Elizabeth Fischer

The ubiquity of denim

A photographic essay taken in the streets of Marseille for a micro-anthropological study on why people wear jeans in 2009 shows people of all generations, male and female, wearing denim trousers -in some instances shorts (worn by women, with stockings and boots) and a jeans overall; however, no denim skirts appear in this field research 1 . The denim wearers summed up in one word their main reason for wearing jeans. These related above all to the values of comfort, practicality and sturdiness. However, other motives were offered: "honest", "original", "timeless", "goes with everything", "America". Some of these assertions sound like contradictions in terms. A pair of jeans is perceived as an item of clothing that is original i.e. unique, and honest -it has nothing to hide, it lets the wearer be his/her true self -even though it seems everyone owns at least one pair of jeans. It is timeless, situated beyond the realm of fashion trends yet suits any outfit, can be dressed up or down. It is produced by cheap fast fashion and expensive design labels alike. Moreover, jeans all over the world are often linked with the mythology of the American way of life yet also reflect regional sociocultural practices. The adoption of jeans on a daily basis by such a large part of the population the world over is a phenomenon well charted by Daniel Miller and Sophie Woodward in their global research project A Manifesto for the Study of Denim 2 .

The variety of values attributed to this single item of clothing might explain why jeans are often qualified as unisex, i.e. clothing that is suitable for both men and women. The OED even gives jeans as a prime example of unisex dress 3 . Equally, denim trousers always figure in what is labelled as "normcore" in contemporary Western dress, a style of dressing that involves the deliberate choice of unremarkable, non-fashionable casual clothes. Both of these styles are primarily founded on items of clothing that stem originally from the masculine wardrobe.

The well-known popular tale of denim from its origins to its contemporary ubiquity is dominated by the masculine figures of gold diggers, miners, farmers, cowboys, Hollywood legends, counterculture rebels and rock stars. This short essay aims to highlight some lesser-known actors in the story of denim that help to explain its special place in the interplay between gender and dress.

Land Army women wear denim during WW1 and WW2

One of the first instances of a large part of the female population wearing an outfit of denim was during WWI, when women were recruited to tend the land and farms instead of the men who were fighting in the army at the front. They were called the Land Girls (also Land Lassies) in Great Britain and the Women's Land Army of America (WLAA) in the United States. The majority of this female workforce was seasonal labour consisting of white urban students, soldiers' spouses, clerks, teachers, secretaries and other office workers who had never done agricultural work before. A uniform of brimmed hat, gloves, men's overalls and a blue work shirt, inspired by masculine denim overalls and work shirts, was provided and required in the USA. War effort campaigns popularised the denim-clad girls' land army as well as propaganda about the good cause of women of all classes undertaking agricultural work. Widely circulated press pictures showed groups of glowing and smiling young women, eagerly wielding farm tools, an image which contributed to romancing the link between jeans and the countryside. Denim was henceforth equated with the valuable cause of hard physical labour on the land to feed the population and stand in for the men. The overall worn by both men and women for farm work, and the Lee Union-Alls jeans which were standard issue for all war workers were considered functional, practical and comfortable. Jeans thus gained respectability as the uniform outfit embodying a collective social effort in hard times. The WLAA workers eventually became wartime icons, much as Rosie the Riveter would in WW2. The same type of outfit -denim overalls, a blue shirt, blue jacket and cap -was worn by the women's land army during WW2, and by women working in factories, since dresses, liable to get caught in machinery, were considered a work hazard.

After WW1, Hollywood helped further romanticize blue jeans in the 1920s and 1930s by putting the trousers on handsome cowboy types played by the likes of John Wayne and Gary Cooper. This glamorous new image spoke to consumers who sought casual leisurewear for weekends and holidays. Publicity photos of actresses like Ginger Rogers and Carole Lombard wearing jeans convinced women that the style was for them, too. Popular magazines advertised dungarees for women as proper attire both on and off the farm, for chores and leisure. In the 1930s, Vogue had given its seal of approval, calling jeans "Western chic"4 .

Claire McCardell's 1942 denim "Popover" wrap dress combines utility and chic

In 1942, the American designer Claire McCardell sold more than 75,000 of her recent design, a denim Popover wrap dress for the woman at home. Its classification as a "utility garment" explained its low price of $6.95 (instead of $20 to $30) because McCardell's manufacturer, Adolf Klein, was able to make a special deal with labour5 . Lauded at the Coty American Fashion Critics' Awards, McCardell's popover housedress was described by Vogue as a major invention "born of necessity." In utility achieved with ingenuity, McCardell found a synergy. The modern woman could both be chic and do the cooking.

Claire Mc Cardell, praised for her practical outlook, translated the fundamentals of the denim wear she liked so much into fashion. She made coats and suits of it for town wear, using details from men's work clothing such as large pockets, blue-jeans topstitching, metal rivets and trouser pleats as a form of decoration. The style she created was a "no frills no fuss" one with clean lines in contrast to a good part of female fashions. This blending of male attributes of attire with female dress shapes and outfits enhanced her clothes with the legitimacy and functionality always granted to the masculine wardrobe but never to the feminine one. She also contributed to giving a fashionable edge to the Americana flavour of jeanswear, focussed on everyday lifestyle rather than the glamour of movie stars. The designer managed to create a lasting style many designers have striven for in their designs: "I have often said that I wish I had invented blue jeans. They have expression, modesty, sex appeal, simplicity -all that I hope for in my clothes 6 ."

Denim overalls in the fight for Blacks' civil rights

In the early 1960s Black American activists' adoption of denim overalls in their fight for civil rights has been largely overlooked in the story of denim7 . Slave owners bought denim for their enslaved workers, partly because the material was sturdy, and partly because it helped contrast them against the linen suits and lace parasols of plantation families. They used to be referred to as 'Negro clothes' and their inclusion in the civil rights movement suggested that pointed societal divide [START_REF] Komar | What the Civil Rights Movement Has to Do With Denim[END_REF] . SNCC activists wore overalls out of practicality -they were tired of mending tears from attacking dogs and highpressure hoses, as denim could withstand the abuse. Young black women civil rights workers with the Student Nonviolent Coordinating Committee (SNCC) "swapped their 'respectable' clothes and processed hairstyles in order to adopt jeans, denim skirts, bib-and-brace overalls, and 'natural' hair." These young women used this newfound "uniform consciously to transgress a black middle-class worldview that marginalized certain types of women and particular displays of blackness and black culture." More importantly though, writes scholar Tanisha C. Ford, jeans quickly became the SNCC uniform, a "cultural and political tool deployed to create community and to represent SNCC's vision for a new American democracy. SNCC women's decision to wear denim was politically effective because it demonstrated the need to break down class barriers within the black community to gain greater freedoms." The images also contributed to building an image of female youth advocating for a better future in a uniform that broke the gender barrier [START_REF] Ford | SNCC Women, Denim, and the Politics of Dress[END_REF] .

A garment transcending class and gender barriers for youthful protesters

As the social revolution continued to dominate the 1960s, denim-clad rebels adopted jeans to embody the changing times. Jeans were an essential wardrobe item for the youth that protested the Vietnam War, questioned their parent's values and the government, redefined gender roles and fought for social progress -or simply the right to be different. By the 1960s, jeans had come to symbolise the counterculture, refusal of capitalist and bourgeois society and pressures, while feminists and women's lib organisers chose blue jeans as a way to demonstrate gender equity. In protest marches and camps, in hippie groups and cultural venues, both men and women wore the same style of jeans, as well as other similar items of apparel. Film characters and rock'n'roll stars helped cement the style as cool, encouraging some high schools to ban the garment, which only served to further enhance its status [START_REF] Komar | What the Civil Rights Movement Has to Do With Denim[END_REF] . Rock stars and marginal youth groups all over Western society embraced the wearing of denim. In Switzerland for example, biker clubs, school drop-outs, rock fans and other young marginals adopted jeans and their American paraphernalia to signal their dissent with bourgeois values 11 . In R. Queneau's 1959 novel Zazie dans le métro, the heroine Zazie yearns to buy a pair of blue jeans from an American surplus shop in Paris, because "that is what girls wear now". However, a man from her small hometown confuses her with a boy when he sees her wearing them [START_REF] Queneau | Zazie dans le métro[END_REF] . The novel constantly plays on the gender confusions arising in the adoption of similar dress by both sexes, notably denim.

