Aude Lacourt

Elisabeth Cardis

Javier Pintos

Lesley Richardson

Laurel Kincl

Geza Benke

Sarah Fleming

Martine Hours

Daniel Krewski

Dave Mc

Dave Mclean

Marie-Elise Parent

Siegal Sadetzki

Klaus Schlaefer

Brigitte Schlehofer

Jerome Lavoue

Martie Van Tongeren

Jack Siemiatycki
email: j.siemiatycki@umontreal.ca

à la diffusion de documents scientifiques de niveau

Keywords: Brain cancer, Case-control study, International, Job exposure matrix, Risk factor

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

INTEROCC case-control study: lack of association between glioma tumors and occupational exposure to selected combustion products, dusts and other chemical agents.

Background

Brain tumors constitute a mixed group of intracranial neoplasms which could be benign or malignant. Gliomas account for almost 80% of primary malignant brain tumours [START_REF] Schwartzbaum | Epidemiology and molecular pathology of glioma [Review][END_REF]. Of the six major histological subtypes of glioma, glioblastoma constitutes the highest grade and the most common type, and is associated with very poor survival [START_REF] Schwartzbaum | Epidemiology and molecular pathology of glioma [Review][END_REF][START_REF]Pathology and Genetics of Tumours of the Nervous System[END_REF]. The incidence of glioma differs according to age, gender, race, ethnicity and country; in many regions slight increases have been reported between the 1970s and 2000, though it is unclear whether this ostensible increase simply reflected diagnostic improvement [START_REF] Fisher | Epidemiology of brain tumors [Review][END_REF]. Except for ionizing radiation, which is a well-established risk factor for glioma [START_REF] Bondy | Brain tumor epidemiology: Consensus from the Brain Tumor Epidemiology Consortium[END_REF][START_REF] Ohgaki | Epidemiology and etiology of gliomas [Review][END_REF], the aetiology of this tumor is largely unknown [START_REF] Bondy | Brain tumor epidemiology: Consensus from the Brain Tumor Epidemiology Consortium[END_REF][START_REF] Connelly | Environmental risk factors for brain tumors[END_REF][START_REF] Ohgaki | Epidemiology of brain tumors[END_REF]. Some case-control studies have found a lower risk of glioma among subjects reporting allergies or other atopic conditions [START_REF] Bondy | Brain tumor epidemiology: Consensus from the Brain Tumor Epidemiology Consortium[END_REF]. There is inconsistent and largely negative evidence regarding possible associations between smoking, diet or alcohol and gliomas [START_REF] Fisher | Epidemiology of brain tumors [Review][END_REF][START_REF] Bondy | Brain tumor epidemiology: Consensus from the Brain Tumor Epidemiology Consortium[END_REF]. Concerns have been raised about possible effects of exposure to non-ionizing radiation including radiofrequency (RF) fields due to mobile phones or electromagnetic fields (EMF) in the extremely low frequency range (ELF), but the evidence remains inconsistent and uncertain [START_REF] Interphone | Brain tumour risk in relation to mobile telephone use: results of the INTERPHONE international case-control study[END_REF][START_REF] Coble | Occupational exposure to magnetic fields and the risk of brain tumors[END_REF][START_REF]SCENIHR Scientific Committee on Emerging and Newly Identified Health Risks: (SCENIHR). Health effects of exposure to EMF[END_REF][START_REF] Sienkiewicz | Report on Risk analysis of human exposure to electromagnetic fields (revised). Report D2. European Health Risk Assessment Network on Electromagnetic Fields Exposure (EFHRAN)[END_REF][START_REF] Villeneuve | Canadian Cancer Registries Epidemiology Research G: Brain cancer and occupational exposure to magnetic fields among men: results from a Canadian population-based case-control study[END_REF].

A few studies have reported on glioma risks in relation to occupations or industries, but there has not been any strong consistent pattern across studies. Slight increased risks have been reported among workers in synthetic rubber manufacturing [START_REF] Preston-Martin | Risk factors for gliomas and meningiomas in males in Los Angeles County[END_REF], petrochemical refineries [START_REF] Carozza | Occupation and adult gliomas[END_REF] and pulp and paper industries [START_REF] Andersson | Gliomas among men employed in the Swedish pulp and paper industry[END_REF], as well as among physicians [START_REF] Carozza | Occupation and adult gliomas[END_REF][START_REF] De Roos | Occupation and the risk of adult glioma in the United States[END_REF][START_REF] Krishnan | Occupation and adult gliomas in the San Francisco Bay area[END_REF], firefighters [START_REF] Carozza | Occupation and adult gliomas[END_REF][START_REF] Krishnan | Occupation and adult gliomas in the San Francisco Bay area[END_REF], farmers [START_REF] De Roos | Occupation and the risk of adult glioma in the United States[END_REF][START_REF] Zheng | Occupational risk factors for brain cancer: A population-based case-control study in Iowa[END_REF] and legal and social services workers [START_REF] Carozza | Occupation and adult gliomas[END_REF]. More studies have reported on occupational risks for all brain cancers combined [START_REF] Zheng | Occupational risk factors for brain cancer: A population-based case-control study in Iowa[END_REF][START_REF] Inskip | Etiology of brain tumors in adults[END_REF][START_REF] Wrensch | Epidemiology of primary brain tumors: Current concepts and review of the literature[END_REF]. There have been reports of increased risk of brain tumors for polyvinyl chloride production workers [START_REF] Lewis | Vinyl chloride and liver and brain cancer at a polymer production plant in Louisville, Kentucky[END_REF][START_REF] Mundt | Historical cohort study of 10 109 men in the North American vinyl chloride industry, 1942-72: update of cancer mortality to 31 December 1995[END_REF], anatomists, pathologists and embalmers [START_REF] Hall | Cancer by industry: analysis of a population-based cancer registry with an emphasis on blue-collar workers[END_REF][START_REF] Stroup | Brain cancer and other causes of death in anatomists[END_REF][START_REF] Walrath | Cancer and other causes of death among embalmers[END_REF], painters, machinists, industrial mechanics and plumbers [START_REF] Demers | Occupation, socioeconomic status, and brain tumor mortality: a death certificate-based case-control study[END_REF][START_REF] Thomas | Occupational risk factors for brain tumors[END_REF]. Few studies have delved into the possible associations between specific occupational exposures in those occupations/industries and the risk of brain cancer. There is a need for studies that go beyond the job or industry title to investigate brain cancer risks in relation to chemical exposures.

The INTEROCC study was built on two important databases: the INTERPHONE study, the largest international collaborative case-control study of brain cancer yet conducted [START_REF] Cardis | The INTERPHONE study: design, epidemiological methods, and description of the study population[END_REF], and FINJEM, a database permitting the translation of job histories into occupational exposure histories [START_REF] Kauppinen | From cross-tabulations to multipurpose exposure information systems: a new job-exposure matrix[END_REF]. The INTERPHONE study involved interviews with over 2,400 meningioma cases, 2,700 glioma cases and over 5,600 controls in regions of 13 countries, with a primary focus on use of mobile phones. A subset of INTERPHONE study investigators were interested in investigating occupational risk factors and had agreed to include in the interview questionnaire a module to elicit a detailed lifetime job history. That module was used in seven of the INTERPHONE countries, and the subjects in those countries comprise the INTEROCC study subjects. No further data collection was required from the study subjects. The second pillar for this project, the FINJEM, allowed us to infer exposure to many occupational agents. Following examination of the agents that were available in FINJEM, and a review of previous epidemiologic evidence concerning occupations and brain cancer risk, we established a list of 28 agents that would be worthwhile to evaluate in relation to brain cancer. These 28 substances comprised solvents, metals, combustion products, dusts and other chemical agents. For most of them there is little epidemiologic evidence, and little or no literature on possible mechanisms of brain cancer carcinogenesis. This should not be an impediment to studies such as the present one, since most known carcinogens were discovered by epidemiologic or clinical observation, before there were evidence-based biological plausibility arguments [START_REF] Siemiatycki | The causes of cancer: Occupation[END_REF]. The present article focuses on risks of glioma in relation to 10 of the agents, namely combustion products (diesel exhaust emissions, gasoline exhaust emissions, benzo(a) pyrene), dusts (asbestos, crystalline silica, wood dust, animal dust) and certain other chemical agents (formaldehyde, oil mist, sulphur dioxide). Subsequent papers will address risks related to other agents and to meningioma.

Methods

The INTERPHONE study

INTERPHONE was a population-based case-control study of brain cancer carried out in 17 centers in regions of 13 countries [START_REF] Interphone | Brain tumour risk in relation to mobile telephone use: results of the INTERPHONE international case-control study[END_REF][START_REF] Cardis | The INTERPHONE study: design, epidemiological methods, and description of the study population[END_REF]. The main purpose was to investigate possible associations with use of mobile phones. For most centers, eligible cases were all patients aged between 30 and 59 years and diagnosed with a glioma or meningioma tumor between 2000 and 2004. In Germany and the UK, the upper age limit was 69 years. Israel had no upper age limit for recruitment; however, only subjects aged 30 to 69 years were included from this country in these analyses. Population controls were frequency or individually matched to cases by sex, age (within 5 years) and center. The sampling frame for population controls differed from center to center. Each of the following was used in at least one center: electoral list, population registry, random digit dialing, general practice patient lists and national health insurance plan lists. When possible, a face-to-face interview with the study subjects was conducted by a trained interviewer using a computer-assisted questionnaire. When the study subject h addiedorwast ooil ltob ei nt erviewed,th einterviewwas conducted with a proxy respondent, usually the spouse, but occasionally an offspring. While this occurred not infrequently among cases, it occurred rarely among controls when the control was selected from the sampling frame, but was not well enough to conduct the interview. Detailed information was collected on socio-demographic characteristics, lifetime use of mobile phones and medical history. Details of each center's study methods are described elsewhere [START_REF] Interphone | Brain tumour risk in relation to mobile telephone use: results of the INTERPHONE international case-control study[END_REF][START_REF] Cardis | The INTERPHONE study: design, epidemiological methods, and description of the study population[END_REF].

