∈ H,

U -1 ϕ = λ∈Λp m λ κ=1 ϕ(λ, κ) Φ λ,κ + Λc×S 1 ϕ(λ, κ) Φ λ,κ dλ ds κ . (iii) U diagonalizes A in the sense that Aϕ = U -1 λ U ϕ ∀ϕ ∈ D(A), ( 18
)
where λ denotes here the operator of multiplication by λ in H.

Operator U is often called a generalized Fourier transform associated with A. Item (i) shows that the perturbation process described above yields a generalized spectral basis of L 2 (R 2 ). Item (ii), which is not used in the present paper, extends the well-known expression of the usual inverse Fourier transform. And item (iii) provides the key for a functional calculus of A since any function of A can be simply defined by f (A) = U -1 f (λ) U .

Analyticity of the generalized eigenfunctions

In addition to the latter proposition, we shall need a basic property of the generalized eigenfunctions expressed as follows.

Proposition 4. Let Ω be a bounded open subset of R 2 and κ ∈ S 1 . The function (-k 2 ∞ , +∞) ∋ λ → Φ λ,κ | Ω ∈ L 2 (Ω) extends to a meromorphic function of λ in the complex half plane Re λ > -k 2 ∞ .
And the same holds for the function

(-k 2 ∞ , +∞) ∋ λ → Φ λ,κ ∈ L 2 (Ω).
Proof. For λ ∈ (-k 2 ∞ , +∞), let G λ denote the outgoing Green's function associated with equation ( 13), which is given by

G λ (y) := i 4 H (1) 0 k 2 ∞ + λ |y| ,
where H

(1) 0

is the Hankel function of the first kind and order 0. It is well-known (see, e.g., [START_REF] Colton | Inverse Acoustic and Electromagnetic Scattering Theory[END_REF]) that the definition ( 14)-( 16) of the generalized eigenfunctions is equivalent to the Lippmann-Schwinger equation

Φ λ,κ (y) - R 2 G λ (y -y ′ ) (k 2 uni (y ′ ) -k 2 ∞ ) Φ λ,κ (y ′ ) dy ′ = Φ ∞ λ,κ (y) ∀y ∈ R 2 . ( 19
)
In this equation, the convolution term represents the scattered wave. The associated operator V λ defined by

V λ ϕ(y) := R 2 G λ (y -y ′ ) (k 2 uni (y ′ ) -k 2 ∞ ) ϕ(y ′ ) dy ′ ∀y ∈ R 2 , is continuous from L 2 (R 2 ) to H 2 (Ω) for every bounded domain Ω ⊂ R 2 . Hence, if Ω contains the support of k 2 uni -k 2 ∞ , V λ appears as a compact operator in L 2 (Ω). Moreover, as H (1)
0 is analytic in C\R -, the above definition of V λ extends to complex values of λ. Using for instance the definition (5) of the complex square root, one can readily show that {V λ ; Re λ > -k 2 ∞ } defines an analytic family of compact operators in L 2 (Ω) and that {Φ ∞ λ,κ ; Re λ > -k 2 ∞ } is an analytic family of vectors of L 2 (Ω). The analytic Fredholm theory [START_REF] Steinberg | Meromorphic families of compact operators[END_REF][START_REF] Colton | Inverse Acoustic and Electromagnetic Scattering Theory[END_REF] then tells us that either I -V λ is not invertible for any λ or its inverse (I -

V λ ) -1 is meromorphic in Re λ > -k 2 ∞ .
The former situation must be excluded. Indeed, if Im λ > 0, then G λ (y) is exponentially decreasing as |y| → ∞, thus it is easily seen that equation ( 19) is equivalent to solve

(A -λ I) Φ sc λ,κ = (k 2 uni -k 2 ∞ )Φ ∞ λ,κ in L 2 (R 2 ),
which is a well-posed equation since λ / ∈ Λ. As a consequence, the solution Φ λ,κ to ( 19) is meromorphic and its poles, which are related to the so-called 'leaky modes' (see, e.g., [START_REF] Vassalo | Optical Waveguide Concepts[END_REF]), are necessarily contained in Im λ < 0.

For the function λ → Φ λ,κ (y), we can use the same arguments starting with the conjugate of the Lippmann-Schwinger equation [START_REF] Steinberg | Meromorphic families of compact operators[END_REF]. In this case, the poles are contained in Im λ > 0.

Proof of Theorem 1

The hardest part of the job is now done! Thanks to the generalized Fourier transform U , the proof of Theorem 1 becomes easy. For the sake of clarity, it is divided into three steps.

First step: getting rid of the defect

The starting point of the proof simply consists in noticing that if u ∈ L 2 (R 3 ) is a solution to the homogeneous Helmholtz equation (1), then

-∆u -k 2 uni u = f (u) in R 3 , where f (u) := (k 2 -k 2 uni ) u ∈ L 2 (R 3
) is compactly supported. The aim of the next sections is to prove the following property related to the uniform waveguide.

Theorem 5. Let f ∈ L 2 (R 3 ) be compactly supported. If u ∈ L 2 (R 3 ) satisfies -∆u -k 2 uni u = f in R 3 , (20)
then u = 0 outside the support of f.