For the historian Eric Hobsbawm, it is "protest denim" that marks the ushering in of a new era. "What has really transformed the western world is the cultural revolution of the 1960s. The year 1968 may prove to be less of a turning point in 20 th -century history than 1965, which has no political significance whatever, but was the year in which the French clothing industry for the first time produced more women's trousers than skirts, and when numbers training for the Roman Catholic priesthood began visibly to collapse. (…) It may be argued that the really significant index of the history of the second half of the 20 th century is not ideology or student occupations, but the forward march of blue jeans." [START_REF] Hobsbawm | Interesting Times: A Twentieth-Century Life[END_REF] This statement is echoed by writer François Bon, in his "Autobiography of things", remembering the acquisition of his first pair of bell-bottom jeans. That single act changes his world forever, from small town to big commercial centre and, more significantly, links him with the Beatles, then the most popular music group. «J'avais acheté à Poitiers mon premier pantalon à pattes d'eph. Je ne crois pas que le scandale ait valu pour paraître devant les copains : le pantalon à pattes d'eph, on avait tous le même. Je ne crois pas non plus que le scandale, vis-à-vis de ma mère, ait concerné les pattes d'eph elles-mêmes : plutôt l'intuition qu'en trahissant la couturière de village, puis le magasin qui avait dans la ville l'exclusivité du tissu en commerce, une rupture bien plus violente et essentielle du monde s'amorçait, qui tuerait la petite ville, ferait des centre-villes (les plus grosses), une infinie boutique à fringues jetables, et des périphéries un entassement de sous-langues (Kiabi et les autres). Mais, sur la pochette d'Abbey Road où traversaient les quatre Beatles (1969), nous savions -et je sais toujours -derrière Lennon tout de blanc mais portant cheveux jusque sur les épaules lequel (Harrison) était chaussé de « clarks », lequel (Ringo) de boots, lequel pieds nus (McCartney) et donc, marchant le dernier, Harrison avec ce blue-jean à pattes d'eph par lequel nous nous égalions à eux tous d'un seul coup de ceinture bouclée. » 14

The fabric of our lives

The twill cotton cloth used for trousers at the end of the 19 th century became conflated with the garment it was commonly used for. This alone explains denims' ubiquity and polysemy, as no other garment conflates with the textile it is made out of. The consistency of denim in people's wardrobe refutes the fashion tenet of constant change: blue twill trousers will be blue twill trousers. Capitalism may account for designer jeans but not for the ubiquitous wearing of jeans. Wear (and tear) make blue jeans intensely personal as well as comfortable. The more you wear denim, the more it adapts to your body and movements, thus becoming one of the most personal and intimate items of clothing. Jeans relieve the anxiety of what to wear for many people, they are the easy / obvious / comfortable / "normal" choice. Most people have more than one pair of jeans, as well as several other denim garments (or imitation denim). Jeans provide a solution to daily dressing whatever the occasion, creating all on their own a homogenised contemporary look, whether for work or leisure [START_REF] Miller | A Manifesto for the Study of Denim[END_REF] .

It might be argued that denim's popular and accepted association with specific groups of women in the 20 th century enables men and women to wear and embody something from the other sex without confusing or neutralizing gender. Tight jeans, usually associated with the female figure, have been worn by gays to show off their bodies and as a code signalling their sexual orientation. They were also a general masculine fashion which even Umberto Eco succumbed to and wrote his opinion about in the Corriere della Sera in 1976 16 . This style fuses the feminine with the masculine in a single pair of figure-hugging trousers. Denim skirts and dresses are often made out of trousers 17 , whether in DIY homemade garments or intricate high fashion, and the allure of these garments showing the masculine and the feminine is abiding. Pregnant women's adoption of the denim bib overall as a very common choice of comfortable and practical maternity outfit (at least in France, Switzerland, Germany) is the most telling instance of the suspension of gender binaries provided by jeans in contemporary western dress in the second half of the 20 th century. In this instance, pregnancy, the one biological trait that uniquely distinguishes a female from a male, is clad in a garment that started out as, and still is, a masculine work uniform.

Jeans are the single most conservative item commonly worn by people of all ages and genders, with a long and respectable history as work wear for both women and men, transcending generational, sexual, class and racial barriers. These same reasons encouraged their adoption as a garment of protest and counterculture. They embody many paradoxes: individualism and conformity, intimacy and ubiquity, anti-establishment and respectability, conservatism and the quest for modernity, fashionable trends and a longevity which transcends fashion. With their diverse history of adoption by various social groups over time, different generations and all genders, for personal or community purposes, jeans enable the individual who so wishes to play more subtly with gender norms in a temporary suspension of binaries eschewing the masculine oriented styles of the unisex or the normcore. 16 Umberto Eco, "La pensée lombaire", La Guerre du Faux, Paris, 1976 ; « Lumbar Thought », Fashion Theory. A Reader, ed. Malcolm Barnard, p. 3 17 See: "Hippie" skirt made from a pair of jeans and adorned with patchwork of various fabrics by Elainya Rainbow in 1967 at the Museum of Boulder https://www.historycolorado.org/denim-hippie-skirt; jeans skirt decorated with sequined flowers by Tom Ford for Gucci in 1999; denim sun dress made out of jeans by Junya Watanabe in 2002; Alexander McQueen jeans dress worn by Katy Perry for American Idol in 2022.

https://www.vogue.fr/fashion/article/vogue-encyclopaedia-the-history-of-denim-jeans

Sally Kirkland, in All-American: A Sportwear Tradition, New York, 1985, cf. https://www.metmuseum.org/art/collection/search/84029 as well as https://agnautacouture.com/2014/08/10/claire-mccardell-originated-the-american-look-part-1/ and https://agnautacouture.com/2014/08/17/claire-mccardell-ones-named-the-high-priestess-of-understatementpart-2/

Yves Saint Laurent in New York Magazine, November 1983

Tanisha C. Ford, "SNCC Women, Denim, and the Politics of Dress", The Journal of Southern History, Vol. 79, No. 3 August 2013, pp. 625-658


Ágoston Török
email: torok.agoston@ttk.mta.hu

Daniel Mestre

Ferenc Honbolygó

Pierre Mallet

Jean-Marie Pergandi

Valéria Csépe

Realistic sound source simulation in multimodal virtual environments

Keywords: Surround system, Ventriloquist illusion, Multisensory integration, Multilevel-modeling, Spatial audio

come     It sounds real when you see it.

Introduction

For a long time, virtual reality designers and researchers have been using auditory stimulation to support visualization [START_REF] Nacke | More than a feeling: measurement of sonic user experience and psychophysiology in a first-person shooter game[END_REF][START_REF] Zhou | An experimental study on the role of 3D sound in augmented reality environment[END_REF]. Even a single loudspeaker is sufficient to change the quality of experience, albeit this type of setup has at least one important limitation: it does not provide any spatial information, other than its own position. Spatial audio requires therefore a more complex approach, but the question is raised what is the minimal complexity that could provide the feeling of audio space (e.g. how many speakers are enough for the perception of a realistic sound environment). In the current study, instead of the traditional approach where researchers experimented with various sound generation methods (e.g. [START_REF] Hu | HRTF personalization based on artificial neural network in individual virtual auditory space[END_REF][START_REF] Seeber | A system to simulate and reproduce audio-visual environments for spatial hearing research[END_REF][START_REF] Wenzel | Localization using nonindividualized head-related transfer functions[END_REF]), we investigated whether, in multimodal environments, a horizontal surround speaker setup was capable of effectively creating the illusion of two dimensional audio environment when perceived in the presence of visual objects. This illusion is called ventriloquism in the cognitive neuroscience literature [START_REF] Vroomen | Perceptual effects of cross-modal stimulation: ventriloquism and the freezing phenomenon[END_REF]. The underlying mechanism is that the brain relies mostly on the visual loca-tion of an object for its localization. Therefore even large distances between the auditory and visual stimulus locations (up to 20 • ) (e.g. [START_REF] Colin | Electrophysiology of spatial scene analysis: the mismatch negativity (MMN) is sensitive to the ventriloquism illusion[END_REF]) are barely perceived by the user, even when s/he is warned of the possibility of discrepancies [START_REF] Török | Effect of stimulus intensity on response time distribution in multisensory integration[END_REF].

Concerning sound, the human brain uses binaural and monaural cues to localize sound sources [START_REF] Middlebrooks | Sound localization by human listeners[END_REF]. Binaural cues are based on the fact that our ears are placed on both sides of the head. Therefore they receive auditory information from the same sound source at slightly different times (interaural time difference, ITD) and at different levels (interaural level difference, ILD). The duplex theory of hearing [START_REF] Middlebrooks | Sound localization by human listeners[END_REF] states that sound localization is based on ITDs for low frequency sounds (under 1500 Hz), where phase differences are big enough to be perceived. For higher intensities, the shadowing effect of the skull serves as the basis for sound localization, attenuating the sound while it spreads through it (ILD). These two cues allow good localization in the horizontal plane (azimuth).

In the vertical plane however, sound localization is more difficult. Sounds on the medial plane cause no ITDs, because they are at the same angle and distance from both ears [START_REF] Middlebrooks | Sound localization by human listeners[END_REF]. Vertical sound localization relies on the characteristics of the pinna, in that its shape and structure modifies the sound's spectrum as it reaches the inner ear. Nevertheless evidence shows that sound source localization in the vertical plane is poorer than that in the horizontal plane [START_REF] Middlebrooks | Sound localization by human listeners[END_REF][START_REF] Thurlow | Certain determinants of the "ventriloquism effect[END_REF].

