FrédéricBerthommier Ganesh Attigoduchandrashekara

Jean Luc Olhanahorna
email: olha.nahorna@gipsa-lab.grenoble-inp.fr

Schwartz

Effect of context, rebinding and noise, on audiovisual speech fusion

Keywords: audiovisual speech perception, McGurk effect, unbinding, rebinding, perception in noise

In a previous set of experiments we showed that audio-visual fusion during theMcGurk effect may be modulated by context. Ashort context (2 to 4 syllables) composed of incoherent auditory and visual material significantly decreases theMcGurk effect. We interpreted this as showing the existence of an audiovisual "binding" stagecontrolling the fusionprocess, and we also showed the existence of a"rebinding" process when an incoherent material isfollowed by a short coherent material. In this work we evaluate the role of acoustic noise superimposed to the context and to the rebinding material. We use either a coherent or incoherent context, followed, if incoherent, by a variable amount of coherent "rebinding" material, with two conditions, either silent or with superimposed speech-shaped noise. The McGurktarget is presented with no acoustic noise. We confirm the existence of unbinding (lower McGurk effect with incoherent context) and rebinding (the McGurk effect is recovered with coherent rebinding). Noiseuniformly increases the rate of McGurk responses compared to the silentcondition. We conclude on the role of audiovisual coherence and noise in the binding process, in the framework of audiovisual speech scene analysis and the cocktail party effect.

Introduction

It is known since long that the human brain combines visual and auditory information to better understand spoken language, particularly in the case of perception in noise [START_REF] Binnie | Auditory and visual contributions to the perception of consonants[END_REF][START_REF] Erber | Auditory and audiovisual reception of words in low-frequency noise by children with normal hearing and by children with impaired hearing[END_REF][START_REF] Grant | The use of visible speech cues for improving auditory detection of spoken sentences[END_REF][START_REF] Sumby | Visual contribution to speech intelligibility in noise[END_REF]. A classical paradigm to demonstrate audiovisual fusion is provided by the "McGurk effect" in which a conflicting visual input modifies the perception of an auditory input, e.g. visual /ga/ added on auditory /ba/ leading to the percept of /da/ [START_REF] Mcgurk | Hearing lips and seeing voices[END_REF]. Audiovisual fusion in speech perception has long been considered as automatic [START_REF] Massaro | Speech perception by ear and eye[END_REF][START_REF] Soto-Faraco | Assessing automaticity in audiovisual speech integration: evidence from the speeded classification task[END_REF]. However a number of recent experiments have provided evidence that it is in fact under the control of attention in a broad sense, consideringthat various cognitive variables can modulate audiovisual integration [START_REF] Alsius | Audiovisual integration of speech falters under high attention demands[END_REF][START_REF] Buchan | The Effect of a Concurrent Cognitive Load Task and Temporal Offsets on the Integration of Auditory and Visual Speech Information[END_REF][START_REF] Colin | Top-down and bottom-up modulation of audiovisual integration in speech[END_REF][START_REF] Mozolic | Modality-specific selective attention attenuates multisensory integration[END_REF][START_REF] Navarra | Assessing the role of attention in the audiovisual integration of speech[END_REF][START_REF] Tiippana | Visual attention modulates audiovisual speech perception[END_REF].

Binding andunbinding in audiovisual fusion

While evidence for the non-automaticity of the fusion mechanism stays compatible with a one-stage architecture, some data suggest that audiovisual interactions could intervene at various stages in the speech decoding process [START_REF] Bernstein | Audiovisual speech binding: convergence or association?[END_REF][START_REF] Grant | The use of visible speech cues for improving auditory detection of spoken sentences[END_REF][START_REF] Van Wassenhove | Visual speech speeds up the neural processing of auditory speech[END_REF]. Actually, audiovisual fusion could be conceived as a two-stage process, beginning by binding together the appropriate pieces of audio and video information, followed by integration per se [START_REF] Berthommier | A phonetically neutral model of the low-level audiovisual interaction[END_REF]. The binding stage would occur early in the audiovisual speech processing chain enabling the listener to extract and group together the adequate cues in the auditory and visual streams, exploiting coherence in the dynamics of the sound and sight of the speech input. To demonstrate the existence of this "binding" process we defined an experimental paradigm possibly leading to "unbinding". In thisparadigm(Figure 1) incongruent "McGurk" (A/ba/ + V/ga/) or congruent "ba" (A/ba/ + V/ba/) targets were preceded by coherent or incoherent audiovisual contexts [START_REF] Nahorna | Binding and unbinding the auditory and visual streams in the McGurk effect[END_REF]. The experimental results showed that the McGurkeffect (displaying the role of the visual input on phonetic decision)depends on the previous audiovisual context. Indeed, various kinds of incoherent contexts,such asacoustic syllables dubbed on video sentences, or phonetic or temporal modifications of the acoustic content of a regular sequence of audiovisual syllables, can significantly reduce the McGurk effect.Short incoherent context durations (even 1syllable long) were sufficient to produce a significant amount of unbinding.On the contrary, coherent contexts let the McGurk effect stable, which suggests that there is possibly a "default mode" in which binding occurs (and hence produces the McGurk effect in isolation).

Evidence for a rebinding process

Then we explored possible"rebinding" effects, searching forparadigmsable to reset the system and put it back in its default mode in which the McGurk effect would recover from unbinding. For this aim we prepared stimuli in which an incoherent context was followed by variable durations of a "reset" stimulus, either acoustic silence dubbed on fixed image, or coherent audiovisual material before the McGurktarget (Fig. 1). The results [START_REF] Nahorna | Audiovisual speech binding[END_REF] showed thatthe "silence + fixed image" reset did not provide any rebinding (the decrease in McGurk effect was not removed by reset). On the contrary, the coherent reset did produce rebinding, that is a significant increase in theMcGurkeffect, coming back to its "default" state for a coherence period of 4 syllables. A first objective of the present study is to confirm the existence of "unbinding-by-incoherence" and "rebinding-by-coherence" processes on the McGurk effect.

The role of noise in audiovisual fusion

The role of visual speech is particularly important in noise [START_REF] Binnie | Auditory and visual contributions to the perception of consonants[END_REF][START_REF] Erber | Auditory and audiovisual reception of words in low-frequency noise by children with normal hearing and by children with impaired hearing[END_REF][START_REF] Grant | The use of visible speech cues for improving auditory detection of spoken sentences[END_REF][START_REF] Sumby | Visual contribution to speech intelligibility in noise[END_REF]. Noise also seems to modulate decision in the case of incongruent stimuli. Indeed, the McGurkeffectdecreases when the extraneous noise is visual, whereas it increases when the noise is auditory [START_REF] Sekiyama | McGurk effect in non-English listeners: Few visual effects for Japanese subjects hearing Japanese syllables of high auditory intelligibility[END_REF][START_REF] Sekiyama | Inter-language differences in the influence of visual cues in speech perception[END_REF][START_REF] Sekiyama | Difference in auditory-visual speech perception between Japanese and Americans: McGurk effect as a function of incompatibility[END_REF][START_REF] Fixmer | The influence of quality of information on the McGurk effect[END_REF][START_REF] Kim | Audiovisual speech processing in visual speech noise[END_REF]. In the well-known "Fuzzy-Logical Model of Perception" (FLMP [START_REF] Massaro | Speech perception by ear and eye[END_REF]) this is interpreted as due to the increasing ambiguity of the noisy component, which would automatically decrease its role in the fusion process. However, it could also be envisioned that there is a specific weighting process in which a given modality would be positively or negatively modulated in the fusion process depending on the noise in this modality [START_REF] Heckmann | Noise Adaptive Stream Weighting in Audio-Visual Speech Recognition[END_REF][START_REF] Huyse | Degradation of labial information modifies audiovisual speech perception in cochlear-implanted children[END_REF].In the first case fusion would only depend on stimuli while on the second case there would be in addition an evaluation of the conditions of perception resulting in a modification of the fusion process per se. Our reasoning here is that if noise is applied in the (context + reset) part of the stimulus in Fig. 1

Method

Globally

Stimuli

The stimuli are described in Fig. 2. They are typically made (Fig. 2, top) of: -an incoherent context (2 or 4 acoustic syllables superimposed on excerpts of video sentences matched for equal duration); -followed by a reset stimulus consisting in 0, 1, 2 or 3 coherent audiovisual syllables; -followed by a target which can be either a congruent audiovisual "ba" or a McGurkstimulus consisting in an audio "ba" dubbed on a video "ga". A control stimulus, aimed at providing a reference for the McGurk effect, is provided by (Fig. 2,bottom): an coherent context (2 or 4 coherent audiovisual syllables; -followed by a target which can be either a congruent audiovisual "ba" or a McGurkstimulus. A series of audiovisual films were presented to participants in two blocks, one without acoustic noise and the other one with acoustic noise superimposed on all the context and reset parts of the stimuli. Noise consisted in speech-shaped noise0 dB SNR. The target parts always remained without noise. Coherent context and reset material was constructed by pairing audiovisual syllables randomly selected within the following syllables ("pa", "ta", "va", "fa", "za", "sa", "ka", "ra", "la", "ja", "cha", "ma", "na"). In the incoherent context material, the auditory content was same, but the visual content was replaced by excerpts of video sentences matched in duration. The congruent "ba" target was used to ensure that participants were performing the speech task correctly and to serve as a baseline to contrast with the McGurk effect. The incongruent McGurktarget was produced by carefully synchronizing an auditory /ba/ with a video /ga/, precise temporal localization of the acoustic bursts of the original "ba" and "ga" stimuli providing the cue for synchonisation.