In addition to the core questionnaire, a subset of INTERPHONE study investigators had agreed to include in the interview questionnaire a module to elicit a detailed lifetime job history, and to participate in an analysis of occupational risk factors.

All participating study centres obtained the appropriate Institutional Review Board authorisations. Written informed consent for participation in the study was obtained from all participants.

The INTEROCC study

The detailed occupational history module was used in seven of the INTERPHONE countries (Australia, Canada, France, Germany, Israel, New Zealand and the UK), and the subjects in those countries comprise the INTEROCC study subjects.

The lifetime work history requested information for all jobs held by participants for more than 6 months, and included job title, description of tasks, company name, description of activities of the company, and the start and end year for each job. Each job held by subjects was coded according to international occupation and industry classifications: the International Standard Classification of Occupations editions 1968 (ISCO68) [START_REF]International Labour Office: International Standard Classification of Occupations[END_REF] and 1988 (ISCO88) [32], and the International Standard Industrial Classification of All Economic Activities, revision 2 (ISIC71) [START_REF]Nations U: International Standard Industrial Classification of All Economic Activities[END_REF]. Common coding guidelines were provided to each center in order to ensure homogeneity in the coding. An interrater trial was conducted at the start of coding and results were discussed with each center in a further effort to ensure consistency of coding [START_REF] Mclean | Evaluation of the quality and comparability of job coding across seven countries in the INTEROCC study[END_REF].

Occupational exposure assessment

Occupational exposure to the agents was assessed using a modified version of the Finnish job exposure matrix FINJEM [START_REF] Kauppinen | From cross-tabulations to multipurpose exposure information systems: a new job-exposure matrix[END_REF]. Based on the Finnish occupation classification system containing 311 major occupational groups and covering the calendar period from 1945 to 2003 divided into several sub-periods, FINJEM can translate an occupational history into a history of exposure to about 90 chemical, physical, behavioral, microbiological, ergonomic and psychosocial factors, including the 10 agents under investigation in this paper. FINJEM provides two exposure estimates for each combination of occupation, calendar sub-period, and agent: the proportion of workers in that occupation who were considered to be exposed to the agent (P) and the mean level of exposure among the exposed (L) expressed in concentration units. When P was considered to be less than 5%, the level in FINJEM was set to zero. The estimates of P and L were based on exposure measurements, hazard surveys, and the judgements by Finnish occupational hygienists. The fraction of daily or weekly worktime during which the worker is thought to be exposed to the agent is implicitly taken into account because L was constructed as a timeweighted average. All 10 agents considered in this paper are dusts or gases or fumes that entailed respiratory exposure.

Since FINJEM uses the Finnish occupational coding system and the INTEROCC work histories were coded according to international classifications, it was necessary to develop a "crosswalk" between the Finnish codes and the ISCO68. Furthermore, for our purposes, FINJEM was modified in three ways [START_REF] Van Tongeren | Assessing Occupational Exposure to Chemicals in an International Epidemiological Study of Brain Tumours[END_REF].

First, the time window 1960-1984 was split into pre and post-1974 periods, and some specific FINJEM entries were modified for some of the agents of interest in order to increase consistency and specificity of exposure assessment. Second, exposure to benzo(a)pyrene was modified to include exposure to environmental tobacco smoke in the workplace. Third, meetings were held with occupational exposure experts from the different centers in order to consider possible differences in the meaning of different job titles across countries and to consider possible occupations and industries where the FINJEM might not be generalizable to other countries. The resulting changes were embodied in a revised version of FINJEM that we call INTEROCC JEM [START_REF] Van Tongeren | Assessing Occupational Exposure to Chemicals in an International Epidemiological Study of Brain Tumours[END_REF].

Statistical methods

In this article, we are focusing on glioma only. All interviewed controls in the seven INTEROCC countries were used in these analyses, including those recruited for meningioma cases in the centers which used individual matching.

For each agent, three exposure indices were defined for the main analysis: i) ever vs. never; ii) lifetime cumulative exposure; and iii) total duration of exposure. As noted above, for each job, the INTEROCC JEM provides the probability (P) that a worker in that occupation was exposed to that agent. There are several approaches for deriving an "ever exposure" variable from the INTEROCC JEM. If we include in the definition all those who had a P greater than 5%, it would be very sensitive but most of the subjects labelled as exposed would have had a low probability of exposure. At the other extreme, if we define a high threshold, say 95%, then it would be very specific, but a large fraction of workers truly exposed would be labelled as unexposed. The trade-off between sensitivity and specificity also has implications for the estimated prevalence of exposure. In order to give greater weight to sensitivity than specificity, but not to exaggerate this choice unduly, we used as the a priori threshold P ≥ 25%. Thus, ever exposure to a given agent was defined as having held at least one job with a probability of exposure of at least 25% and for at least one year. Subjects who had held jobs with a probability of exposure of less than 25% but greater than 5%, or for less than one year were considered to be of "uncertain" exposure status and were not included in the analyses. The lifetime cumulative exposure index was defined among ever exposed (corresponding to the ever definition criteria) as the sum of the product of the probability of exposure (P), the level of exposure (L) and the duration for each job held by a subject. The continuous cumulative exposure index was categorized according to tertiles of the distribution among exposed controls. The total duration of exposure was defined as the sum of the duration of exposure for each job held by a subject (corresponding to the ever definition criteria) minus the possible overlap period between two jobs. The total duration of exposure was also categorized, but categories were defined a priori: 1-4years,5-9 years, and 10 or more years of exposure.

To allow sufficient time between occupational exposure and disease initiation, all exposures that had occurred within five years of the reference date (age at diagnosis for cases and age at interview for controls) were not taken into account, therefore establishing a lag period of 5 years.

For all analyses, the reference category included subjects who had never been exposed to the specific chemical agent of interest, e.g. subjects never exposed or exposed with a probability lower than 5%. Because of the exploratory nature of these analyses, each chemical agent was considered independently of exposure to other agents. That is, the possibility of mutual confounding among these agents was ignored.

Associations between glioma and each of the 10 chemical agents were estimated using conditional logistic regression stratified by sex, age (5-year categories) and center. Further, all analyses were a priori adjusted for the following variables: i) age as a continuous variable (in order to remove any residual confounding due to age in the strata definition), ii) the maximum education level attained by the subject or her/his spouse (primary, intermediate college, tertiary), iii) the Standard International Occupational Prestige Scale (SIOPS) [START_REF] Treiman | Occupational Prestige in Comparative Perspective[END_REF], iv) history of atopy, defined as ever diagnosed with allergy, asthma and/or eczema (recognized as associated with glioma) [START_REF] Brenner | History of allergies and autoimmune diseases and risk of brain tumors in adults[END_REF][START_REF] Schlehofer | Role of medical history in brain tumour development. Results from the international adult brain tumour study[END_REF], and v) the respondent status (subject himself vs. proxy respondent). Education level and SIOPS provide two different measures of socio-economic status.

Missing values for the maximum level of education attained in the household were imputed to the middle category "intermediate college" (14 subjects). Missing values for the SIOPS variable were imputed to the median value in the corresponding subject strata of age (5-year categories), sex, center and maximum level of education attained in the household (104 subjects).

Sensitivity analyses

Our definition of "ever exposed" embodied a priori decisions about probability of exposure (P ≥ 25%), duration (D ≥ 1 year), and lag period (5 years). To evaluate the sensitivity of results to these decisions, we tried different thresholds. For probability of exposure, we implemented thresholds of 5%, 25% (default) and 50%. For duration, we implemented thresholds of 1 year (default) and 5 years. Finally, for lag period, we implemented thresholds of 1 year, 5 years (default) and 10 years. Altogether there were 18 combinations (3×2×3), including the main default combination.

In addition to the 17 sensitivity analyses embodied in those alternative definitions of ever exposure, additional sensitivity analyses were conducted: i) among males and females separately; ii) restricting cases to the subset of glioblastoma cases, iii) restricting the study sample to those who responded for themselves (i.e. excluding proxies), iv) including two additional a priori confounders in each model: smoking status (ever, ex, never smokers) and marital status (married vs. others); v) excluding each of the a priori confounders from the default model; vi) excluding subjects with missing values, instead of imputing values; vii) for benzo(a)pyrene (which had a high proportion of subjects in the uncertain category), exposure assessment according to the original version of FINJEM, which excluded benzo(a)pyrene exposure from environmental tobacco smoke [START_REF] Kauppinen | From cross-tabulations to multipurpose exposure information systems: a new job-exposure matrix[END_REF].

Results

Of the eligible subjects who completed the interview, 54 cases and 36 controls were removed from the analyses because it was technically impossible to link part of their occupational history to the INTEROCC JEM. An additional set of 9 cases and 6 controls were removed due to a family history of neurofibromatosis and/or tuberous sclerosis. Following these removals, the study sample consisted of 1,800 glioma cases and 5,160 controls. Response rates for cases and controls were 68% and 50%, respectively.

Table 1 describes the main characteristics of subjects included in the analysis. Germany, Israel, UK, and Australia are the main contributor countries in terms of number of glioma cases, accounting for more than 80% of the total. The interviews were conducted with proxy respondents for 14% of the cases and less than 1% of the controls. The distribution of medical history of atopic conditions, smoking, marital status, education level, and mean values for SIOPS were similar among cases and controls.