Currently, the most precise way of providing simulated spatial sounds is to use binaural recordings. If sounds from different spatial locations are recorded by microphones inserted into the ear canals of listeners, based on the recordings one can estimate a person's individual head related transfer function (HRTF, [START_REF] Hu | HRTF personalization based on artificial neural network in individual virtual auditory space[END_REF][START_REF] Wenzel | Localization using nonindividualized head-related transfer functions[END_REF][START_REF] Raykar | Extracting the frequencies of the pinna spectral notches in measured head related impulse responses[END_REF]). This HRTF can then be used to generate localized spatial sounds.

Although this is the most adequate way, it takes time and sophisticated equipment, and in most cases researchers and engineers use dummy heads (e.g. KEMAR, head acoustics) to simulate an individual's head. This approach has certain limitations; the ears of these dummy head models are made to be exactly identical, contrary to the human ears, where slight but significant differences certainly provide important cues for localization [START_REF] Middlebrooks | Sound localization by human listeners[END_REF]. Setting up an accurate and individually tailored simulated binaural sound environment takes a lot of time and the process has to be repeated whenever a new person would like to experience it.

So far we have mentioned two ways for adding spatial sounds to a visual scene: while the one-speaker solution is cost-friendly and easy to use, it does not provide spatial information about the scene. Binaural recordings are rather expensive, difficult to setup and use but enable excellent spatial resolution. We can visualize these approaches on a two dimensional graph, where one dimension denotes simplicity of installation and use and the other the resolution of spatial information. In this scale the one speaker solution has high value on the easiness and low score on the spatial resolu-Fig. 1 Visualization of the two approaches in a two dimensional coordinate system, where one dimension is spatial information and the other is simplicity of installation and use. Binaural systems are difficult to set up properly, but provide a high degree of spatial information (localization of sound source). A single speaker is very easy to install, but does not provide spatial information other than its own position tion dimension, while the binaural solution has the opposite values in both dimensions (see Fig. 1).

The question arises whether there are any solutions having high values on both dimensions. If we take only audio, it is unlikely that there is a better possibility than an expensive installation, using a high number of speakers in all possible spatial directions (e.g. [START_REF] Seeber | A system to simulate and reproduce audio-visual environments for spatial hearing research[END_REF]). However, from the innovative point of view, price is also an important parameter of technical solutions. We suppose that there is a golden way to achieve the desired performance on a reasonable cost. In order to define this, however, one must take into account how human perception works in multimodal situations.

The human sensory systems are different in their respective performance and limits. Hearing is very good at detecting transient changes in our environment [START_REF] Shams | Illusions. What you see is what you hear[END_REF], but spatial localization based solely on audition is poorer than spatial localization based on vision. Vision, in contrast, is superior regarding spatial resolution, but often fails to detect transient changes in the environment [START_REF] Henderson | Is there a mismatch negativity during change blindness?[END_REF]. Not only the senses themselves are different, but the brain is also adapted to their relative reliability in certain domains [START_REF] Alais | The ventriloquist effect results from nearoptimal bimodal integration[END_REF][START_REF] Ohshiro | A normalization model of multisensory integration[END_REF]. In multimodal situations the cognitive system combines information from different modalities in a weighted manner [START_REF] Körding | Causal inference in multisensory perception[END_REF]. A wellknown effect, called ventriloquism, demonstrates the relative importance and integration of auditory and visual information for spatial localization [START_REF] Thurlow | Certain determinants of the "ventriloquism effect[END_REF][START_REF] Raykar | Extracting the frequencies of the pinna spectral notches in measured head related impulse responses[END_REF][START_REF] Shams | Illusions. What you see is what you hear[END_REF][START_REF] Henderson | Is there a mismatch negativity during change blindness?[END_REF][START_REF] Alais | The ventriloquist effect results from nearoptimal bimodal integration[END_REF][START_REF] Ohshiro | A normalization model of multisensory integration[END_REF][START_REF] Körding | Causal inference in multisensory perception[END_REF][START_REF] Howard | Human spatial orientation[END_REF]. Ventriloquism refers to the phenomena where vision "captures" the apparent location of a concurrent sound. Sounds can be ventriloquized easily, without particular effort, that is to say without consciousness [START_REF] Bertelson | Automatic visual bias of perceived auditory location[END_REF] or attentive access [START_REF] Stekelenburg | Illusory sound shifts induced by the ventriloquist illusion evoke the mismatch negativity[END_REF]. This effect is observed for offsets between the visual and auditory locations of stimuli up to 20 • [7]. This could lead to the conclusion that the actual spatial information of a sound has low weight in multimodal spatial localization and localization is mostly defined by the position of the visual stimulus component.

However sounds have an exquisite role in directing attention [START_REF] Brewster | The design of sonically-enhanced widgets[END_REF][START_REF] Mcdonald | Involuntary listening aids seeing: evidence from human electrophysiology[END_REF], so one of the greatest drawbacks of not having auditory spatial information is the weakness in directing the user's attention to certain places of the environment. The question could be raised whether this attention driving effect would be served adequately if sounds just approximated the exact location of the visual object by containing only horizontal information. As it was previously shown, the ventriloquism effect is even stronger in the vertical direction [START_REF] Thurlow | Certain determinants of the "ventriloquism effect[END_REF][START_REF] Hartnagel | Eye position affects audio-visual fusion in darkness[END_REF][START_REF] Werner | Vertical sound source localization influenced by visual stimuli[END_REF]. Taking this into account, one may assume that visual stimuli catch the vertical location of sounds in virtual environments with audio-visual properties. If this is true, a surround setup could be a golden mean between binaural and one speaker solutions, especially because surround speaker systems are easy to install and broadly available in the consumer market. Moreover, this kind of audio stimulation is readily available in most VR labs. Thus it would be important to see how realistic these audio-visual environments are for the human perception, and not from an objective point of view. Our current study aimed to serve this purpose.

Our hypotheses were the following:

1. Sounds can be ventriloquized in the vertical plane, therefore it is not necessary to provide vertical auditory spatial cues, and a sound system of good horizontal resolution is enough to provide a realistic audio-visual environment. 2. Sounds can be ventriloquized in the horizontal plane, thus a small mismatch or slight scarcity of sound simulation (e.g. because of asymmetric room reverberation characteristics) does not lead to measurable changes in perception. 3. In multimodal situations, sound source localization in environments using surround systems is as good as in environments using free field speakers. 4. The ventriloquist effect differs in the horizontal and vertical plane when using surround speakers.

In order to test these hypotheses, we designed two experiments in a Cave Automatic Virtual Environment (CAVE) setup, installed in the Mediterranean Virtual Reality Center (http://www.crvm.eu). Participants were asked to locate sound sources occurring with or without simple visual stimuli (Gaussian blobs). The paradigm was based on that reported in the study of [START_REF] Besson | Bayesian networks and information theory for audiovisual perception modeling[END_REF]. Sound sources were either free field speakers (free field condition) or their simulated copies delivered through a stereo speaker set (surround condition). We used left, middle and right sound directions to test whether there was any difference in the ventriloquism effect, depending on from where the participants heard the sound. Visual stimuli were placed on the vertical plane in the first experiment and on the horizontal plane in the second experiment.

Methods

Participants

All participants were recruited as volunteers from Aix-Marseille University, Marseilles, France. The study involved exclusively non-invasive perceptual measurements, and was approved by the Institute of Movement Science Laboratory Review Board. The experiment was conducted in accordance with the Declaration of Helsinki.

Apparatus

The experiment took place in a dimly lit hall designed for the virtual reality equipment. The walls were painted black and the hall had no windows. The CAVE had a set of four screens; a 3 × 4 m frontal display, two 3 × 4 m lateral displays and a 3 × 3 m fiber optic screen on the ground. Participants sat in a comfortable chair at a 1.2 m distance from the frontal screen with their eye-level at about 1.15 m from the ground level of the CAVE. For the experiments we defined our setup so that all visual stimuli were on the frontal screen, this way we avoided any bias caused by brightness transitions on the edges of the screens. The frontal screen's resolution was 1400 × 1050 pixels. Visual stimuli were lightblobs (visual angle 7.6 • ) with a Gaussian envelope. Blobs were presented for 16.67 ms (one frame). The baseline luminance of the screen was 0.006 cd/m 2 and the luminance of the visual stimuli was 0.35 cd/m 2 . Participants wore passive stereo-glasses (Infitec) and the projectors used static stereo image rendering. It should be noted that, in the case of such short exposures and the kind of visual stimuli used, no actual conscious depth perception occurs. At the beginning of each trial, a fixation cross appeared on the screen in the center, at 1.1 m height. The acoustic stimuli of 16.67 ms duration were broadband noises high pass filtered at 250 Hz. Sounds were delivered via seven identical speakers of a 7.1 surround system (Creative Inspire p7800), sound pressure level at the participants position was 65 dB SPL. Speakers were placed on a 2.99 m radius circle with its center at 0.76 m from the ground level of the CAVE in the participants' position. The seven speakers were placed 10.5 • from each other, with speakers 2 (-21 • ), 4 (0 • ) and 6 (+21 • ) as free field speakers, and speakers 1 and 7 for the surround condition to simulate speaker position 2, 4 and 6. Speakers 3 and 5 were not used in the current study. We used panning (inter-speaker sound level differences) to create the stereo sounds projected at the position of the free field speakers, sound levels were matched between the free field and surround conditions. The participants used a Flystick (ART Flystick 2) to respond. The flystick's position was logged by infrared cameras (ART), this way participants could easy and naturally locate sounds. The 3D orientation of the flystick was used as an indication of the perceived sound direction.

