Adela Castelló

PhD Isabel Río

PhD María Sandín-Vázquez

PhD Francisco Bolúmar
email: francisco.bolumar@uah.es

Adela Catelló-Pastor

Shortening of gestational length among native-born and immigrants in Spain (1997-2008)

Keywords: Duration of pregnancy, Gestational age, Immigrants

    (1997-2008)

Shortening of gestational length among native-born and immigrants in Spain

Introduction

Length of gestation is a key reproductive indicator because the survival and health status of the newborns depend largely on their maturity at birth. In humans, fetal development is considered normal in duration when birth occurs between the 37 th and the 41 st gestational week. Longer, and above all, shorter gestational times are associated to higher health risk and lower probability of survival. A wide range of factors can harmfully influence the duration of development in uterus. In industrialized countries, the greater access to contraceptive methods and the massive incorporation of women into the labor market have contributed significantly to the reduction of birth rates and to a later age of motherhood. In this regard, it is known that primiparae and women with first motherhood at >30 years have shorter pregnancies than multiparae and younger women, respectively [START_REF] Bergsjø | Duration of human singleton pregnancy. A population-based study[END_REF][START_REF] Delpisheh | Pregnancy late in life: a hospitalbased study of birth outcomes[END_REF][START_REF] St-Laurent | Biopsychosocial determinants of pregnancy length and fetal growth[END_REF][START_REF] Petridou | Modulators of length of gestation. A study in Greece[END_REF] . Emotional stress and anxiety, high workload and exposure to toxics are variables with a known negative impact on the duration of pregnancies [START_REF] Hosseini | Trait anxiety in pregnant women predicts offspring birth outcomes[END_REF][START_REF] Tegethoff | Maternal Psychosocial Adversity During Pregnancy Is Associated With Length of Gestation and Offspring Size at Birth: Evidence From a Population-Based Cohort Study[END_REF][START_REF] Mamelle | Prematurity and occupational activity during pregnancy[END_REF][START_REF] Saurel-Cubizolles | Employment, working conditions, and preterm birth: results from the Europop case-control survey[END_REF][START_REF] Guendelman | Maternity leave in the ninth month of pregnancy and birth outcomes among working women[END_REF][START_REF] Stillerman | Environmental exposures and adverse pregnancy outcomes: a review of the science[END_REF] . At the clinical setting, programs to reduce post-term pregnancies, better survival of increasingly premature births, increase of fertility treatments and obstetrical procedures such as elective caesarean section or elective induced labor are all factors that contribute to reduce the average duration of gestational age at birth in a population [START_REF] Bergsjø | Duration of human singleton pregnancy. A population-based study[END_REF][START_REF] Allen | Reproductive Endocrinology Infertility Committee of the Society of Obstetricians and Gynaecologists of Canada (SOGC)[END_REF][START_REF] Welmerink | Infertility treatment use in relation to selected adverse birth outcomes[END_REF][START_REF] Oshiro | Decreasing elective deliveries before 39 weeks of gestation in an integrated health care system[END_REF] . The coexistence of many of these factors in current industrialized societies could be the explanation for the shortening of the gestational length identified in some recent studies [START_REF] Roberts | Trends in births at and beyond term: evidence of a change?[END_REF][START_REF] Davidoff | Changes in the gestational age distribution among U.S. singleton births: impact on rates of late preterm birth, 1992 to 2002[END_REF][START_REF] Astolfi | Gestational age shortening in single births at term. Italy 1990-1998[END_REF][START_REF] Donahue | Trends in birth weight and gestational length among singleton term births in the United States: 1990-2005[END_REF] .

In Spain, there have been significant socio-demographic changes during the last decade with impact on reproductive outcomes. On the one hand, the process of incorporating women into the labor market has consolidated. The main consequence of this process has been a delay of the first maternity, increasing the number of primiparae older mothers [START_REF] Bosch | Investigating the reasons for Spain's falling birth rate[END_REF] and the number of women having to resort to infertility treatments [START_REF] Matorras | Fertility policy in Spain[END_REF] .

On the other hand, over the last decade there has been a massive arrival of immigrants to the country [START_REF]Population in 2009 by country of birth[END_REF] . Immigrants are a population largely in reproductive age and an important increase in the number of births from immigrant mothers has taken place [START_REF]Births by mother's nationality in[END_REF] .

Several studies have found significant differences in duration of gestational length amongst different ethnic groups [START_REF] Papiemik | Racial differences in pregnancy duration and its implications for perinatal care[END_REF][START_REF] Wilcox | Birthweight from pregnancies dated by ultrasonography in a multicultural British population[END_REF][START_REF] Patel | Does gestation vary by ethnic group? A London-based study of over 122,000 pregnancies with spontaneous onset of labour[END_REF] , and the higher proportion of children with shorter gestational age born to immigrant mothers has been proposed as one possible explanation for changes in the gestational length in European countries [START_REF] Nielsen | Births and infant mortality in the Nordic countries[END_REF][START_REF] Stoltenberg | Children with low birth weight and low gestational age in Oslo, Norway: immigration is not the cause of increasing proportions[END_REF] . Therefore, the aim of this study was to describe the evolution of the duration of gestational age during the period 1997-2008 in Spain, separately for native-born and immigrant population in order to assess their contribution to the overall pattern of gestational length.

Methods

Registration of newborns is mandatory for all births occurring in Spain, regardless of the mother's nationality. All single births during the period 1997-2008 were selected from the Spanish Vital Statistics Registry. Data on gestational age at birth (weeks), parity, age, occupation and on country of origin of the mother were available.

