Anne Ganteaume
email: anne.ganteaume@irstea.fr

Marlène Long-Fournel
email: marlene.long@irstea.fr

Driving factors of fire density can spatially vary at the local scale in SE France Running head title: Spatial variation of fire density driving factors at local scale

Keywords: wildfire, fire ignition, département Bouches du Rhône, spatial analysis, kernel density, multivariate analysis

Southeastern France is the most wildfire prone region of the country. To identify the main driving factors of fire density (defined as the number of fires per hectare) at the local scale (cluster of communities which are homogeneous in terms of land cover, climate and WUI) and to assess their spatial variation at this scale, the fire density was investigated in the département Bouches du Rhône using georeferenced fire ignitions. To assess relationships between fire density and explanatory factors, statistical analyses and spatial evaluation were performed on each clusters taking into account climate conditions, topography, land cover, Wildland-Urban Interface (defined as a buffer of 100 m around housing located at less than 200 m from natural vegetation), minor road and population densities as well as fire density as dependent variable. High fire density was mainly related to high proportion of WUI in the study area. The proportion of natural vegetation and steep slope were also among the most important drivers of fire density. Depending on the cluster, some biophysical factors can in turn enhance or mitigate fire density but coolest and wettest climate conditions related to highest elevations as well as low housing density always mitigated fire density. This work showed that, at the local scale, the identification of fire density driving factors could improve fire prevention as they can better be targeted.

Short general summary

Spatial relationships between fire density and biophysical and human-related factors were found in the study area, pointing to WUI as the principal driver of fire density but at a smaller scale, some of these factors spatially varied. A better understanding of their influence on fire density will help to improve fire prevention at a fine scale.

Introduction

Wildfires have been an important disturbance in the Mediterranean region for thousands of years and, more recently, fire risk has become a more and more serious concern, especially in the Wildland-Urban Interface (WUI) where the stakes are the greatest. Since the past decades, WUIs have expanded in many Mediterranean countries, leading to an increase in the fire ignition as population and human infrastructure facilities are disseminated throughout the forested zones, especially in the vicinity of large cities and tourist resorts [START_REF] Viegas | Fire Spread and the Urban Wildland Interface Problem[END_REF]. This trend is also due to afforestation of abandoned agricultural lands located close to settlements [START_REF] Galiana-Martin | A wildland-urban interface typology for forest fire risk management in Mediterranean areas[END_REF]. In these areas, most fires are human-caused and human development as well as activity patterns (both starting and putting out fires) are increasingly overriding the biophysical factors that historically controlled fire regimes (Syphard et al. 2007a). Thus in these places, human-related factors such as housing density, population density or road networks have all been identified as important influences in the spatial distribution of wildfires [START_REF] Maingi | Factors influencing wildfire occurrence and distribution in eastern Kentucky, USA[END_REF]Sturtevant and Cleland 2007;[START_REF] Calef | Human influences on wildfire in Alaska from 1988 through 2005: an analysis of the spatial patterns of human impacts[END_REF][START_REF] Catry | Modelling and mapping the wildfire ignition risk in Portugal[END_REF][START_REF] Grala | Spatial and temporal characteristics of wildfires in Mississippi, USA[END_REF] while their relationships might vary in space [START_REF] Koutsias | Do factors causing wildfires vary in space? Evidence from geographically weighted regression[END_REF]. However, apart from the human component, specific traits of vegetation, topography, and meteorology are considered critical variables in fire ignition, spread and behavior in the Mediterranean Basin [START_REF] Catry | Modelling and mapping the wildfire ignition risk in Portugal[END_REF][START_REF] Koutsias | Where did the fires burn in Peloponnisos, Greece the summer of 2007? Evidence for a synergy of fuel and weather[END_REF].

Numerous works evaluating the relationships between fire metrics and a large range of explanatory factors (mostly socio-environmental factors such as land cover or climate and human factors such as population density) have been carried out at different spatial scales such as the global scale [START_REF] Chuvieco | Global characterization of fire activity: towards defining fire regimes from earth observation data[END_REF][START_REF] Chuvieco | Relations Between Human Factors and Global Fire Activity[END_REF]Ganteaume et al. 2012), national scale (Vazquez and Moreno 1998;[START_REF] Catry | Characterizing and modelling the spatial patterns of wildfire ignitions in Portugal: fire initiation and resulting burned area[END_REF]Martinez et al. 2009[START_REF] Thompson | Integrated national-scale assessment of wildfire risk to human and ecological values[END_REF][START_REF] Verdu | [END_REF], regional scale (Syphard et al. 2007b[START_REF] Koutsias | What is common in wildland fire occurrence in Greece and Switzerland? -Statistics to study fire occurrence pattern[END_REF][START_REF] Verdu | [END_REF][START_REF] Miranda | Spatial and temporal drivers of wildfire occurrence in the context of rural development in northern Wisconsin, USA[END_REF]Ganteaume andJappiot 2012, 2013) or ecoregions [START_REF] Padilla | On the comparative importance of fire danger rating indices and their integration with spatial and temporal variables for predicting daily humancaused fire occurrences in Spain[END_REF], local scale (Badya-Perpinyá et al. 2006[START_REF] Romero-Calcerrada | GIS analysis of spatial patterns of human-caused wildfire ignition risk in the SW of Madrid (central Spain)[END_REF]Syphard et al. 2007b[START_REF] Syphard | Predicting spatial patterns of fire on a southern California landscape[END_REF], and even at several spatial scales [START_REF] Lozano | A multiscale approach for modeling fire occurrence probability using satellite data and classification trees: a case study in a mountainous Mediterranean region[END_REF].

Southeastern France, which is characterized by Mediterranean climate conditions, is the most wildfire prone region of the country. Previous studies carried out at the regional scale (Ganteaume andJappiot 2012, 2013) showed that this region accounted for 55% of the total number of fires in 2010, and fire ignition as well as burned area varied spatially within this region for the small as for the large fires. However, the spatial variation of the fire density, defined as the number of fires per surface area unit (hectare) and which is one of the fire characteristics commonly grouped under the general term of fire regime [START_REF] Morgan | Mapping fire regimes across time and space: understanding coarse and fine-scale fire patterns[END_REF][START_REF] Stocks | Large forest fires in Canada, 1959-1997[END_REF], has not been investigated at a fine scale yet. As already noted by Martinez et al. (2009), the use of a density index reduces the bias of comparing the absolute number of fires per municipality, since they have a wide range of sizes. Moreover, at the regional level in SE France, the spatial variation of the fire patterns influenced by anthropogenic and environmental factors has been highlighted in previous works (Ganteaume andJappiot 2012, 2013), as has the need for further analyses at a smaller scale which better takes into account the environment of fire ignitions.

To fill this gap, the current work focuses on two fine spatial scales: (i) an administrative district of SE France (called "département") and (ii) the different subsections identified within this section (clusters of communities). As stated by [START_REF] Padilla | On the comparative importance of fire danger rating indices and their integration with spatial and temporal variables for predicting daily humancaused fire occurrences in Spain[END_REF], refining the spatial resolution of the work can help to take into account the spatial variation of fire metrics over a large study area; the spatial patterns of fire being better integrated. In this framework, the first aim of this work was to identify the main driving factors of fire density (predictor variables) at the two fine scales previously identified. Another aim of this work was to assess whether the relationships between underlying drivers and ignition density varied within these scales using both a statistical descriptive analysis (multivariate analysis) and a spatial approach (hereafter named spatial analysis). This latter was identified by [START_REF] Díaz-Delgado | Spatial patterns of fire occurrence in Catalonia, NE, Spain[END_REF] to lead to a better understanding of the processes underlying spatial trends and of the relative importance of variables involved in wildfire patterns. Accomplishing these objectives will help improving fire prevention at fine scale and, consequently, improve the local fire risk management.

Material and methods

Study area

The study area was the département Bouches du Rhône (Northwestern coordinates: 43.655°N, 5.495°E; Southeastern coordinates: 43.832°N, 5.672°E; total area: 508 700 ha), one of 15 administrative districts composing the Southeastern part of France (Fig 1). This département is among the areas most affected by wildfires (54 fires year -1 and 1247.3 hectares burned per year in the 2000-2010 period according to the regional forest fire database Prométhée). The main fuel types of the study area, located mostly on limestone-derived soils, are Pinus halepensis forests [START_REF] Quézel | Taxonomy and biogeography of Mediterranean pines (Pinus halepensis and P. brutia)[END_REF] and mixed pine-oak (Quercus ilex and Q. pubescens) forests (respectively 42% and 18% of the forest stands; database of Inventaire National Forestier), often the pre-forest vegetation type before oak forests (Quézel and Barbéro 1992). Shrublands, called "garrigue" on limestone-derived soils, are another dominant fuel type in the study area (43% of the natural vegetation; database of Inventaire National Forestier) that corresponds to the predominant successional stage after woodland degradation [START_REF] Barbéro | Pines of the Mediterranean Basin. Ecology and Biogeography of Pinus[END_REF]. Wildfires occur frequently in the whole area and overall, the study area is a mosaic of all the previously mentioned types of natural vegetation and of agricultural areas. The Southwestern part of the département Bouches du Rhône is occupied by a large wetland area and by irrigated crops like rice paddies corresponding to the Rhône's delta.