Theorem 1 is actually a corollary of the latter: it suffices to apply the unique continuation principle [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators[END_REF] to equation [START_REF] Aubin | Applied Functional Analysis[END_REF], which tells us that u also vanishes inside the support of f. So, by reducing the proof of Theorem 1 to that of Theorem 5, we got rid of the defect. It remains to prove Theorem 20. In the sequel, f denotes a function of L 2 (R 3 ) compactly supported and u ∈ L 2 (R 3 ), a solution to [START_REF] Vassalo | Optical Waveguide Concepts[END_REF]. From the definition (6) of A, we can rewrite this equation in the following abstract form:

- ∂ 2 u ∂x 2 + Au = f in R. ( 21
)
3.2 Second step: searching for finite-energy solutions

As both functions u and f belong to L 2 (R 3 ), we know that u(x, •) and f (x, •) belong to L 2 (R 2 ) for almost every x ∈ R so that we can define their partial generalized Fourier transforms by

u λ,κ (x) := (U u(x, •))(λ, κ) and f λ,κ (x) := (U f (x, •))(λ, κ).
Thanks to the notion of tensor products of Hilbert spaces (see, e.g., [START_REF] Aubin | Applied Functional Analysis[END_REF]) and the fact that U is unitary from L 2 (R 2 ) to H (Proposition 3), we know that these functions, considered as functions of (x, λ, κ), belong to L 2 (R) ⊗ H . This implies in particular that u λ,κ ∈ L 2 (R) for a.e. λ ∈ Λ and κ ∈ K λ .

Note that the expression 'for a.e. λ ∈ Λ' (for almost every) is relative to the spectral measure: it means 'for all λ ∈ Λ p ∪ (Λ c \ Λ 0 )' where Λ 0 ⊂ Λ c has Lebesgue measure zero. Applying U to equation ( 21) and using ( 18) then yields

- ∂ 2 u λ,κ ∂x 2 + λ u λ,κ = f λ,κ in R, for a.e. λ ∈ Λ and κ ∈ K λ . (23)
The solutions to this differential equation can be decomposed as

u λ,κ = u gen λ,κ + u par λ,κ
where u gen λ,κ denotes the general solution to the homogeneous equation (i.e., when f λ,κ = 0) which is given by u gen λ,κ (x) := α - λ,κ e - √ λ x + α + λ,κ e + √ λ x with α ± λ,κ ∈ C, and u par λ,κ is a particular solution to (23) defined by

u par λ,κ (x) := R γ λ (x -x ′ ) f λ,κ (x ′ ) dx ′ where γ λ (x) := e - √ λ |x| 2 √ λ (24)
is a Green's function of the operator -∂ 2 /∂x 2 + λ. Let ℓ > 0 such that the support of f is contained in the strip |x| ≤ ℓ. We deduce from the above expression that

u par λ,κ (x) = β ± λ,κ e - √ λ |x| if ± x > ℓ, where β ± λ,κ := +ℓ -ℓ e ± √ λ x ′ 2 √ λ f λ,κ (x ′ ) dx ′ .
As a consequence

u λ,κ (x) =    α - λ,κ e - √ λ x + α + λ,κ + β - λ,κ e + √ λ x if x < -ℓ, α - λ,κ + β + λ,κ e - √ λ x + α + λ,κ e + √ λ x if x > +ℓ.
First consider the case of propagative modes, that is, negative values of λ. In this case, the above formula tells us that u λ,κ is a linear combination of two linearly independent oscillating functions outside [-ℓ, +ℓ]. So it belongs to L 2 (R) (see [START_REF] Weder | Spectral and Scattering Theory for Wave Propagation in Perturbed Stratified Media[END_REF]) only if both coefficients of the combination vanish, which yields α ± λ,κ = β ± λ,κ = 0 for a.e. λ ∈ Λ ∩ (-∞, 0) and κ ∈ K λ .

On the other hand, for positive values of λ, function u λ,κ belongs to L 2 (R) only if the coefficients of the unbounded exponential vanish, which yields α ± λ,κ = 0 for a.e. λ ∈ (0, +∞) and κ ∈ K λ = S 1 .

To sum up, we have proved that if u is a finite-energy solution to [START_REF] Weder | Absence of eigenvalues of the acoustic propagator in deformed wave guides[END_REF], then its generalized Fourier transform writes as

u λ,κ (x) = R γ λ (x -x ′ ) f λ,κ (x ′ ) dx ′ for x ∈ R and a.e. λ ∈ Λ and κ ∈ K λ , (25)
and must be such that u λ,κ (x) = 0 for |x| > ℓ and a.e. λ ∈ Λ ∩ (-∞, 0) and κ ∈ K λ .

The latter condition means that outside a section containing the support of f, all spectral components of u associated with propagative modes must vanish. This is not surprising. Indeed, the propagative modes are those which transport energy along the waveguide axis. Hence if the corresponding spectral components did not vanish, the energy of u would be infinite.