Table 2 shows the main results for each of the 10 agents analysed, using the a priori cut-point definition of ever exposure: P ≥ 25%, duration ≥ 1 year; lag period of 5 years. Subjects whose exposure probability was greater than 5% but less than 25% were removed from the analysis (number of removed subjects varied by agent).

Combustion products

Lifetime prevalence of exposure to benzo(a)pyrene, diesel exhaust emissions and gasoline exhaust emissions ranged from 8% to 20%. In all cases, prevalence of exposure was much higher for males than females (data not shown).

The patterns of results were similar for the three combustion products. There was no indication of excess risk when analysing ever/never exposure status, and there were no clear dose-response patterns, although the top category of duration to diesel exhaust emissions did manifest a borderline significant OR of 1.3 (95% CI: 1.0-1.8).

We observed generally similar results in all sensitivity analyses, with slight variations in some (data not shown).

Dusts

The lifetime prevalence of exposure ranged from 4% for wood dust to 16% for asbestos, with intermediate prevalences for crystalline silica and animal dust. As expected, prevalences were much higher for males than females.

None of these agents manifested any association with gliomas, neither in the ever/never analyses, nor in the dose-response analyses, with either cumulative exposure or duration. All sensitivity analyses gave similar results (data not shown).

Other chemical agents

Prevalence of exposure ranged from 0.4% for sulphur dioxide to 3.5% for oil mist to 5.5% for formaldehyde. Prevalences were much higher for males than females.

ORs for glioma amongst ever exposed compared to never exposed were around 1.0 for both formaldehyde and oil mist, and there were no indications of higher risk in the higher categories of cumulative exposure or duration. For sulphur dioxide, the least prevalent of the 10 agents analysed (thus providing the most imprecise estimates), the OR of glioma for ever exposure was 2.0 (95% CI: 1.0-3.8), but the excess risk was concentrated in the lowest tertile category of cumulative exposure and the lowest duration category.

All sensitivity analyses gave similar results to those presented in Table 2 (data not shown).

To explore whether the borderline significant result regarding sulphur dioxide was in fact reflecting a clearer excess risk in some subset of workers who incurred their exposures in particular occupations or industries, we reviewed the list of jobs of each subject ever exposed to sulphur dioxide. There was no perceptible difference between cases and controls in the distributions of industries and occupations leading to sulphur dioxide exposure. Most of these were in the iron and steel industry. This exploration did not reveal any noteworthy cluster of cases in any particular occupation or industry. Since a borderline significant OR was also observed for one of the subcategories of diesel exhaust emissions, we also examined the lists of jobs of subjects ever exposed to diesel exhaust emissions. There were no discernible differences in occupational profiles between cases and controls, with motor vehicle drivers or mechanics representing the main occupations with diesel exhaust exposure. Here again we did not detect any noteworthy cluster of cases. All sensitivity analyses are available from the corresponding author upon request.

Discussion

We believe INTEROCC is the largest study ever conducted on occupational chemical exposure and risk of glioma. We examined the association between glioma and 10 different chemical agents using three different exposure indices. A large number of sensitivity analyses were conducted, varying the definition of exposure as derived from the INTEROCC JEM, covariates and other analytic tactics and stratifying on the nature of the tumor, sex of subjects, respondent status (self/proxy). Considering all the sensitivity analyses, we analyzed over 200 different models for each chemical agent and thus more than 2,000 models in total. Despite the large sample size, none of the 10 agents demonstrated a pattern of findings that would be persuasive of a causal association with glioma. The handful that were borderline statistically significant could most likely be attributed to multiple testing and random chance.

Although there have been few previous studies on occupational risk factors for glioma, in particular, there have been many previous studies on possible associations between brain cancer and occupational circumstances. However, most of these studies have suffered from some combination of low statistical power due to few study subjects combined with rare exposures, and inadequate assessment of occupational exposures. While there have been no consistently and persuasively reported associations, there are a few leads that deserve attention and that have motivated the present analyses.

There have been some reports of possible excess risks of brain cancer among anatomists, pathologists and embalmers [START_REF] Hall | Cancer by industry: analysis of a population-based cancer registry with an emphasis on blue-collar workers[END_REF][START_REF] Stroup | Brain cancer and other causes of death in anatomists[END_REF][START_REF] Walrath | Cancer and other causes of death among embalmers[END_REF] and this has led to hypotheses regarding a possible role for formaldehyde. But other studies of formaldehyde-exposed workers have failed to support an etiologic association between formaldehyde and brain cancer [START_REF] Bosetti | Formaldehyde and cancer risk: a quantitative review of cohort studies through 2006[END_REF][START_REF] Coggon | Extended follow-up of a cohort of british chemical workers exposed to formaldehyde[END_REF][START_REF] Hauptmann | Mortality from solid cancers among workers in formaldehyde industries[END_REF][START_REF] Hauptmann | Mortality from lymphohematopoietic malignancies and brain cancer among embalmers exposed to formaldehyde[END_REF]. Our results also failed to support such an association.

While there have been some reports linking PAH exposure to brain cancer risk [START_REF] Rodvall | Glioma and occupational exposure in Sweden, a case-control study[END_REF][START_REF] Wesseling | Cancer of the brain and nervous system and occupational exposures in Finnish women[END_REF], such an association was not observed in all studies [START_REF] Navas-Acien | Occupation, exposure to chemicals and risk of gliomas and meningiomas in Sweden[END_REF]. In our study, we focused specifically on benzo(a)pyrene, and our results do not support a possible role of such exposure in the aetiology of gliomas. Still, exposure to PAHs is ubiquitous, occurring in many different occupations and industries, and often in environments where workers are simultaneously exposed to many other agents. Thus, analysing the specific role of PAHs is challenging.

Whereas some previous results suggested a possible increase of brain cancer risk among subjects exposed to wood dust [START_REF] Andersson | Gliomas among men employed in the Swedish pulp and paper industry[END_REF][START_REF] Band | Cohort mortality study of pulp and paper mill workers in British Columbia. Canada[END_REF][START_REF] Pan | Occupational risk factors for brain cancer in Canada[END_REF][START_REF] Stellman | Cancer mortality and wood dust exposure among participants in the American Cancer Society Cancer Prevention Study-II (CPS-II)[END_REF], two large studies, one a cohort study [START_REF] Szadkowska-Stanczyk | Cancer risk in workers of the pulp and paper industry in Poland. A continued follow-up[END_REF] and the other a large international multi-center case-control study [START_REF] Schlehofer | Occupational risk factors for low grade and high grade glioma: results from an international case control study of adult brain tumours[END_REF], failed to detect any risk. In our study, we did not find any association between glioma and wood dust.

For diesel exhaust emissions and gasoline exhaust emissions, an increased risk of brain cancer has been reported among subjects employed as motor vehicle operators [START_REF] Krishnan | Occupation and adult gliomas in the San Francisco Bay area[END_REF][START_REF] Cocco | Occupational risk factors for cancer of the central nervous system: a case-control study on death certificates from 24 U.S. states[END_REF], but our results did not support such an association. The borderline significant OR for one subset analysis of diesel exhaust emissions is most likely due to random chance since we did not observe any excess risks in relation to cumulative exposure. An association between asbestos exposure and brain cancer risk has been suggested by some studies [START_REF] Pan | Occupational risk factors for brain cancer in Canada[END_REF][START_REF] Seidman | Mortality of brain tumors among asbestos insulation workers in the United States and Canada[END_REF], but other investigations, including a recent large multicenter case-control study of brain tumors in adults, did not find any association between gliomas and asbestos or insulation products in both men and women [START_REF] Schlehofer | Occupational risk factors for low grade and high grade glioma: results from an international case control study of adult brain tumours[END_REF]. These findings are in agreement with our results, with ORs near the null.

There has not been previous research on sulphur dioxide in relation to glioma. While we found a borderline statistically significant OR between sulphur dioxide, the least prevalent of the 10 agents, and glioma, this was due to an excess of cases with very short duration, and it is likely to be explained by random variability.

Finally, previous evidence for an association between occupational exposure to oil products and/or animal products and glioma is sparse and inconclusive [START_REF] Rodvall | Glioma and occupational exposure in Sweden, a case-control study[END_REF][START_REF] Schlehofer | Occupational risk factors for low grade and high grade glioma: results from an international case control study of adult brain tumours[END_REF], and our results fail to support any such association.

Participation rates were 68% among glioma cases and 50% among controls. While these rates are not atypical for contemporary population-based case-control studies, they are low enough to raise concerns about a potential participation bias. This issue has been explored in some depth in the parent INTERPHONE study with regard to mobile phone use and brain cancer [START_REF] Vrijheid | Quantifying the impact of selection bias caused by nonparticipation in a case-control study of mobile phone use[END_REF], and it was estimated that differential participation rates could those estimated ORs by 10% to 15%. Such biases, related to the differential cellphone usage of participants vs. non-participants, are unlikely to be applicable to the present analyses of occupational exposures because whereas cellphone usage is a prevalent behavioural characteristic that might conceivably be related to the behaviour of being willing and available to participate in a study, it is less likely that being exposed to chemicals at work is similarly correlated with study participation. E veniftherewasabiasinthesameorderofmagnitude, it would hardly change the overall inferences of the present analysis, as the ORs are far from being statistically elevated.