Procedure

Each trial began with a fixation cross, participants were asked to move the cursor with the flystick to the fixation cross, to ensure that at the start of each trial their hand was in the same position, and that they fixated the central cross. The fixation cross disappeared after 1000 ms and the test stimulus occurred with a 50 ms onset delay. Each trial consisted of either single auditory or audio-visual stimulus presented for 16.67 ms followed by 420 ms blank screen and then an "x" sign as participants used the flystick's cursor as a response tool by the participants. According to the instruction given, they had to respond as accurately as possible by moving the cursor to the location of the auditory stimuli. A new trial started after they responded. The participants never had to respond to the visual stimuli, but were asked to always keep their eyes open till the end of the experiment.

In Experiment 1, visual stimuli had a vertical offset relative to sounds. There were three sound positions (left, middle, right), two sound types (free field, surround) and six visual positions (no visual, -21 • , -10.5 • , 0 • (same position as the sound), 10.5 • , 21 • relative to the sounds on the vertical axis).

In Experiment 2 a similar design was used, only the six visual positions were spread out horizontally, not vertically.

Figure 2 illustrates the possible stimulus presentations for the left sound position in the two experiments. In both experiments each trial was repeated 20 times, making a total of 750 presentations. The experiments lasted1hwithoneor,if participants needed, two 5 min long breaks.

Data analysis

We first inspected the data for outliers. We rejected every response where RT was less than 300 ms or more than 4000 ms. Due to significant time uncertainty (variable delay) caused by the computer cluster system, we used response times just for filtering. After the removal of outliers, on average 92 % of each participant's data in Experiment 1 and 82 % in Experiment 2 remained and were entered in the analyses. Errors as projection of response bias in the direction of the distractor stimulus were also measured. Multimodal errors (perceived shift) were calculated for each condition relative to its respective average perceived unimodal response direction. We used this approach because some participants tended to mislocalize sounds on the vertical plane, thus analyzing relative bias to veridical sound positions would distort our results [START_REF] Wozny | Recalibration of auditory space following milliseconds of cross-modal discrepancy[END_REF]. Since our data were collected from a relatively small sample (six and five participants) and sphericity was violated, our data structure is not well suited for standard ANOVA analyses [START_REF] Hoffman | Multilevel models for the experimental psychologist: foundations and illustrative examples[END_REF]. The averaging of responses across conditions also caused considerable data loss, so we decided to run multilevel modeling, to be able to deal with possible differences between individuals and with the whole range of responses. We used sound type (real, surround); sound direction (left, center, right); visual stimulus direction (1-5) as fixed factors and participants' ID as random effect in the models, we used Restricted Maximum Likelihood (REML) estimation in SPSS.

Results

Results of Experiment 1

We built five models to explore the underlying effects in our study. First we built an empty model where we entered only the Participants' ID as a random factor. It was also the baseline to evaluate further models, so we compared the information criteria to the initial model's [Schwarz's Bayesian Criterion (BIC) = 26,079.46]. Wald Z test for the τ 00 was not significant, indicating that there was no systematic difference between subjects. The residual variance (σ 2) was high enough to reach significance (Wald Z = 41.49; p < 0.001), meaning that it was justified to enter individual level predictors in the model. Therefore we entered sound type, sound direction, and visual stimulus direction in the model as fixed factors. We handled visual stimulus direction as scalar variable, sound type and sound direction as dummy variables. We used this approach because prior inspection of raw data indicated a possible difference between the center and the two eccentric positions. Comparison of the model estimating linear fit and nominal (dummy) estimates favored the handling of Sound direction as nominal variable (BIC: 22,348.23 < 22,376.90).

First we only took the main effects into consideration. We ran the model with variance components specifications because the covariance was too small to estimate [START_REF] Heck | Multilevel and longitudinal modeling with IBM SPSS (Google eBook)[END_REF]. This model had far better goodness of fit values (BIC = 22,348.23) and revealed a significant effect for Visual stimulus direction. Linear fitting to the data showed a 5.18 • difference in pointing response bias from level to level of Visual stimulus direction (see Fig. 3). The model's intercept was the estimated bias for the surround speaker set simulating a sound in the right position presented with a light blob in the downmost position. We did not find significant variations of Visual stimulus direction factor between subjects. Based on the estimations and their standard errors it can be said that localizations were clearly affected by the visual distractors, and localization bias showed linear connection with visual stimulus distance.

In order to take into account interactions that may modulate the results, we entered first all two ways and then the Fig. 3 Estimated attraction for each level of sound type and direction. The strong capture of visual stimuli is visible in the graph. Free-field sounds especially in the central position are localized lower possible three way interaction in our model. These effects brought further improvement (BIC 2way 22,241.07, BIC 3way 22,240.24; see supplementary Table 1) to our model, revealing a significant interaction between sound direction and sound type (F (2, 3414.00) = 26.25; p < 0.001) and between Sound direction and Visual stimulus direction (F (8, 3414.00) = 5.90; p < 0.001). An overall lower localization of sounds for the real left (p < 0.01) and for the real central sound sources (p < 0.001) were behind the sound direction and sound type interaction. The interaction of visual stimulus direction and sound direction was caused by a significantly (p < 0.001) stronger visual capture in the central presentations (see Fig. 3). Because the two way interaction of Sound type and Visual stimulus direction and the three way interaction were not significant (p > 0.7) we left them out from our final model. This model had better information criteria than the one containing only main effects (BIC 22,239.02). Summary of the final model can be seen in Table 1.

Summarizing the results of Experiment 1, we found strong visual capture for all sound positions, especially for central sounds. We also found a slight difference in sound localization errors between Real and Surround conditions, although the effect of visual capture did not differ between sound gen-

Results of Experiment 2

We ran a multilevel modeling on the data of Experiment 2 using the same factors as in Experiment 1. First we built an empty model as our baseline. We entered visual stimulus direction, sound direction and sound type in the second model both as fixed and as random effects. We built a model where sound direction was entered as an ordinal variable and one where it was a three level nominal variable. The later had better fit criteria (BIC 17,534.72 < 17,761.48) so we decided to use it as nominal. Model 2 had better fit parameters to the data, and revealed a tendentious effect of Visual stimulus direction (estimation 2.41, SE 1.083; see Fig. 4). Besides the fixed effect of visual stimulus direction, the random effect of Sound direction was significant (p < 0.05), indicating that the effect of sound direction differed between subjects. We entered first the two way interactions in the third and then all interactions in the fourth model. The fourth model had better fit parameters than the third. The three way interaction was not significant, but two way interactions revealed that visual capture is weakest for the right sounds (estimate 1.53, SE 1.12), slightly stronger for left sounds (estimate 2.22, SE 1.46) and the strongest for sounds in the center (estimate 3.38, SE 1.46; supplementary Table 2).

Because neither the three way interaction nor the interaction between sound type and visual stimulus direction were significant we decided to leave them out of our final model. This model had the best fit parameters to the data. The effect of visual stimulus direction was not significant, but the interaction between visual stimulus direction and sound direction

-20°- 10°0 °10°2 0°- 21°-10.5°0°10.5°21°) e l g n a l a t n o z i r o h ( s a i b n o i t a z i l a c o l d n u o S

Visual stimulus position

Real Left Real Center Real Right Surround Left Surround Center Surround Right Fig. [START_REF] Seeber | A system to simulate and reproduce audio-visual environments for spatial hearing research[END_REF] Estimated attraction for each level of sound type and direction. Strong capture of visual stimuli is visible only for central sounds. Surround sounds in the center are localized more to the left was significant again (l eft < 0.1 (0.083); c enter < 0.001). The significant interaction of visual stimulus direction and sound direction meant that the effect of visual capture was most salient for central sounds, regardless of sound type. Interaction between sound type and sound direction indicated that simulated and real sound sources were perceived somewhat differently. The random effect of sound direction was also significant, so sound localization was variable between subjects. Table 2 summarizes the parameter estimates of our final model.

Summarizing the results of Experiment 2, we found that visual capture was strongest in the center, and this effect was not different for real and surround sounds. As in Experiment 1, we found that participants localized real and surround sounds differently in bimodal situations in the horizontal plane, however no clear trend was apparent, rather a slight variance in the perception of sound sources.

Discussion

In the present experiments, we investigated the audio source localization ability of 11 subjects by measuring their per- formance in multimodal situations. We aimed to evaluate the usability of surround systems in supporting visualization and creating realistic perceptual situations. We first hypothesized that sounds can be ventriloquized on the vertical plane.