McGurktargetswere presented three times more than congruent "ba" targets, which served as controls. For each (context+reset) condition (2 context durations; 4 reset durations for incoherent context; 2 noise conditions; hence altogether 20 context conditions) there were 4 occurrences of a "ba" target and 12 occurrences of a McGurk target. Hence there were 320 sequences in total spread over 2 blocks of 10 min each.

Procedure

All experiments were carried out in a soundproof booth. Stimulus presentation and recording of responses were controlled by the Presentation software. The experiment consisted of two possible responses"ba" or "da" (with one button for "ba" and one for "da,") and the participants were instructed to constantly look at the screen and, each time a "ba" or a "da" was perceived, to immediately press the corresponding button. The films werepresented on a computer monitor with high-fidelity headphones set at a comfortable fixed level. The video stream was displayed at a rate of 25 images per second, the subject being positioned at about 50 cm from the screen. There were 5 different orders of the stimuli in the films, the order of the two blocks "silent" and "noise" was counterbalanced between subjects, and the response button was also interchanged between subjects.

Participants

Twenty participants (13 women and 7 men; 34 years) participated in the experiment. All of them were French native speakers, without any reported history of hearing disorders and with normal or corrected-to-normal vision. Written consent was obtained from each participant and all procedures were approved by the Grenoble Ethics Board (CERNI).

2.4Assumptions andanalyses

The experiment was focused on the role of context, reset and noise on the McGurk effect. For each (context, reset and noise) condition, each target and each subject, the amount of "ba" responses against "ba+da" responses was computed and used as an index of the subject's perception. Though response times were systematically recorded and processed, they will not be presented here. We had three main assumptions, all involving McGurk stimuli (let us recall that "ba" targets are just there as controls.

-

Results

Global effect of target, noise and context duration in the incoherent context condition

On Fig. 3 we display the global scores (percentage of "ba" responses relative to "ba" + "da" responses) for all "ba" and McGurk targets, averaged over all incoherent context conditions (whatever context and reset duration), for silent vs. noise blocks. As expected (and as in previous experiments) the "ba" target leads to 100% "ba" responses. This is the case also in the coherent context.

Assessing the effect of rebinding and noise onMcgurk targets

On Fig. 4 we display "ba" scores for McGurk targets in all conditions (averaging over the two context durations 2 and 4 syllables). Three major facts emerge from this figure.

-Unbinding: Focusing on the "without noise" condition (black bars), the score is higher (less McGurk effect) for incoherent context without reset (most left) than for coherent context (most right). This replicates previous findings in [START_REF] Nahorna | Binding and unbinding the auditory and visual streams in the McGurk effect[END_REF]. -Rebinding: Still focusing on the "without noise" condition (black bars), the score decreases (more McGurk effect) with increasing reset duration. For all reset durations higher than 1 the score has actually reached a value equal to or even lower than for coherent context. This replicates previous findings in [START_REF] Nahorna | Audiovisual speech binding[END_REF]. -Modulation by noise: Comparing black (with noise) and grey (without noise) bars, noise decreases scores (increases the McGurk effect) for all conditions. To assess the effects of rebinding and noise, we performed an analysis of variance with the factors subject (random-effect), reset (with noise & without noise) and reset duration (0, 1, 2 & 3 syllables). An arc(sin(sqrt)) transformation was applied on "ba" scores to ensure Gaussianity of the dependent variable.Subject, reset and reset duration factors are both statistically significant [subject: F(19,18)=13.46, P<0.001; reset: F(1,19)=6.12, P<0.05]; reset duration [F(3,57)=14.82, P<0.001]. There was no significant interaction between any pair of factors. Therefore this confirms our previous results in unbinding and rebinding [START_REF] Nahorna | Binding and unbinding the auditory and visual streams in the McGurk effect[END_REF][START_REF] Nahorna | Audiovisual speech binding[END_REF]. It also shows that noise applied in the context part modifies the results of audiovisual fusion, with a global and more or less stable effect leading to an increase of about 15% in the McGurk effect whatever the context.

Discussion

Unbinding, rebinding and noise in the audiovisual fusion process

This experiment enabled us confirm that context may modify the McGurk effect, through a process that we described by general binding principles, with "unbinding" with incoherent context and "rebinding" with coherent reset [START_REF] Berthommier | A phonetically neutral model of the low-level audiovisual interaction[END_REF][START_REF] Nahorna | Binding and unbinding the auditory and visual streams in the McGurk effect[END_REF][START_REF] Nahorna | Audiovisual speech binding[END_REF].

In the present study, the rebinding process was evaluatedwithout noise vs.with noise applied on the "context+reset" portions of the stimuli, while McGurk targets were systematically silent. It appears that noise systematically increases the Mcgurkeffect. Toour knowledge this is the first time such a result is obtained. This strongly suggests that noise in the McGurk effect, already displayed with noise applied on the target itself [START_REF] Sekiyama | McGurk effect in non-English listeners: Few visual effects for Japanese subjects hearing Japanese syllables of high auditory intelligibility[END_REF][START_REF] Sekiyama | Inter-language differences in the influence of visual cues in speech perception[END_REF][START_REF] Sekiyama | Difference in auditory-visual speech perception between Japanese and Americans: McGurk effect as a function of incompatibility[END_REF][START_REF] Fixmer | The influence of quality of information on the McGurk effect[END_REF][START_REF] Kim | Audiovisual speech processing in visual speech noise[END_REF], intervenes not only at the level of the stimuli, but also at the level of the fusion process itself. At this level, it is possible to come back to the models of audiovisual fusion available in the literature. Classical models consider that phonetic decision operates at a given representational stage and produces an integrated percept combining auditory and visual cues in a given way, possibly mediated by general attentional mechanisms. Our data on the binding process led us suggest that an additional computational stage should be incorporated before decision operates, involving online computation of some assessment of the coherence/ incoherence of the auditory and visual inputs, resulting in a "two-stage model" of audiovisual speech perception [START_REF] Berthommier | A phonetically neutral model of the low-level audiovisual interaction[END_REF] (see Fig. 5). The present results first add some information about the way coherence could be computed, involving a dynamics made of unbinding and rebinding stages with short constant times: indeed, less than one second of incoherent (2 syllables or less) suffice to produce unbinding, and less than one second on coherence (2 syllables or less) suffice to produce complete rebinding. Furthermore, the results about noise suggest that noise, and probably more generally knowledge about the conditions of communication, also participate to the decision process by providing an enhancement of "efficient" modalities, not contaminated by noise, versus modalities where noise could contaminate the decision process (Fig. 5). The present data suggest that the role of unbinding/rebinding on one hand, and noise-based selective weighting of each modality on the other hand, could play additional independent roles, according to the lack of interaction between noise and reset in Section 3.2. This will have to be confirmed in future experiments specifically dealing with this question.

4.2Future experiments

A number of further experimentswill have to extend the present data in various directions, involving e.g. more about the dynamics of unbinding and rebinding. Various proposals could also deal with reset mechanisms (such as changing speaker or the global communication setting), or specificity of the binding mechanism (could non-speech incoherent audiovisual material also produce unbinding?). The role of noise could also be further assessed by usingvisual noise. Indeed, some studies [START_REF] Colin | Top-down and bottom-up modulation of audiovisual integration in speech[END_REF][START_REF] Kim | Audiovisual speech processing in visual speech noise[END_REF] have manipulated the size or position of the face and found influence on the McGurk effect, showing in both that visual noise maydecrease the McGurkeffect just as auditory noise increases it. If our conjecture on the role of noise in Fig. 5 is correct, thus effect should also occur for visual noise added on the contextual part of the stimuli in the present paradigm. Another very important extension concerns intelligibility in noise. The present paradigm was also an aim to progress towards the next important question that is to know if unbinding mechanisms would also decrease the beneficial effect of lipreading in noise. Future experiments will deal with targets consisting in ambiguous though coherent stimuli and test if an incoherent audiovisual context is able to remove the visual benefit. This will enable us incorporate the two-stage model inside a general question concerning the cocktail-party effect and what we propose to call "audiovisual speech scene analysis" [START_REF] Nahorna | Binding and unbinding the auditory and visual streams in the McGurk effect[END_REF].

Figure 1 :

 1 Figure 1: Experimental paradigm for displaying unbinding or rebinding mechanisms modulating the McGurk effect

  but not on the target itself, if fusion only depends on stimuli, then the McGurk effect should not change since the McGurk target stays clear. If however fusion depends on a weighting process depending on the environment, then application of acoustic noise in the context part should result in increasing the role of vision in fusion, hence increasing the McGurk effect. The second objective of the present study is to test the role of noise on context, and its interaction with the binding/unbinding/rebinding processes.

  , the experiment consisted in testing the McGurk effect in various kinds of contexts including: (i) a coherent vs. incoherent part to replicate unbindingdecrease of the McGurk effectwith incoherent contexts; (ii) in case of incoherent contexts, a coherent reset component to replicate rebindingrecovery of the McGurkeffect; (iii) addition of acoustic noise in one set of conditions, to test if noise added to the (context+reset) part could increase globally the McGurk effect.

Figure 2 .

 2 Figure 2.Description of the audiovisual material

  Firstly, incoherent context should produce unbinding and decrease the McGurk effect (hence increase the amount of "ba" responses) in respect to coherent context, whatever the context duration (2 or 4 syllables). -Secondly, for incoherent context, coherent reset should produce rebinding and increase the McGurk effect (hence decrease the amount of "ba" responses), from 0 to 3 syllables of duration of the reset coherent stimulus. -Thirdly, noise should enhance the role of vision and hence globally increase the McGurk effect (decrease the score of "ba" responses) whatever the context and reset.