The only solidly established risk factors for glioma are rare (ionizing radiation, rare medical conditions), probably rarer than the agents that we are examining. Also, these risk factors are probably not strongly correlated with the agents we are examining. These conditions militate against the likelihood that there was confounding of our results by known risk factors. Further, since our observed ORs were mainly around the null value, if confounding had occurred it would have distorted a true causal association to a null one, and this would require quite strong negative correlation between the known risk factor(s) and the agents that we are examining. Since there has been some debate about the possible association between socioeconomic status and brain cancer risk [START_REF] Preston-Martin | Risk factors for gliomas and meningiomas in males in Los Angeles County[END_REF][START_REF] Demers | Occupation, socioeconomic status, and brain tumor mortality: a death certificate-based case-control study[END_REF][START_REF] Navas-Acien | Occupation, exposure to chemicals and risk of gliomas and meningiomas in Sweden[END_REF][START_REF] Chakrabarti | A population-based description of glioblastoma multiforme in Los Angeles County, 1974-1999[END_REF][START_REF] Inskip | Sociodemographic indicators and risk of brain tumours[END_REF], we adjusted all results for socioeconomic status using two different markers, one based on the Standard International Occupational Prestige Scale and another one based on the maximum level of education attained in the household. In some sensitivity analyses, we also adjusted for marital status (married vs. other) and this additional adjustment did not change results. While it is theoretically possible that our essentially null findings were biased by other unmeasured confounders, this seems unlikely, as such factors would have to be quite strongly negatively associated with the occupational agents we examined, conditional on the socio-economic factors we did adjust for.

The exposure assessment was based on the subjects' reported job histories, complemented by the data in the INTEROCC JEM. The assessments did not rely on self-reports of exposures, which have been shown to be unreliable [START_REF] Fritschi | Self-assessed versus expert-assessed occupational exposures[END_REF]. Besides the advantage of providing occupational exposure assessments quickly and with modest cost, especially considering the large number of study subjects that had to be evaluated (approximatively 7,000 subjects), the use of a JEM has the advantage of providing a standardized tool to assign exposure and thereby minimizing the possibility of recall and reporting bias. On the other hand, a JEM allocates the same exposure estimate to each subject in a given occupation in a given era, without taking into account inter-individual variability in performed tasks and job environments. The error involved in applying a JEM is non-differential with respect to case/control status and therefore can lead to some attenuation of true odds ratios [START_REF] Bouyer | Performance of odds ratios obtained with a job-exposure matrix and individual exposure assessment with special reference to misclassification errors[END_REF]. Furthermore, in implementing the INTEROCC JEM, we had to deal with values of probability of exposure ranging from 0% to 100%. It was not self-evident how to use this information in developing an index of exposure for our study. We decided to create an ever/never exposed variable by dichotomising the continuous probability entries at 25% probability of exposure as assessed by the INTEROCC JEM. Such a decision, favouring sensitivity over specificity, is debatable and can lead to further exposure misclassification [START_REF] Burstyn | Aggregation of Exposure Level and Probability into a Single Metric in Job-Exposure Matrices Creates Bias[END_REF]. In sensitivity analyses, we used 5% and 50% as alternative cut-points, and the results were almost identical. Higher values of the cut-point were impractical since the number of subjects considered exposed would be too low to sustain valid meaningful analyses.

A final source of uncertainty and error relates to the generalizability of the exposure profiles from Finland to other countries. There must surely be some variability between countries in the exposures of workers. Firstly, the industrial profiles differ between different countries. But this in itself is not a source of error in our study. What would cause error is the fact that even within a given occupation, workers in different countries may experience different exposures. All of the countries in INTEROCC have modern industrial economies where, a priori, we would not expect industrial processes to be radically different within occupations as would be the case if we were comparing countries at very different levels of industrialisation. Two comparisons have been carried out between FINJEM estimates and those derived by local experts, one in Australia [START_REF] Benke | Comparison of occupational exposure using three different methods: hygiene panel, job exposure matrix (JEM), and self reports[END_REF] based on a set of exposure agents that were not part of the present analysis, and one in Canada [START_REF] Lavoue | Comparison of exposure estimates in the Finnish jobexposure matrix FINJEM with a JEM derived from expert assessments performed in Montreal[END_REF], based on a set of exposures that included those analysed here. Both comparisons demonstrated varying degrees of concordance between exposure estimates derived from FINJEM compared with those provided by local experts. For some agents there was high concordance, while for other agents, the concordance was rather low. An indirect indication that our results were not compromised by inter-country patterns of exposure in relation to occupations is the fact that when we carried out analyses within countries, the results were the same as when we combined countries (data not shown).

Statistical power to detect risks was limited by the low prevalence of exposure for some agents, by the measurement error inherent in our exposure assessment, and by the fact that many of the exposed subjects may have experienced relatively low concentrations of exposure. Thus, the essentially null results presented here should not be interpreted as strong evidence for an absence of any risk. Still, this is the strongest evidence to date on glioma risks from these agents.

Conclusion

This is the largest study to date to investigate associations between selected occupational exposures and glioma. Our findings do not support an association between any of the chemical agents examined and risk of glioma.

Table 1

 1 Main characteristics of glioma cases and controls in the seven-country INTEROCC study

		Cases	Controls
		(1,800)	(5,160)
		n%n%
	Country
	Australia	277	15.4	665	12.9
	Canada	169	9.4	649	12.6
	France	93	5.2	471	9.1
	Germany	366	20.3 1494 29.0
	Israel	282	15.7	698	13.5
	New Zealand	75	4.2	160	3.1
	United Kingdom	538	29.9 1023 19.8
	Sex
	Males	1116 62.0 2308 44.7
	Females	684	38.0 2852 55.3
	Age (years)
	30-39	328	18.2	791	15.3
	40-49	460	25.6 1388 26.9
	50-59	678	37.7 2022 39.2
	60-69	334	18.6	959	18.6
	Interview type
	Self respondent	1543 85.7 5137 99.6
	Proxy respondent	257	14.3	23	0.4
	Atopy a
	Yes	416	23.1 1380 26.7
	No	1384 76.9 3780 73.3
	Smoking status
	Current	508	28.2 1427 27.7
	Ex	386	21.4 1212 23.5
	Never	906	50.3 2521 48.9
	Marital Status
	Married	1422 79.0 4035 78.2
	Single, divorced or widowed	363	20.2 1115 21.6
	Unknown	15	0.8	10	0.2
	Maximum level of education attained
	Primary, secondary	766	42.6 2253 43.7
	Intermediate college	382	21.2 1080 20.9
	Tertiary	643	35.7 1822 35.3
	Unknown	9	0.5	5	0.1

a Atopy: any medical history of allergy, asthma and/or eczema.

Table 2

 2 Association between 10 occupational agents and glioma in the seven-country INTEROCC studyfive year lag period

				Formaldehyde					Oil mist
		Cases (1,800)	Controls (5,160)	OR d	95% CI	Cases (1,800)	Controls (5,160)	OR	95% CI
		n% c	n%			n	%	n%
	Non exposed	1586	88.1	4639	89.9	1.0	-	1657	92.1	4810	93.2	1.0	-
	Ever exposed a	96	5.3	283	5.5	0.8	0.6-1.1	63	3.5	151	2.9	0.8	0.6-1.1
	Cumulative exposure b
	Lowest tertile	32	1.8	99	1.9	0.9	0.6-1.3	12	0.7	50	1.0	0.5	0.3-1.0
	Middle tertile	28	1.6	88	1.7	0.8	0.5-1.4	24	1.3	49	0.9	1.0	0.6-1.7
	Highest tertile	34	1.9	94	1.8	0.7	0.5-1.1	23	1.3	50	1.0	0.9	0.5-1.6
	Duration (years)
	1 -4	28	1.6	106	2.1	0.7	0.5-1.1	14	0.8	52	1.0	0.5	0.3-0.9
	5 -9	22	1.2	85	1.6	0.6	0.4-1.1	22	1.2	47	0.9	1.1	0.6-1.8
	≥ 10	44	2.4	90	1.7	1.1	0.7-1.7	23	1.3	50	1.0	0.9	0.5-1.6
				Diesel exhaust emissions			Gasoline exhaust emissions
	Non exposed	1458	81.0	4446	86.2	1.0	-	1558	86.6	4654	90.2	1.0	-
	Ever exposed a	201	11.2	427	8.3	1.0	0.8-1.2	191	10.6	415	8.0	1.0	0.8-1.2
	Cumulative exposure b
	Lowest tertile	47	2.6	140	2.7	0.7	0.5-1.0	48	2.7	135	2.6	0.8	0.5-1.1
	Middle tertile	73	4.1	139	2.7	1.2	0.9-1.7	68	3.8	135	2.6	1.1	0.8-1.5
	Highest tertile	77	4.3	143	2.8	1.1	0.8-1.5	71	3.9	139	2.7	1.0	0.7-1.4
	Duration (years)
	1 -4	57	3.2	153	3.0	0.8	0.6-1.2	51	2.8	135	2.6	0.8	0.6-1.2
	5 -9	55	3.1	126	2.4	0.9	0.6-1.2	56	3.1	137	2.7	0.8	0.6-1.2
	≥ 10	85	4.7	143	2.8	1.3	1.0-1.8	80	4.4	137	2.7	1.3	0.9-1.8
				Benzo(a)pyrene					Sulphur dioxide
	Non exposed	221	12.3	667	12.9	1.0	-	1775	98.6	5124	99.3	1.0	-
	Ever exposed a	367	20.4	1056	20.5	0.8	0.6-1.0	20	1.1	19	0.4	2.0	1.0-3.8
	Cumulative exposure b
	Lowest tertile	94	5.2	344	6.7	0.7	0.5-0.9	11	0.6	7	0.1	3.1	1.2-8.5
	Middle tertile	118	6.6	344	6.7	0.8	0.6-1.1	3	0.2	6	0.1	1.2	0.3-5.0
	Highest tertile	149	8.3	353	6.8	0.9	0.7-1.2	6	0.3	6	0.1	1.4	0.4-4.7
	Duration (years)
	1 -4	99	5.5	383	7.4	0.6	0.5-0.9	15	0.8	7	0.1	4.4	1.7-11.2
	5 -9	126	7.0	305	5.9	1.0	0.8-1.4	2	0.1	6	0.1	0.3	0.0-2.2
	≥ 10	136	7.6	353	6.8	0.8	0.6-1.1	3	0.2	6	0.1	1.1	0.3-4.6
				Asbestos					Crystalline silica
	Non exposed	1352	75.1	4204	81.5	1.0	-	1579	87.7	4712	91.3	1.0	-
	Ever exposed a	292	16.2	621	12.0	0.9	0.8-1.1	200	11.1	413	8.0	1.0	0.8-1.2
	Cumulative exposure b
	Lowest tertile	89	4.9	210	4.1	0.8	0.6-1.1	54	3.0	135	2.6	0.8	0.3-1.2
	Middle tertile	100	5.6	200	3.9	1.1	0.8-1.4	68	3.8	135	2.6	1.1	0.8-1.5
	Highest tertile	103	5.7	211	4.1	1.0	0.7-1.3	74	4.1	138	2.7	1.1	0.8-1.5
	Duration (years)
	1 -4	84	4.7	219	4.2	0.9	0.6-1.1	52	2.9	135	2.6	0.9	0.6-1.2
	5 -9	104	5.8	191	3.7	1.0	0.7-1.3	61	3.4	136	2.6	0.9	0.7-1.3
	≥ 10	104	5.8	211	4.1	1.0	0.8-1.3	83	4.6	137	2.7	1.2	0.9-1.7