According to this information mothers were grouped by parity (primiparae, one previous delivery, two or more previous deliveries) and 3 Births from mothers coming from developed countries showed a similar reduction in gestational weeks (1.12 days) than the Spanish for the whole period. Therefore, the analyses were carried out considering two groups: Spanish and Immigrant (considered as those coming from countries economically disadvantaged)

For both natives and immigrants, we compared the evolution of the mean gestational age by year and the statistical significance of the differences was assessed using ANOVA test. To assess differences in the distribution of the gestational age along the study period within each of the two population groups, we classified deliveries in 4 periods (1997-1999, 2000-2002, 2003-2005 and 2006-2008) To quantify the risk of having very preterm, moderate preterm and post-term deliveries along the 4 periods considered and the influence of variables with known impact on the duration of gestation we fitted crude and adjusted multinomial logistic regression models (with 37-41 weeks as the baseline category) separately for natives and immigrants. Additionally, separately for each group, we explored the evolution of the odds of delivering at term babies with 37-39 weeks by means of crude and adjusted binary logistic regression models (with 40 weeks as the reference category). In all crude and adjusted models Odds Ratios (OR) and their 95% Confidence Intervals (95%C I) were obtained.

Results

5,018,229 singleton births were registered between 1997 and 2008 in Spain, 88% of them corresponding to Spanish women and 12% to immigrants (10.63% economic immigrants and 1.37% non economic immigrants). Data on parity, age, area of origin and occupation of the mother were complete, but gestational age data were missing for 13.36% of newborns from Spanish mothers and for 20.69% of newborns from immigrants.

The evolution of the mean gestational age over time for all births (native born and immigrants jointly) showed a gradual decrease, going from 39.28 weeks in 1997 to 39.07 in 2008, a mean decrease of 1.47 days. As Figure 1 shows, among native-born women the mean gestational age decreased from 39.27 weeks in 1997 to 39.09 in 2008, a mean decrease of 1.26 days. Economic Immigrant women showed a more noticeable gestational shortening over time than Spanish, going from 39.32 weeks in 1997 to 38.99 in 2008, a mean decrease of 2.31 days (1.79 days for Latin-American, 3.24 for East European, 2.02 for North-African, 2.24 for Sub-Saharan and 3.03 for Asian).

Differences in mean gestational age along the years were statistically significant in both groups.

The proportion of deliveries by gestational age along the 4 periods considered are summarised in Table 1 and During this time, the decrease in the prevalence of preterm births was marginal. The shortening of the mean gestational age is due to a striking decrease in the proportion of at term births with 40 weeks in favor of an increase of births with 37-39 weeks. A decrease of post-term deliveries over time also contributed to the shortening of gestational age in Spanish newborns. Among immigrants, the number of deliveries has experienced a marked 7-fold increase from first to last period. In contrast to Spanish women, shortening of the mean gestational age observed in immigrant women is partly consequence of a clear increase of preterm births over time, mainly attributable to moderate preterm but also to very preterm deliveries. However, the main reason for gestational shortening in immigrants was a more marked decrease than in Spanish in the proportion of 40 weeks deliveries in favor of births with 37-39 weeks. This more pronounced shift to the left in the length of at term gestation among immigrants is seen in Figure 2. A higher decrease of post-term deliveries was also evident among immigrants along the period.

Table 2 shows the proportion of deliveries according to maternal characteristics. The proportion of primiparae mothers among Spanish has slightly increased over time. More noticeable was the increase in the proportion of older mothers practically doubling the percentage of women giving birth after 35 from 1997-1999 to 2006-2008. Unlike Spanish mothers, the proportion of primiparae in immigrants decreased and no remarkable changes occurred in the proportion of deliveries from ≥35 years mothers.

Both, Spanish and immigrants showed a reduction in the proportion of deliveries from non working mothers, parallel to an increase of deliveries from mothers working in services and manufacturing and in managerial and clerical jobs but also in farm work or related in immigrants.

Analyses showing the evolution of the risk of gestational shortening in preterm and post-term deliveries are shown in table 3. The adjusted risk of very preterm delivery for Spanish mothers was an 11% bigger in 2006-2008 than in 1997-1999 while the adjusted risk of moderate preterm birth seems to have decreased slightly along the period (5% smaller). More noticeable was the progressive reduction of the risk of post-term delivery, a 17%, 24% and 34% lower in 2000-2002, 2003-2005 and 2006-2008, respectively, than in 1997-1999. For immigrant women, the adjusted risk of very preterm delivery was 34% higher in 2006-2008 than in 1997-1999, clearly higher than for Spanish women. Moreover, unlike Spanish, the risk of moderate preterm delivery increased for immigrants being 18% higher in the last period. Analysis of post-term showed, as for Spanish, a progressive reduction in the risk over time, 19%, 20% and 42% lower in 2000-2002, 2003-2005 and 2006-2008, respectively, than in 1997-1999.

Finally, the evolution of the odds of gestational shortening among at term deliveries is shown in table 4. A gradual increase over time in the adjusted risk of being born at 37-39 weeks at the expense of deliveries with 40 weeks occurred both in Spanish and immigrants, but more pronounced in the latter group.

Discussion

Our results show that in the last 12 years a progressive shortening in the mean gestational age of live births has taken place in Spain. While the overall decrease of gestational length along the period was 1.47 days, closer to that in Spanish women (1.26 days), it was 2.31 days among immigrants (3.24 days in Eastern European). In both Spanish and immigrants this shortening was mainly due to an increase in the proportion of 37-39 weeks deliveries at the expense of a substantial decrease in those with 40 weeks, but also to a reduction in the prevalence of post-term deliveries. However, among immigrant women, an increase of very and moderate preterm deliveries over time has also contributed to the gestational shortening. Those effects remained after controlling for known confounding variables such as maternal age, parity and occupation.