Figure 3 .

 3 Figure 3. Percentage of "ba " responses for " ba " (in dark grey) and " McGurk " (in light grey) targets, in the "without noise" (left) vs. "with noise" (right) conditions for incoherent context.

Figure 4 .

 4 Figure 4. Percentage of "ba " responses for "McGurk" targets, in the "without noise" vs. "with noise" conditions for incoherent context with the four reset durations, compared with coherent context. The ANOVAwas performed only for the four reset durations and the two noise levels in the incoherent context.

Figure 5 :

 5 Figure 5: A two-stage model of audiovisual speech perception

Acknowledgements

This project has been supported by ANR Multistap (MULTISTability and binding in Audition and sPeech: ANR-08-BLAN-0167 MULTISTAP) and by Academic Research Community "Quality of life and ageing" (ARC 2) of the Rhône-Alpes Region, which provided a doctoral funding for Ganesh AttigoduChandrsheakra.


Olha Nahorna
email: olha.nahorna@gipsa-lab.grenoble-inp.fr

Attigodu Ganesh
email: ganesh.attigodu@gipsa-lab.grenoble-inp.fr

Frédéric Chandrashekara

Jean Luc Berthommier
email: frederic.berthommier@gipsa-lab.grenoble-inp.fr

Schwartz

Modulating fusion in the McGurk effect by binding processes and contextual noise

Keywords: audiovisual speech perception, McGurk effect, unbinding, rebinding, perception in noise

In a series of experiments we showed that the McGurk effect may be modulated by context: applying incoherent auditory and visual material before an audiovisual target made of an audio "ba" and a video "ga" significantly decreases the McGurk effect. We interpreted this as showing the existence of an audiovisual "binding" stage controlling the fusion process. Incoherence would produce "unbinding" and result in decreasing the weight of the visual input in the fusion process. In this study, we further explore this binding stage around two experiments. Firstly we test the "rebinding" process, by presenting a short period of either coherent material or silence after the incoherent "unbinding" context. We show that coherence provides "rebinding", resulting in a recovery of the McGurk effect. In contrary, silence provides no rebinding and hence "freezes" the unbinding process, resulting in no recovery of the McGurk effect. Capitalizing on this result, in a second experiment including an incoherent unbinding context followed by a coherent rebinding context before the target, we add noise all over the contextual period, though not in the McGurk target. It appears that noise uniformly increases the rate of McGurk responses compared to the silent condition. This suggests that contextual noise increases the weight of the visual input in fusion, even if there is no noise within the target stimulus where fusion is applied. We conclude on the role of audiovisual coherence and noise in the binding process, in the framework of audiovisual speech scene analysis and the cocktail party effect.

Introduction

It is known since long that the human brain combines visual and auditory information to better understand spoken language, particularly in the case of perception in noise [START_REF] Binnie | Auditory and visual contributions to the perception of consonants[END_REF][START_REF] Erber | Auditory and audiovisual reception of words in low-frequency noise by children with normal hearing and by children with impaired hearing[END_REF][START_REF] Grant | The use of visible speech cues for improving auditory detection of spoken sentences[END_REF][START_REF] Sumby | Visual contribution to speech intelligibility in noise[END_REF]. A classical paradigm to demonstrate audiovisual fusion is provided by the "McGurk effect" in which a conflicting visual input modifies the perception of an auditory input, e.g. visual /ga/ added on auditory /ba/ leading to the percept of /da/ [START_REF] Mcgurk | Hearing lips and seeing voices[END_REF]. Audiovisual fusion in speech perception has long been considered as automatic [START_REF] Massaro | Speech perception by ear and eye[END_REF][START_REF] Soto-Faraco | Assessing automaticity in audiovisual speech integration: evidence from the speeded classification task[END_REF]. However a number of recent experiments have provided evidence that it is in fact under the control of attention in a broad sense, considering that various cognitive variables can modulate audiovisual integration [START_REF] Alsius | Audiovisual integration of speech falters under high attention demands[END_REF][START_REF] Buchan | The Effect of a Concurrent Cognitive Load Task and Temporal Offsets on the Integration of Auditory and Visual Speech Information[END_REF][START_REF] Colin | Top-down and bottom-up modulation of audiovisual integration in speech[END_REF][START_REF] Mozolic | Modality-specific selective attention attenuates multisensory integration[END_REF][START_REF] Navarra | Assessing the role of attention in the audiovisual integration of speech[END_REF][START_REF] Tiippana | Visual attention modulates audiovisual speech perception[END_REF].

Binding and unbinding in audiovisual fusion

While evidence for the non-automaticity of the fusion mechanism stays compatible with one-stage architecture, some data suggest that audiovisual interactions could intervene at various stages in the speech decoding process [START_REF] Bernstein | Audiovisual speech binding: convergence or association?[END_REF][START_REF] Grant | The use of visible speech cues for improving auditory detection of spoken sentences[END_REF][START_REF] Van Wassenhove | Visual speech speeds up the neural processing of auditory speech[END_REF]. Actually, audiovisual fusion could be conceived as a two-stage process, beginning by binding together the appropriate pieces of audio and video information, followed by integration per se [START_REF] Berthommier | A phonetically neutral model of the low-level audiovisual interaction[END_REF]. The binding stage would occur early in the audiovisual speech processing chain enabling the listener to extract and group together the adequate cues in the auditory and visual streams, exploiting coherence in the dynamics of the sound and sight of the speech input. To demonstrate the existence of this "binding" process we defined an experimental paradigm possibly leading to "unbinding". In this paradigm (Figure 1) incongruent "McGurk" (A/ba/ + V/ga/) or congruent "ba" (A/ba/ + V/ba/) targets were preceded by coherent or incoherent audiovisual contexts [START_REF] Nahorna | Binding and unbinding the auditory and visual streams in the McGurk effect[END_REF]. The experimental results showed that the McGurk effect (displaying the role of the visual input on phonetic decision) depends on the previous audiovisual context. Indeed, various kinds of incoherent contexts, such as acoustic syllables dubbed on video sentences, or phonetic or temporal modifications of the acoustic content of a regular sequence of audiovisual syllables, can significantly reduce the McGurk effect. Short incoherent context durations (even 1syllable long) were sufficient to produce a significant amount of unbinding [START_REF] Nahorna | Audiovisual speech binding[END_REF]. On the contrary, coherent contexts let the McGurk effect stable, which suggests that there is possibly a "default mode" in which binding occurs (and hence produces the McGurk effect in isolation).

Experiment 1-From unbinding to rebinding

Our previous studies clearly show that an incoherent context results in a decrease of the McGurk effect, which is due in our interpretation to an "unbinding" mechanism. An unanswered question is to know what kind of information is able to reset the system and put it back in its supposedly bound default state [START_REF] Nahorna | Binding and unbinding the auditory and visual streams in the McGurk effect[END_REF]. The objective of the first experiment in the present paper is to attempt to answer this question. For this aim, we tested whether applying a period of either coherent material or silence after the incoherent "unbinding" context would enable to recover the McGurk effect (Figure 1).

Experiment 2-The role of noise in audiovisual fusion

The role of visual speech is particularly important in noise [START_REF] Binnie | Auditory and visual contributions to the perception of consonants[END_REF][START_REF] Erber | Auditory and audiovisual reception of words in low-frequency noise by children with normal hearing and by children with impaired hearing[END_REF][START_REF] Grant | The use of visible speech cues for improving auditory detection of spoken sentences[END_REF][START_REF] Sumby | Visual contribution to speech intelligibility in noise[END_REF]. Noise also seems to modulate decision in the case of incongruent stimuli. Indeed, if one applies noise during a McGurk stimulus, the McGurk effect decreases when the extraneous noise is visual, whereas it increases when the noise is auditory [START_REF] Sekiyama | McGurk effect in non-English listeners: Few visual effects for Japanese subjects hearing Japanese syllables of high auditory intelligibility[END_REF][START_REF] Sekiyama | Inter-language differences in the influence of visual cues in speech perception[END_REF][START_REF] Sekiyama | Difference in auditory-visual speech perception between Japanese and Americans: McGurk effect as a function of incompatibility[END_REF][START_REF] Fixmer | The influence of quality of information on the McGurk effect[END_REF][START_REF] Kim | Audiovisual speech processing in visual speech noise[END_REF]. In the well-known "Fuzzy-Logical Model of Perception" (FLMP [START_REF] Massaro | Speech perception by ear and eye[END_REF]) this is interpreted as due to the increasing ambiguity of the noisy component, which would automatically decrease its role in the fusion process. However, it could also be envisioned that there is a specific weighting process in which a given modality would be positively or negatively modulated in the fusion process depending on the noise in this modality [START_REF] Heckmann | Noise Adaptive Stream Weighting in Audio-Visual Speech Recognition[END_REF][START_REF] Huyse | Degradation of labial information modifies audiovisual speech perception in cochlear-implanted children[END_REF]. In the first case fusion would only depend on stimuli while in the second case there would be in addition an evaluation of the perception conditions resulting in a modification of the fusion process per se. Our reasoning here is that if noise is applied in the (context + reset) part of the stimulus in Figure 1

Stimuli

The stimuli are described in Figure 2. They were typically made (Fig. 2, top) of: -an incoherent context (2 or 4 acoustic syllables superimposed on excerpts of video sentences matched for equal duration) for both experiments ("incoherent context"); -followed by a reset stimulus consisting in 0, 1, 2 or 3 coherent audiovisual syllables ("coherent reset") or audio silence with fixed image of duration 0, 480, 1000, 1480 ms ("fixed reset") in Experiment 1, while in Experiment 2 we used only the coherent reset; -finishing by a target which could be either a congruent audiovisual "ba" or a McGurk stimulus consisting in an audio "ba" dubbed on a video "ga". A control stimulus, aimed at providing a reference for the McGurk effect, was provided by (Fig. 2,bottom): a coherent context (2 or 4 coherent audiovisual syllables) ("coherent context"); -followed by a target which could be either a congruent audiovisual "ba" or a McGurk stimulus.