Table 2

 2 Association between 10 occupational agents and glioma in the seven-country INTEROCC studyfive year lag period (Continued) Probability of exposure in the modified version of FINJEM ≥ 25%; duration ≥ 1 year; lag period of 5 years. b Calculated among exposed subject as ∑Probability × Level × Duration of each job held by a subject; cut-points at tertiles among exposed controls; expressed in mg/m 3 (oil mist, diesel and gasoline exhaust emissions, crystalline silica, wood dust and animal dust); ppm (formaldehyde, sulphur dioxide); μg/ m 3 (benzo(a) pyrene) or in fibers/m 3 (asbestos). c Percentages may not add up to 100 due to the exclusion of subjects classified as having 'uncertain' exposure. d Odds ratios stratified by sex, age (5-year categories) and center and adjusted for age as a continuous variable, the maximum education level attained in the household, the standard international occupational prestige scale, atopy and the respondent status.

				Wood dust					Animal dust
	Non exposed	1706	94.8	4933	95.6	1.0	-	1733	96.3	5006	97.0	1.0	-
	Ever exposed a	88	4.9	204	4.0	1.1	0.8-1.5	66	3.7	147	2.8	0.6	0.6-1.2
	Cumulative exposure b
	Lowest tertile	31	1.7	66	1.3	1.1	0.7-1.8	18	1.0	49	0.9	0.7	0.4-1.4
	Middle tertile	31	1.7	66	1.3	1.1	0.7-1.8	24	1.3	49	0.9	0.9	0.5-1.5
	Highest tertile	26	1.4	68	1.3	1.1	0.6-1.7	23	1.3	49	0.9	0.9	0.5-1.5
	Duration (years)
	1 -4	32	1.8	69	1.3	1.1	0.7-1.7	25	1.4	55	1.1	0.8	0.4-1.5
	5 -9	27	1.5	63	1.2	1.1	0.7-1.8	17	0.9	43	0.8	0.9	0.5-1.6
	≥ 10	29	1.6	68	1.3	1.1	0.7-1.8	23	1.3	49	0.9	0.8	0.4-1.4

a

Acknowledgements

We thank Timo Kauppinen of the Finnish Institute of Occupational Health for allowing us to use and modify the FINJEM as well as sharing his valuable expertise on the exposure assessment committee in the company of Nils Plato of the Karolinska Institute in Stockholm. Jordi Figuerola and Rodrigo Villegas of CREAL have been the backbone of the team in cleaning and preparing the data for analysis. Michelle Turner provided useful comments on the manuscript. We thank Louise Nadon (Canada), Hélène Tardy (France), Florence Samkange-Zeeb (Germany), Avital Jarus-Hakak (Israel), and Anne Sleeuwenhoek (UK) who coded the occupations and assisted in the data clean-up. We are grateful to the following colleagues and researchers who allowed us to use the occupational data from their respective INTERPHONE study centres: Bruce Armstrong (Australia), Mary McBride (Canada -Vancouver), Maria Blettner and Joachim Schutz (Germany), Alistair Woodward (New Zealand) and Patricia McKinney (UK). Funding This work was funded by the National Institutes for Health (NIH) Grant No. 1R01CA124759-01. Coding of the French occupational data was in part funded by AFSSET (Convention N°ST-2005-004). The INTERPHONE study was supported by funding from the European Fifth Framework Program, 'Quality of Life and Management of Living Resources' (contract 100 QLK4-CT -1999901563) and the International Union against Cancer (UICC). The UICC received funds for this purpose from the Mobile Manufacturers' Forum and GSM Association. In Australia, funding was received from the Australian National Health and Medical Research 5 Council (EME Grant 219129) with funds originally derived from mobile phone service licence fees; a University of Sydney Medical Foundation Program; the Cancer Council NSW and The Cancer Council Victoria. In Canada funding was received from the Canadian Institutes of Health Research (project MOP-42525); the Canada Research Chair programme; the Guzzo-Cancer Research Society Chair in Environment and Cancer; the Fonds de la recherche en sante du Quebec; the Canadian Institutes of Health Research (CIHR), the latter including partial support from the Canadian Wireless Telecommunications Association; the NSERC/SSHRC/ McLaughlin Chair in Population Health Risk Assessment at the University of Ottawa. In France, funding was received by l'Association pour la Recherche sur le Cancer (ARC) (Contrat N85142) and three network operators (Orange, SFR, Bouygues Telecom). In Germany, funding was received from the German Mobile Phone Research Program (Deutsches Mobilfunkforschungsprogramm) of the German Federal Ministry for the Environment, Nuclear 45 Safety, and Nature Protection; the Ministry for the Environment and Traffic of the state of Baden-Wurttemberg; the Ministry for the Environment of the state of North Rhine-Westphalia; the MAIFOR Program (Mainzer Forschungsforderungsprogramm) of the University of Mainz. In New Zealand, funding was provided by the Health Research Council, Hawkes Bay Medical Research Foundation, the Wellington Medical Research Foundation, the Waikato Medical Research Foundation and the Cancer Society of New Zealand. Additional funding for the UK study was received from the Mobile Telecommunications, Health and Research (MTHR) program, funding from the Health and Safety Executive, the Department of Health, the UK Network Operators (O2, Orange, T-Mobile, Vodafone, '3') and the Scottish Executive.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions AL: analysis and manuscript writing. JP: involved in exposure assessment, analysis strategy for the INTEROCC study. LR: coordinator involved in study design, exposure assessment, data management, statistical analysis, and manuscript development. EC: responsible for the coordination of the INTEROCC study, supervision of database management, validations, validation of exposure coding, estimation of exposure for study subjets, participation in analyses and in interpretation of results of INTEROCC, JS: overall supervision of all aspects of this manuscript, including analytical approach, choice of exposure parameters and interpretation of the findings. Other co-authors: responsible for original data collection and/or for the decisions taken about exposure assessment and analysis strategy, as well as review of the manuscript. All authors read and approved the final manuscript.

Author details 1 University of Montreal Hospital Research Centre (CRCHUM), Montreal, Canada. 2 Center for Research on Environmental Epidemiology (CREAL), Barcelona, Spain. 3 Oregon State University, Corvallis, USA. 4 Monash University, Melbourne, Australia. 5 University of Leeds, Leeds, UK. 6 INRETS, Lyon, France. 7 University of Ottawa, Ottawa, Canada. 8 Massey University, Wellington, New Zealand. 9 INRS-Institut Armand-Frappier|, Montreal, Canada. 10 Gertner Institute, Chaim Sheba Medical Center & Tel Aviv University, Tel Aviv, Israel. 11 DFKZ, Heidelberg, Germany. 12 Centre for Human Exposure Science, Institute of Occupational Medicine, Edinburgh, UK.


Xavier Desquesnes
email: xavier.desquesnes@unicaen.fr

Abderrahim Elmoataz
email: abderrahim.elmoataz-billah@unicaen.fr

Olivier Lézoray
email: olivier.lezoray@unicaen.fr

Propagation de fronts généralisée sur graphes pondérés

Dans cet article, nous considérons la méthode des ensembles de niveaux pour la propagation de fronts telle qu'elle a été introduite par les travaux d'Osher-Sethian et proposons de l'étendre au domaine des graphes. L'adaptation que nous proposons repose sur le formalisme des Équations aux différences Partielles (EdPs) via une famille d'opérateurs gradients pondérés sur graphes. Puis, en profitant des avantages liés à la nature discrète des graphes, nous proposons d'étendre la formulation initiale (qui ne considère qu'un unique front évoluant selon une vitesse positive) pour obtenir une formulation très générale et qui permet de considérer la propagation simultanée de plusieurs fronts évoluant sans restrictions sur le signe de leurs vitesses. Le potentiel de notre approche pour le traitement d'images et de données est également présenté.