Trends of gestational shortening have been reported in other countries [START_REF] Roberts | Trends in births at and beyond term: evidence of a change?[END_REF][START_REF] Davidoff | Changes in the gestational age distribution among U.S. singleton births: impact on rates of late preterm birth, 1992 to 2002[END_REF][START_REF] Astolfi | Gestational age shortening in single births at term. Italy 1990-1998[END_REF][START_REF] Donahue | Trends in birth weight and gestational length among singleton term births in the United States: 1990-2005[END_REF] . A study carried out in Australia [START_REF] Roberts | Trends in births at and beyond term: evidence of a change?[END_REF] including births from 1990 to 1996 showed a decline in 40 and ≥42 weeks' births and a concomitant increase of term births with 38-39 weeks of gestation. Although the pattern was similar to that found in our study, theirs was confined to ≥ 37 weeks' deliveries and carried out only in the region of New South Wales. In the United States, the analysis of births in 1992, 1997 and 2002 from mothers of different racial/ethnic groups [START_REF] Davidoff | Changes in the gestational age distribution among U.S. singleton births: impact on rates of late preterm birth, 1992 to 2002[END_REF] showed a shift to the left in gestational age, from 40 weeks as the most frequent length of gestation in 1992 to 39 weeks in 2002. This shift was consequence, again, of a decline in the proportion of post-term and 40 weeks' births, but also in those at 41 weeks. A more recent study including at term births from 1990 to 2005 in the United States 20 confirms a decrease of more than 2 days in the mean length of gestation as a consequence of the decline of births with 40-41 weeks. In the only, to our knowledge, European study which has analyzed the evolution of gestational age, a shift toward shorter times of gestation was identified among Italian at term births (≥37 weeks) born between 1990 and 1998 [START_REF] Astolfi | Gestational age shortening in single births at term. Italy 1990-1998[END_REF] . Along this period the mean gestational age was reduced by 1.33 days, quite similar to the reduction observed in Spain.

Our study was nationwide, looking at term births but also preterm and post-term. The period considered was recent and longer than in most previous studies.

One significant problem of the study is the large number of cases with missing information on gestational age, especially for deliveries from immigrant mothers.

Although in Spain there is universal access to health care, even for undocumented immigrants, we can not rule out the possibility that cases with no data for gestational age might represent births of immigrant women with limited access to health care services. Limited access to health care has been related with worse reproductive results (prematurity, low birth weight) in immigrant populations [START_REF] Malin | Maternal care and birth outcomes among ethnic minority women in Finland[END_REF][START_REF] Manzanares Galan | Results of poor prenatal care in the immigrant population[END_REF][START_REF] Puig Sola | Hospital admission in newborns according to ethnicity and parents' country of origin in an urban area of Barcelona [Spain]][END_REF][START_REF] Tierney-Gumaer | Risk factors for low birth weight infants of Hispanic, African American, and White women in Bexar County, Texas[END_REF] . Even under this hypothetical scenario, it is doubtful that missing data on gestational age explains the shortening of gestational age identified in this study, although the estimated difference between Spanish and immigrants could be higher. In order to check this hypothesis and considering the correlation between gestational age and birth weight, we have checked the information about this last variable among cases with missing gestational age.

Information on newborn's weight was missing for only 4.96% of cases and we checked that 95.82% (95.7% for Spanish and 96.22% for immigrants) of births with missing value for gestational age were babies with weight over 2500grs (mean birth weight 3,246.47 and 3,316.86 respectively). Therefore it is unlikely that infants who do not have data for gestational age are the most premature.

On the other hand, the possibility of gestational shortening as an artifact stemming from a more accurate reporting of the duration of pregnancies over time needs to be considered. The Spanish Vital Statistics Registry registers information on gestational age provided by parents or relatives at the time of birth registration. A change over time in the accuracy of reporting seems unlikely. The fluctuation in the degree of completeness of this variable throughout the period analyzed does not explain consistently the trends identified in the study.

A change along the period in the method used for gestational age determination might have an impact on the estimated gestational age. In fact, several studies have consistently shown that, compared to the last menstrual period dating, early ultrasound dating results in a shift to the left in the gestational age distribution [START_REF] Yang | How does early ultrasound scan estimation of gestational age lead to higher rates of preterm birth?[END_REF][START_REF] Ananth | Menstrual versus clinical estimate of gestational age dating in the United States: temporal trends and variability in indices of perinatal outcomes[END_REF] . However, no changes in the method for determination of the gestational age have occurred in Spain during the period considered.

Data shows that the mean gestational age for immigrants exceeds the Spanish until 2005, but dropped thereafter. Immigration in Spain is a recent phenomenon. Latin American and North-African women have been the largest immigrant population groups during the last decade, but in recent years there has been a very rapid increase in the affluence of women from Eastern Europe and Sub-Saharan Africa with a notable increase of births to mothers from these geographical areas. Similar or better reproductive outcomes than for native women have been reported for immigrant women from Latin-America or North-African in different contexts [START_REF] Guendelman | Birth outcomes of immigrant women in the United States, France, and Belgium[END_REF][START_REF] Vahratian | Birthweight differences among infants of North African immigrants and Belgians in Belgium[END_REF][START_REF] Buekens | High birthweights among infants of north African immigrants in Belgium[END_REF] , although a worsening in their reproductive indicators parallel to the increased time of residence in the host country has been reported [START_REF] Callister | Acculturation and perinatal outcomes in Mexican immigrant childbearing women: an integrative review[END_REF][START_REF] Page | Positive pregnancy outcomes in Mexican immigrants: what can we learn?[END_REF][START_REF] Urquia | Immigrants' duration of residence and adverse birth outcomes: a population-based study[END_REF] . In Spain, a recent study shows that more recent immigrant women from Eastern-Europe and Sub-Sahara have a much higher risk of prematurity than other immigrant groups or Spanish-born women [START_REF] Río | Reproductive and perinatal health indicators in immigrant and Spanish-born women in Catalonia and Valencia (2005-2006)[END_REF] and our data shows that those two immigrant groups have experienced the greatest decrease in gestational age. Collectively, all this evidence could explain the increased difference between Spanish and immigrants in the decline of the mean gestational age in most recent years.