Third step: using an analyticity argument

Consider the expression (25) of u λ,κ for λ in the continuous spectrum Λ c = [-k 2 ∞ , +∞). From (24) and the definition (17) of the generalized Fourier transform, this expression becomes

u λ,κ (x) = R e - √ λ |x-x ′ | 2 √ λ R 2 f (x ′ , y) Φ λ,κ (y) dy dx ′ for x ∈ R, λ ∈ Λ c , κ ∈ S 1 .
Recall that f is compactly supported. Hence the above integral is actually defined on a bounded domain, say

L × Ω ⊂ R × R 2 . Noticing that • for all κ ∈ S 1 , the function (-k 2 ∞ , +∞) ∋ λ → Φ λ,κ ∈ L 2 (
Ω) extends to a meromorphic function of λ in the complex half plane Re λ > -k 2 ∞ (see Proposition 4), • λ → √ λ is analytic outside the branch cut (see ( 5)), we infer that for all x ∈ R and κ ∈ S 1 , the function (-k 2 ∞ , +∞) ∋ λ → u λ,κ (x) ∈ C extends to a meromorphic function of λ in the complex half plane Re λ > -k 2 ∞ outside the branch cut of the complex square root. As (26) tells us that if |x| > ℓ, this function vanishes on the interval (-k 2 ∞ , 0), we deduce that it vanishes everywhere. To sum up, outside a section containing the support of f, the spectral components of u associated with evanescent modes also vanish.

By the inverse generalized Fourier transform, u vanishes outside a section containing the support of f. From the unique continuation principle [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators[END_REF] applied to equation [START_REF] Vassalo | Optical Waveguide Concepts[END_REF], we conclude that u vanishes outside the support of f. This completes the proof of Theorem 5, and thus also of Theorem 1.

Possible extensions of Theorem 1

In the mathematical model introduced in section 1.2, we have considered the case of a penetrable defect in a uniform waveguide. The absence of trapped modes holds for non-penetrable defects, that is, if one replaces the Helmholtz equation (1) set in the whole space by

-∆u -k 2 u = 0 in R 3 \ D, (27)
where D denotes a compact subset of R 3 which may represent the non-penetrable part of the defect. Any kind of boundary condition can be imposed on the boundary of D. Actually, it is not even necessary to impose a boundary condition to prove that Theorem 1 holds true in this context with the same assumptions ( 2 ). As f (u) is compactly supported, Theorem 5 shows that u vanishes outside the support of f (u). And the fact that it vanishes also inside this support is again a consequence of the unique continuation principle. This shows that the statement of Theorem 1 remains valid for the exterior Helmholtz equation ( 27).

Instead of the Laplace operator, one can consider a second-order differential operator ∇ • (α ∇) with a variable coefficient α = α(x, y) appearing as a localized perturbation of a function α uni = α uni (y) where α and α uni satisfy the same assumptions ( 2)-( 4) as k and k uni . The absence of trapped modes then holds for the equation -∇ • (α ∇u) -k 2 u = 0 in R 3 , provided that the unique continuation principle applies. The proof is the same. The only change lies in the generalized Fourier transform U which must diagonalize the transverse part of the above operator.

A more involved extension of Theorem 1 can be obtained for the junction of two semi-infinite open waveguides. In this case, k appears as a local perturbation of a function k abr (in the sense that k -k abr is compactly supported), where k abr = k abr (x, y) represents the wavenumber function of an abrupt junction of two uniform waveguides described respectively by two functions k ± uni = k ± uni (y) :

k abr (x, y) = k ± uni (y) if x ∈ R ± .
In this context, the general strategy of the proof of Theorem 1 still applies but its implementation becomes far more intricate, notably because of the use of two different generalized Fourier transforms on both sides of the junction. One can follow the same steps as in [START_REF] Bonnet-Ben Dhia | Mathematical analysis of the junction of two acoustic open waveguides[END_REF] (which deals with the case of a two-dimensional junction of step-index waveguides).

On the other hand, the question of the existence of trapped modes in open curved waveguides seems to be open, as well as the case of locally perturbed periodic waveguides. Works aiming to apply the ideas explained in the present paper to these situations are in progress.

  )-(4) considered outside D. The only change in the proof lies in the first step which allows us to get rid of the defect. We simply have to chooseR > 0 such that D ⊂ B R := {(x, y) ∈ R 3 ; x 2 +|y| 2 < R 2 } and a function χ ∈ C ∞ (R 3 ) such that χ = 0 in B R and χ = 1 outside B R+1 . If u ∈ H 1 (R 3 \ D) is a solution to (27), then function χu which is initially defined in R 3 \ D can be naturally continued by 0 inside D so that χu ∈ H 1 (R 3 ). Moreover we have-∆(χu) -k 2 uni (χu) = f (u) in R 3 where f (u) := (k 2 -k 2 uni ) (χu) -2∇χ • ∇u -(∆χ)u ∈ L 2 (R 3

Funding POEMS lab (Propagation d'Ondes: Etude Mathématique et Simulation) is a mixed research team (UMR 7231) funded by CNRS (Centre National de la Recherche Scientifique), ENSTA ParisTech (Ecole Nationale Supérieure de Techniques Avancées) and INRIA (Institut National de Recherche en Informatique et en Automatique).