The Mediterranean climate prevailing in the study area is characterized by short and wet winter periods and by prolonged hot and dry summer periods, with strong drying wind (called Mistral) which favours fire incidence (Ruiz de la Torre 1999). The mean maximum temperature ranged from 9.8°C to 13.2°C in winter and from 27.9°C to 30.6°C in summer. Mean annual precipitation ranged from 472 mm to 820 mm (Météo France database). In general, topography is not very rough and the altitude ranges from sea level to 1038 m.

The total population of the study area is estimated to about 2 000 550 inhabitants, with a density of 388 inhabitants per km 2 , most concentred around big cities like Marseilles. Initially the finer scale used in this work was the community (Local Administrative Unit 2 according to Eurostat 1 ), which was defined as a city and its outskirts. The study area is composed of 119 communities (Fig. 1) among which 107 have been affected by wildfires since 1960; however, in order to get as many fire ignitions as required for the analyses, homogeneous geographical Author-produced version of the article published in International Journal of Wildland Fire 24(5) 650-664 The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209 areas, defined as "cluster of communities", were identified in the study area according to climate and land cover variables as well as to the proportion of WUI.

Data description

Dependent variables: We compiled the georeferenced forest fire database of the French Forest Office (ONF) that records since 1960 all the georeferenced forest fires with a burned area larger than 1 ha in Southeastern France. In total, 2557 georeferenced fire ignitions were recorded between 1960 and 2011 in the département Bouches du Rhône (Fig. 1). Fire densities, calculated in each community, were taken into account as dependent variables in the analyses. For the statistical analyses, mean fire density was calculated in each cluster, as the number of fire ignitions divided by the vegetation area in the cluster, and for the multivariate analysis, this variable was classified in three groups: high (14.89 -89.05 fires ha -1 ), medium (4.69 -14.89 fires ha -1 ), and low (0 -4.69 fires ha -1 ) fire density. For the spatial approach, fire density was calculated, in each cluster, as the number of fire ignitions divided by the area occupied by each explanatory factor.

Explanatory factors (recorded in each community):

 Climate conditions: climate factors such as precipitation and temperature, which were important in the models of Cardille et al. (2001), were often not significant in other studies [START_REF] Diez | Prediction of the daily number of forest fires[END_REF][START_REF] Padilla | On the comparative importance of fire danger rating indices and their integration with spatial and temporal variables for predicting daily humancaused fire occurrences in Spain[END_REF][START_REF] Miranda | Spatial and temporal drivers of wildfire occurrence in the context of rural development in northern Wisconsin, USA[END_REF]Ganteaume and Jappiot 2012) likely because they operate on a broader scale. Thus, the climate factors used in this work were derived from the bioclimatic index of [START_REF] Vennetier | Un nouveau modèle bioclimatique pour la forêt méditerranéenne[END_REF] which combined several climate and topographic variables (elevation, exposure, mean annual temperature, mean annual precipitation, mean summer precipitation, number of dry months and number of cold months, distance to the sea). This index, previously coded in nine classes, has been recoded into four classes used as proxy variables for climate conditions, from warm and dry conditions to cooler and wetter conditions as defined in [START_REF] Vennetier | Un nouveau modèle bioclimatique pour la forêt méditerranéenne[END_REF]: (i) coastal, (ii) intermediate, (iii) inland and (iv) relief (Fig. 2a). These data were given in percentage of the total community area.

 Topography: The topographic factors were generated from a 1:25 000 digital terrain model from the National Geographic Institute using geographical information systems (ArcGIS 10). Three types factors were thus derived:

(i) Elevation was coded in five classes in the spatial analysis (Fig. 2b) and the mean elevation in each community was used in the statistical analyses. In the département Bouches du Rhône, the topographic elevation gradient increases from the South-West to the North-East.

(ii) Slope was coded in four classes in the spatial analysis (Fig 2c) and the maximum slope was used in the statistical analyses.

(iii) Exposure to the dominant wind (Mistral, blowing from the North-West) was coded in three classes in the spatial analysis (Windward:270° to 360°, leeward: 90° to 180°, and intermediate: 0° to 90° and 180° to 270°). This factor was derived from the direction of the descending slope and from the maximum variation rate of the values obtained in each pixel compared to the neighbouring pixels (Fig. 2d). Only the proportion of area exposed to the dominant wind in each community was used in the statistical analyses.

Author-produced version of the article published in International Journal of Wildland Fire 24(5) 650-664 The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209  Land cover: We used Corine Land Cover maps from 2006 at a 1:100 000 scale. In the spatial analysis, land-cover categories were simplified into 3 thematic classes by reclassifying the original categories: "Natural vegetation" (merging the different classes of fuels corresponding to forests, shrublands and pastures), "Agriculture" and "Others". The latter merged the urban and other artificialized areas (such as dump areas, mines, communication networks, industrial areas) where fire ignitions can occur, but did not take into account wetlands or water bodies (Fig. 2e). Each land cover type was given in percentage of the total community area.

 Socio-economic factors: Mean population density (number of inhabitant per ha per community; INSEE 2011) was used in the statistical analyses, however, as this variable was not available as a spatial layer, housing density (calculation of the number of housing per ha) was used as a proxy of population density in the spatial approach of the results. This variable was coded in four classes extracted from BD TOPO® 2009 and was calculated by the Kernel density method in ArcGis 10 (Fig. 2f).

 Wildland-Urban Interfaces areas were defined as a buffer of 100 m around housing located at less than 200 m from natural vegetation (Lampin-Maillet et al. 2010)(Fig. 2g).

 Network density: The spatial data related to all types of networks (electric lines, railways, main roads, and minor roads in km ha -1 ) were extracted from BD TOPO® 2009. A buffer of 50 m around the network, which corresponded to the area where the maximum fire density occurred [START_REF] Romero-Calcerrada | GIS analysis of spatial patterns of human-caused wildfire ignition risk in the SW of Madrid (central Spain)[END_REF], was calculated around each type of network in order to have the corresponding network area. This calculation was done using the command 'Distance' in the ArcGIS software, which computes the Euclidian distance from each point to its closest road. The ignition frequency occurring along each type of network was calculated in order to highlight the type of network that concentrated most fire ignitions. Minor road network showed the highest fire occurrence (78% of ignitions occurred within the minor network buffer) compared to the other types of networks. Thus, in the spatial analysis we only focused on minor roads (Fig. 2h).

Data Analysis

The identification of homogeneous clusters of communities according to descriptive factors (climate conditions, land covers and WUI in each community) was performed using Hierarchical Cluster Analysis (R 2.11-1, ADE-4 1.5-1 package). This analysis allowed the 119 communities to be grouped into five homogeneous clusters of communities according to these factors (Fig. 1 and tab. 1).

Partial least squares (PLS) regression analysis was used to examine the influence of the explanatory variables on fire density (x1000), quoted "FD", in the whole study area as well as in each cluster (in each case, the sample was the communities belonging to the whole study area or to each cluster). This analysis can be used as an exploratory analysis tool to select suitable predictor variables. Because of the too small number of communities, PLS regression was not performed on cluster 4. The dependent variable (FD) was normalized using a logarithmic transformation (Ln (FD + 1)) prior to calculations and the explanatory variables were centred and scaled to unit variance to give all variables of the same relative importance.

The significance of components for the models was determined by uncertainty tests carried

Author-produced version of the article published in International Journal of Wildland Fire 24(5) 650-664 The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209 out within a full cross-validation. PLS was carried out using Statgraphics Centurion XV software. Then, using R statistical packages, a bootstrap procedure (bootsize=1000) was performed on the set of variables that presented the highest regression coefficients to rebuild the model, followed by a backwards elimination process until all predictor variables in the model were significant with P values< 0.05.

Co-inertia analysis [START_REF] Dolédec | Co-inertia analysis; an alternative method for studying speciesenvironment relationships[END_REF] is fitted when the dataset is comprised of a large number of variables compared with a small number of data, and avoids the problems of co-linearity between variables. In the present work, this analysis was performed on the dependent variables (high, medium and low fire density) and on the explanatory factors (climate, land-cover, WUI, network density, topographic, socio-economic variables) to examine associations between these two types of data. The complete matrix of data was transferred to the statistical package under R 2.11-1 (R Development Core Team 2005) then analysed using the ADE-4 1.5-1 package [START_REF] Thioulouse | ADE-4: a multivariate analysis and graphical display software[END_REF]. The first step of this multivariate analysis (Ter Braak and Schaffers 2004) was to conduct a correspondence analysis on the dependent variables, then a principal component analysis on the explanatory factors. A factorial plane was thus created and enabled a new ordination of each data set. The statistical significance of each effect or combination of effects has been tested using the Monte-Carlo permutation test with 1000 permutations using the "randtest" in the 'coin' package on R.