A series of audiovisual films were presented to participants in two blocks in both experiments. In Experiment 1, there was one block with coherent reset and the other one where the reset consisted in the silence with fixed image. In Experiment 2 there was one block without acoustic noise ("silent") and the other one with acoustic noise superimposed on all context and reset parts of the stimuli ("noise"). Noise consisted in speechshaped noise at 0 dB SNR. The target parts always remained without noise.

Coherent context and coherent reset material was constructed by pairing audiovisual syllables randomly selected within the following syllables ("pa", "ta", "va", "fa", "za", "sa", "ka", "ra", "la", "ja", "cha", "ma", "na"). In Experiment 1, the "fixed reset" was obtained by dubbing auditory silence on fixed image with durations 0, 480, 1000 or 1480 ms. In the incoherent context material, the auditory content was same, but the visual content was replaced by excerpts of video sentences matched in duration. The congruent "ba" target was used to ensure that participants were performing the speech task correctly and to serve as a baseline to contrast with the McGurk effect. The incongruent McGurk target was produced by carefully synchronizing an auditory /ba/ with a video /ga/, precise temporal localization of the acoustic bursts of the original "ba" and "ga" stimuli providing the cue for synchronization. McGurk targets were presented three times more than congruent "ba" targets, which served as controls.

For each (context+reset) condition (2 context durations; 4 reset durations for incoherent context; 2 reset types in Experiment 1, and 2 noise conditions in Experiment 2; hence altogether 20 context conditions) there were 4 occurrences of a "ba" target and 12 occurrences of a McGurk target. Hence there were 320 sequences in total spread over 2 blocks of 10 min each.

Procedure

All experiments were carried out in a soundproof booth.

Stimulus presentation and recording of responses were controlled by the Presentation software. The experiment consisted of two possible responses "ba" or "da" (with one button for "ba" and one for "da,") and the participants were instructed to constantly look at the screen and, each time a "ba" or a "da" was perceived, to immediately press the corresponding button. The films were presented on a computer monitor with high-fidelity headphones set at a comfortable fixed level. The video stream was displayed at a rate of 25 images per second, the subject being positioned at about 50 cm from the screen. There were 5 different orders of the stimuli in the films, and the order of the two blocks "fixed reset" and "coherent reset" in the case of Experiment 1 and "silent" and "noise" in the case of Experiment 2 was counterbalanced between subjects. The response button was also interchanged between subjects.

Participants

Twenty subjects participated in Experiment 1 (9 women and 11 men; mean 25.7 years) and twenty in Experiment 2 (13 women and 7 men; mean 34 years). All of them were French native speakers, without any reported history of hearing disorders and with normal or corrected-to-normal vision.

Written consent was obtained from each participant and all procedures were approved by the Grenoble Ethics Board (CERNI).

Assumptions and analyses

The experiment was focused on the role of context, reset and noise on the McGurk effect. For each (context, reset and noise) condition, each target and each subject, the amount of "ba" responses against "ba+da" responses was computed and used as an index of the subject's perception (between 0 and 1). An arc(sin(sqrt)) transformation was applied on these relative "ba" scores to ensure Gaussianity of the dependent variable in the analyses of variance that will be presented in Section 3. Though response times were systematically recorded and processed, they will not be presented here.

We had three main assumptions, all involving McGurk stimuli (let us recall that "ba" targets are just there as controls).

-Firstly, incoherent context should produce unbinding and decrease the McGurk effect (hence increase the amount of "ba" responses) in respect to coherent context, whatever the context duration (2 or 4 syllables). -Secondly, for incoherent context, reset should produce rebinding and increase the McGurk effect (hence decrease the amount of "ba" responses), from 0 to 3 syllables of duration of the reset stimulus. However, we had no expectation at the beginning of Experiment 1 whether "coherent" or "fixed" reset would both provide efficient reset. -Thirdly, noise in Experiment 2 should enhance the role of vision and hence globally increase the McGurk effect (decrease the score of "ba" responses) whatever the context and reset.

Results

Preliminary remarks

As expected, the "ba" target leads to 100% "ba" responses in both experiments and in all conditions. Therefore, for now on, we shall concentrate on McGurk targets.

Preliminary analyses of the role of context duration in the incoherent context conditions in both experiments showed that the incoherent context duration (2 vs. 4 syllables) has only little effect on the McGurk effect, hence we shall average data for the two context durations in the next analyses.

Assessing the efficiency of coherent vs. fixed resets in Experiment 1

On Figure 3 3, corresponding to the "fixed reset" condition, it appears that this reset (made of acoustic silence + fixed image) provides almost no rebinding, since the "ba" score only slightly decreases from 0 to 1-syl (that is 480ms duration), then remains stable and stays much larger than the score for coherent context even for the longest reset duration (3-syl corresponding to 1480 ms). -Good rebinding with coherent reset. On the contrary, looking at the bars in dark grey corresponding to the "coherent reset" condition, we observe that the "ba" score regularly decreases with reset duration and reaches the same value as for coherent context, coming back to its "default" state for the largest coherence period of 3 syllables.

To assess the significance of the rebinding effects, we performed an analysis of variance with the factors "subject" (random-effect), "reset" (coherent vs. fixed) and "reset duration" (0, 1, 2 & 3 syllables / 0, 480, 1000, 1480 ms). The three factors were statistically significant ("subject": F(19,18)=6.88, P<0.001; "reset": F(1,19)=5.45, P<0.05]; "reset duration" [F(3,57)=14.9, P<0.001]. The interaction between "reset" and "reset duration" was also significant [F(3,57)=7.65, P<0.001], which is in agreement with the difference between variations of scores with reset duration for fixed vs. coherent reset. Post-hoc analyses with Bonferroni corrections show that in the fixed reset condition, there is no difference between scores for the four reset durations. In the case of coherent reset syllables, the score at 0 was significantly higher than with 2 or 3 syllables, and the score at 1 or 2 syllables was significantly higher than with 3 syllables (P <0.05).

Figure 3. Results for Experiment 1. Percentage of "ba " responses for "McGurk" targets, in the "coherent reset" vs. "fixed reset" conditions for incoherent context with the four reset durations, compared with coherent context. The ANOVA was performed only for the four reset durations in the incoherent context.

Assessing the effect of noise in Experiment 2

On Figure 4 we display "ba" scores for McGurk targets in all conditions for Experiment 2 (averaging over the two context durations 2 and 4 syllables). Two major facts emerge from this figure.

-Unbinding/rebinding: Focusing on the "without noise" condition (black bars), we replicate the results of Experiment 1 in the "coherent reset" condition (remember that only coherent reset is used in Experiment 2). Indeed, the "ba" score is higher (less McGurk effect) for incoherent context without reset (most left) than for coherent context (most right) (unbinding). But it decreases when reset duration increases from 0 to 3 syllables (rebinding).

The effects are quantitatively different from Experiment 1 to Experiment 2, which is not unexpected considering the large inter-individual differences in the McGurk effect. But the portrait is qualitatively similar.

-Modulation by noise: Comparing black bars (with noise) and grey bars (without noise), it appears that noise decreases "ba" scores (and hence increases the

McGurk effect) for all conditions, and by a large amount (12 to 20%). To assess the effects of rebinding and noise, we performed an analysis of variance with the factors subject (random-effect), noise (silent vs. noise) and reset duration (0, 1, 2 & 3 syllables). The three factors are statistically significant ("subject": F (19, 18) =13.46, P<0.001; "noise": F(1,19)=6.12, P<0.05; "reset duration": F(3,57)=14.82, P<0.001). There was no significant interaction between any pair of factors. The effect of reset duration confirms the result of Experiment 1 for coherent reset. The effect of noise confirms that applying noise in the context+reset part modulates the target perception even though there is no noise during the target. The lack of interaction between noise and reset duration shows that the role of noise seems more or less stable whatever the reset duration. Altogether, it appears that noise applied in the context part modifies the results of audiovisual fusion, with a global and more or less stable effect leading to an increase of about 15% in the McGurk effect whatever the context.

Discussion

Unbinding, rebinding and noise in the audiovisual fusion process

This set of experiments confirms that context may modify the McGurk effect, through a series of mechanisms, which combine unbinding (through incoherent context decreasing the role of the visual input), rebinding (through coherent reset setting back the weight of the visual input) and noise (increasing the role of the visual input). In Experiment 1, it appeared that a fixed reset has almost no rebinding effect, with the consequence that even for the longer duration (around 1.5s) the subjects stay "frozen" in an unbound state where the McGurk effect is largely decreased. On the contrary, a coherent reset of 3 syllables is enough to completely recover from unbinding and restore the default binding stage.