In order to investigate this, in the first experiment we looked at how well participants localized sound sources when they occur with synchronous, but vertically displaced visual distractors. We found that the visual distractor positions greatly affected the subjects' localization judgments, but the effect was slightly stronger for centrally presented sounds. This effect was persistent both in free field speaker and in surround situations. Our second hypothesis was that horizontal ventriloquist illusion could compensate sound source simulation scarcity. Thus, in the second experiment we presented sounds and visual distractors with horizontal offsets. We found that the participants' judgment of sound source locations was affected by visual distractors. Furthermore, the effect of visual distractors was greater for sounds in the center.

The third hypothesis stated that sound source localization is not different between surround and free field speakers. In contrast, we observed slight differences between surround and free field speakers in both experiments. One likely explanation for the variance is that there was some difference in the speakers' characteristics. Alternatively, the asymmetry in the reverberation structure of the experimental hall could alter the reverberation properties of sounds. Because we used identical speakers and sound levels were measured for each speaker separately, it is more likely that the asymmetry of the experimental hall contributed to the differences in localization. This further highlights the importance of visual capture and multimodal stimulation to prevent perceptual changes caused by imperfection of sound source modeling and rendering.

Handling of visual stimulus positions as scalar variables led to a better fit than using them as dummy variables, and the effect of visual stimuli depended on the sound's direction. We found the strongest capture in both experiments for central sound sources. One possible explanation for this is that the shifts in sound source position judgments were strongly affected by the size of the visual stimulus' cortical representation. Visual stimuli closer to the focus (and so the fovea) have greater cortical representation [START_REF] Rovamo | An estimation and application of the human cortical magnification factor[END_REF]. ITDs and ILDs are more pronounced farther away from the medial plane [START_REF] Middlebrooks | Sound localization by human listeners[END_REF]. Participants reported similar impressions in the debriefing after the experiments.

The observed effects are comparable to those of [START_REF] Hartnagel | Eye position affects audio-visual fusion in darkness[END_REF][START_REF] Besson | Bayesian networks and information theory for audiovisual perception modeling[END_REF][START_REF] Besson | A comprehensive model of audiovisual perception: both percept and temporal dynamics[END_REF]. Although our methodology was based on these earlier studies, important differences exist. References [START_REF] Besson | Bayesian networks and information theory for audiovisual perception modeling[END_REF][START_REF] Wozny | Recalibration of auditory space following milliseconds of cross-modal discrepancy[END_REF][START_REF] Hoffman | Multilevel models for the experimental psychologist: foundations and illustrative examples[END_REF][START_REF] Heck | Multilevel and longitudinal modeling with IBM SPSS (Google eBook)[END_REF][START_REF] Rovamo | An estimation and application of the human cortical magnification factor[END_REF][START_REF] Besson | A comprehensive model of audiovisual perception: both percept and temporal dynamics[END_REF]used only one movable near sound source (cca. 50 cm) in a soundproof chamber whereas in our experiments sound sources were much farther (3 m) away from the viewpoint in a reverberating hall. This difference is even more important since near and far sound sources are localized differently [START_REF] Moore | Auditory perception: the near and far of sound localization[END_REF]. This could be also important when we consider why the effects were different for sound directions. Another important difference is that while [START_REF] Besson | Bayesian networks and information theory for audiovisual perception modeling[END_REF] used a led array as visual stimuli, positioned at the distance of the sound sources, in our case Gaussian blobs were projected to the frontal screen at a distance of 1.5 m from the participant's viewpoint. Moreover the screen was not curved, but the blobs were stereo-projected to a virtual sphere at 2.99 m from the viewpoint. The last important difference was that in contrast to the earlier studies we allowed participants to respond freely both horizontally and vertically simply by moving their hands, this way we could avoid artifacts caused by unnatural response methods, such as choice from a button array, or button rotation.

Our fourth hypothesis claimed that vertical and horizontal ventriloquist effects are different in size. We decided not to compare the data of the experiments in one analysis, because the random effects in the models were different for the two experiments indicating sample variability, although indirect comparison is possible. The fact that MLM showed more consistent effect of visual stimuli for vertical arrangements indicates that visual capture is stronger in the vertical plane. Earlier, with different methodology, [START_REF] Thurlow | Certain determinants of the "ventriloquism effect[END_REF] reached very similar conclusions.

The present study has certain limitations. Based on the methodology we used, we cannot decide whether the sounds were really perceived close to the visual stimuli or the effect was caused by post perceptual response strategies. After the experiments, the participants reported that they felt sometimes that sounds and flashes were coming from elsewhere. The fact that responses fell between the visual and sound positions and were not centered at the place of the visual position might mean that the participants did try to locate the sounds and not simply chose the position of the visual stimuli. Our methodology was based on standard ventriloquism paradigms, which were also affected by this criticism [START_REF] Vroomen | Perceptual effects of cross-modal stimulation: ventriloquism and the freezing phenomenon[END_REF]. However, there are other studies showing that the ventriloquist effect occurs in non-transparent (i.e. where the discrepancy is so little that it is not possible to differentiate consciously the audio and visual signal's location) paradigms as well, so it cannot be solely defined by response strategies. It is also important to note that the brain responses elicited by ventriloquized and non-ventriloquized sounds differ at early processing stages [START_REF] Bonath | Neural basis of the ventriloquist illusion[END_REF]. A preattentive brain response, the mismatch negativity observed in EEG studies, is sensitive to the ventriloquist effect [START_REF] Colin | Electrophysiology of spatial scene analysis: the mismatch negativity (MMN) is sensitive to the ventriloquism illusion[END_REF][START_REF] Stekelenburg | Illusory sound shifts induced by the ventriloquist illusion evoke the mismatch negativity[END_REF], further supporting the suggestion that response strategies do not cause the effect by themselves. Although the above limitations must be kept in mind, it is highly unlikely that the effects found can be fully attributed to conscious, decision-related processes.

Our research fits well within the scope of cognitive infocommunications [START_REF] Baranyi | Definition and synergies of cognitive infocommunications[END_REF]. It shows that our brain does not just co-evolve with infocommunication devices but we can use the specific features of our cognitive system in the design of better and/or cheaper infocommunication tools. Similar approaches, already present in the literature, demonstrated how perceptual based illusions can benefit multimodal user interfaces [START_REF] Lee | Feeling what you hear: task-irrelevant sounds modulate tactile perception delivered via a touch screen[END_REF]. Furthermore, we think that our approach shows a channel where knowledge from explorative psychophysical research could flow into the field of applied cognitive ergonomics.

To know more about how multimodal integration works in virtual reality, further studies are needed, utilizing brain imaging and electrophysiological methods. The question of how the brain perceives virtual environments is already a major topic in neuroscience research [START_REF] Haans | Embodiment and telepresence: toward a comprehensive theoretical framework[END_REF][START_REF] Kober | Cortical correlate of spatial presence in 2D and 3D interactive virtual reality: an EEG study[END_REF]. However, studies involving recordings of brain activity in interactive conditions are mostly lacking, mainly due to signal processing limitations.

To sum-up, in the present experiments we found that (1) the ventriloquist effect works in virtual reality, (2) sounds can be ventriloquized both vertically and (3) horizontally, and (4) there is a slight deterioration in the sound source position judgments when using surround system compared to free field speakers. In conclusion, researchers and virtual reality designers should be able to use surround systems efficiently in the generation of realistic audio-visual scenes, because the human perceptual system is well adapted to the experienced mismatches in audio and visual positions. This is important also for the design of multimodal user interfaces [START_REF] Ghirardelli | Auditory-visual interactions[END_REF] where the scope is to build an intuitive and natural UI that frees up the visual system [START_REF] Brewster | The design of sonically-enhanced widgets[END_REF], with the help of audio stimulation that directs and lowers the load on the attentional system.

Fig. 2

 2 Fig. 2 Possible presentation sets for a left sound. Dark squares indicate the sound positions and grey circles mark the place of the synchronously presented visual stimuli. In the figure, audio and visual stimuli positions are presented with a separator line for illustrative purposes

  6 participants (1 female, mean age 32.4 years, min 25 years, max 48 years) took part in the (1st) and 5 (1 female, mean age 27 years, min 21 years, max 41 years) in the (2nd) experiment. Participants had normal hearing and had normal or corrected to normal vision. Because of population variability in the ability for stereoscopic perception, participants were tested for stereo vision using stereoscopic random dot figures (Randot Stereotests, Stereo Optical Co.). Stereo vision was adjusted for each participant based on their interocular distances.

Every participant gave written informed consent prior to the experiment. Each participant took part in one experiment. They did not receive any compensation for the experiments.