According to these considerations, the trend identified in our study seems to reflect a real change. Underlying causes for a true shortening in the mean length of gestation might be concomitant changes in maternal characteristics [START_REF] Bergsjø | Duration of human singleton pregnancy. A population-based study[END_REF][START_REF] Delpisheh | Pregnancy late in life: a hospitalbased study of birth outcomes[END_REF][START_REF] St-Laurent | Biopsychosocial determinants of pregnancy length and fetal growth[END_REF][START_REF] Petridou | Modulators of length of gestation. A study in Greece[END_REF] and/or in the obstetric practice [START_REF] Allen | Reproductive Endocrinology Infertility Committee of the Society of Obstetricians and Gynaecologists of Canada (SOGC)[END_REF][START_REF] Welmerink | Infertility treatment use in relation to selected adverse birth outcomes[END_REF][START_REF] Oshiro | Decreasing elective deliveries before 39 weeks of gestation in an integrated health care system[END_REF] along the period studied. Our results show an increasing proportion of primiparae over time only among Spanish women. Given the positive association between primiparity and the risk of reduced length of gestation [START_REF] Bergsjø | Duration of human singleton pregnancy. A population-based study[END_REF][START_REF] Delpisheh | Pregnancy late in life: a hospitalbased study of birth outcomes[END_REF][START_REF] St-Laurent | Biopsychosocial determinants of pregnancy length and fetal growth[END_REF][START_REF] Petridou | Modulators of length of gestation. A study in Greece[END_REF] , this difference between Spanish and immigrant women might predict a more pronounced shortening among Spanish. However, adjusted analyses showed a major risk of gestational shortening for immigrants, parity not being an explanatory variable of shortening neither in the Spanish nor in the immigrant group. Deliveries from women ≥35 years increased over time only among natives and adjusted analyses of risk suggest that gestational shortening occurs independently of mother's age in both population groups.

Regarding the assessment of the impact of changes in obstetrical procedures throughout the period, our study have a limitation because the Spanish Vital Statistics Registry do not provide information about the type of delivery (spontaneous or induced) and only since recently (2007) about the mode of delivery (vaginal or caesarean section). The increase in induced deliveries and caesarean sections along the period analyzed is well documented in Spain [START_REF]Indicadores de actividad hospitalaria en España[END_REF] , and therefore their possible influence on our results cannot be ruled out. However, the studies previously mentioned show that shortening of gestational length is quite independent of type and mode of delivery and of maternal age. The study carried out in New South Wales [START_REF] Roberts | Trends in births at and beyond term: evidence of a change?[END_REF] found that the shift to earlier gestational ages also occurred among women less than 35 years and among births with spontaneous onset. In one of the studies performed in the USA [START_REF] Davidoff | Changes in the gestational age distribution among U.S. singleton births: impact on rates of late preterm birth, 1992 to 2002[END_REF] the shift to the left in the distribution of gestations occurred among all three categories of delivery: spontaneous, induced/caesarean and those with premature rupture of membranes. The more specific analysis of the evolution of deliveries [START_REF] Donahue | Trends in birth weight and gestational length among singleton term births in the United States: 1990-2005[END_REF] conducted in the other USA study also confirmed that neither statistical adjustment by maternal and neonatal characteristics or by obstetric practices nor restriction of the sample to a homogeneous low-risk subgroup -none of whom were induced or delivered by caesarean sectionattenuated observed declines in gestational duration of term births. Finally, the study performed in Italy [START_REF] Astolfi | Gestational age shortening in single births at term. Italy 1990-1998[END_REF] showed that the shortening experienced in pregnancies from mothers <30 years was more relevant than that occurred in pregnancies among older mothers independently of the educational level.

Another factors with known negative impact on the duration of pregnancy but not recorded in birth records are the nutritional status of the women, the exposition to toxics, the emotional stress and anxiety or the presence of specific medical conditions 5- 10 . We cannot rule out temporal changes in some of these factors that might contribute to the identified decline in the gestational length. In this regard, detailed studies would be needed to explore their influence on delivery at the population level. There is also a positive association among infertility treatments and reduced length of gestation [START_REF] Allen | Reproductive Endocrinology Infertility Committee of the Society of Obstetricians and Gynaecologists of Canada (SOGC)[END_REF][START_REF] Welmerink | Infertility treatment use in relation to selected adverse birth outcomes[END_REF] . In Spain, an increase of infertility treatments over the period analyzed has occurred [START_REF] Matorras | Fertility policy in Spain[END_REF] , but we consider that it does not explain neither shortening among at term pregnancies or the higher gestational shortening experienced by immigrant women.

The mean gestational age was quite similar between native-born and immigrants for the first three periods considered and differed markedly for the last period. This difference was due not only to a significant increase in preterm births but also in early at term births. Thus, given that the number of births from immigrant mothers increased notably in this last period, the differentiated contribution of immigrants to the overall gestational shortening was bounded to the period 2006-8.

Neither the underlying reasons for the gestational age shortening nor the consequences of such a decline are definitely established. Since the duration of gestation is a predictor of short and long-term disease risk among preterm but also among term newborns, it seems necessary to carry out further research to monitor this trends, with particular attention to differences among population groups. Comparisons of the percentages along the 4 periods in each category were statistically significant (p-value <0.001)

  categories of age (<20, 20-34 and 35 or more years), 4 types of work (managerial or clerical, services or manufacturing, farm work or related and no job) and as Spanish, economic immigrants (coming mainly from Latin America [Ecuador, Colombia, Bolivia, Peru], Eastern Europe [Romania, Bulgaria], Maghreb [Morocco and Algeria], and Sub-Saharian Africa [Senegal and Gambia]) or non economic immigrants (coming from developed countries).

represented in Figure 2 .