In Experiment 2, it appeared that noise applied on the contextual partbut NOT on the targetsystematically increases the McGurk effect, whatever the content of context and reset. To our knowledge it is the first time that such a result is obtained. This strongly suggests that noise in the McGurk effect, already displayed with noise applied on the target itself [START_REF] Sekiyama | McGurk effect in non-English listeners: Few visual effects for Japanese subjects hearing Japanese syllables of high auditory intelligibility[END_REF][START_REF] Sekiyama | Inter-language differences in the influence of visual cues in speech perception[END_REF][START_REF] Sekiyama | Difference in auditory-visual speech perception between Japanese and Americans: McGurk effect as a function of incompatibility[END_REF][START_REF] Fixmer | The influence of quality of information on the McGurk effect[END_REF][START_REF] Kim | Audiovisual speech processing in visual speech noise[END_REF], intervenes not only at the level of the stimuli, but also at the level of the fusion process itself. From there on, it is possible to come back to the models of audiovisual fusion available in the literature. Classical models consider that phonetic decision operates at a given representational stage and produces an integrated percept combining auditory and visual cues in a given way, possibly mediated by general attentional mechanisms. Our data on the binding process led us suggest that an additional computational stage should be incorporated before decision operates, involving online computation of some assessment of the coherence/ incoherence of the auditory and visual inputs, resulting in a "two-stage model" of audiovisual speech perception [START_REF] Berthommier | A phonetically neutral model of the low-level audiovisual interaction[END_REF] (see Fig. 5). The present results first add some information about the way coherence could be computed, involving a dynamics made of unbinding and rebinding stages with short constant times: indeed, less than one second of incoherence (2 syllables or less) suffices to produce unbinding, and less than one second of coherence (2 syllables or less) suffices to produce complete rebinding. Furthermore, the results about noise suggest that noise, and probably more generally knowledge about the conditions of communication, also participate to the decision process by providing an enhancement of "efficient" modalities, not contaminated by noise, versus modalities where noise could contaminate the decision process (Fig. 5). The present data suggest that the role of unbinding/rebinding on one hand, and noise-based selective weighting of each modality on the other hand, could play additional independent roles, according to the lack of interaction between noise and reset in Section 3.3. This will have to be confirmed in future experiments specifically dealing with this question.

Future experiments

A number of further experiments will have to extend the present data in various directions, involving e.g. more about the dynamics of unbinding and rebinding. Various proposals could also deal with reset mechanisms (such as changing speaker or the global communication setting), or specificity of the binding mechanism (could non-speech incoherent audiovisual material also produce unbinding?). The role of noise could also be further assessed by using visual noise. Indeed, some studies [START_REF] Colin | Top-down and bottom-up modulation of audiovisual integration in speech[END_REF][START_REF] Kim | Audiovisual speech processing in visual speech noise[END_REF] have manipulated the size or position of the face and found influence on the McGurk effect, showing in both that visual noise may decrease the McGurk effect just as auditory noise increases it. If our conjecture about the role of noise in Figure 5 is correct, this effect should also occur for visual noise added on the contextual part of the stimuli in the present paradigm. Another important extension concerns intelligibility in noise. The present paradigm was an aim to progress towards the next important question that is to know if unbinding mechanisms would also decrease the beneficial effect of lipreading in noise. Future experiments will deal with targets consisting in ambiguous though coherent stimuli and test if an incoherent audiovisual context is able to remove the visual benefit. This will enable us incorporate the two-stage model inside a general question concerning the cocktail-party effect and what we propose to call "audiovisual speech scene analysis" [START_REF] Nahorna | Binding and unbinding the auditory and visual streams in the McGurk effect[END_REF].

Figure 1 :

 1 Figure 1: Experimental paradigm for displaying unbinding or rebinding mechanisms modulating the McGurk effect

Figure 2 .

 2 Figure 2. Description of the audiovisual material

Figure 4 .

 4 Figure 4. Results for Experiment 2. Percentage of "ba " responses for "McGurk" targets, in the "silent" vs. "noise" conditions for incoherent context with the four reset durations, compared with coherent context. The ANOVA was performed only for the four reset durations and the two noise levels in the incoherent context.

Figure 5 :

 5 Figure 5: A two-stage model of audiovisual speech perception

Acknowledgements

This project has been supported by ANR Multistap (MULTISTability and binding in Audition and sPeech: ANR-08-BLAN-0167 MULTISTAP) which provided a doctoral funding for Olha Nahorna, and by Academic Research Community "Quality of life and ageing" (ARC 2) of the Rhône-Alpes Region, which provided a doctoral funding for Ganesh Attigodu Chandrashekara.


Denis Roegel
email: roegel@loria.fr

The "Villarceau circles" in Uhlberger's staircase (ca. 1580)

The city of Strasbourg is the home of many architectural masterpieces, and in the first place the magnificent cathedral. Its spire was completed in 1439, but some of its parts were added during the later Renaissance. The case of today's astronomical clock, for instance, was built between 1571 and 1574, and its stone structure, as well as the winding staircase that allows access to its various mechanisms, were built by Hans Thomann Uhlberger, the then architect of the cathedral.

. This is a winding staircase with a hollow central newel (figures 1 and 2). A visit to the Musée de l'OEuvre Notre-Dame will lead through it, and it should not be missed.

The circular structures of the newel

The staircase of the OEuvre Notre-Dame is a sinistral (left-handed) staircase with a hollow newel. More precisely, the newel is made of three columns which are joined at the top by two circular constructions (figure 2).

The lowest of the circular constructions is a relatively simple sector, with grooves reminiscent of other parts of the staircase. This sector joins the three columns, one of which has a corinthian capital.

The three columns extend above this sector and reach a fully circular structure, before continuing to the sexpartite-like rib vault. This last circular structure, which is in fact the end of the ramp, is one of the most interesting features of the staircase.

A close examination reveals that the curves which make up this circular structure are nearly circular rings. (We assume "ring" to mean an annular shape, but not necessarily a circular one.) The fact that the structure contains rings was of course known by Uhlberger and those who studied the staircase afterwards, but this feature got renewed interest a few years ago when the French mathematician Marcel Berger interpreted these rings as so-called "Villarceau circles" [START_REF] Berger | Mathématiques élémentaires approfondies : Géométrie[END_REF][START_REF] Berger | Géométrie[END_REF][START_REF] Berger | Géométrie[END_REF][START_REF] Berger | Les cercles de Villarceau[END_REF][START_REF] Berger | Géométrie vivante, ou l'échelle de Jacob de la géométrie[END_REF][START_REF] Berger | Geometry revealed: A Jacob's ladder to modern higher geometry[END_REF]7]. We will go into more detail about these rings in the next section, but for now, what is important for us is that these rings are interlaced and non-intersecting. These rings were again mentioned in the recently published guide of the museum [11, p. 179], albeit without crediting Berger.

How many rings does this structure contain? Berger identified three rings (figure 3, left) [START_REF] Berger | Les cercles de Villarceau[END_REF]7]. It is true that the photograph used by Berger seems to show three circles, but the picture contains at least one inconsistency, namely that there seems to be something else towards the right, which we marked in red in figure 3. Moreover, Berger's 2002 article has four different pictures of the circular structure, and the one in figure 3 (right) is even more explicit.

There are in fact not three, but four rings in that circular structure! The fact that Berger missed the fourth ring may be due to the assumption that the number of rings equals the number of columns in the newel, or to an indirect examination. And the fact that there are three columns is itself determined by the rib vault, and hence by the number of sides of the tower containing the staircase.

Figure 4 shows another view of the circular structure, and an identification of these four rings. It appears that Berger merged two circles, and missed the circle protruding towards the right. The fact that the blue circle (in our reconstruction and in Berger's picture) cannot be seen from below is obvious when the circular structure is examined from above (figure 5).

Villarceau circles

In his lectures on geometry published in 1972-1974 [START_REF] Berger | Mathématiques élémentaires approfondies : Géométrie[END_REF], and perhaps before, Berger identified the rings of the top circular structure as "Villarceau circles." Berger's statement was repeated in subsequent editions of his lectures, in particular in 1977 [START_REF] Berger | Géométrie[END_REF] and 1990 [START_REF] Berger | Géométrie[END_REF]. In all these works, Berger included a picture drawn from Hans Haug's book [START_REF] Haug | L'art en Alsace[END_REF]. Haug was the curator of the Strasbourg museums and the founder of the museum of the OEuvre Notre-Dame. Berger's (brief) description of the staircase was reprinted (with the picture) in the journal L'Ouvert in 1998 [START_REF]Les cercles de Villarceau[END_REF], and it eventually made its way to a larger audience in 2002 [START_REF] Berger | Les cercles de Villarceau[END_REF] Berger's claim is that the top of the staircase is a torus sculpted in such a way that its edges are Villarceau circles. This analysis, in turn, is used to support the claim that these circles were known long before Villarceau.

So, what are these "Villarceau circles"? Villarceau circles are certain circles which can be traced on a torus. A torus is the surface generated by a circle rotating around an axis lying in the circle's plane and not intersecting it (see figure 6). Through each point of the torus, it is possible to find four circles lying on the torus. One circle has its axis parallel to the generation axis, another one is the generating circle, and the last two are known as Villarceau circles, as they were first described by Antoine Yvon Villarceau (1813-1883) in 1848 [START_REF] Yvon | Extrait d'une note communiquée à M. Babinet par M. Yvon Villarceau[END_REF]. Conversely, if a circle is rotated, it may generate a torus. However for that to be the case, the offset of the circle needs to be related to its tilt and to its radius, so that most tilted and offset circles will not generate a torus, although they will generate a similar surface. The circle of radius r first has to be tilted by an angle α around the axis X, then offset along that same axis by s = r sin α, before eventually being rotated around the axis Z in order to generate the torus.