The analysis of spatial fire patterns (called spatial analysis in this work), here fire density, studying the spatial relation according to different explanatory factors, was performed in order to identify the socio-environmental variables that are important determinants of the spatial distribution of fire density. All spatial analysis and cartographic production were made with the previously commented fire metrics, as well as with the explanatory factors using geographical information systems (Arc Gis 10). We used the Spatial Analyst Extension of ArcGIS, for our GIS analysis and data processing. The spatial relationships between fire density and explanatory variables were assessed through the spatial distribution of the fire ignition. In order to decrease the uncertainty of the location of fire ignitions (accuracy at ± 10 m), the layer of fire ignitions was, at first, converted in raster with a 10 m resolution in order to combine it to the other raster spatial data. Then this layer was transformed in point density using the Kernel density method in order to have continuous data on the whole study area.

The resolution of the statistical analyses was the community as data is available for each community whereas spatial data, which is continuous in the study area, is available at different resolutions according to the spatial explanatory variables. Analyzing the spatial fire density was thus performed taking into account these spatial resolutions.

Author-produced version of the article published in International Journal of Wildland Fire 24(5) 650-664 The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209 Author-produced version of the article published in International Journal of Wildland Fire 24(5) 650-664

The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209

Results

Spatial distribution of fire density at the different fine scales

The fire density obtained using the Kernel method in the whole study area showed an increase along a West-East gradient with values up to 0.052 ignitions per hectare (Fig. 3). Regarding fire density cluster by cluster, clusters 2 and 3 presented the highest fire densities (up to 0.055 and 0.048 ignitions per hectare respectively) and cluster 4 the lowest values (up to 0.023 ignitions per hectare). Higher fire densities were located in the Southern part of the clusters 1 and 4, in the centre of cluster 3 and in the East of cluster 5 whereas, in cluster 2, fire density was quite homogeneous in the whole area (Fig. 3).

Characterization of the main explanatory factors of fire density

At the level of the entire study area, results of the Partial Least Squares regressions, including the dataset of explanatory variables (climate conditions, land covers, WUI, minor road and population densities, mean elevation, maximum slope and exposure to dominant wind), showed that fire density was enhanced by population and minor road densities, WUI, topographic variables (all of them), land covers "natural" and "others" as well as the climate condition "intermediate" (2 components, p<0.0001, r 2 =0.49). On the contrary, fire density was mitigated by the climate condition "relief" and the land cover "agriculture" (Fig. 4). After the bootstrap procedure, only the explanatory variables climate condition "relief", WUI (both highly significant), land cover "agriculture", population density and the three topographic variables remained significant predictors of the fire density (Tab. 2).

At the cluster level, results of the regression analysis, including the dataset of explanatory variables according to the fire density, showed that fire density was significantly correlated to most explanatory variables always mitigated by the land cover "agriculture" and, according to the cluster, either mitigated or enhanced by climate conditions "relief" and "inland" as well as by minor road density (Fig. 4). After the bootstrap procedure, only one explanatory variable remained a significant predictor of the fire density in clusters 1 and 5: exposure to dominant wind (cluster 1) and WUI (cluster 5). The variables WUI, land cover (all the classes), population density and the climate condition "inland" remained significant in cluster 2 whereas population density, WUI, land cover "others", slope and elevation were significant predictors in cluster 3 (in this cluster, the climate condition "coastal" has not been taken into account in the analysis because cluster 3 does not contain any coastal community) (Tab. 2). This analysis was not carried out on cluster 4 because of the too small sample (four communities).

Revealing the relationships between explanatory factors and fire density

At the level of the study area, the spatial approach showed that fire density was among the lowest in the agricultural areas (0.0026 fires ha -1 ) whereas it was highest in the "natural" areas (0.0091 fires ha -1 ) even if this land cover represents only 36% of the study area. "Relief" was the only significant climate condition correlated to fire density (Tab. 2), however, fire density values were not high in this "climate" class. Although WUI areas only represented 15% of the territory, fire density was highest (0.014 fires ha -1 ) in these areas. Author-produced version of the article published in International Journal of Wildland Fire 24(5) 650-664 The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209

Regarding the topographic variables, fire densities were highest in areas presenting medium slopes (0.012 fires ha -1 ), at lowest elevation (0.01 fires ha -1 ) and in windward areas (0.008 fires ha -1 ). Regarding the socio-economic factors, fire density was highest in dense housing areas (2 to 3 housing ha -1 ). Minor road network, whose area represented 31% of the study area, presented a mean fire density of 0.009 fire ha -1 .

In each cluster of communities, the link between the different explanatory factors and the fire density was investigated using multivariate analysis (co-inertia analysis) and spatial analysis (Tab. 3). In cluster 1, results obtained using co-inertia analysis showed that high proportions of WUI, of the climate condition "inland", and to a lesser extent, of natural vegetation, as well as steep slopes and high elevations, were linked to high fire density (Fig. 5). This result was confirmed by the spatial approach (Fig. 6). For low fire density, both types of analysis identified the high proportion of agricultural lands and the low population/housing density as mitigating factors. Besides, spatial analysis highlighted the proportion of the climate condition "intermediate", low and higher elevations and low slopes as mitigating factors of fire density. Low proportions of the land cover "others" and low network density were also linked to low fire density according to co-inertia analysis.

In cluster 2, the results obtained using co-inertia analysis showed that high proportions of the land cover "others" (here the urban areas where fires are concentrated around), of the climate condition "coastal", of WUI as well as steep slopes were linked to high fire density (Fig. 5). Spatial analysis also identified high proportion of WUI and steep slope as linked to high fire density but added to these factors the high proportion of natural vegetation (Fig. 6). For low fire density, both types of analysis identified the high proportion of agricultural lands and low population/housing density as mitigating factors. Besides, spatial analysis highlighted the proportion of the climate condition "coastal" and low slopes, as mitigating factors of fire density. Low proportions of the land cover "natural" and of areas exposed to dominant wind as well as low network density were linked to low fire density according to co-inertia analysis.

In cluster 3, the results obtained using co-inertia analysis showed that high proportions of the land cover "others" and of WUI and to a lesser extent, of the climate condition "intermediate", as well as high population and network densities, were linked to high fire density (Fig. 5). Except for the land cover "others", this result was confirmed by spatial analysis which also added medium elevations and slopes, natural vegetation as well as exposure to dominant wind to the factors linked to high fire density (Fig. 6). For low fire density, both analyses identified the proportion of the climate condition "relief" as a mitigating factor. Besides, spatial analysis also highlighted higher elevation and low housing density as linked to low fire density whereas low proportion of the climate condition "inland" and low elevations were linked to low fire density according to co-inertia analysis.

In cluster 4, results obtained using co-inertia analysis were highly significant (randtest, p < 0.001) in spite of the small sample. They showed that high proportion of agricultural lands, high population density as well as low proportion of the land cover "natural" were linked to high fire density (Fig. 5). This result was confirmed by spatial analysis which also added the high proportion of WUI to the factors linked to high fire density (Fig. 6). For low Author-produced version of the article published in International Journal of Wildland Fire 24(5) 650-664 The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209 fire density, this latter analysis identified higher elevation and low housing density as mitigating factors whereas high proportion of the climate condition "relief" as well as high elevations and steep slopes were linked to low fire density according to co-inertia analysis.

In cluster 5, results obtained using co-inertia analysis showed that high proportions of natural vegetation and of WUI as well as of areas exposed to the dominant wind were linked to high fire density (Fig. 5). Spatial analysis identified medium elevations and slopes as enhancing factors of fire density (Fig. 6). For low fire density, both analyses identified the proportion of agricultural lands as a mitigating factor. Besides, spatial analysis also highlighted the proportions of the land cover "others" and of the climate condition "coastal", low slopes, low and higher elevations and low housing density as mitigating factors of fire density. Low network density was also linked to low fire density according to co-inertia analysis.