Table 1

 1 Summary of the final model in Experiment 1

	Fixed effects	Est	SE
	Intercept (surround.right.1)	-17.07*	5.16
	Sound type (real)	0.92	0.62
	Sound type (surround)	0.00	0.00
	Sound direction (left)	1.04	0.94
	Sound direction (center)	-0.82	0.95
	Sound direction (right)	0.00	0.00
	Visual stim dir (1-5)	4.86*	1.36
	Sound type (r) × sound direction (l)	-1.45**	0.49
	Sound type (r) × sound direction (c)	-4.50***	0.50
	Sound direction (l) × visual stim dir	0.11	0.17
	Sound direction (c) × visual stim dir	0.87***	0.18
	Random effects
	Residual	35.45***	0.86
	Intercept (subject = ID)	156.18	99.55
	Sound type (subject = ID)	0.77	0.57
	Sound direction (subject = ID)	1.49*	0.75
	Visual stim dir (subject = ID)	11.06	7.02
	Fit statistics
	AIC	22,208.31
	BIC	22,239.02
	*p< 0.05
	** p < 0.01
	*** p < 0.001

Est estimate, SE standard error, AIC akaike information criterion, BIC Schwarz's Bayesian information criterion; visual stim dir: 1, downmost; 5, upmost; l left, c center, r real

Table 2

 2 Summary of the final model in Experiment 2

	Fixed effects	Est	SE
	Intercept (surround.right.1)	-8.12*	4.32
	Sound type (real)	1.84+	0.82
	Sound type (surround)	0.00	0.00
	Sound direction (left)	4.71	3.30
	Sound direction (center)	-2.98	3.31
	Sound direction (right)	0.00	0.00
	Visual stim dir (1-5)	1.71	1.10
	Sound type (r) × sound direction (l)	-3.47***	0.68
	Sound type (r) × sound direction (c)	1.83**	0.69
	Sound direction (l) × visual stim dir	0.42+	0.24
	Sound direction (c) × visual stim dir	1.75***	0.25
	Random effects
	Residual	50.33***	1.41
	Intercept (subject = ID)	64.81	52.78
	Sound type (subject = ID)	1.11	0.93
	Sound direction (subject = ID)	25.37*	12.82
	Visual stim dir (subject = ID)	5.93	4.23
	Fit statistics
	AIC	17,393.01
	BIC	17,422.23
	Est estimate, SE standard error, AIC akaike information criterion, BIC
	Schwarz's Bayesian information criterion; visual stim dir: 1, leftmost;
	5, rightmost; l left, c center, r real
	*p< 0.05
	** p < 0.01
	*** p < 0.001

Acknowledgments

The research leading to these results has received funding from the European Community's Research Infrastructure Action-grant agreement VISIONAIR 262044-under the 7th Framework Programme (FP7/2007-2013). Ágoston Török was supported by a Young Researcher Fellowship from the Hungarian Academy of Sciences. Thanks to Dénes Tóth for his help during the statistical analyses. Thanks to Orsolya Kolozsvári for her help in the preparation of the manuscript.


Javier Belinchon-Moreno

Nanopore

Nanopore Adaptive sampling: A powerful approach to decipher the genetic diversity of the NLRome/resistome in melon

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Bioanalyses

Expertise and consulting in Genomics

WetLab expertise

High quality data LIMS

What is Nanopore adaptive sampling?

• Accept/reject molecules based on small initial part of sequence (~500 bp)

• Enrich/deplete specific regions of interest.

• Increase on-target data, reduce time-toanswer.

• Real time target selection: easy library preparation.

Our objective: Nanopore adaptive sampling to decipher the genetic diversity of the NLRome in melon • Variable between experiences

• But consistent between experiences for clusters under 20X

• Average ~equal between experiences

Genome assemblies of target regions: statistics

Filter out reads mapping to the mitochondria and chloroplast What Nanopore adaptive sampling offers?

• Nanopore adaptive sampling is a simple, reliable, efficient and cost-saving approach.

In conclusion

What can we do?

1. Adaptive sampling allows to efficiently retrieve (mapping and assembly) our 15 ROI in melon.

2. It will allow to explore the genetic diversity of melon NLR genes using a large amount of varieties.

Nanopore

  Adaptive sampling: A powerful approach to decipher the genetic diversity of the NLRome/resistome in melon EPGV -INRAE: plant genomic platform located at Genoscope,

(•

  Adapted fromBarragan & Weigel, 2021) High number of varieties (~150 varieties)Need for selective sequencingVariable number of NLR genes between plant species Usually grouped into clusters High level of presence/absence polymorphisms (PAV) Low frequency in the Cucurbitaceae family C.melo variety Anso77 84 NLR genes in 15 clusters NLRGenomeSweeper (Toda et al., 2020) Adapted fom (Barragan & Weigel, 2021) Our target regions: 15 NLR-gene clusters Reference genome Variety Anso77 (draft genome produced at INRAE-GAFL/EPGV) Reference genome (.fasta) + Target regions (.bed) Genome length ~380Mb Target regions length ~6.16Mb Clusters: NLR genes + 20kb-extra flanking Variable in size Variable in number of genes Our first experience: Variety Anso-77 (same as provided reference) Increased accuracy, improved sequencing speed • Reactive consumption reduced, adaptors capture improved • Duplex sequencing, better throughput

March 2023 / Javier BELINCHON-MORENO

Long-term perspectives:

1. Perform a similar experience using a diploid heterozygote species with a higher number of NLR genes and more complex clusters: Vitis vinifera.


Charles H Dossal
email: charles.dossal@math.u-bordeaux1.fr

Gabriel Peyré
email: gabriel.peyre@ceremade.dauphine.fr

Jalal M Fadili
email: jalal.fadili@greyc.ensicaen.fr

Challenging Restricted Isometry Constants with Greedy Pursuit

This paper proposes greedy numerical schemes to compute lower bounds of the restricted isometry constants that are central in compressed sensing theory. Matrices with small restricted isometry constants enable stable recovery from a small set of random linear measurements. We challenge this compressed sampling recovery using greedy pursuit algorithms that detect ill-conditionned sub-matrices. It turns out that these sub-matrices have large isometry constants and hinder the performance of compressed sensing recovery.

I. COMPRESSED SAMPLING RECOVERY

Compressed sampling paradigm consists in acquiring a small number of linear measurements y = Ax, where x ∈ R N is the high resolution signal to recover, and y ∈ R P is the vector of measurements with P N . Using the underdetermined sensing matrix A that is drawn from a proper random ensemble, and having knowledge of the measurements y, the sparsest vector giving rise to these measurements is sought. Throughout this paper, we consider the case where the entries of A are independent and identically distributed N (0, 1/P ).

For noiseless measurements y = Ax, the recovery of a sparse vector x is achieved by solving the convex program

min x∈R N ||x|| 1 subj. to Ax = y, (1)
where

||x|| 1 = i |x i |.
The vector x is said to be identifiable if the solution x to (1) is unique and coincides exactly with x. To ensure identifiability, several sufficient conditions on x were considered in the literature. Of particular interest are those relying on the sparsity (or cardinality of the support) ||x|| 0 = |I(x)|, where the support of x is

I(x) = {i \ x i = 0} .
With high probability on the sampling matrix A, compressed sampling theory [START_REF] Donoho | Compressed sensing[END_REF], [START_REF] Candès | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[END_REF] asserts that any vector satisfying ||x|| 0 ρ(P/N )P

is identifiable for ρ(η) > 0.

Restricted isometry based criteria. The seminal work of Donoho [START_REF] Donoho | Compressed sensing[END_REF], Candès, Romberg and Tao [START_REF] Candès | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[END_REF] has focused on the stability of the compressed sampling decoder. Towards this goal, these authors introduced the restricted isometry property (RIP), with the RIP constants 0 < δ min s δ max s < 1. These constants are the smallest numbers such that for every vector x ∈ R N with ||x|| 0 s,

(1 -δ min s )||x|| 2 ||Ax|| 2 (1 + δ max s )||x|| 2 . ( 3
)
Let A I = (a i ) i∈I be a sub-matrix of s = |I| columns extracted from A. Then ( 3) is equivalent to saying that for all I such that |I| = s, the smallest and largest eigenvalues λ min (A I ) and λ max (A I ) of the Gram matrix A * I A I are respectively lowerbounded and upper-bounded by 1 -δ min s and 1 + δ max s . Thus, the RIP constants are equivalently defined as

δ min s = max |I|=s δ min (A I ) and δ max s = max |I|=s δ max (A I ) where δ min (A I ) = 1 -λ min (A I ), δ max (A I ) = λ max (A I ) -1 .
The original work of Candès et al. [START_REF] Candès | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[END_REF] considers a symmetric RIP constant δ min s = max(δ max s , δ s ). These authors proved that a small enough value of δ 2s ensures identifiability of all s-sparse vectors. For instance, it is proved in [START_REF] Candès | The restricted isometry property and its implications for compressed sensing[END_REF] that δ 2s √ 2 -1 guarantees identifiability of all s-sparse vectors. This is achieved with high probability on A if s CP/ log(N/P ), which corresponds to condition (2) with ρ(η) C/ log(η -1 ) with η -1 = N/P the redundancy of the matrix A.