So, one should be cautious, because if the rings of the staircase are planar and circular, it does still not entail that these rings are Villarceau circles of some imaginary torus. Figure 7 shows the surfaces generated by correctly offset circles (a), and those generated by other circles (b-d). But when the tilt of the circles is not the right function of its offset, the section of the revolution surface is not a circle, hence the circles are not Villarceau circles. In (b), the tilt is too large. In (c), the tilt is too small. And in (d), the absolute value of the offset is correct, but not in the right direction.

Now that we know what Villarceau circles are, can we conclude about the rings in the staircase?

The first thing that we should notice is that the upper circular structure does not contain any obvious torus. There are rings, and these rings are joined, but the structure does not have the appearance of a torus with superimposed rings. Of course, we can consider that the initial piece of stone was some kind of torus, and that the rings were carved out, but that does not mean that the stone cutter or architect had the understanding that the rings could embrace a torus.

The second observation is that such carved out rings can be designed as rotated rings, that is, one ring can be obtained by the rotation of the previous one. In that case, the architect may have found out that by shifting the tilted circles and rotating them, these circles would not intersect. But that again does not entail that there was a knowledge of an underlying torus.

And finally, we have seen that rotated circles do only produce a torus when the offset is appropriately related to the tilt of the circles.

In summary, nothing in this circular structure supports that these four rings are Villarceau circles of some imaginary torus. They may be such circles, but that can only be ascertained by proper measurements. It seems very unlikely that the rings do exactly lie on a torus.

But if that is nevertheless the case, should we conclude that Uhlberger knew about Villarceau circles? If the torus was visible, I would probably answer yes to that question. But since this is not the case, I consider that it is a historical mistake to interpret these four rings as Villarceau circles, even if they are (close to) Villarceau circles of an imaginary torus. I can therefore not subscribe to the statement that "these rings are the practical expression of a theory stated five centuries later by the mathematician who gave them his name." [11, p. 179]. Apart from the fact that there are only three centuries between Uhlberger and Villarceau, this statement is a historical distortion, a projection of the future into the past.

We have also assumed that the four rings are circles. But are they? Berger wrote that they are planar, hence that they are Villarceau circles. We have seen above that this is a hastily drawn conclusion since the planarity is a necessary, but not sufficient, condition for the curves to be Villarceau circles. In fact, when we examined the rings, their planarity was far from obvious! In figure 5(a), for instance, the blue ring seems to have a bent and the projected curve does not seem convex. The same may perhaps be observed with the other rings. In other words, it is very likely that the rings are not even circular, which should definitely exclude the interpretation as Villarceau circles, and certainly the suggestion that Uhlberger may have known about such a property of the torus.

Twisted beams

We have seen that if the rings at the top of the staircase are circles, they could be obtained by rotating one of them. If the offset is related to the tilt and radius of the circles, then the circles generate a torus. Uhlberger certainly did not have a torus in mind, but one might argue that he wanted to experiment with rotated circles and came to a construction which was close to the one which would be obtained by starting from a torus and adding a number of Villarceau circles. This assumption, however, does not explain why the rings are not all planar.

There is however another explanation to the construction of the capital of staircase. And that is that the capital is actually an extension of the ramp. The staircase is lefthanded (sinistral staircase) and the ramp therefore also twists towards the left. At some point the ramp comes very close to the capital and the capital seems to extend the ramp, but with a zero lead. What is more interesting is that the ramp can be seen as a twisted beam with four (helicoidal) edges. And the capital basically shows four edges, twisted towards the left in the continuation of the ramp. This both explains the orientation of the "circles", and their number.

Indeed, a square beam twisted 360 degrees after one turn gives four interlaced curves (figure 8). A pentagonal beam would give five interlaced curves, and so on. However a beam twisted regularly only provides an approximation of Villarceau circles, because the density of Villarceau circles varies within a section of the torus (see figure 7(a)). In particular, the curves produced by the edges are not planar (figure 9). They are nevertheless a good approximation to circles, as shown in figure 10.

Conclusion

The staircase of the OEuvre Notre-Dame turns out to be most interesting, and we have discussed the claim that it embeds Villarceau circles. Instead, we consider that the architect Uhlberger had no knowledge about these circles and that these curves (which are only approximate circles) appeared accidentally as an extension of the construction of the ramp of the staircase.

It would now be interesting to have a closer examination of the rings, and in particular to measure the dimensions of the circles. More precisely, the planarity of the four curves should be checked, as well as their size, their tilt, their offset from center, and their spread around the circumference, to ascertain how close these curves are to Villarceau circles of an imaginary torus.

Figure 1 :Figure 2 :

 12 Figure 1: The Musée de l'OEuvre Notre-Dame in Strasbourg. The staircase is located within the hexagonal tower between the two buildings. (Wikipedia)

Figure 3 :

 3 Figure 3: Two pictures used in Berger's article [4].

Figure 4 :Figure 5 :

 45 Figure 4: (a) The interlaced rings seen from below. (Photograph by the author.) (b) The identification of the four rings. (c) A reconstruction of the rings, using tilted circles, offset, and rotated 90, 180 and 270 degrees around the yellow axis. (d) The picture used in Berger's article [4], with the blue circle incorrectly drawn. The wrong part is marked in red, and corresponds to the fourth circle.

  and a few years later in Berger's Géométrie vivante [5, 6]. Berger's 2002 text was further expanded in 2010 for an online text on the bibnum site [7]. Only in 2002 and 2010 did Berger consider that there are three circles. The number of circles is apparently not mentioned in Berger's other works.

Figure 6 :

 6 Figure 6: A torus with two of the circles discovered by Villarceau (in red and green), in addition to the white and yellow circles which were well known before Villarceau. The view on the right shows the lower half of the torus cut by the plane containing two of the Villarceau circles.

Figure 7 :

 7 Figure 7: The section of a set of 20 Villarceau circles is shown in (a) and it is circular, although one can observe that the intersections of the circles are not evenly spaced.But when the tilt of the circles is not the right function of its offset, the section of the revolution surface is not a circle, hence the circles are not Villarceau circles. In (b), the tilt is too large. In (c), the tilt is too small. And in (d), the absolute value of the offset is correct, but not in the right direction.

Figure 8 :

 8 Figure 8: A square beam twisted regularly provides a good approximation of Villarceau circles. It is only an approximation, because the density of Villarceau circles in a section of the torus is not constant.

Figure 9 :

 9 Figure 9: The edges of the twisted beam (in black) superimposed with the Villarceau circles (in color). The two sets of curves are close, but not identical. Moreover, the edges of the twisted beam are not a planar curve, as displayed with the blue circle which lies in the viewing direction.

Figure 10 :

 10 Figure 10: The twisted beam superimposed with the Villarceau circles.


T Tormos

S Dupuy

K Van Looy

E Barbe

P Kosuth

AN OBIA FOR FINE-SCALE LAND COVER SPATIAL ANALYSIS OVER BROAD TERRITORIES: DEMONSTRATION THROUGH RIPARIAN CORRIDOR AND ARTIFICIAL SPRAWL STUDIES IN FRANCE

Keywords: Top-down classification tree construction, Rules-based classification, Large mapping, Classification automation, Riparian Corridor, Artificial sprawl, Broad scale spatial analysis

Spatial analysis using fine-scale information over broad territories is essential to define efficient restoration strategies from local to national scale. We designed an OBIA dedicated to produce operationally reliable fine-scale information over broad territories. The originality of our OBIA lies particularly in the top-down approach for the construction of the classification tree and the use of "knowledge-based rules" classification technique. The implementation of this OBIA over the two study areas -(i) the Normandy region for riparian area land cover mapping (5600 km² riparian area) and (ii) fours departments over the Languedoc-Roussillon region (22644 km²)demonstrates the operability of our approach (time-efficient, reproducible, transferable, portable). Broad scale spatial analysis conducted from resulting maps demonstrate the interest of using fine-scale information and highlight that OBIA, following our approach, will be at very short run a broadly applicable method to carry out such analysis.

INTRODUCTION 1.1 VHRS and OBIA

Spatial analysis is a key feature to the understanding of structures, dynamics and changes in the environment, societies and territories. Furthermore it is essential for the exchange between stakeholders in society (citizens, government, socioeconomic) to define coordinated management strategies and implement them. Therefore a huge step forward in the remote sensing for Earth Observation has been seen in the last two decades for producing spatial information required to carry out spatial analysis (Turner et al., 2001). Nowadays, with the progresses of satellite and airborne imagery in High and Very High Spatial Resolution (H&VHSR), fine-scale land cover spatial analysis over broad territories could be conducted. However, due to the heterogeneity and the volume of information within H&VHSR images, extracting it efficiently and with lower costs over broad territories is quite challenging. The last five years, OBIA have revolutionized the processing of H&VHRS remote sensing data by providing effective computerassisted classification techniques whose results come close to the quality of manual photo-interpretation, while being much faster and cheaper and much more reproducible (e.g., [START_REF] Durieux | A method for monitoring building construction in urban sprawl areas using object-based analysis of Spot 5 images and existing GIS data[END_REF][START_REF] Tiede | Transferability of obia rulesets for idp camp analysis in darfur[END_REF]. As a result, we proposed in this paper an operational OBIA procedure designing for the extraction of spatial information from H&VHSR remotelysensed data over broad territories. Two case studies (CS) supporting public politics in France are exhibited: riparian corridor and artificial sprawl management.