Discussion

In the study area, fire density increased along a gradient from the West to the East and it varied at the finer scale (cluster of communities). The low fire density recorded in the Western part of the study area could be explained by the large flat area of marshes and irrigated crops which are not favourable to fire ignition. At the level of the study area, the climate condition "relief" and the land cover "agriculture" mitigated fire density contrary to population density, topographic variables (all of them) and especially WUI. This latter variable strongly enhanced fire density even if the WUI area represented only 15% of the study area (from 4% to 35% according to the cluster) and its influence on fire metrics has been already highlighted in several works [START_REF] Prestemon | Understanding broadscale wildfire risks in a human-dominated landscape[END_REF][START_REF] Romero-Calcerrada | GIS analysis of spatial patterns of human-caused wildfire ignition risk in the SW of Madrid (central Spain)[END_REF][START_REF] Badia-Perpinyá | Spatial distribution of ignitions in Mediterranean periurban and rural areas: the case of Catalonia[END_REF]. In areas where most fires are humancaused, ignitions are often more frequent along transportation networks [START_REF] Vega-García | A logit model for predicting the daily occurence of human caused forest fires[END_REF][START_REF] Abhineet | Forest fire risk modelling using remote sensing and geographic information system[END_REF][START_REF] Thompson | Using forest fire hazard modelling in multiple use forest management planning[END_REF]Cardille et al. 2001;[START_REF] Vasconcelos De | Spatial prediction of fire ignition probabilities: comparing logistic regression and neural networks[END_REF][START_REF] Keeley | Impact of past, present, and future fire regimes on North American Mediterannean shrublands[END_REF][START_REF] Stephens | Forest fire causes and extent on United States Forest Service lands[END_REF]. Several works showed that these networks should be an important descriptor of the effect of human population [START_REF] Vasconcelos De | Spatial prediction of fire ignition probabilities: comparing logistic regression and neural networks[END_REF]Cardille et al. 2001;[START_REF] Chou | Mapping probability of fire occurrence in San Jacinto Mountains, California, USA[END_REF][START_REF] Romero-Calcerrada | GIS analysis of spatial patterns of human-caused wildfire ignition risk in the SW of Madrid (central Spain)[END_REF][START_REF] Badia-Perpinyá | Spatial distribution of ignitions in Mediterranean periurban and rural areas: the case of Catalonia[END_REF][START_REF] Dickson | Mapping the probability of large fire occurrence in northern Arizona, USA[END_REF]). However, depending on the authors, population and road densities were either positively (Cardille et al. 2001;[START_REF] Keeley | Reexamining fire suppression impacts on brushland fire regimes[END_REF] or negatively [START_REF] Mercer | Comparing production function models for wildfire risk analysis in the wildland-urban interface[END_REF]) correlated to fire; some authors also found no correlation between these variables [START_REF] Prestemon | Understanding broadscale wildfire risks in a human-dominated landscape[END_REF]Ganteaume and Jappiot 2012). As Syphard et al. (2007a) already showed, we found that the minor road density was not a significant predictor of fire density (since roads are narrow and linear features, the influence of this variable may not be detected at the scale used in this work) in contrast of the population density. [START_REF] Catry | Modelling and mapping the wildfire ignition risk in Portugal[END_REF] also found that population density was among the main driving factors of fire occurrence in Portugal.

At finer scale (cluster of communities), a spatial variation of the driving factors of fire density has been highlighted and the proportion of WUI in the cluster was one of the most important factor positively linked to high fire density (except in cluster 4) as previously found at the scale of the study area. However, according to the regression analysis, WUI was not a significant variable in cluster 1 (despite it was positively linked to high fire density

Author-produced version of the article published in International Journal of Wildland Fire 24(5) 650-664 The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209 Fig. 6 Fire density in the five clusters of communities according to the different explanatory factors using a spatial approach.

Author-produced version of the article published in International Journal of Wildland Fire 24(5) 650-664 The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209 according to both multivariate and spatial analyses) which could be explained by the lower proportions of WUI in this cluster as well as in cluster 4 (respectively 6 and 4% of the study area). This result agreed with several works which showed that the influence of WUI on fire metrics varied spatially [START_REF] Keeley | Reexamining fire suppression impacts on brushland fire regimes[END_REF]Cardille et al. 2001;[START_REF] Prestemon | Understanding broadscale wildfire risks in a human-dominated landscape[END_REF]Syphard et al. 2007a).

Topographic variables were significant predictors of fire density only in two clusters (in cluster 3, elevation and slope enhanced fire density, so was the exposure to dominant wind in cluster 1). Moreover, exposition to dominant wind was linked to high fire density in cluster 3 according to the spatial analysis as well well as in cluster 5 according to both co-inertia and spatial analyses. Usually, wind appeared to be an important factor, as highlighted in the Southwestern part of the USA and in southern California [START_REF] Cohen | The Big Burn -The Northwest's Forest Fire of 1910[END_REF][START_REF] Schoennagel | The interaction of fire, fuels, and climate across Rocky Mountain forests[END_REF][START_REF] Keeley | Impact of antecedent climate on fire regimes in coastal California[END_REF]) but this variable was often related to the burned area. On the whole, medium and steep slopes were linked to high or medium fire density, which agreed with the work of [START_REF] Syphard | Predicting spatial patterns of fire on a southern California landscape[END_REF] and [START_REF] Lampin-Maillet | Mapping wildland-urban interfaces at large scales integrating housing density and vegetation aggregation for fire prevention in the South of France[END_REF], and spatial analysis revealed that medium to high elevations were also linked to high fire density in clusters 1 and 3. In Portugal, [START_REF] Catry | Modelling and mapping the wildfire ignition risk in Portugal[END_REF] found that higher fire occurrence at high elevation (from 500 to 1000 m) was linked to renovation of pastures for livestock; however this explanation did not fit with our context as the natural vegetation in clusters 1 and 3 was only forests not pastures. In most clusters, higher elevations, corresponding to "mountainous" areas, were not favorable to fire density and low slopes, mostly corresponding to agricultural or wetland areas, also mitigated fire density. In contrast, Badia-Perpinyá and Pallares- [START_REF] Badia-Perpinyá | Spatial distribution of ignitions in Mediterranean periurban and rural areas: the case of Catalonia[END_REF] found that fires were mostly situated in easily accessible areas on gentle slopes in some part of Catalonia (Spain) and other authors [START_REF] Vega-García | Practical application of a daily prediction model for the occurrence of human-caused forest fires in Catalonia[END_REF][START_REF] Padilla | On the comparative importance of fire danger rating indices and their integration with spatial and temporal variables for predicting daily humancaused fire occurrences in Spain[END_REF] showed trends suggesting that more ignitions were likely to occur in low elevation areas.

At the cluster level, after the bootstrap procedure, climate conditions were significant predictors of fire density only in cluster 2 in which "inland" mitigated fire density. Before this procedure, "relief", characterized by the coolest and wettest climate conditions, significantly mitigated fire density in the other clusters. Both multivariate and spatial analyses showed that this climate variable was also positively linked to high fire density in cluster 1 and that the climate variable "relief" was positively linked to low fire density in cluster 3. [START_REF] Syphard | Predicting spatial patterns of fire on a southern California landscape[END_REF] noted that coastal areas in Southern California which were warmer and dryer than remote interior regions were more conducive to fire; however, in our work, different climate conditions could enhance fire density according to the cluster: warm and dry climate conditions ("intermediate"), conducive to fire ignition, were positively linked to high fire density in cluster 3 but cool and wet climate conditions ("inland") did the same in cluster 1, likely increasing the fuel load. Indeed, for this latter cluster, high fire density was also linked to high elevation and steep slopes, often corresponding to the climate condition "inland". However, the coolest and wettest conditions (relief), which are not favourable to fire ignition, were linked to low fire density in the most "mountainous" areas (clusters 3 and 4).

The land cover classes were significant predictors of fire density only in cluster 2 in which the proportions of land covers "natural vegetation" and "others" enhanced fire density in contrast of the proportion of agricultural land. However, the proportion of natural vegetation was positively linked to high fire density in cluster 1 and 5 according to both

Author-produced version of the article published in International Journal of Wildland Fire 24(5) 650-664 The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209 multivariate and spatial analyses. This result agreed with [START_REF] Syphard | Predicting spatial patterns of fire on a southern California landscape[END_REF] who showed that, usually, the vegetation type is related to ignition. In cluster 3, only the land cover "others" (mostly corresponding to the urban areas) enhanced fire density, indicating that fires were concentrated around urban areas; this significant variable was also associated to population density in this cluster. Our analyses also revealed that the proportion of agricultural land was negatively linked to high fire density in clusters 1, 2 and 5 and positively linked in cluster 4 in contrast of the proportion of natural vegetation. This result agreed with several works, carried out in Mediterranean countries or in the upper Midwest of the USA, which showed a tendency of fire to occur on or near non-forested areas (Cardille and Ventura 2001;[START_REF] Badia-Perpinyá | Spatial distribution of ignitions in Mediterranean periurban and rural areas: the case of Catalonia[END_REF][START_REF] Catry | Modelling and mapping the wildfire ignition risk in Portugal[END_REF]. Indeed, this behavior was probably due to the relative ease of igniting non-forest fuels, which could be agricultural residues. However, the impact of agriculture on fire can differ in the literature; for instance, [START_REF] Chuvieco | Relations Between Human Factors and Global Fire Activity[END_REF] showed that crops were negatively correlated to fire density but Martinez et al. (2009) found that, in Spain, agricultural variables entailed an increase in fire ignition due to sparks emitted by engines or stubble burning, given that most agricultural practices in this country imply frequent use of fire. This could be the case in cluster 4 where the fire was related to agricultural areas.