Introduction

De nombreuses applications impliquent des données définies sur des domaines de topologie complexe. Ces données peuvent être définies sur des variétés ou des domaines de forme irrégulière, définies sur des structures de réseaux, ou encore définies comme des nuages de points de grandes dimensions par une collection de vecteurs caractéristiques. Ces données, qu'elles soient organisées ou non, peuvent être représentées sous la forme de graphes dont les sommets représentent les données et dont les arêtes représentent les interactions entre ces données. Par conséquent, il est très important d'adapter des outils initialement développés pour traiter des données définies sur des espaces euclidiens et dont l'efficacité a été prouvée, aux graphes (voir la récente revue parue dans [START_REF] Shuman | The Emerging Field of Signal Processing on Graphs : Extending High-Dimensional Data Analysis to Networks and Other Irregular Domains[END_REF]).

Dans cet article, nous considérons la méthode des ensembles de niveaux pour la propagation de fronts telle qu'elle a été introduite par les travaux d'Osher-Sethian [START_REF] Sethian | Level Set Methods and Fast Marching Methods : Evolving Interfaces in Computational Geometry[END_REF] et proposons de l'étendre au domaine des graphes. L'adaptation que nous proposons repose sur le formalisme des Équations aux différences Partielles (EdPs) via une famille d'opérateurs gradients pondérés sur graphe. Puis, en profitant des avantages liés à la nature discrète des graphes, nous proposons d'étendre la formulation initiale (qui ne considère qu'un unique front évoluant selon une vitesse positive) pour obtenir une formulation très générale et qui permet de considérer la propagation simultanée de plusieurs fronts évoluant sans restrictions sur le signe de leurs vitesses.

Au delà de l'intérêt théorique d'une formulation discrète de l'équation du mouvement de fronts, cette adaptation permet d'étendre significativement le domaine d'application de cette approche. Notamment, nous présentons dans cet article un certain nombre d'applications pour le calcul de géodésiques sur maillages, la segmentation d'images naturelles (combinant graphes non-locaux et patchs), mais également la classification semi-supervisée de données via une implémentation simple des contours actifs sur graphes.

Graphes et opérateurs pondérés

Nous considérons que tout ensemble discret peut-être représenté par un graphe pondéré G = (V, E, w), composé de deux ensembles finis : V = {u 1 , ..., u n } un ensemble de n sommets et E ⊂ V × V un ensemble d'arêtes pondérées. Une arête (u, v) ∈ E connecte deux sommets adjacents u et v. Le poids w uv d'une arête est défini par une fonction w : V × V → IR + si (u, v) ∈ E, et w uv = 0 sinon. On note N (u), l'ensemble des noeuds adjacents à u. Dans cet article, les graphes sont supposés simples, complets et non-orientés.

Gradients morphologiques sur graphes Dans ce paragraphe, nous rappelons quelques opérateurs sur les graphes précédemment définis dans [START_REF] Ta | Nonlocal pdes-based morphology on weighted graphs for image and data processing[END_REF] et références connexes. Soient G = (V, E, w) un graphe et f : V → IR une fonction définie sur ses sommets. Les gradients morphologiques externes et internes sur graphe sont définis par :

(∇ ± w f )(u) = ( √ w uv |(Df (u)) ± | p ) v∈V , où Df (u) = f (v) -f (u) , (x) + = max(x, 0) et (x) -=
min(x, 0). Les normes associées à ces gradients sont alors données pour les normes L p , p ∈ {1, 2}, respectivement L ∞ , par

(∇ ± w f )(u) p = v∼u w p/2 uv |(Df (u)) ± | p 1/p (∇ ± w f )(u) ∞ = max v∼u ( √ w uv |(Df (u)) ± |), (1)
avec l'égalité suivante :

(∇ w f )(u) p p = (∇ + w f )(u) p p + (∇ - w f )(u) p p . ( 2
)
Construction des graphes Il existe de nombreuses méthodes pour transformer un ensemble discret de données en un graphe pondéré, où le voisinage reflète les interactions entre les différents éléments par le biais d'une mesure de distance µ : V × V → IR + . Dans le cas général, on peut citer : -Les ε-graphes où deux éléments u et v sont connectés si µ(u, v) ε, avec ε > 0. -Les graphes des k plus proches voisins (k-ppv) où chaque noeud u est connecté à ses k plus proches voisins. Dans le cas de données structurées, on peut citer les graphes de régions adjacentes (RAG). Un RAG peut être construit à partir de n'importe quelle structure de données représentée par un graphe, où une région R i est définie comme un ensemble connexe de noeuds avec

R i = V et R i = ∅. Deux ré- gions R i et R j sont connectées si ∃v i ∈ R i et v j ∈ R j |v i ∼ v j .
Un RAG peut facilement être augmenté d'un kppv afin de permettre l'ajout d'arêtes supplémentaires entre des régions non-adjacentes. Finalement, quelle que soit la construction du graphe, la pondération des arêtes est effectuée à partir d'une fonction de similarité telle que

g 0 (u, v) = 1, ou g 1 (u, v) = (µ(F u , F v ) + ε) -1 avec ε > 0, ε → 0, ou g 2 (u, v) = exp(-µ(F u , F v )/σ 2 ) avec σ > 0, où F u , u ∈ V est un vecteur caractéristique associé au noeud u.

Adaptation de l'approche par ensembles de niveaux

Soit Γ un front évoluant sur un domaine euclidien Ω ⊂ IR n sous l'influence d'une fonction vitesse F : Ω × IR → IR et représenté à tout instant t ∈ [0, ∞[ par une courbe paramétrique γ(t). L'idée de l'approche par ensembles de niveaux consiste à représenter le front comme le niveau zéro d'une fonction d'ensembles de niveaux φ(x, t) tel qu'à chaque instant t la position du front soit donnée par l'ensemble des points x ∈ IR n vérifiant φ(x, t) = 0.

Suivant cette approche, l'équation du mouvement de Γ s'exprime sous la forme du Hamiltonien suivant

H(x, ∇φ) = ∂φ(x, t) ∂t -F(x, t) ∇φ(x, t) = 0, (3)
et recouvre également, dans le cas où la vitesse F : Ω → IR est toujours positive, la célèbre équation eikonale

F(x) ∇T (x) -1 = 0, x) ∈ Ω\S T (x) = ψ(x), x ∈ S, (4)
où T : Ω → IR + est la fonction d'instant d'arrivée du front Γ en chaque point d' Ω.

Notre adaptation discrète de cette équation au domaine des graphes pondérés repose sur l'interprétation de l'équation continue non-stationnaire (3) comme un processus morphologique d'érosion (vitesse négative) ou de dilatation (vitesse positive). À partir de cette interprétation, et en reprenant les travaux de Ta [START_REF] Ta | Nonlocal pdes-based morphology on weighted graphs for image and data processing[END_REF] sur l'adaptation de la morphologie mathématique sur graphe, nous obtenons l'équation suivante

∂φ ∂t (u, t) = F(u) (∇ + w φ)(u, t) p si F(u) > 0 F(u) (∇ - w φ)(u, t) p si F(u) < 0, (5)
qui correspond à un processus morphologique d'érosion ou de dilatation discrète sur graphe en fonction du signe de la vitesse. Dans ce cas, la fonction d'ensembles de niveaux discrète φ : V × t → IR est définie par analogie avec son pendant continu, telle que

φ(u, t) > 0 si u ∈ Ω t < 0 si u ∈ Ω c t , (6)
où Ω t désigne l'ensemble des sommets situés à l'intérieur du front à l'instant t et Ω c t l'ensemble des sommets situés à l'extérieur du front à l'instant t. La position du front étant donnée à tout instant t comme l'interface entre ces deux sous ensembles.

Nous avons montré [START_REF] Desquesnes | Eikonal equation adaptation on weighted graphs : fast geometric diffusion process for local and non-local image and data processing in[END_REF] que pour un graphe particulier représentant une grille régulière, cette équation coïncide exactement avec le schéma de discrétisation d'Osher de l'équation du mouvement continue [START_REF] Ta | Nonlocal pdes-based morphology on weighted graphs for image and data processing[END_REF].

Lien avec l'équation eikonale et l'algorithme du Fast Marching. Si nous nous restreignons maintenant au cas où le signe de la vitesse F est toujours positif, notre équation (5) devient

∂φ(u, t) ∂t = F(u) ∇ + w φ (u, t) p . (7)
Soit T : V → IR la fonction qui associe à chaque sommet du graphe l'instant d'arrivée du front Γ en ce sommet. Par analogie avec la formulation continue, où la relation entre la fonction d'ensembles de niveaux φ et la fonction d'instant d'arrivée tient au changement de variable φ(x, t) = t -T (x), nous proposons une version stationnaire de l'équation (7) telle que

∂φ(u, t) ∂t = F(u) ∇ + w (t -T ) (u) p = F(u) ∇ - w T (u) p , (8)
qui correspond à une adaptation discrète de l'équation eikonale sur graphe.

Finalement, l'équation du mouvement d'un front évoluant toujours dans la direction de sa normale positive est donnée par le schéma numérique suivant

S u (T , V, F) = 0 ∀u ∈ Ω 0 T (u) = 0 ∀u ∈ Ω 0 , (9)
où S u (T , V, F) = ∇ - w T (u) p -P(u) et P = 1/F est une fonction de potentiel.

Nous avons apporté une preuve de l'existence et de l'unicité de la solution à cette équation, et également proposé un algorithme rapide de résolution de cette équation et qui recouvre en le généralisant aux graphes l'algorithme du Fast Marching [START_REF] Desquesnes | Eikonal equation adaptation on weighted graphs : fast geometric diffusion process for local and non-local image and data processing in[END_REF].