CS1: Riparian corridor management

Maintaining and restoring good riparian buffer area conditions could constitute a major action to improve the freshwaters" ecological status required by the European Water Framework Directive (WFD) by 2015. Riparian vegetation, being an interface between terrestrial and aquatic systems, influences the biodiversity and water quality of stream and river ecosystems in many specific ways (Naiman et al., 2005). As a result, the first case study (CS1) is dedicated to the fine classification of Riparian Area Land Cover (RALC). RALC maps will serve as a basis for calculating spatial indicators related to human pressures along rivers and riparian vegetation characteristics (e.g. composition, continuity and strip width). These indicators will help in better understanding and predicting mechanisms influencing river ecological status at riparian scale, in order to prioritize and design efficient river corridor restoration strategies [START_REF] Gergel | Landscape indicators of human impacts to riverine systems[END_REF][START_REF] Tormos | Improving the quantification of land cover pressure on stream ecological status at the riparian scale using High Spatial Resolution Imagery[END_REF]. The classification must, as a minimum requirement, extract: "water surfaces" and "semi-natural bare soils" required to better delimitated the river bed for building spatial indicators; "arable areas", "urban areas" considered to be the main causes of stream ecological status alteration [START_REF] Allan | Landscapes and Riverscapes: The Influence of Land Use on Stream Ecosystems[END_REF]; "tree vegetation", "semi-natural herbaceous and shrub vegetation" and "permanent agricultural grassland" constituting the main natural elements of the river corridor landscape that maintain biodiversity and regulate non-point source pollution (Naiman et al., 2005).

CS2: Artificial sprawl management

In France, in the peri-urban context, artificial sprawl (i.e., sprawl of urban and other artificial land development such as roads, quarries, landfills...) dynamics are particularly strong with huge population growth as well as a land crisis. The increase and spreading of built-up areas from the city centre towards the periphery takes place to the detriment of natural and agricultural spaces. The conversion of land with agricultural potential is all the more worrying as it is usually irreversible [START_REF] Pointereau | Abandon et artificialisation des terres agricoles[END_REF]. As a result, the second case study (CS2) is dedicated to the fine extraction of artificial objects at a given time. The production of such map on different time steps will serve as a basis for calculating spatial indicators related to artificial sprawl. These indicators will help in better localizing and quantifying loss of land at both local and regional scales in order to prioritize and define efficient restoration strategies dedicated to the conversion of land with agricultural potential. Two classes are required: "artificial areas" (i.e., urban and fabric areas, transport infrastructures…) and "non artificial areas".

DATA AND METHODS

2.1 Study areas, datasets and pre-processing 2.1.1 For the CS1 We classified RALC on a part of the Normandy river network (25 000 km² basin; 6000 km long river network, 5600 km² riparian area) located in the North of France (see Figure 1). This area is dominated by a strong agricultural occupation, primarily focused on field crops and livestock, with specific structures in the riparian buffer, such as tree vegetation and grassland near the river. This agricultural landscape is dotted with urban zones, from small to large. Datasets were chosen according to both their availability and their cost effectiveness on the overall French territories for managers. A comparative economic analysis of the different data led to the collection of two types of H&VHSR multispectral remotely sensed data: orthophotos and SPOT 5 satellite images. The characteristics of these image and preprocessing data are summarized in table 1. Orthophotos (0.5 m pixel) provide the textural information required to detect narrow and fragmented cover types along rivers [START_REF] Müller | Mapping riparian vegetation along rivers: old concepts and new methods[END_REF]. SPOT-5 XS (10 m pixel) is also acquired in order to get information in the NIR band essential to discriminate vegetation classes [START_REF] Johansen | Mapping Structural Parameters and Species Composition of Riparian Vegetation Using IKONOS and Landsat ETM+ Data in Australian Tropical Savannahs[END_REF]. In addition existing spatial thematic data, relevant for RALC classification were collected. This data gives a precise (metric precision) spatial information on artificial continuous areas (city centre), roads, hydrographic surfaces (lakes and reservoirs) and the majority culture of farms declared in the frame of the European common agricultural policy. The European land cover database, CORINE Land Cover (CLC), was collected also to perform the spatial analysis of this case study. CLC was built by visual interpretation of both Landsat and SPOT satellite images acquired from 1990 and 2000 years. Interpretation of the images is based on transparencies overlaid on 1/100.000 hard copy prints of satellites images (Bossard et al., 2000). It is based on a standard nomenclature organized into a 3-level hierarchy containing 44 classes. CLC features, characterized by a 25 ha minimum area, are either homogenous areas or combinations of land cover types with a certain recognizable structure.

For the CS2

We extracted artificial objects over the four coastal departments (Aude, Gard, Hérault and Pyrénées-Orientales) of the Languedoc-Roussillon region located in the South of France. The region covers 22644 km² and had a population of 2548000 in 2007. This is one of the most productive regions in wine in France. Over the last three decades, population pressure in Languedoc-Roussillon has led to rapid and poorly managed urbanization of the coastal plain which comprises notably the most productive land in the region (Abrantes et al., 2010). This phenomenon was increased by the successive crises that occurred in the wine making sector in recent years [START_REF] Jarrige | Mutation du foncier agricole en frange urbaine. Elaboration et mise à l'épreuve d'une politique de régulation territoriale[END_REF]. RapidEye images, HSR multi-spectral satellite images, were collected in order to extract artificial object at summer 2009. Summer period is a suitable period for this work because vegetation is in active growth on this region that limits confusion between bare soils (natural or agricultural) and artificial areas. These images are produced by the remote sensors in a five-satellite constellation in use since 2008. This constellation can quickly provide homogenous and recent data covering large areas. Images are delivered in 25 km large orthorectified blocks with a spatial resolution of 5 m. Given the difficulties involved in extracting roads from remotely sensed images, we collected this information from available French topographic data. An earlier map informing on artificial objects in 1997 was collected also to perform the spatial analysis of this case study. This map was produced by [START_REF] Dupuy | An Object-Based Image Analysis Method for Monitoring Land Conversion by Artificial Sprawl Use of RapidEye and IRS Data[END_REF] from indian remote sensing images (5.8 m pixel for panchromatic image and 23 m for multi-spectral image).

OBIA approach

The originality of our OBIA approach lies in (i) the employment of the thematic spatial information into the classification, (ii) the classification tree; and (iii) the definition of the classification rules. (i) Thematic spatial data contain fully reliable information to make maps. Therefore, this is the first information that is exploited by our OBIA procedure. A first level segmentation is created from this information. The study area is segmented according to the boundaries of thematic data entities. Then, the image objects that result are affected -or not -to a thematic class using Boolean rules. Within these boundaries, other(s) segmentation level(s), suitable for extracting objects of interest, are built. The classification tree is implemented on the resulting hierarchical image object network. 1. Image data characteristics for the two case studies (B = Blue, G = Green, R = Red, RE = Red Edge, NIR = Near InfraRed, SWIR = Shortwave Infrared) (ii) The construction of the classification tree can be categorized as top-down (i.e., low to high level) image interpretation. The tree starts from the classes that are the easiest to extract, according to available data sources, to the classes of interest (e.g. "water surface" / "land surface", and then within "land surface", "soils with high vegetation" / "soils with low vegetation"…). (iii) For each decision in the classification, rules are developed using fuzzy or crisp membership function based on one or several relevant spectral, spatial and contextual features selected by either expert knowledge or trial and error runs or sole visual judgment. Tree classification is implemented over the specific hierarchical image object network. A complete description of OBIA implemented over the two cases studies can be found in [START_REF] Tormos | Object-based image analysis for operational fine-scale regional mapping of land cover within river corridors from multispectral imagery and thematic data[END_REF] for CS1 and [START_REF] Dupuy | An Object-Based Image Analysis Method for Monitoring Land Conversion by Artificial Sprawl Use of RapidEye and IRS Data[END_REF].

Classification automation

The study zone is first divided into homogeneous mapping regions according to the image acquisition date. Then, a master ruleset is designed according to the OBIA approach on a pilot zone. Finally, the master ruleset is implemented for each mapping region in adjusting classification rules and eventually added some classes to the classification tree for taking into account regions specificities. Master ruleset and classification rules adjustment have been developed using eCognition 8 developer. OBIA processing for each mapping region was performed with four licenses of eCognition 8 server (for running 4 CPU on server) using tiling and stitching process.

Classification validation

As a metric accuracy of boundary location was not crucial for the final aim of the studies, we did not assess planimetric accuracy of boundaries but focused on assessing the semantic quality of the classification (i.e. assessment of object nature). Obviously, the semantic quality depends on planimetric accuracy. The semantic quality assessment was performed using a confusion matrix [START_REF] Foody | Status of land cover classification accuracy assessment[END_REF]. The confusion matrixes were computed after grouping classes according to the typology targeted for each study (see 1.2 for CS1 and 1.3 for CS2). Given that the features used for classification are calculated at the object scale, objects or polygons have been chosen as sampling units for the selection of control data [START_REF] Grenier | Accuracy assement method for wetland object-based classification In: Geographic Object Based Image Analysis[END_REF][START_REF] Tiede | Supervised and forest type-specific multiscale segmentation for a one-level-representation of single trees[END_REF]. However, confusion matrixes were computed using the area of the selected control objects (expressed as a number of 0.5 m pixels) because the land-cover maps result from the implementation of the multilevel OBIA scheme and contains objects of different sizes, from fine to large. For each CS, to select control data with the objective of a spatially and thematically well distributed sample over the studied zone, as suggested by [START_REF] Congalton | Putting the map back in map accuracy assessment. A peer-reviewed chapter[END_REF],the entire sampling frame is divided into N equal grid cells. Next, a stratified random sampling is performed using grid cells as geographical strata (equal area for all strata). The number of cells (N) is equal to the desired sample size, in order to have at least one object of each class per grid cell. As suggested by [START_REF] Congalton | A review of assessing the accuracy of classifications of remotely sensed data[END_REF], 50 samples are collected for each class in order to build the confusion matrix. Considering the size of the study areas (and mapping regions), collecting field data for the control sample would be extremely labour intensive and time consuming. As suggested by [START_REF] Zhu | Accuracy Assessment for the U.S. Geological Survey Regional Land-Cover Mapping Program: New York and New Jersey Region[END_REF], selected control objects are photo-interpreted using the image with the highest spatial resolution as control data. To maintain objectivity of photointerpretation, the classified maps were not viewed during the process.