Regarding socio-economic variables, minor road density was not a significant predictor of fire density regardless of the cluster but, according to multivariate analysis, this variable was linked to high fire density at least in cluster 3. In the literature, most works did not detail the type of transportation network as we did in the current study but they showed that humancaused fires were more likely to occur in areas of high road density improving access to remote forested locations (Swetnam 1990;Cardille et al. 2001;[START_REF] Dellasala | An ecologically based strategy for fire and fuels management in national forest roadless areas[END_REF], and near those roads [START_REF] Chou | Mapping probability of fire occurrence in San Jacinto Mountains, California, USA[END_REF]. Population density significantly enhanced fire density in clusters 2 and 3 and this variable was also linked to high fire density in cluster 4 according to multivariate and spatial analyses. It is worth noting that low housing density, often corresponding to agricultural areas, was linked to low fire density regardless of the cluster, and that medium housing density was linked to high fire density (especially in clusters 3 and 4) as previously found by Syphard et al. (2007a). Usually, the distribution of fires is influenced by such human-related factors (Martinez et al. 2009;Vega-Garcia et al. 1995[START_REF] Thompson | Using forest fire hazard modelling in multiple use forest management planning[END_REF][START_REF] Vasconcelos De | Spatial prediction of fire ignition probabilities: comparing logistic regression and neural networks[END_REF][START_REF] Syphard | Predicting spatial patterns of fire on a southern California landscape[END_REF]) and these factors have been identified as influencing the spatial distribution of wildfires in places where people tend to start most fires (Cardille et al. 2001;[START_REF] Maingi | Factors influencing wildfire occurrence and distribution in eastern Kentucky, USA[END_REF]Sturtevant and Cleland 2007;Syphard et al. 2007a;[START_REF] Calef | Human influences on wildfire in Alaska from 1988 through 2005: an analysis of the spatial patterns of human impacts[END_REF][START_REF] Catry | Modelling and mapping the wildfire ignition risk in Portugal[END_REF][START_REF] Grala | Spatial and temporal characteristics of wildfires in Mississippi, USA[END_REF]. However, Badia-Perpinyá and Pallares- [START_REF] Badia-Perpinyá | Spatial distribution of ignitions in Mediterranean periurban and rural areas: the case of Catalonia[END_REF] showed that, according to the population density, the influence of roads on ignition varied; this factor did not influence the spatial distribution of ignitions in areas with sparse population.

Conclusions

There were significant relationships between fire density and biophysical and human-related factors in the study area. The spatial distribution of high fire density at this scale was clearly related to a combination of abiotic, biotic, and human factors: no single factor or factor type dominates in this area. At a finer scale (cluster of communities), there was also a spatial variation of fire density. The main explanatory variable in the different analyses pointed to

Author-produced version of the article published in International Journal of Wildland Fire 24(5) 650-664 The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209 WUI as the principal factor enhancing fire density regardless of the scale. Hence, better fire prevention must target WUIs in the whole study area with land use planning taking into account the lessening of sprawling development of WUIs. Moreover, controls of the implementation of the mandatory brush-clearing should be reinforced in these areas. Sometimes, this variable was associated with population/housing density (clusters 2 and 3) so people awareness should also be reinforced in these areas as well. Indeed, humans are altering both spatial and temporal pattern of fire regime and the fire risk is a function of the spatial arrangement of housing development and fuels (Syphard et al. 2007a;[START_REF] Lampin-Maillet | Land cover analysis in wildland-urban interfaces according to wildfire risk: A case study in the South of France[END_REF]. In cluster 4 only, agricultural practices seemed to enhance fire density; thus, reinforcing farmer awareness, especially regarding the practice of residues burnings, would help improve fire prevention in this area. Depending on the geographical location, climate conditions and topography parameters can enhance (inland, steep slopes, high elevations, exposure to dominant wind) or mitigate (relief, low slopes, low and higher elevations) fire density, often associated with the natural vegetation factor. In the study area, and probably at a finer scale, most fires are intentional, arsonists selecting the right place at the right moment regarding meteorological conditions to set fires. However, these climate conditions will be changing in the future according to the global change; areas in altitude presenting cool and wet conditions becoming warmer and dryer. This change, combined with the change in land use, will play on the spatial variation of fire density. With this perspective, more management of fire risk and of fire prevention planning will be needed for instance in the elevated part of the study area which are currently less at risk.

The different analyses we used mostly gave similar or complementary results and the insights they provided allowed better understanding of the influence of these explanatory factors on fire density at fine scale. As already noted by [START_REF] Syphard | Conservation threats due to human-caused increases in fire frequency in mediterranean-climate ecosystems[END_REF], the fine scale spatial analysis of relationships between people and fires may help identify areas where increases in fire metrics will threaten valuables areas.

In further works, the temporal variation of the driving factors of fire density as well as the driving factors of the burned area have to be checked at fine scale in the framework of the global change. Moreover, as several authors pointed out that economic difficulties may promote conflicts manifested by the increase in intentional fires (Martinez et al. 2009;Ganteaume and Jappiot 2012), socio-economic factors such as the unemployment rate should be added to the analyses if they are available in the requested format.

1Fig. 1

 1 Fig. 1 Map of the study area (Département Bouches du Rhône) in Southeastern France showing the five clusters of communities and the fire ignition points (BD Carto®).

  Maps showing the distribution of the different explanatory factors of fire density in the study area.

Fig. 3 Fig. 4

 34 Fig. 3 Maps showing the fire density (FD: number of ignitions per ha) calculated by the kernel density method in the whole study area (FD) and in the five clusters of communities (FD1, FD2, FD3, FD4, FD5).

Fig. 5

 5 Fig. 5 Identification of the explanatory variables (in black with box) of fire density (in grey) in the five clusters by means of co-inertia analysis: FD log-transformed and x 1000): POPD: population density, COAST: coastal climate conditions, INT: intermediate climate conditions, INL: inland climate conditions, REL: relief climate conditions, WUI: wildland-urban interface, AGR: agriculture land cover, NAT: natural vegetation land cover, OTH: other land covers, R2D: minor road density, ELEV: elevation, SLO: slope, WIND: exposure to dominant wind. In lower case: communities composing each cluster.

Table 1 : Characteristics of each cluster of communities according to the descriptive variables (proportions of climate conditions, WUI and land covers)

 1 Author-produced version of the article published in International Journal of Wildland Fire 24(5) 650-664 The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209

	CLUSTER 1 CLUSTER 2 CLUSTER 3 CLUSTER 4 CLUSTER 5

Table 2 : Regression coefficients and p-values (*: 0.05, **:0.01, ***:0.001) of the significant explanatory variables (PLS model with 2 components with bootstrap procedure)

 2 Author-produced version of the article published in International Journal of Wildland Fire 24(5) 650-664 The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209Author-produced version of the article published in International Journal of Wildland Fire 24(5) 650-664 The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209 Low values of: minor road density Coastal, low and higher elevation, low slope, low housing density,Author-produced version of the article published in International Journal of Wildland Fire 24(5) 650-664The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209

	Percentile	0.1th	1th	5th	95th	99th	99.9th	P-value
	Study area
	Population density	-0.067	-0.0261	0.0147	0.224	0.2606	0.2834	*
	% Relief	-0.3876	-0.3563	-0.3126	-0.0965	-0.0617	-0.018	***
	% WUI	0.2003	0.2335	0.2613	0.4377	0.4645	0.5193	***
	% Agricultural land	-0.4248	-0.3576	-0.3131	-0.0814	-0.0292	0.0267	**
	Elevation	-0.0788	-0.0228	0.02	0.2829	0.3313	0.375	*
	Slope	-0.055	-0.023	0.0155	0.2032	0.2361	0.2599	*
	Exposure to dominant wind	-0.0585	-0.0322	0.0207	0.2592	0.3028	0.3498	*
	Cluster 1
	Exposure to dominant wind	-0.2932	-0.1108	0.0376	0.5937	0.7134	0.8275	*
	Cluster 2
	Population density	-0.0513	-0.0281	0.0073	0.1468	0.1805	0.2138	*
	% Inland	-0.4669	-0.3541	-0.3013	-0.0541	-0.0114	0.022	**
	%WUI	0.0532	0.0938	0.1341	0.3341	0.3806	0.4281	***
	% Agricultural land	-0.4394	-0.4117	-0.3786	-0.242	-0.2022	-0.1726	***
	% Natural vegetation	0.1027	0.1604	0.1963	0.3435	0.3697	0.3954	***
	% land cover "Others"	-0.0499	-0.0143	0.0229	0.2029	0.2392	0.2606	*
	Cluster 3
	Population density	0.014	0.0715	0.1339	0.2959	0.3326	0.3824	***
	%WUI	0.0625	0.162	0.2113	0.4577	0.5387	0.6841	***
	% land cover "Others"	-0.0344	0.0332	0.0881	0.2822	0.3106	0.3489	**
	Elevation	0.0862	0.1416	0.219	0.4745	0.5642	0.6597	***
	Slope	-0.0471	-0.0097	0.0712	0.2833	0.318	0.3579	*
	Cluster 5

http://epp.eurostat.ec.europa.eu Author-produced version of the article published in International Journal of Wildland Fire

24(5) [650][651][652][653][654][655][656][657][658][659][660][661][662][663][664] The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209

Acknowledgments

This work was funded by the European Commission in the framework of the collaborative project FUME n° 243888.