 2 The number of deliveries from Spanish mothers increased slightly from 1997-1999 to 2006-2008 in 127,663 births.

Figure 2 .

 2 Figure 2. Evolution of the proportion of births by gestational age among Spanish and immigrant women

TABLES Table 1 . Proportion of deliveries by gestational age (GA) in 4 periods among Spanish and immigrant women

 TABLES1

			SPANISH n (%)			IMMIGRANTS n (%)
		1997-1999 a	2000-2002 a	2003-2005 a	2006-2008 a	1997-1999 a	2000-2002 a	2003-2005 a	2006-2008 a
	BIRTHS	1036431	1082448	1131506	1164094	34293	84966	164167	249801
	BIRTHS WITH GA 885037 (85.39%) 949150 (87.69%) 1010997 (89.35%) 979735 (84.16%) 27140 (79.14%) 69979 (82.36%) 137835 (83.96%) 183924 (73.63%)
	PRETERM b	55747 (6.3%)	59776 (6.3%)	62460 (6.18%)	59551 (6.08%)	1601 (5.9%)	4521 (6.46%)	8811 (6.39%)	13321 (7.24%)
	VPT (≤ 32 ) b	7354 (0.83%)	7968 (0.84%)	8420 (0.83%)	8971 (0.92%)	323 (1.19%)	822 (1.17%)	1585 (1.15%)	2668 (1.45%)
	MPT (33-36) b	48393 (5.47%)	51808 (5.46%)	54040 (5.35%)	50580 (5.16%)	1278 (4.71%)	3699 (5.29%)	7226 (5.24%)	10653 (5.79%)
	AT TERM b	786154 (88.83%) 850511 (89.61%) 911329 (90.14%) 889212 (90.76%) 23793 (87.67%) 61420 (87.77%) 121630 (88.24%) 163590 (88.94%)
	37-39 b	345299 (39.02%) 398883 (42.03%) 452426 (44.75%) 461434 (47.1%) 9693 (35.71%) 27609 (39.45%) 58370 (42.35%) 84545 (45.97%)
	40 b	341166 (38.55%) 343780 (36.22%) 336506 (33.28%) 284358 (29.02%) 11511 (42.41%) 27066 (38.68%) 48708 (35.34%)	53156 (28.9%)
	41 b	99689 (11.26%) 107848 (11.36%) 122397 (12.11%) 143420 (14.64%)	2589 (9.54%)	6745 (9.64%) 14552 (10.56%) 25889 (14.08%)
	POST-TERM b	43136 (4.87%)	38863 (4.09%)	37208 (3.68%)	30972 (3.16%)	1746 (6.43%)	4038 (5.77%)	7394 (5.36%)	7013 (3.81%)

a Figures do not add up because of missing values. b Comparisons of the percentages along the 4 periods in each GA category were statistically significant (p-value <0.001)

Table 2 . Proportion of deliveries by maternal characteristics in 4 periods among Spanish and immigrant women.

 2

			SPANISH n (%)			IMMIGRANTS n (%)
		1997-1999	2000-2002	2003-2005	2006-2008	1997-1999	2000-2002	2003-2005	2006-2008
	PARITY
	Primiparae a	546824 (52.76%) 595784 (55.04%) 635362 (56.15%) 655839 (56.34%) 20145 (58.74%) 49982 (58.83%) 96177 (58.58%) 140295 (56.16%)
	1 previous birth a	382697 (36.92%) 388807 (35.92%) 402509 (35.57%) 411960 (35.39%) 8996 (26.23%) 22021 (25.92%) 43932 (26.76%) 72421 (28.99%)
	≥2 previous births a	106910 (10.32%)	97857 (9.04%)	93635 (8.28%)	96295 (8.27%) 5152 (15.02%) 12963 (15.26%) 24058 (14.65%) 37085 (14.85%)
	AGE
	<20 a	30947 (2.99%)	29047 (2.68%)	26758 (2.36%)	27313 (2.35%)	1808 (5.27%)	5320 (6.26%)	10381 (6.32%)	14953 (5.99%)
	20-34 a	827056 (79.8%) 827871 (76.48%) 834307 (73.73%) 819345 (70.38%) 27304 (79.62%) 68347 (80.44%) 131704 (80.23%) 199374 (79.81%)
	≥35 a	178428 (17.22%) 225530 (20.84%) 270441 (23.9%) 317436 (27.27%) 5181 (15.11%) 11299 (13.3%) 22082 (13.45%)	35474 (14.2%)
	OCCUPATION
	Managerial & Clerical a 307985 (35.29%)	351911 (39%) 408732 (43.56%) 479983 (48.17%) 2900 (13.15%) 6862 (13.03%) 11853 (11.99%) 22409 (12.45%)
	Services or manufact. a 177508 (20.34%) 222358 (24.64%) 266332 (28.39%) 271588 (27.25%) 4926 (22.34%) 13759 (26.12%) 31274 (31.65%) 56792 (31.55%)
	Farmwork or related a	9990 (1.14%)	10996 (1.22%)	11213 (1.2%)	11911 (1.2%)	150 (0.68%)	877 (1.67%)	1504 (1.52%)	3207 (1.78%)
	No work a	377333 (43.23%) 317182 (35.15%) 251982 (26.86%) 233044 (23.39%) 14078 (63.83%) 31169 (59.18%) 54194 (54.84%) 97574 (54.21%)

a

Table 3 . Evolution of the risk of Very preterm (VPB), moderate preterm (MPB) and post-term birth. SPANISH IMMIGRANTS OR * c (CI95%) a OR * a (CI95%) b OR * c (CI95%) a OR * a (CI95%) b