Généralisation de l'approche

Forts de cette adaptation, nous en proposons ensuite une généralisation pour considérer maintenant l'évolution simultanée de plusieurs fronts concurrents évoluant sur un même graphe, puis le cas où ces fronts sont soumis à des vitesses qui peuvent être à la fois positives et négatives [START_REF] Desquesnes | Generalized fronts propagation on weighted graphs in proceedings of Algoritmy[END_REF].

Généralisation à la propagation simultanée de plusieurs fronts. Pour cette première généralisation, nous considérons des fronts évoluant simultanément sur un même graphe qui ne peuvent se chevaucher et qui n'ont aucune influence mutuelle excepté qu'ils sont mutuellement bloquants.

Soient Γ 1 , ..., Γ N , N fronts évoluant sur un graphe G. Ces fronts sont respectivement définis à l'instant initial par les sous ensembles Ω 1 0 , ..., Ω N 0 . Nous proposons de représenter ces fronts, comme un front unique (ou global) Γ, représenté par l'ensemble Ω 0 = Ω i 0 , dont l'évolution est réalisée par une version modifiée de l'équation du mouvement d'un front sur graphe (9). Dans ce cas, le schéma de résolution en un sommet u est modifié pour ne plus tenir compte de l'ensemble des sommets du graphe, mais uniquement des sommets atteints par le même front que u, que nous notons V u . Ainsi, l'équation du mouvement du front global devient

S u (T , V u , F u ) = 0 ∀u ∈ Ω 0 T (u) = 0 ∀u ∈ Ω 0 , ( 10
)
où F u désigne la vitesse associée au front atteignant u. Cette équation peut être résolue par le même algorithme que dans le cas d'un front unique.

Généralisation sans restrictions sur le signe de la vitesse des fronts. Dans un second temps, et en nous basant sur la généralisation précédente, nous étendons une nouvelle fois notre approche pour considérer le cas où les fronts sous soumis à des vitesses positives et négatives. Dans ce cas, nous proposons de découper chaque front en deux : D'un côté la partie du front soumise à une vitesse positive, et de l'autre la partie du front soumise à une vitesse négative. Nous montrons que l'évolution de cette dernière partie de front selon une vitesse négative F coïncide avec l'évolution de son complémentaire (au sens des ensembles de sommets) selon une vitesse positive -F. Ainsi, nous proposons de réaliser l'évolution de nos N fronts, comme l'évolution simultanée et selon des vitesses positives de 2 × N fronts, composés des parties positives des fronts originaux et des complémentaires des parties négatives des fronts originaux.

L'évolution de ces fronts peut donc être simplement réalisée grâce à l'algorithme précédent pour la propagation simultanée de plusieurs fronts sur un même graphe.

Résultats

Dans cette section, nous présentons quelques résultats de notre approche, ceci afin d'illustrer son potentiel et surtout son adaptabilité à une large gamme de données d'origines et de topologies variées.

Calcul de géodésiques La fonction d'instant d'arrivée T se comportant comme une fonction de distance aux labels initiaux, notre formulation permet tout naturellement le calcul de géodésiques sur graphes. La Figure 1 illustre cette application sur le graphe d'un maillage 3D (représentant une amphore). Dans le cas d'une fonction de poids constante (au milieu), le front se propage de manière homogène et la carte de distance ne dépend que de la topologie du graphe. Dans le cas où la fonction de poids dépend de la texture de l'amphore (à droite), la propagation est perturbée voire presque stoppée au niveau des discontinuités de cette texture. On remarquera dans ce dernier cas qu'un simple seuillage permet d'extraire l'objet d'intérêt de la texture. Classification non-supervisée de données Enfin, la Figure 3 illustre une dernière application de notre approche pour la classification non-supervisée. Cette application repose avant tout sur la dernière extension de notre approche qui permet de considérer l'évolution simultanée de plusieurs fronts soumis à des vitesses signées et évoluant au cours du temps. Ceci permettant aux fronts de se dilater ou de se rétracter sur le graphe, en fonction du signe de leur vitesse à chaque instant t. Dans cet exemple, l'ensemble des données est étiqueté de manière aléatoire selon deux classes. Afin de corriger cette répartition aléatoire, un front est associé à chaque classe et propagé sur le graphe sous l'influence d'une vitesse dépendant en chaque sommet de sa similarité à la moyenne des sommets du front, tel que : Si un sommet est plus similaire à la moyenne d'un front que du reste du graphe, alors la vitesse de ce front en ce sommet est positive, sinon négative. Ainsi, la vitesse F i d'un front Γ i est exprimée par

F i (u, t) = D µ(f, v ∈ Ω i ), f (u) -D µ(f, v ∈ Ω i ), f (u)
(11) où µ désigne la moyenne et D la distance euclidienne et f : V → IR n associe à chaque sommet un vecteur caractéristique. Ce schéma correspond en fait à une généralisation du modèle de Chan et Vese pour les contours actifs [START_REF]Active Contours without Edges in[END_REF].

FIGURE 1 -

 1 FIGURE 1 -Calcul de géodésique sur un maillage 3D texturé. De gauche à droite : le maillage original avec l'origine de la géodésique, la géodésique avec une fonction de poids constante et enfin la géodésique avec une fonction de poids dépendant de la texture.

FIGURE 2 -

 2 FIGURE 2 -Segmentation d'images. La segmentation est réalisée par la propagation simultanée de plusieurs fronts(à partir de germes) sur deux graphes construits à partir de l'image. Le premier graphe au centre est un classique graphe de 4-connexité, et le second à droite est un graphe non-local ou chaque pixel est connecté grâce à un voisinage étendu et pondéré avec des patchs, ce qui permet de mieux caractériser la texture.

FIGURE 3 -

 3 FIGURE 3 -Contours actifs sur graphe pour la classification de données (des imagettes de 0 et de 1). La classification est réalisée grâce à la propagation simultanée de deux fronts initialisés aléatoirement (un par classe) sur le graphe construit à partir des données (une base d'imagettes). Le graphe est un graphe des 7 plus proches voisins et la fonction de poids dépend de la similarité entre les pixels des imagettes. Segmentation d'image naturelle Lorsque l'on traite des images, un des premiers avantages de l'utilisation de graphes pondérés est qu'ils permettent de faire abstraction de leur représentation naturelle (une grille de pixels en 4 ou 8 adjacence). Il est possible par exemple d'utiliser un graphe non local où chaque pixel bénéficie du voisinage étendu des pixels appartenant à une fenêtre centrée sur lui même et est représenté par un patch également centré sur lui même. Combinée à notre adaptation du Fast Marching sur graphe, cette approche permet de tirer avantage de la représentation non locale (qui caractérise bien mieux la texture), tout en bénéficiant de l'efficacité reconnue du Fast Marching. La Figure 2 illustre l'intérêt de l'utilisation d'un graphe non local comparée à l'utilisation d'un graphe local pour la segmentation d'une image naturelle. Enfin, grâce à la propagation simultanée de plusieurs fronts, les différentes zones d'intérêt de l'images sont extraites en une seule passe.

Dans cette section, nous présentons notre adaptation des équations de l'approche par ensembles de niveaux pour la propagation d'un front se propageant dans la direction de sa normale positive. Cette approche, introduite par Osher-Sethian, peut aisément être liée avec l'équation eikonale bien connue et largement utilisée dans le domaine du traitement d'image.Adaptation de l'approche par ensembles de niveaux. Nous rappelons l'approche par ensembles de niveaux telle qu'elle a été introduite par Sethian[START_REF] Sethian | Level Set Methods and Fast Marching Methods : Evolving Interfaces in Computational Geometry[END_REF] pour décrire le déplacement d'un front et son implémentation par l'algorithme du Fast Marching.


Gang Liu

Zifeng Wang

Gui-Song Xia

Wen Yang

SAR IMAGE SEGMENTATION VIA NON-LOCAL ACTIVE CONTOURS

. The idea is to partition a SAR image via computing the patch similarity in the SAR image non-locally, and formulize the segmentation problem with an active contour model. More precisely, after computing the statistical features of SAR images, non-local comparisons between feature patches are used to calculate the active contour energy, which is defined by integrating the interactions between pairs of patches inside and outside the segmented region. A level set method is finally used to minimize the non-local energy. Compared with existing approaches for SAR image segmentation, the only requirement of this method is a local similarity between patches, and it is less sensitive to initial segmentation. The experimental results show the effectiveness and feasibility of the proposed method.

INTRODUCTION

Synthetic aperture radar (SAR) systems have been widely used in remote-sensing application for many years due to its advantage of strong penetrability and all-weather acquisition capability. As one of the fundamental problems to understand SAR data, the segmentation of SAR images is challenging because of the speckle effects, which can be described by a multiplicative noise model. This problem has been widely investigated in the literature [START_REF] Zhang | Spectral clustering ensemble applied to sar image segmentation[END_REF][START_REF] Xia | A rapid and automatic mrfbased clustering method for sar images[END_REF], among which active contour models show many advantages.