Author-produced version of the article published in International Journal of Wildland Fire 24(5) 650-664 The original publication is available at http://www.publish.csiro.au/?paper=WF13209 doi : 10.1071/WF13209


Santiago Velasco-Forero

Inner-Cheeger Opening and Applications

The aim of this paper is to study an optimal opening in the sense of minimize the relationship perimeter over area. We analyze theoretical properties of this opening by means of classical results from variational calculus. Firstly, we explore the optimal radius as attribute in morphological attribute filtering for grey scale images. Secondly, an application of this optimal opening that yields a decomposition into meaningful parts in the case of binary image is explored. We provide different examples of 2D, 3D images and mesh-points datasets.

Introduction

Many of the methods in mathematical morphology study the relationship between size/shape of objects via max-plus convolution by structuring elements (SE) [START_REF] Najman | Mathematical morphology: from theory to applications[END_REF]. One can realize that openings and closings with square, disk or hexagon SEs, are often good enough for some filtering tasks. However, if the structuring elements are able to adapt their shapes and sizes to the image content, some enhancement properties are improved [START_REF] Angulo | Structurally adaptive mathematical morphology based on nonlinear scale-space decompositions[END_REF][START_REF] Ćurić | Adaptive mathematical morphology-a survey of the field[END_REF]. This intuition leads to propose area openings [START_REF] Vincent | Grayscale area openings and closings, their efficient implementation and applications[END_REF], and more generally, to introduce attributes openings [START_REF] Breen | Attribute openings, thinnings, and granulometries[END_REF] . Recently many other attribute filters have been proposed to more specific problems [START_REF] Talbot | Efficient complete and incomplete path openings and closings[END_REF][START_REF] Serna | Attribute controlled reconstruction and adaptive mathematical morphology[END_REF][START_REF] Morard | Efficient geodesic attribute thinnings based on the barycentric diameter[END_REF].

In this paper, we introduce a new attribute named "Inner-Cheeger opening" which is, for a binary shape S, the size t, such that an opening by a disk of t minimize the relationship perimeter over area: t is named the Cheeger constant of S in the theory of Variational Calculus [START_REF] Maggi | Set of Finite Perimeter and Geometric Variational Problems[END_REF]. Additionally, we explore the application of this operator in the context of decomposition for two-dimensional (2D) and three dimensional (3D) shapes into meaningful parts, which is a challenging problem in image processing [START_REF] Mehtre | Shape measures for content based image retrieval: a comparison[END_REF], pattern recognition [START_REF] Cerri | Shrec'13 track: retrieval on textured 3d models[END_REF][START_REF] Velasco-Forero | Statistical shape modeling using morphological representations[END_REF], remote sensing [START_REF] Gueguen | Classifying compound structures in satellite images: A compressed representation for fast queries[END_REF], and computer vision [START_REF] Younes | Spaces and manifolds of shapes in computer vision: An overview[END_REF]. Thus, we propose a new shape decomposition method, called Inner-Cheeger shape decomposition, denoted by ICSD. This method is characterized by a sequence of Inner-Cheeger openings. Finally, the experimental section includes some examples of 2D, 3D images and mesh-points datasets to illustrate the interest of our method.

Inner-Cheeger opening

The theory of sets of finite perimeter provides a particularly well-suited framework for studying the existence, symmetry, regularity, and structure of minimizers in those geometric variational problems in which surface area is minimized under a volume constraint [START_REF] Maggi | Set of Finite Perimeter and Geometric Variational Problems[END_REF]. In this paper, we consider the follow variational problem and its application to image processing and shape characterization.

Formulation

Given a domain S of R d , with d ≥ 2, one is asked to find the Cheeger constant of S, defined as:

h(S) := min X⊂S Per(X) Area(X) , (1)
where Area(X) is the (d)-dimensional volume of X and Per(X) is the perimeter of X or (d-1)-dimensional measure. It is important to note that the minimum in (1) is taken over all nonempty sets of finite perimeter contained in S. Thus, any minimiser of (1) is named Cheeger set of S and it is denoted by Ch(S). Despite this simple formulation of (1), many no trivial questions arise, for instance, the computation, and estimation of h(S) and the characterization of Cheeger sets of S. They have been the main subject of many recent research papers [START_REF] Kawohl | Characterization of Cheeger sets for convex subsets of the plane[END_REF][START_REF] Alter | Uniqueness of the cheeger set of a convex body[END_REF].

The following are some of the basic properties which are well-known about the Cheeger problem in [START_REF] Najman | Mathematical morphology: from theory to applications[END_REF], the proofs of them can be found for instance in [START_REF] Kawohl | Characterization of Cheeger sets for convex subsets of the plane[END_REF][START_REF] Alter | Uniqueness of the cheeger set of a convex body[END_REF][START_REF] Stredulinsky | Area minimizing sets subject to a volume constraint in a convex set[END_REF][START_REF] Leonardi | On the cheeger sets in strips and non-convex domains[END_REF]. X ⊆ S such that ∂X intersect ∂S. 5. (Closure with union and intersection) If X 1 and X 2 are Cheeger in S, then X 1 ∪ X 2 and X 1 ∩ X 2 (if it is not empty) are also Cheeger in S.

Many other properties can be found in [START_REF] Alter | Uniqueness of the cheeger set of a convex body[END_REF][START_REF] Leonardi | On the cheeger sets in strips and non-convex domains[END_REF]. However, we have included only the most relevant for our work.

Convex case

In the case of S is convex, we can add to the list of properties in Proposition 1, the uniqueness and convexity of Cheeger sets, which has been proved in [START_REF] Alter | Uniqueness of the cheeger set of a convex body[END_REF]. However, finding the Cheeger set of a given S is a difficult task. This task is simplified if S is a convex set and d = 2. In this situation is follows from [START_REF] Stredulinsky | Area minimizing sets subject to a volume constraint in a convex set[END_REF], Theorem 3.32 that Cheeger = Ch(S) is convex and uniquely defined as the union of a set of disks of suitable radius [START_REF] Kawohl | Characterization of Cheeger sets for convex subsets of the plane[END_REF]. Thus, the Cheeger set in S is unique and is identified by the following theorem.

Theorem 1. [START_REF] Kawohl | Characterization of Cheeger sets for convex subsets of the plane[END_REF][START_REF] Alter | Evolution of characteristic functions of convex sets in the plane by the minimizing total variation flow[END_REF] Let S ⊆ R 2 be a nonempty bounded convex set. There exist a unique value t > 0 such that Area(S t ) = πt 2 . Then h(S) = 1 t and the Ch(S) = δ B(t) (S t ), where B(t) denotes a ball of radius t and S t = {x ∈ S : dist(x, ∂S) > t} is called the inner Cheeger set of S.

The proof of Theorem 1 is basically based on the Steiner's formulae in the case of d = 2 [START_REF] Breen | Attribute openings, thinnings, and granulometries[END_REF] and it is included in Appendix 1.

Corollary 1. For a nonempty bounded convex set S ∈ R 2 , the Cheegger set Ch(S) is an opening with a disk of radius given by the Theorem 1 as structuring element.

In the sequel, we use the name "Inner-Cheeger opening" to the opening defined in Corollary 1.

Small perturbations

Additionally, it is important to note that the uniqueness of Cheeger sets holds up to arbitrary small perturbations of S [START_REF] Caselles | Some remarks on uniqueness and regularity of cheeger sets[END_REF].