It turns out that the largest and smallest eigenvalues λ min (A I ) and λ max (A I ) do not deviate from 1 at the same rate. Using asymmetric RIP constants, Foucart and Lai [START_REF] Foucart | Sparsest solutions of underdetermined linear systems via via q -minimization for 0 < q 1[END_REF] show that

(4 √ 2 -3)δ min 2s + δ max 2s < 4( √ 2 -1) (4)
implies identifiability of all s-sparse vectors. Blanchard et al. [START_REF] Blanchard | The restricted isometry property and q -regularization: Phase transitions for sparse approximation[END_REF] determine ρ 0 such that with high probability on A ||x|| 0 ρ 0 (P/N )P

ensures that condition (4) is in force. Condition [START_REF] Blanchard | The restricted isometry property and q -regularization: Phase transitions for sparse approximation[END_REF] guarantees not only identifiability, but also robustness to noisy measurements. This however causes the function ρ 0 (η) to be quite small, and for instance ρ 0 (1/2) = 0.003 and ρ 0 (1/4) = 0.0027.

Topological noiseless criteria. More precise identifiability criteria exist that do not rely solely on the sparsity level ||x|| 0 of x. Donoho [START_REF] Donoho | High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension[END_REF] devises a topological recovery criterion that allows him to derive an explicit sharp value for ρ(η) in [START_REF] Candès | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[END_REF].

For instance ρ(1/2) ≈ 0.089 and ρ(1/4) ≈ 0.065. This is better than the RIP-based bound ρ 0 given above, since the RIP condition imposes a stronger metric constraint on the sensing matrix. These bounds based on topological criteria are however not robust to noise, and we restrict in this paper our analysis to the RIP condition. This topological criterion is local and ensures recovery of a given x alone. This allows Donoho to devise a bound that ensures the recovery of most sparse vectors. This bound is better than (5) since it does not constrain all s-sparse vectors but only most of these vectors. In contrast, recovery conditions based on restricted isometry constants are globals. The recovery of some x ∈ R N does not only rely on the conditioning of the matrix A I for I = I(x). In particular, a single matrix A I , with |I| 2s may be a problem for the identifiability of many s-sparse vectors x if δ min (A I ) or δ max (A I ) is too large.

II. GREEDY SINGULAR VALUE PURSUIT

The exact computation of the RIP constants δ min s and δ max s is combinatorial since it requires an exhaustive enumeration of all sub-matrices A I for |I| = s, which might take an exponential time in N . We here propose to compute good approximate lower-bounds δmin The set of all supports is a lattice ordered by inclusion, that is visualized using a graph structure, as depicted in Figure 1. A small sub-set of this lattice is computed by several traversals, starting from the singleton supports I of size |I| = 1. Lattice of the set of supports, for N = 4. Dashed: a possible path followed by the algorithm to select a matrix A I with I = {0, 2, 3}.

A step of the traversal, that follows an edge in the lattice, corresponds to a greedy extension I ← I ∪ {i} computed by adding a properly selected index i / ∈ I to increase the size of a support I. This new index is added so as to maximize the value of δ min (A I∪i ) or δ max (A I∪i ). The corresponding brute force greedy search is described in Algorithm 1 for the computation of δmax Initialization: set I (1) = {{0}, {1}, . . . , {N -1}} ; for k = 2, . . . , s do Initialization:

I (k) = ∅ ; for I ∈ I (k-1) do Compute i = argmax i / ∈I δ max (A I∪{i} ) ; Set I (k) ← I (k) ∪ {I ∪ {i }} ; Set k ← k + 1. Return: δmax s = max I∈I (s) δ max (A I ).

III. BOUNDING THE SINGULAR VALUES

The brute force greedy scheme is still too computationally intensive to be applicable to large scale sampling matrices. This is because the evaluation of the isometry constant δ min (A I∪{i} ) is required for all candidate extensions i / ∈ I. This necessitates the computation of a large amount of minimum and maximum singular values of P × s matrices for an increasing value of s.

To alleviate this computational difficulty, we propose a family of criteria that bounds the minimum and maximum singular values of each sub-matrix indexed by I by using the "radius" of clustering of its columns, see Proposition 1. A specific instantiation of these bounds leads to one that is easy to compute, and that can be approximately optimized by computing simple inner products with a properly chosen vector. This is made formal in Proposition 2.

Bounding singular values by clustering. We define two regions of R P associated to a non-zero vector d ∈ R P

C d = v ∈ R P \ | d, v | 1 , C c d = v ∈ R P \ | d, v | 1 . Remark 1.
In the normalized case-columns of A have unit 2 -norm-C d and its complement have a nice geometrical interpretation. In such a case, C d corresponds to a doublespherical cap, whose radius r satisfies r 2 = 1 -1/||d|| 2 . Similarly, C c d defines a band on the unit sphere in R P . The following proposition shows that clustering the vectors {a i } i∈I within these regions allows one to lower-bound the maximum or minimum RIP contants of A I .

Proposition 1. If {a i } i∈I ⊂ C d , then δ max s s/||d|| 2 -1. ( 6
)
If {a i } i∈I ⊂ C c d and if d ∈ Span(a i ) i∈I then δ min s 1 -s/||d|| 2 . ( 7
)
Proof: We prove ( 6) and (7) can be proved similarly. The orthogonal projection d of d onto Span(a i ) i∈I reads

d = A I A + I d = (A + I ) * A * I d where A + I = (A * I A I ) -1 A * I is the pseudo-inverse of A I .
Since {a i } i∈I ⊂ C d , we have

∀ i ∈ I, d, a i = d, a i 1.
This shows that

||A * I d|| 2 = i∈I | d, a i | 2 = ||A * I d|| 2 s,
and hence

||d|| 2 || d|| 2 λ min ((A + I ) * )||A * I d|| 2 s λ max (A I )
.

Given a sub-matrix A I , a precise estimate of δ max (A I ) is obtained by maximizing the right hand side of (6). This is achieved by identifying the region C d that encloses the columns of A I , and corresponding to the smallest ||d||.

Clustering with appropriate d. Finding such an optimal cluster of points is however difficult in large dimension. It is thus desirable to compute an approximate clustering using a well chosen vector d for the region C d .

In the following, we consider a vector d = d(σ) defined by the vectors {σ i a i } I , where

{σ i } i∈I ∈ {+1, -1} |I|
is an appropriate set of signs. The following proposition ensures that this vector can be easily computed.

Proposition 2. For a given set of signs {σ i } i∈I such that A I has full rank, the vector

d(σ) = A I (A * I A I ) -1 σ, (8)
is such that This region C d(σ) is an optimal choice to estimate δ max (A I ) using ( 6) if we restrict the choice to regions whose boundary contains the vectors {σ i a i } i∈I , which corresponds to condition (9). Better estimates might be obtained using another region C d that pass only through a subset of these vectors, or that is defined using different signs, but it is not obvious how to compute them efficiently. Once a set of signs σ is fixed, the vector d(σ) (and corresponding region) is fast to compute as it only requires inverting an overdetermined linear system. Remark 2. When the columns of A have unit norm, it is worth noting the following geometrical facts:

∀ i ∈ I, σ i a i , d(σ) = 1. (9
• Any other choice of d in Proposition 2 leads to a larger spherical cap C d . • The cap C d(σ) draws a circle on the unit sphere which is a circumcircle cap since it passes through all the points {σ i a i } i∈I . The vector d(σ) intersects the circumcircle at its circumcenter.

IV. GREEDY PURSUIT USING d(σ)

Proposition 1, applied to the region C d(σ) defined from Proposition 2 leads to the following lower bounds on the RIP constants of

A I δ max s s/||d(σ)|| 2 -1, δ min s 1 -s/||d(σ)|| 2 , (10)
and these bounds are expected to be reasonably tight since d(σ) pass through the points {σ i a i } i∈I .

Minimal and maximal extensions. To extract a sub-matrix A I with a large isometry constant δ max (A I ), we thus propose a greedy scheme that iteratively extends both the support I and the set of signs. An elementary step of the scheme extends the sign vector σ into σ = σ + ζ∆ i with ζ ∈ {+1, -1}, for i / ∈ I, where ∆ i is the Dirac vector at location i. The support is thus extended from I = I(σ) to Ĩ = I(σ) = I ∪{i}. In view of (10), the choice of i and ζ should be made in order to minimize or maximize ||d(σ)||. The following proposition gives essential guidelines to reformulate and solve this optimization problem. Proposition 3. Let ãi ∈ Span(a j , j ∈ Ĩ) be the dual vector such that ∀ j ∈ I, ãi , a j = 0 and ãi , a i = 1 .

Then ||d(σ)|| 2 = C + ||ã i ||| d(σ), a i -ζ| ,
where C does not depend on i.