 3 ORc= Crude Odds Ratio and 95% confidence intervals b ORa=Odds Ratio adjusted by parity, age and occupation of the mother and 95% confidence intervals. * Baseline category 37-41 weeks

	VPB (≤32 weeks)
	1997-1999	1	1	1	1
	2000-2002	1.00(0.97,1.03)	1.02(0.98,1.05) 0.99(0.87,1.12) 1.11(0.94,1.31)
	2003-2005	0.99(0.96,1.02)	1.00(0.97,1.04) 0.96(0.85,1.08) 1.00(0.85,1.17)
	2006-2008	1.08(1.05,1.11)	1.11(1.07,1.15) 1.20(1.07,1.35) 1.34(1.15,1.55)
	MPB (33-36 weeks)
	1997-1999	1	1	1	1
	2000-2002	0.99(0.98,1.00)	1.00(0.99,1.02) 1.12(1.05,1.20) 1.13(1.05,1.22)
	2003-2005	0.96(0.95,0.98)	0.98(0.97,1.00) 1.11(1.04,1.18) 1.10(1.02,1.18)
	2006-2008	0.92(0.91,0.94)	0.95(0.94,0.96) 1.21(1.14,1.29) 1.18(1.10,1.26)
	POST-TERM (≥42 weeks)
	1997-1999	1	1	1	1
	2000-2002	0.83(0.82,0.84)	0.83(0.82,0.85) 0.90(0.85,0.95) 0.81(0.76,0.87)
	2003-2005	0.74(0.73,0.75)	0.76(0.75,0.77) 0.83(0.78,0.87) 0.80(0.75,0.85)
	2006-2008	0.63(0.63,0.64)	0.66(0.65,0.67) 0.58(0.55,0.62) 0.58(0.54,0.62)

a

Table 4 . Evolution of the risk among at term births of being born with 37-39 weeks. SPANISH IMMIGRANTS OR * c (CI95%) a OR * a (CI95%) b OR * c (CI95%) a OR * a (CI95%) b

 4 ORc= Crude Odds Ratio and 95% confidence intervals b ORa=Odds Ratio adjusted by parity, age and occupation of the mother and 95% confidence intervals.

	1997-1999	1	1	1	1
	2000-2002	1.15(1.14,1.15) 1.14(1.13,1.15) 1.21(1.17,1.25) 1.19(1.15,1.24)
	2003-2005	1.33(1.32,1.34) 1.31(1.30,1.32) 1.42(1.38,1.47) 1.39(1.35,1.45)
	2006-2008	1.60(1.59,1.61) 1.56(1.54,1.57) 1.89(1.83,1.94) 1.79(1.73,1.85)

a * Baseline category 40 weeks

Acknowledgments

Spanish Ministry of Science and Innovation (SEJ 2005/07679); CIBER of Epidemiology and Public Health (Ciberesp, Spain).


Jean-Luc Dion

Sylvie Le Moyne

Gaël Chevallier

Hamidou Sebbah

Gear Impacts And Idle Gear Noise: Experimental Study And Non Linear Dynamic Model

Keywords: Idle gear noise, hertzian contact, shock, backlash, non linear vibrations, signal processing

This paper develops an experimental and numerical study of dynamic phenomena involving gear impacts with one loose gear (non engaged gear pair) inside an automotive gearbox.

A dedicated test bench was designed for this study. Signal processing tools, based on the Order Tracking Method, were specially developed in order to clarify the underlying phenomena. A Particular attention was paid to the relationship between the drive shaft excitation and the energy and nature of impacts.

A topological model of contact is being proposed, based on a geometric description of solids in contact. The contact between gear meshes is defined using a Single Degree of Freedom, non linear, elastic and dissipative model. The chosen parameters of the model are not updated from measurements, but from the knowledge of the gear topology and tolerance class.

Simulations, compared with experimental results, confirm the accuracy of the model proposed.

Both model and experiments show, for particular excitation conditions, the emergence of repeated impacts on one side of the gear mesh.

As the demand for acoustic comfort is constantly increasing, car manufacturers are concerned by noise annoyance problems in vehicles, and especially by the annoyance generated by gearboxes.

Among numerous gearboxes noises, idle gear noise is perceived as particularly disagreeable and often wrongly associated with an engine problem by consumers. This noise is generated by gear impacts which can appear either in driving conditions ("rattle" noise) or in neutral ("idle" noise or "neutral rattle" noise ). As neutral is the worst configuration, it is the focus of this paper.

The present paper presents an experimental and numerical study of the gear impact. The aim of the study is to define a simple and realistic model of contact able to describe the dynamic impacts between gears with invariant parameters for several frequencies, levels and types of excitations. To this end, the nonlinear Single Degree of Freedom (SDOF) gear pair model with backlash, which has often been investigated in the literature, has only included few parameters for the gear topology and tolerance class. The choice of the model to be developed was governed by a literature study, but also by a preliminary experimental study.

The test bench designed for this study is composed of an off-theshelf simplified gear box (only one gear couple is kept). The choice of measurement and signal processing techniques was governed by the very transient character of the events being observed. This experimental part of the study particularly focuses on the determination of relationships between the excitation kinematics and the nature and the energy content of the impact. Energy contents are analyzed globally and also in terms of frequency distribution.

The model thus formulated endeavors to predict the dynamic behaviour of a gear box by associating a limited number of nonlinear SDOF. After an analytical description of the model, parametric identification techniques are presented.