Comparison with skeleton transformation

The skeleton transformation is a widely used transformation in the field of image processing. The definition of skeleton was introduced by [START_REF] Blum | A transformation for extracting new descriptors of shape[END_REF], proposing the grass fire analysis: the skeleton consists of the points where different fire fronts intersect, or quench points. Later, another formal definition of the skeleton was proposed by Calibi in 1965 [START_REF] Calabi | A study of the skeleton of plane figures[END_REF] which relies on the concept of maximal ball. The skeleton is defined as the set of the maximal balls. In [START_REF] Calabi | A study of the skeleton of plane figures[END_REF], it was proved that the notion of quench points and centers of maximal balls are equivalent. According to Chapter one of [START_REF] Najman | Mathematical morphology: from theory to applications[END_REF], the notion of maximal ball has been known since the 30s [START_REF] Durand | Théprie des ensembles. points ordinaires et point singuliers des enveloppes de sphères[END_REF][START_REF] Bouligand | Introduction à la gémétrie infinitésimale directe[END_REF]. A skeleton transformation has to obey the following properties to make its results convenient for the global representation of objects [START_REF] Malandain | Euclidean skeletons[END_REF]: Homotopy: the skeleton transform must preserve the topology of the initial object; Rotation invariance: the skeleton transform of a rotated shape should be the rotated of the skeleton; Reconstruction: the initial shape should be reconstructed from the skeleton transform. For continuous images, the skeleton sk(S) of a shape S is given by the Lantuéjoul's formulate [START_REF] Lantuejoul | La squelettisation et son application aux mesures topologiques des mosaiques polycristallines[END_REF][START_REF]Skeletonization in quantitative metallography[END_REF] defined as follows:

sk(S) = ρ>0 µ>0 [ε ρB (S) -(γ B (ε ρB (S)))] (2)
A ball B included in S is said to be maximal if and only if there exist no other ball included in S and containing B, ∀B ball, B ⊆ B ⊆ S ⇒ B = B ,i.e., a ball is maximal in the shape if it is not included in any single other ball in the shape. We denote M BT (S) = {x ∈ S; ∃ a maximal ball centred at x}, the set of all the centres of the maximal balls included in S, and r x the corresponding radius of it associated maximal ball. The skeleton sk(S) of a set S is then defined as the set of the centers of its maximal balls:

sk(S) = {x ∈ S, ∃r x ≥ 0, B(x, r x ) maximal ball of S}. (3)
This collection of maximal balls M BT (S) is equivalent to the shape S itself since one has the reconstruction formula:

S = x∈M BT (S) δ rx (x) (4)
In this representation, we can easily see that for S ⊆ R 2 the Inner-Cheeger opening is

Ch(S) = x∈M BT (S),rx≥t δ rx (x), ( 5
)
where t is defined by Theorem 1. An example of the collection of maximal balls and the correspondent Inner-Cheeger set is shown in Fig. 2.

Applications

In this section we present some examples to illustrate our methods. We present two applications: Attribute filtering and shape decomposition in 2D, 3D and mesh shapes.

Inner-Cheeger set as an attribute filter

Many works about attribute filters have been presented in the literature [START_REF] Salembier | Flat zones filtering, connected operators, and filters by reconstruction[END_REF]. Most of them address the problem of filter-out target objects and preserving the contours of non-interesting objects. In this first application, we start with the description of a grey scale image I by its upper level set decomposition:

I(x) = sup(i : I(x) ≥ i is true), (6)
where is discrete images i = {0, 1, . . . , t max } and t max stands by the maximum grey scale value of the image. Now, following [START_REF] Breen | Attribute openings, thinnings, and granulometries[END_REF], we can define an attribute filter on I by the well-known threshold decomposition principle:

ψ(I)(x) = sup(i : Ψ (I(x) ≥ i) is true) (7)
where Ψ is a binary attribute filter. The advantage of this representation is the fact that most of the properties that are known to hold for Ψ are also true for its grey-level counterpart ψ. Mainly, if Ψ is an increasing criterion, it was proved if [START_REF] Breen | Attribute openings, thinnings, and granulometries[END_REF] that ψ is a valid opening in the morphological sense. We propose to use the optimal value of t is Theorem 1 as attribute to describe connected components in the threshold decomposition in Eq. 7. In proposition 1, property 2, we have stated that our attribute is increasing, which allows the use a fast implementation (by reducing the search space in Theorem 1), for instance, by means of a max-tree structure [START_REF] Carlinet | A comparison of many max-tree computation algorithms[END_REF]. Some example of this attribute filter are illustrated in Fig. 3 for different values of t. This attribute opening preserves contours of object with largest Cheeger set, i.e., objects with large sphericity.

Shape Decomposition

Instead of characterizing an object as a whole, part-based representation deals with a representation where an object is decompose it into a number of "natural parts [START_REF] Hoffman | Parts of recognition[END_REF]. The aim of this shape decomposition is to simplify a given shape into meaningful parts to make easier its analysis. Many morphological-based operators approaches [START_REF] Xu | Morphological decomposition of 2-d binary shapes into convex polygons: A heuristic algorithm[END_REF][START_REF] Yu | Shape representation based on mathematical morphology[END_REF][START_REF] Kim | A new shape decomposition scheme for graph-based representation[END_REF] have been formulated with the intuition of finding the "best" opening (with a ball as structuring element) by means of an index considering the size "part/shape" ratio and "part/convex hull" ratio. However, these methods are often hard to implement and/or do require the solution of difficult optimization problem. In this section we propose to use the inner Cheeger decomposition to perform shape decomposition. For a given shape S, we firstly compute the Ch(S) and successively the same transformation is applied to each connected component of the difference between S and Ch(S). We note that many of the connected components can have small areas, so a threshold parameter is included to obtain only meaningful parts of the object. We use the term Inner-Cheeger shape decomposition (ICSD) to refer this sequential separation, due to the fact that in the case of non-convex shapes, one cannot say that this procedure find a valid Cheeger set. However, as it is illustrated in Fig. 4, the decomposition determines relevant parts and as in most of the distance based transformation, the result is robust to articulate transformations. In addition, in our case it is robust to small perturbations of the original shape. At this point, we can use a Colour are associated with the order of discovery in the iteration of ICSD. Note that results are robust to modification in the human shape.

similar approach to 3D shapes in Fig. 5 and 3D points clouds in Fig. 6. However, the optimal value of t in Theorem 1 cannot be applied directly for 3D shapes. Thus, assuming that the inner set of a convex 3D shape if composed by the union of balls, we can obtain an equivalent expression for three dimensional objects by using the Steiner's formula as it is shown in Appendix. Accordingly, in 3D shape, we should look for the radius t that Vol(S t )+MeanB(S t )t 21 2π -1 = 8πr 3 3 , where MeanB(S) stands for mean breadth of the binary shape S. A Fig. 5 illustrates the results of ICSD for three postures for the same 3D human model. Additionally, we explore the application of our approach in mesh of points in three different frames of "Ben Walking Scene" 1 .

Conclusion

We have defined an optimal opening via Inner-Cheeger sets. We have explored some important morphological properties including the robustness against small perturbations. Finally, we have illustrated the interest and behavior of such operators in some problems of image processing and shape decomposition. Future work includes the analysis of similar problem in shapes with holes as it is the case of [START_REF] Liu | Dual-space decomposition of 2d complex shapes[END_REF] and compare our results with the space of shapes from [START_REF] Xu | Morphological filtering in shape spaces: Applications using tree-based image representations[END_REF]. Fig. 6. In red the points identified as part of the Cheeger opening for meshes (black lines) in some frames of "Ben Walking" 4D dataset. The inner-Cheeger set in this case is the thorax of "Ben". The results are quite robust considering that the input data includes the presence of the "chair" and different postures of "Ben"

Proposition 1 .

 1 Let S 1 , S 2 ⊆ R d be bounded, open sets. Then the following properties hold. 1. (Existence) There exists a (possibly non-unique) Cheeger set X ⊆ S. 2. (Decreasing) If S 1 ⊂ S 2 then h(S 1 ) ≥ h(S 2 ) 3. (Isometry) For any λ > 0 and any isometry T : R d → R d , one has h(T (S)) = 1 λ h(S) 4. (Intersection with the boundary) The minimum in (1) is attained at a subset

Theorem 2 .

 2 [START_REF] Caselles | Some remarks on uniqueness and regularity of cheeger sets[END_REF] Let S ⊆ R N be an open set with finite volume. Then for any compact set K ∈ S, there exist a bounded open set S K ⊂ S such that K ⊂ S K and S K has a unique Cheeger set.A visual example of this property is illustrated in Fig.1.

Fig. 1 .

 1 Fig. 1. Example of invariance of Inner-Cheeger opening to small perturbations of the shape. (b) and (d) are perturbations of (a) and (c) respectively.

Fig. 2 .

 2 Fig. 2. Comparison of Inner-Cheeger opening and Skeleton transformation for two images. Colours in (b) and (f) show of the number of balls in the reconstruction of MBT.

Fig. 3 .

 3 Fig. 3. Inner-Cheeger opening for different values of proposed attribute.

Fig. 4 .