Proof: Since d(σ) ∈ Span(a j , j ∈ I) and I ⊂ Ĩ, we have

d(σ) -d(σ), d(σ) = 0. Consequently ||d(σ)|| 2 = ||d(σ)|| 2 + ||d(σ) -d(σ)|| 2 .
We then have that

d(σ) -d(σ) = -ã j ( d(σ), a i -ζ), ∀ j ∈ Ĩ , which implies that ||d(σ) -d(σ)|| = ||ã i ||| d(σ), a i -ζ| .
Finding an extension that maximizes (resp. minimizes) ||d(σ)|| is thus equivalent to maximizing (resp. minimizing) ||ã i ||| d(σ), a i -ζ| over both i and ζ. Calculating ||ã i || for all possible i / ∈ I is computationally demanding since it requires to solve an over-determined system of linear equations for each i.

We thus select an approximately optimal extension by maximizing or minimizing | d(σ), a i -ζ| instead of ||ã i ||| d(σ), a i -ζ|. This optimization can be solved in closed form, and defines the extension σ = σ + ζ∆ i for the approximation of δ max s using s/||d(σ)|| 2 -1 where

i = argmin j / ∈I(σ) |1 -| d(σ), a j ||, ζ = sign( d(σ), a i ). (11)
Similarly, the approximation of δ min s is obtained using

i = argmax j / ∈I(σ) | d(σ), a j |, ζ = -sign( d(σ), a i ). ( 12
)
Greedy pursuit with d(σ). By exploiting Proposition 3, our iterative scheme suggests to compute a lower bound δmax s by iteratively extending a support according to (11). This greedy pursuit is summarised in Algorithm 2. A similar algorithm allows one to estimate δmin s by using the extension rule (12).

Algorithm 2: (Pruned-Weak)-Greedy pursuit with parameters Q and R.

Initialization: set Σ (1) = {∆ 0 , ∆ 1 , . . . , ∆ N -1 } ; for k = 2, . . . , s do Initialization: Σ (k) = ∅ ; for σ ∈ Σ (k-1) do Compute for each i ∈ argmin j / ∈I(σ) [R] |1 -| d(σ), a j || ; Compute ζ = sign( d(σ), a i ) ; Set Σ (k) ← Σ (k) ∪ {σ + ζ∆ i } ; Pruning: set Σ (k) = argmin σ∈Σ (k) [Q] ||d(σ)|| ; Set k ← k + 1. Return: δmax s = max σ∈Σ (s) δ max (A I(σ) ).
The algorithm is accelerated by pruning the candidate set Σ (k) at each iteration. This pruning corresponds to the extraction of the Q signs σ ∈ Σ (k) corresponding to the smallest values of ||d(σ)||, which is formally written as

argmin σ∈Σ (k) [Q] ||d(σ)||.
This keeps only those signs that generate the largest maximum singular value, as approximately measured by s/||d(σ)|| 2 .

As our algorithm is greedy by nature, the selection rules (11)-( 12) may be too stringent, and important candidate extensions can be missed while the algorithm evolves. We then consider the so-called weak selection rule which keeps the R smallest values of |1 -| d(σ), a j || for δmax s , and largest values of | d(σ), a j | for δmin s . For δmax s , this is written

argmin j / ∈I(σ) [R] |1 -| d(σ), a j ||,
and similarly for δmin s . Corresponding variants of greedy pursuit are called weak greedy pursuit with parameter R.

Empirical evaluation of our greedy pursuit bound. To assess the performance of our greedy pursuit, Algorithm 2 (with R = Q = 1), we compare it with the brute force pursuit, Algorithm 1, that is expected to perform better since at each of its step, it maximizes the RIP constants. Figure 2 shows that this is indeed the case, but the gap between the two estimates of the RIP constants provided by the two methods is rather small. This proves numerically that the heuristic (10) is accurate in practice.

We also compared the performance of the different variants of Algorithm 2: greedy pursuit with no pruning (Q = N, R = 1), and pruned weak greedy pursuit with Q = N/4 and R = 4. The increase in performance brought by the pruned weak greedy variant is slighty more salient as the sparsity level increases. Figure 4 shows that for a fixed value of η = P/N and s/P = 10 -2 , the estimates of the RIP constants provided by Algorithm 2 are close to being constant when the size (P = 100s, N = P/η) of the sensing matrix varies. This Table I OUR NUMERICALLY COMPUTED CRITICAL SPARSITY LEVELS s 0 (η, P ) VERSUS THE THEORETICAL UPPER-BOUND OF [START_REF] Blanchard | The restricted isometry property and q -regularization: Phase transitions for sparse approximation[END_REF] ρ 0 (1/4)P ∼ 0.0027P . is consistent with the asymptotic upper bound of restricted isometry constants provided by Blanchard et al. [START_REF] Blanchard | The restricted isometry property and q -regularization: Phase transitions for sparse approximation[END_REF]. A slight drift is however observed as the sparsity increases, which might be a consequence of the greedy nature of the algorithm. This numerical result tends to prove that the existence of illconditionned sub-matrices at such small sparsity levels is not restricted to small dimensions. Empirical sparsity bounds for RIP condition. For each underdeterminacy value η = P/N 1, we computed s 0 (η, P ), the minimum sparsity s for which our empirical estimates invalidate condition (4), hence 1 -identifiability,

(4 √ 2 -3) δmin 2s + δmax 2s 4( √ 2 -1) . (13)
Figure 5 depitcs our numerical estimate of the bound (4) for varying s. Table I reports our numerically computed critical sparsity levels s 0 (η, P ) for η = 1/4, and compares this numerical evidence with the theoretical bound of Blanchard et al. [START_REF] Blanchard | The restricted isometry property and q -regularization: Phase transitions for sparse approximation[END_REF] ρ 0 (1/4) ∼ 0.0027. Computational speed. A chief advantage of our (weak) greedy algorithm over brute force pursuit is that it has a much lowest computational load while leading to comparable estimates of the RIP constants. This is clearly testified by the exectution times reported in Table II for two typical problem sizes. Note also that the pruned weak greedy variant is faster than the greedy version owing to the pruning step.

V. CONCLUSION

We have proposed in this paper a new greedy algorithm to find sub-matrices with a small number of columns that are for (N, P ) = (8000, 2000), as a function of s on the x axis, computed using our greedy pursuit with no pruning (circles) and pruning (asterisks, Q = N/4). The solid curve corresponds to the asymptotic upper bound of [START_REF] Blanchard | The restricted isometry property and q -regularization: Phase transitions for sparse approximation[END_REF]

  a small sub-set of the whole set of supports.

Figure 1 .

 1 Figure 1.Lattice of the set of supports, for N = 4. Dashed: a possible path followed by the algorithm to select a matrix A I with I = {0, 2, 3}.

  s. A similar algorithm computes δmin s by adding at each step the index i that maximizes δ min (A I∪{i} ).

Figure 2 ,

 2 dashed curve, shows examples of the lower bounds δmin s and δmax s computed with this brute-force greedy algorithm for several values of s.

Algorithm 1 :

 1 Brute force greedy pursuit.

)

  Any other region C d with another vector d that satisfies this property leads to a worse lower bound on δ max s . Proof: By definition a i , d(σ) = σ i , and any other vector d with this property satisfies ||d|| > ||d(σ)||. Indeed, a i , d -d(σ) = 0, and since d(σ) ∈ Span(a i ) i∈I , we have d(σ), d -d(σ) = 0 implying that ||d|| 2 = ||d -d(σ)|| 2 + ||d(σ)|| 2 > ||d(σ)|| 2 . In view of the right hand side of (6), the conclusion follows.

Figure 2 .

 2 Figure 2. Comparison of the brute force pursuit and our greedy pursuit with Q = N, R = 1 for (N, P ) = (2000, 500). Solid and dashed lines correspond respectively to δmax s and δmin s , as a function of s on the x axis.

Figure 3 .

 3 Figure 3. Comparison of the different variants of Algorithm 2 for (N, P ) = (8000, 2000). Circles: greedy pursuit Q = R = 1. Asterisks: pruned weak greedy pursuit Q = N/4, R = 4. The curve without markers corresponds to the asymptotic upper bound of [5]. Solid and dashed lines correspond respectively to δmax s and δmin s , as a function of s on the x axis.

Figure 4 .

 4 Figure 4. Plot of δmax s (solid curves) and δmin s (dashed curves) as a function of s on the x axis, for two values of η = P/N and for P = 100s. The curves are obtained by averaging the value of δmax s and δmin s for 5 realizations of the random matrix A.

  . The dashed line corresponds to the limit y = 4( √ 2 -1) bellow which condition (13) is in force. N = 800, P = 200, s = 10 Algorithm Time (s) δmax s Singular value pursuit 2145 0.596 Greedy pursuit (Q = N, R = 1) 11.1 0.595 Pruned weak greedy pursuit (N/Q = 10, R = 4) 6.0 0.597 Pruned weak greedy pursuit (N/Q = 100, R = 4) weak greedy pursuit (N/Q = 10, R = 4) 113 0.634 Pruned weak greedy pursuit (N/Q = 100, R = 4) OF COMPUTATION TIMES FOR THE EVALUATION OF δmax s . ill-conditionned. This allows us to check numerically sparsitybased criteria for compressed sampling recovery based on the restricted isometry property.