The only model parameters that are identified with experimental techniques are the dissipative coefficients. All other parameters of the model are determined in accordance with technical design data.

In the final section, before drawing our conclusions, numerical simulations and experimental measurements will be compared.

Literature

Most previous studies of such phenomena propose analytical models for the vibroacoustic behaviour of the entire mechanical system (gears, transmission chain). In [START_REF] Comparin | Analysis of automotive neutral gear rattle[END_REF], R.Singh et al. retain a Lumped Model, non-linear, and find analytical solutions, derived piecewise, section after section, from a linear analysis. In this particular case, the stiffness is described with functions which have non continuous derivatives. In [START_REF] Velex | An analytical study of tooth friction excitations in errorless spur and helical gears[END_REF], the authors improved the latter model by the introduction of friction. Moreover the authors derive analytical solutions based on the Harmonic Balance Method.

Many investigations have been performed to define contact characteristics of simple or multi-mesh gear trains, taking into account friction phenomena and the influence of backlash. The first experiments focused on the detection and the analysis of nonlinear phenomena [START_REF] Pan | Intelligent joint fault diagnosis of industrial robots[END_REF].

The instantaneous characteristics in the signals can be obtained from the ridges and skeletons of the Wavelet Transform [START_REF] Tjahjowidodo | Experimental dynamic identification of backlash using skeleton methods[END_REF]. Such non-linear systems can also be described with a Volterra Functional Series which generalise the Superposition Principle and allows Impulse Responses and Transfer Functions of various orders to be obtained [START_REF] Tawfiq | Contribution to the extension of modal analysis to non-linear structure using volterra functional series[END_REF]. Other Detection Methods are based on the speed variations, [START_REF] Jeffrey | Automatic detection of clearance in mechanical systems: Experimental validation[END_REF]; in this paper, the authors show that a low cost conventional tachometer is sufficient for measuring the induced speed variations due to backlash. They can also be predicted via simulation. For example, T. Tjahjowidodo et al., [START_REF] Tjahjowidodo | Experimental dynamic identification of backlash using skeleton methods[END_REF],

uses a detailed multibody simulation to develop and test the effectiveness of the proposed detection approach.

The idle gear noise is chaotic [START_REF] Lin | Chaotic vibration of mechanical systems with backlash[END_REF]. The chaotic behaviour gets stronger with increasing backlash, see [START_REF] Trendafilova | Non-linear dynamics tools for the motion analysis and condition monitoring of robot joints[END_REF]. In [START_REF] Tjahjowidodo | Quantifying chaotic responses of mechanical systems with backlash component[END_REF], T. Tjahjowidodo et al., quantify the chaotic responses and correlate them to the parameters of the non-linear system, in particular the magnitude of the backlash.

A. and M. Ajmi et al. [START_REF] Al | Non-linear dynamic analysis of multi-mesh gear train using multi-term harmonic balance method: sub-harmonic motions[END_REF]) proposed models to introduce non-linearity in the contact elasticity. They also take into account the structural elasticity of gears. M. Ajmi et al. [START_REF] Ajmi | model for simulating the quasi-static and dynamic behaviour of solid wide-faced spur and helical gears[END_REF] considers a continuous non-linear model concentrated on contact lines in gears with a Finite Element global description. A.Alshyyab et al. [START_REF] Al | Non-linear dynamic analysis of multi-mesh gear train using multi-term harmonic balance method: sub-harmonic motions[END_REF] chose Lumped Models, non-linear with analytical solutions derived from decompositions in frequency domain.

In a similar manner, J.P.Raclot et al. [START_REF] Raclot | Simulation of dynamic behaviour of single and multi-stage geared systems with shape deviations and mounting errors by using a spectral method[END_REF] built a combination of flexible bodies in contact with Spectral Methods for a wide excitation spectrum.

Various experimental studies, see for example ( [START_REF] Blomqvist | Detection of rattle sources in gearbox using correlation technique[END_REF]), involve source location techniques by means of multiple Impulse Response Functions to detect sources of gearbox rattle using correlation between several excitations and several IRFs.

Other studies, of the complete gear box were performed taking into account the influence of manufacturing error on the rattle noise (see for example [START_REF] Rigaud | Allocation of gear tolerances to minimize gearbox noise variability[END_REF]). The degree of noise variability is assumed to be only induced by tolerances of geometric descriptions of gears. Mechanical models of solids are combined with acoustic models in a global statistical study. In The influence of the harmonic content of engine acyclism is analysed. As this is a main source of gear schoks, this work attempts to complement the work carried out by [START_REF] Barthod | étude de l'influence des caractéristiques de l'acyclisme sur le bruit de "graillonnement" dans les boites de vitesses automobiles[END_REF].

Models, such as the N.Barabanov et al. model [START_REF] Barabanov | A new model for control of systems with friction[END_REF], using Finite Element Methods are frequently employed for specific excitation sources: [START_REF] Ducret | Calcul du bruit rayonné par les carters des transmissions à engrenages : méthode et applications[END_REF] introduces mechanical models for solids periodically excited by a static transmission error and incorporates acoustic models in the finite element method. Likewise, [START_REF] Velex | A mathematical model for analysing the influence of shape deviations and mounting error on gear dynamic behaviour[END_REF] conducted a study with several manufacturing errors (tooth spacing -pitch, eccentricities, misaligned gears ) and variation in excitations using the model described in [START_REF] Ajmi | model for simulating the quasi-static and dynamic behaviour of solid wide-faced spur and helical gears[END_REF].

The vibroacoustic behaviour of the gear box is generally assumed to be linear and described by transfer functions following propa-gation paths for solid transmission of vibration and vibroacoustic coupling with fluid propagation (atmosphere) for acoustic transmission ( [19] [17]).