 4 Fig. 4. Example of decomposition by Iterative Cheeger Openings (2 Levels).Colour are associated with the order of discovery in the iteration of ICSD. Note that results are robust to modification in the human shape.

Fig. 5 .

 5 Fig. 5. Three examples of proposed decomposition for 3D shapes. Colour are associated with the order of discovery in the iteration of ICSD. Note that results are robust to changes in the pose of the human shape.

http://4drepository.inrialpes.fr/pages/home

We follow the results from [START_REF] Kawohl | Characterization of Cheeger sets for convex subsets of the plane[END_REF][START_REF] Alter | Evolution of characteristic functions of convex sets in the plane by the minimizing total variation flow[END_REF] for 2D shapes to obtain: In the case of d = 3, the Steiner's formula gives explicitly for a compact convex:

Sur(δ rB (S)) = Sur(S) + 2πMeanCurv(S)r + 4πr 2 (9)

Additionally, it is important to note that the mean breadth, denoted by MeanB(I), connects the integral of the mean curvature MeanCurv(S) for an object I whose boundary is of class C 2 by the formula, MeanB(I) = 2πMeanCurv(S).

Vol(δ rB (S)) = Sur(S) + 2πMeanCurv(S)r + 4πr 2 Vol(S) + Sur(S)r + MeanCurv(S)r 2 + 4πr 3 3 and after simplification, we obtain 8πr 3 3 = Vol(S) + MeanB(S)r 2 1 2π -1 . We remark that the percolation analysis in three-dimensional objects in [START_REF] Jeulin | Random structures in physics[END_REF] uses similar results in random models with spheres and Poisson polyhedra as primary grains.


Sébastien Forestier

Pierre-Yves Oudeyer Flowers

Towards Hierarchical Curiosity-Driven Exploration of Sensorimotor Models

 and describe some ways to extend those architectures to the exploration of a hierarchy of sensorimotor skills. We developed a simulated robotic setup to evaluate the different architectures, where a robot has to push an object to different locations.

Introduction

We use and extend the Explauto library [START_REF] Moulin-Frier | Explauto: an open-source Python library to study autonomous exploration in developmental robotics[END_REF] that aims at studying autonomous exploration. In the Explauto framework, a sensorimotor model is learned together with an interest model that guides future exploration.

• Sensorimotor model

This model stores the experimented motor commands and their associated sensory experience and builds a mapping between the motor space and the sensory space. We use the fast nearest neighbor algorithm to build this mapping but more powerful regression methods could be used instead.

• Interest model

The interest model estimates how interesting it is to explore given parts of the sensory space. We use the SAGG-RIAC architecture (Self Adaptive Goal Generation -Robust Intelligent Active Curiosity [START_REF] Baranes | Active learning of inverse models with intrinsically motivated goal exploration in robots[END_REF]) with an intrinsic motivation that pushes the agent to explore regions where the progress of the competence to reach self-generated goals in the sensory space is the higher.

• Exploration

The agent selects a goal in its sensory space according to the interest model and infers motor parameters to get close to this point with the sensorimotor model. It adds some exploration noise to discover new motor configurations, executes the command and observes the sensory experience. Finally, it updates the sensorimotor model with the new association, and the interest model with the competence to reach the goal.

Curiosity-Driven Exploration

To study hierarchical exploration, we design a setup where the agent learns two sensorimotor models, with the second one that reuse the first one. In our setup, the first sensorimotor model in the hierarchy is a relation between the 20 motor parameters of a robotic arm and the 9 parameters of the 3D trajectory of the robotic hand. The second sensorimotor model is a relation between the trajectory of the hand and the 2D position of a block at the end of the movement. Control architectures will have to learn directly a relation between the motor parameters and the position of the block.

20D 2D 9D M Sh So 20D M 2D So

Hierarchical Exploration

• Motor Babbling Control Exploration of a mapping between M and S o with random motor actions.

• Goal Babbling Control Exploration of a mapping between M and S o with a competence-based intrinsic motivation (SAGG-RIAC).

• Simplest First Exploration of a mapping between M and S h for the first half of the trials and of a mapping between S h and S o for the second half. Both explorations are made with SAGG-RIAC.

• Top-Down Guidance Exploration of a mapping between S h and S o . When asking to explore a hand's movement s h , this goal s h is used to guide the exploration of a mapping between M and S h which is given a certain amount of iterations to try motor configurations to reach that goal (with a black-box optimization technique [START_REF] Hansen | The CMA evolution strategy: a comparing review[END_REF]).

Learning Architectures

• The robotic setup is the left arm of the Poppy robot [START_REF] Lapeyre | Poppy Project: Open-Source Fabrication of 3D Printed Humanoid Robot for Science, Education and Art[END_REF], with 4 degrees of freedom, simulated with the V-REP simulator based on the Bullet physics engine.

• We use Dynamical Movement Primitive [START_REF] Ijspeert | Dynamical movement primitives: learning attractor models for motor behaviors[END_REF] to control the arm's movement as this framework allows the production of a diversity of arm's trajectories with few parameters. Each arm's motor is controlled by a DMP with a starting and a goal position equal to the rest position of the motor. Each DMP is parameterized by one weight on each of 5 basis functions whose centers are distributed homogeneously throughout the movement of duration 4s. M is the 20D space of the motor parameters.

• S h is a 9D space representing the 3D trajectory of the hand. We also use the DMP framework to project each of the X, Y and Z movements on a sensory DMP with 3 basis functions.

• A 3cm wide block is located near the robot's hand and can be moved in different complex ways, e.g. with the hand pushing on the top of the block or on a side. S o is the 2D space representing the position of the block at the end of the simulation. The initial position of the block is (X = 0.225m, Y = 0.135m), X axis points in front of the robot, Y axis on its left.

Poppy Torso in the V-rep simulator

We ran different trials for each conditions (n M B = 13, n GB = 9, n SF = 16, n T DG = 18). Each trial is made of 5000 learning iterations. Each iteration takes about 1s on 3, 06 GHz Xeon x5675 nodes. Exploration is evaluated each 500 iterations with 2 different measures. Statistical tests are performed after 5000 iterations to compare conditions.

Experiments

• Exploration Measure This measures how diverse outcomes have been found by the agent in its goal space. It makes no assumptions about the interest of exploring different regions of the goal space from the viewpoint of the engineer.

To compute the quantity of exploration we consider the goal space as an unbounded grid where we count the cells that have been reached during training.

Low Robustness

In our setup, the environment is highly stochastic in the sense that when the agent tries the same motor command different times, the variability in the generated arm movement and the collisions between the arm and the block leads to high variances in the end position of the block. Thus the agent might not succeed in reaching again a previously reached position, and in other words, a high exploration does not mean a high learning. We define the following measure to tackle this problem.

• Competent Exploration Measure

This measures how competent is the robot on the explored part of its goal space. We give the centers (black dots) of explored cells as goals to the agent, and we count the cells where the agent manages to get close to the goal (within 2cm). Right: In the competent exploration measure, we plot a red cell if the agent did not manage to get within 2cm of the center, and a green cell if it did. In green cells, black dots are goals and green dots are corresponding reached points. X axis points in front of the robot, Y axis on its left, units are meters.

Results

• Results show that exploration is better in the Top-Down Guidance condition than in the Simplest First condition, with both measures. This implies that the guidance from the top-level model is useful to drive the exploration of the lower-level model on parts of the task space that are more interesting for the exploration of the higher one.

• Results also show comparable exploration for the Top-Down Guidance and the Goal Babbling control conditions, whereas the competent exploration measure shows that the architectures learning an intermediate hand's movement representation allow the agent to put again the object on much more diverse locations. An intuition for this result is that when exploring around motor parameters that lead to a movement of the block, the control architecture modifies directly joints' trajectories leading to less accurate collisions with the block than hierarchical architectures that try to modify hand's Cartesian trajectory. For that reason, hierarchical architectures would produce less diverse but more reproducible interactions than control architectures.

• In this setup, the intermediate hand's movement representation is reused by only one higher-level model, but learning intermediate representations should be even more beneficial for exploration in complex learning hierarchies where more than one models are reusing the learned representations.

• The next step is to integrate social interaction with a human peer that will give demonstrations on the task space and the hierarchy of models to explore.

  square is the initial position of the block, blue dots are reached positions of the block at the end of the movement, blue squares are reached cells.

  U tests at the end of the experiments show that the exploration measures in the Goal Babbling control and Top-Down Guidance conditions are not significantly different but are significantly greater (p < 0.05) than in the Simplest First condition, where it is significantly greater than in the Motor Babbling control condition.Mann-Whitney U tests at the end of the experiments show that the competent exploration measure is significantly greater in the Top-Down Guidance condition than in the Simplest First condition where it is significantly greater than in the control conditions.