Spatial analysis

Building riparian indicators (CS1)

A spatial indicator is invariably defined by aggregating a landscape structure attribute over a delimited area (spatial scale). Some examples of structural attributes are the number of different cover types, the proportion of each cover type, the shape of patches, and the spatial arrangement and connectivity of patches [START_REF] Li | On Definition and Quantification of Heterogeneity[END_REF]. The domain over which spatial indicators are computed at the riparian scale for the modelling is generally defined by combining a lateral distance to the river with longitudinal distances upstream and downstream from the ecological station where stream ecological status is measured [START_REF] Tormos | Improving the quantification of land cover pressure on stream ecological status at the riparian scale using High Spatial Resolution Imagery[END_REF]. We analysed here the land cover composition (area percentage of each land cover category) according to different buffer widths (5,10,15,20,25,30,35,40,45, and 50 m) on 3-km upstream distance from an ecological station. We used the typology targeted for this study (see 1.2). According to the definition, this makes a total of 10 spatial indicators for each land cover category. These indicators were computed from H&VHRS derived map (resulting of OBIA) and CLC database in order to highlight the gain of H&VHRS remotely sensed data for characterizing RALC. This spatial analysis in this study was conducted using ESRI GIS tools.

Building artificial patches (CS2)

To quantify artificial sprawl we need before to compute artificial patches. Artificial patch is defined as a continuous geographic entity of neighbouring artificial objects [START_REF] Dupuy | An Object-Based Image Analysis Method for Monitoring Land Conversion by Artificial Sprawl Use of RapidEye and IRS Data[END_REF]. Artificial patches are built generally from a mathematical morphology operation, the dilatation and erosion algorithm [START_REF] Haralick | Image Analysis Using Mathematical Morphology[END_REF]. Distant objects up to 50 m (Euclidean distance) were considered as neighbouring objects in our study. This distance was chosen by trial and error runs of this algorithm using different Euclidian distances [START_REF] Dupuy | An Object-Based Image Analysis Method for Monitoring Land Conversion by Artificial Sprawl Use of RapidEye and IRS Data[END_REF]. Artificial patches were built from 2009 artificial object maps (resulting of OBIA) and 1997 artificial object maps. The spatial analysis in this study was conducted using ERDAS imagine tools.

RESULT

Classification accuracy and processing time

Fine-scale maps were obtained for the two case studies (see Figure 2 RALC map). Accuracy and time-efficiency of OBIA are summarized in Table 2. A high accuracy was obtained for the two case studies with a relatively operational timeefficiency.

Case studies

Comparison of riparian spatial indicators (CS1)

Figure 3 shows the evolution of the land cover composition (area percentage) according to the lateral distance to the river channel (buffer width from 5 m to 50 m) from H&VHSRderived map and CLC database. Not surprisingly, the spatial resolution of land-cover maps has a strong influence on our ability to quantify landscape indicators [START_REF] Lattin | Influence of remote sensing imagery source on quantification of riparian land cover/land use[END_REF]. We can see that major landscape patterns inside the riparian areas (i.e., semi-natural vegetation types) which are revealed by H&VHSR maps are totally smoothed in CLC maps. According to CLC (Figure 3B), only two land cover category are present in riparian area (grassland and urban areas) and spatial information does not significantly change with the buffer width whatever the land cover category, in contrast to the H&VHSR-derived map (Figure 3A).

Urban areas Arable areas Grassland

Map of artificial sprawl

Figure 4 shows an extract of the artificial sprawl map between 1997 and 2007 years. It demonstrates that this phenomenon occurs in Languedoc-Roussillon region (an artificial sprawl of 18% in total was observed over the study area) localized globally around each town whatever the size.

DISCUSSION

Through these two case studies, we confirm OBIA have revolutionized the processing of H&VHSR remote-sensing data and demonstrated that our OBIA approach is an operational method for producing reliable spatial information in order to conduct fine-scale spatial analysis over broad territories.

Our approach appears (1) reproducible: whatever the mapping region, defined in the two case studies, the total accuracy was good to very good (see Table 2); (2) easy transferable: despite the diversity of landscapes in study areas, it has been possible to use the same parameters of segmentation and the same class hierarchy on all mapping regions, without changing classification features: the major operator task was to adjust the threshold values of the different features used to define classes. This is partly due to the top-down approach for the construction of tree classification (that divides the feature space into finer and finer units) and the use of "knowledge-based rules" classification technique. Top-down approach promotes the use of simple rules that are easier to transpose to other mapping regions and facilitates the appropriation of the methodology by new operators. Moreover specific classes of a given mapping region can be integrated in the tree classification without questioning its overall construction;

(3) quickly applicable over broad areas: the OBIA processing is little time consuming (see Table 2). The adjustment time of rules obviously depends on the experience of the operator and his knowledge of the study area and its land-cover diversity. As observed by [START_REF] Lucas | Rule-based classification of multitemporal satellite imagery for habitat and agricultural land cover mapping[END_REF][START_REF] Tiede | Transferability of obia rulesets for idp camp analysis in darfur[END_REF]) "knowledge-based rules" classification technique appears more flexible: all rules could be refined with the full control of the user, at any time in the classification process and, in most cases, without changing the class allocation of other objects (which is generally not the case with the "supervised" classification technique). Moreover, while the knowledge-based rules" classification technique can be distributed easily on a cluster machine, thus drastically reducing the computation time, the "supervised" classification technique cannot be distributed because it requires the processing of a mapping region as a whole in order to collect a spatially representative training sample.; and (4) Portable and scalable (i.e. able to integrate new data sources): the method can easily manage information from multiple data sources by resolving conflicts between these sources using fuzzy logic. This property has been particularly valuable in the CS1 to resolve conflicts between SPOT 5 XS and orthophotos data sources. For instance, information for assigning an object to one of these two classes, "soil with high vegetation" and "soil with low vegetation", can be contradictory because data were not acquired in the same season on the mapping region. Thus, fuzzy logic appears indispensable to treat multisource information in this OBIA classification. Thanks to fuzzy logic, new data sources could easily be integrated and combined with the initial data sources. This reliable and fine-scale spatial information over broad territory is crucial to better understand the structures, dynamics and changes in the environment, societies and territories.

Through the CS1, we can point out the interest of RALC H&VHRS map over broad territory. This map characterizes more finely and reliably riparian landscape than traditional land cover databases (see Figure 3). It provides accurate and original information on the presence and intensity of pressures close to the stream and on forested riparian strip attributes (uniformity, mean width and continuity). Given these results, new challenges emerge in view of gaining a better understanding of the relationships between land cover pressure and stream ecological status. This better understanding will offer valuable information for managers in order to achieve the WFD aims in prioritizing and designing efficient river corridor restoration strategies.

Through the CS2, we can note that fine-scale artificial sprawl maps over broad territories allow to better quantify and localize from local to regional scale the loss of land (natural or agricultural) to the detriment of built-up areas (in the centre and periphery of the city). Such information is essential for French managers in order to prioritize and define efficient restoration strategies dedicated to the conversion of land with agricultural potential.

CONCLUSION

To conclude, through these two case studies, concerning riparian corridor and artificial sprawl management from local to national scale, this paper demonstrates the drastic need of finescale information for supporting action strategies and the high interest of our OBIA approach for producing this information.

Our approach, based on a top-down approach for the construction of the classification tree and the use of "knowledge-based rules" classification technique appears operational given the high accuracy results and relatively short processing time obtained in the two case studies. OBIA, following our approach, will be probably at very short run a broadly applicable method for producing reliable spatial information dedicated to carry out broad scale spatial analysis supporting public policies.

Figure 1 .

 1 Figure 1. Localization of study areas for the two case studies

  Figure 2. Extract of riparian area land cover map in obtained over the Normandie region (CS1) from OBIA combining SPOT XS, orthophotos and ancillary data.

ShrubFigure 3 .

 3 Figure3. Riparian spatial indicators: Area percentages of a given land cover type according to the lateral distance to the river channel upstream of a stream ecological station (3km upstream). In A, built from High and Very High Spatial Resolution (H&VHSR)-derived map resulting of OBIA; in B built from CORINE Land Cover (CLC) database.

Figure 4 .

 4 Figure 4. Extract of the artificial sprawl map between 1997 and 2007 years.

Proceedings of the 4th GEOBIA, May 7-9, 2012 -Rio de

ACKNOWLEDGMENTS

The riparian corridor mapping study has been made possible by a PhD grant from the French Ministry of Research. The work has been partly realized in the framework of a research contract between ONEMA and Irstea. Acquisition of remote-sensing data benefited from the support of the ISIS programme of the French national space agency CNES (SPOT-5 images, distribution Spot Image), while IGN"s airborne orthophotos were made available by the French Ministry of Agriculture. The artificial sprawl mapping study has been made possible by the DRAAF Languedoc-Roussillon (regional directorate of the French Ministry of Agriculture). Acquisition of remote-sensing data (RapidEyes images) benefited from the support of the GEOSUD project (http://geosud.teledetection.fr/).