The early works on this topic include the method proposed by Germain et al. [START_REF] Germain | Edge location in sar images: Performance of the likelihood ratio filter and accuracy improvement with an active contour approach[END_REF], where in order to segment a SAR image, a likelihood ratio edge detector has been employed to localize the edges of the image and refine them subsequently with the "snake model" [START_REF] Kass | Snakes: Active contour models[END_REF] for achieving the final result. Instead of evolving discretized curves, level set functions has been proposed to solve the active contour models [START_REF] Osher | Fronts propagating with curvaturedependent speed: algorithms based on hamilton-jacobi formulations[END_REF]. The extension of this method to SAR image segmentation includes [START_REF] Ben Ayed | Multiregion level-set partitioning of synthetic aperture radar images[END_REF]. Recently, in order to get a better segmentation of SAR images, the energy function was modified to achieve a stationary global minimum of the optimization problem [START_REF] Shuai | Sar image segmentation based on level set with stationary global minimum[END_REF]. In [START_REF] Hu | Sar image segmentation based on kullback-leibler distance of edgeworth[END_REF], the Kullback-Leibler divergence of Edgeworth was used instead to segment SAR images into homogeneous regions accurately. In [START_REF] Sui | A novel multi-scale level set method for sar image segmentation based on a statistical model[END_REF], a multi-scale level set approach was proposed for segmenting SAR images. However, these methods impose a global homogeneity to a SAR image, meaning that the SAR image inside and outside the regions to be segmented are homogeneous respectively, which are usually not true for SAR images and the results are heavily affected by speckles.

To overcome this problem, this paper proposes to use the non-local active contour model [START_REF] Jung | Non-local active contours[END_REF] for SAR image segmentation, which only requires a local homogeneity to the image, in contrast to other methods. In particular, a non-local energy is defined by using a pairwise interaction of features inside and outside the regions to be segmented. In our context, the pairwise interaction is calculated by evaluating the patch similarity in a non-local way [START_REF] Buades | Image denoising methods. a new nonlocal principle[END_REF]. More precisely, each patch is described by a logarithmic normal distributions and the patch similarity is measured by the Kullback-Leibler divergence between the distributions. As we shall see, this method is robust to speckles in SAR images.

The remainder of this paper is organized as follows. In section II, we present the non-local active contour model for SAR image segmentation. In section III, we show the experimental results and conclude the paper in section IV.

A NON-LOCAL ACTIVE CONTOUR MODEL FOR SAR IMAGE SEGMENTATION

In this section, we formalize the SAR image segmentation problem via non-local active contours.

Non-local active contour model

Let f : Ω → R be an image, with Ω as the image grid and ∂Ω as its boundary. The patch

p s centered on s ∈ Ω is denoted by p s (k) = f (s + k), k ∈ Ω τ = [-τ /2, τ /2]
min φ E(φ) + γ L(φ) (1)
where φ is the level set function. E(φ) is the the energy that measures the dissimilarity inside and outside the contour C. L(φ) is a regularization term and γ is a weight parameter to balance the two terms.

The energy E(φ) is computed in a non-local way as

E(φ) = S×S G α (s -t) • d(p s , p t ) ds dt + S c ×S c G α (s -t) • d(p s , p t ) ds dt, (2)
where p s and p t are the patches centered on pixel s and t respectively. S and S c denotes the regions inside and outside the contour C respectively, with S S c = Ω and S S c = ∅. G α (•) is a Gaussian kernel with the scale of α. The first term of Eq.( 2) measures the internal energy while the second term measures the external energy.

For simplicity, the integration inside and outside the contour C is implemented with a Heaviside function

H(u) = 1 2 + 1 π arctan( u ε ), (3)
where the parameter ε should be small enough for a sharp boundary.

Combining Eq. ( 2) and Eq. (3), we have

E(φ) = ρ H φ(s) , H φ(t) • G α (s -t) • d(p s , p t ) ds dt (4)
where ρ is the indicator function defined as ρ(u, v) = 1 -|u -v|, implying that only pairs of pixels for which φ has the same sign are considered. The segmentation region is penalized by the length of the contour C as

L(φ) = ∇H(φ(s)) ds. (5)
Combining Eq. (1), Eq. ( 4) and Eq. ( 5), one can obtain the optimization function, the minimization of which can be achieved by using an iterative scheme proposed in [START_REF] Jung | Non-local active contours[END_REF].

Pairwise patch interaction

The left problem is how to compute the pairwise patch interactions in the context of segmenting SAR images. The pairwise patch similarities has been recently investigated by Deledalle et al. [START_REF] Deledalle | Iterative weighted maximum likelihood denoising with probabilistic patch-based weights[END_REF], which shows that the statistical distribution characteristics perform the best among others on SAR images.

There are many probabilistic distribution functions available to describe SAR images, such as the logarithmic normal distribution, Rayleigh distribution, Gamma distribution, K distribution, G 0 A distribution, etc [START_REF] Soumekh | Synthetic aperture radar signal processing[END_REF]. Though the logarithmic normal distribution is a simple statistical model for SAR images, our experiments show that it is efficient enough for many SAR image segmentation in our context.

The pixels in image patches are described by the logarithmic normal distribution [START_REF] Soumekh | Synthetic aperture radar signal processing[END_REF], defined as

g(z) = 1 √ 2πσz exp - (ln z -µ) 2 2σ 2 , ( 6
)
where z is the intensity of the pixels, µ and σ are the average and the variance of the logarithmic image patch respectively. After the parameter µ and σ of each patch are estimated, the distribution of pixels in the patch could be obtained by Eq. ( 6).

Given two patches p s and p t centered on s, t ∈ Ω, the Kullback-Leibler divergence between them is implemented as

d(p s , p t ) = g s (z) • log g s (z) g t (z) dz, (7)
where g s (z) and g t (z) denote logarithmic normal distributions, the features of the patch p s and p t respectively. Considering the dissymmetry of d(p s , p t ) and d(p t , p s ), we rewrite Eq. ( 7) by

d(p s , p t ) = g s (z) • log g s (z) g t (z) dz + g t (z) • log g t (z) g s (z) dz, (8)
From Eq.( 6) and Eq.( 8), we have

d(p s , p t ) = 1 2 σ 2 s σ 2 t + σ 2 t σ 2 s + (µ s -µ t ) 2 2 • 1 σ 2 s + σ 2 t , (9)
where µ s and µ t (σ s and σ t resp.) are the average (standard derivation resp.) of the logarithmic image patches log p s and log p t . This divergence is close to zero if the two patches are similar, and it is large if the two patches are dissimilar.

EXPERIMENTAL RESULTS

This section evaluates the proposed method on three kinds of real SAR images. The first includs three MSTAR SAR images of tanks. The second is an X-band Cosmo-Skymed SAR image with a spatial resolution of 3m. The last one is a C-band ENVISAT SAR image with a spatial resolution of 30m. The proposed method is compared with that in [START_REF] Shuai | Sar image segmentation based on level set with stationary global minimum[END_REF].

MSTAR tank extraction

This experiment tests the proposed method on three MSTAR SAR images for tank extraction. The patch width τ is set to be 5 and the non-local region width q is set to be 51. Thus, γ is set to be

5 n×q 2 ×τ 2 = 1 13005n
, with n as the number of pixels in images. The evolution of the contour is not sensitive to the initialization, and usually converges to the boundary of the object after about 50 iterations. Fig. 1 shows the results of MSTAR SAR images. The left column displays the original images, the middle column shows the segmentation results of the method in [START_REF] Shuai | Sar image segmentation based on level set with stationary global minimum[END_REF] and the right column illustrates the results of the proposed approach. We can see that in the results of [START_REF] Shuai | Sar image segmentation based on level set with stationary global minimum[END_REF], some small regions are segmented to be objects duo to speckles, while the proposed method extracts out the tanks excellently and are more robust to speckle. This is mainly because we use the similarity of patches instead of pixels.

Pond extraction

In this experiment, a 500 × 500 Cosmo-Skymed SAR image which contains a pond is used to validate the proposed method. The patch size τ is set to be 9. The non-local region size q is set to be 101. γ is set to 5 n×q 2 ×τ 2 , with n as the number of pixels in images. The results are shown in Fig. 2. The left column shows the original image of a pond, the middle column displays the segmentation result of [START_REF] Shuai | Sar image segmentation based on level set with stationary global minimum[END_REF] and the right column provides our result. In the segmentation result of [START_REF] Shuai | Sar image segmentation based on level set with stationary global minimum[END_REF], a lot of small regions in the homogenous areas are segmented to be object. However, in our result, the pond are detected out successfully.

Oil spill extraction

The proposed method is also tested on a 500 × 500 ENVISAT ASAR image which contains oil spills. We use the same parameter settings as for the Experiment B. Fig. 2 illustrates the results. The left column gives the original SAR image of oil spill, the middle column displays the segmentation result of [START_REF] Shuai | Sar image segmentation based on level set with stationary global minimum[END_REF], and the right column shows our result. In the segmentation result of [START_REF] Shuai | Sar image segmentation based on level set with stationary global minimum[END_REF], many regions in oil spill are not extracted but our method gives much better result, which proves that the proposed method is promising to for the oil spill detection problem.

CONCLUSION

This paper has presented a method of SAR image segmentation via non-local active contour which is focusing on the feature of patches and the metric of similarity of patches in SAR images. The experimental results show that proper feature and metric of similarity are very important in SAR im- ages segmentation. In the future work, we will extend the proposed method to multi-polarization SAR images.

2 ,

 2 with τ to be the width of the patch. Let C be a curve in the image domain Ω to partition the image f into two parts: part inside C and the part outside C. The basic idea of the non-local active contour model[START_REF] Jung | Non-local active contours[END_REF] for segmentation is to evolve the curve C that captures the objects of interest. With a level-set method, the curve C is represented by the zero cross of the level-set function φ : Ω → R, i.e. C = {s|φ(s) = 0}. The non-local active contour model computes the segmentation of an image f as a stationary point of the following optimization problem

Fig. 1 .

 1 Fig. 1. Results on three MSTAR SAR images. From left to right: the original images, the results of [8] and our results.

Fig. 2 .

 2 Fig. 2. Segmenting SAR images of a pond (top) and oil spill (bottom). From left to right: original image, result of [8] and our result.