The only mechanical elements which cannot be described by linear models are bearings ( [START_REF] Lim | Vibration transmission through rolling element bearing, part i : Bearing stiffness formulation[END_REF], [START_REF] Lim | Vibration transmission through rolling element bearing, part ii : System studies[END_REF], [START_REF] Lim | Vibration transmission through rolling element bearing, part iii : Geared rotor system studies[END_REF], [START_REF] Lim | Vibration transmission through rolling element bearing, part iv : Statistical energy analysis[END_REF]) and gears in contact.

4 Test Bench

Gear Box

The test bench must show impacts induced by backlash between two gears. Thus, all other sources of uncontrolled vibrations have to be suppressed.

The solution adopted was to obtain an off-the-shelf gearbox and modify it in order to keep only one gear couple (4 th gear). As a result of the gaps due to the removal of gears and the entire command system, crossbars were added.

Principal gear data are the following:

• 4 th helical gear on the main shaft:

• pitch diameter: 57 mm

• number of teeth: 28

• face width: 20 mm

• helix angle: 30 o

• 4 th loose helical gear:

• pitch diameter: 74 mm

• number of teeth: 37

• face width: 20 mm

• helix angle: 30 o

• Contact ratio: 2.9

• individual pitch error : between 3 µm and 15 µm

The translational oscillations of an electrodynamics shaker rod are transformed into angular oscillations of the main shaft through two perpendicular transmission beams (Fig . 1 and Fig . 2). The motion obtained is pure angular oscillations with an average angular speed equal to zero. The authors chose to carry out the experiment in this way in order to reproduce a gearbox in neutral. Transmission beam dimensions were calculated in order to respect the following conditions:

• The circular trajectory of the joint between the two transmission beams (point A) should not damage the shaker.

• The eigen frequencies of the excitation system should not perturb measurements.

The rigid transmission beam, between the flexible beam and the axe of the main shaft, is 165 mm long.

The excitation reproduces actual engine torque oscillations which are sources of the noise and vibration under study: sinusoidal excitation from 30 Hz to 60 Hz with a maximal oscillation magnitude of 2000 rad/s 2 .

Measurement Setup

The excitation is controlled by a piezoelectric accelerometer fixed on the output of the shaker. In order to verify the non distortion of the signal transmitted to the main shaft of the gearbox, a second accelerometer is fixed at the end of the main shaft.

A rigid support is inserted in the loose gear under study, at 46 mm from the axe. This support is used for the fixation of a piezoelectric accelerometer.

A hole was made in the gearbox house for accessibility and fixation of sensors (Fig . 3 and Fig . 4). The natural eigen frequency of the support was calculated greater than 3200 Hz.

Signal Processing And Analysis Method

The present method uses only instantaneous signals in both time and frequency domains, and thus banishes averaging techniques.

Indeed, preliminary studies involving average signal in time domain, in order to eliminate the random part of the signals and retain the deterministic part, led to significant signal leakage and more particularly to an underestimation of the frequency content of the signals.

After conducting a major experimental study, it was possible to determine the most efficient signal processing representations for those particular phenomena, in both time and frequency domains:

• Instantaneous temporal signals

• Periodgrams representing the magnitude of the loose gear acceleration, according to the period N = t.F ex where t is time and F ex the excitation frequency

• Three dimensional spectral representations of loose gear acceleration magnitude versus frequency (or tracking order) and excitation frequency.

Measures are converted from translation motion to angular motion in order to respect the kinematical properties of the system (Fig . 3).

The system is excited by a sine wave with a variable frequency, the observed responses are then periodic but not harmonic. Experimental results are thus to be normalized to the excitation frequency. A re-sampling technique based on FIR filter algorithm is implemented to avoid signal distorsion.

Empirical Analysis

One can first observe instantaneous time signals (Fig . 5). Different types of behaviour appear, depending of the excitation level:

• for low excitation levels, no impact appears between the teeth of the gears,

• as the excitation increases, an impact occurs only on one side (F+) of the tooth. After the impact, the behaviour of the tooth leads to free vibrations (Fig . 5c),

• for greater excitation, impacts occur on both sides of the tooth (F+ and F-). The impact level on F+ is greater and free vibrations are observed only for this impact (Fig . 5d),

• when the level of excitation increases, two impacts are observable in a very short time (less than 1ms): two pairs of teeth are in contact at the same time (Fig . 5h),

• for the greatest excitation the impact on F-becomes stronger than the one on F+ (Fig . 5g).

Those behaviours and their evolution were observed in the entire range of the excitation frequency ([30 Hz 60 Hz]).

It is particularly interesting to observe that impacts appear several times on the same side F+ before changing the side F-. Indeed, the first impact is not strong enough to push the tooth up to the other side. Two or three impacts then occur before changing the side of impact. This phenomenon appears more clearly on periodgrams (Fig . 6).

Acceleration signals of the loose gear show areas with a constant acceleration near zero (Fig . 7 right). For these short sections, the velocity decreases slowly (Fig . 7 left): the loose gear is in free motion with a small damping. During those sections of "free motion", the decay of the velocity is proportional to the velocity: this energy dissipation can then be described with a classical viscous damping as an angular dashpot. This behaviour is due to the lubricant between the second shaft and the loose gear.

The analysis of the relative displacement on the pitch diameter allows one to estimate the backlash between gears. When the level of excitation increases, this free motion gap presents an apparent increase (from 40 µm to 80 µm): in fact, for the highest level, some parts of the mechanism can not be assumed to be rigid bodies, flexible behaviour has to be introduced. For example, a 1000 N force on the driving arm induces a deformation of 10 µm. The actual backlash is in fact 80 µm.

Frequency Analysis

Spectral analysis is represented in the frequency domain or with the Tracking Order Method. The three dimensional representations in Fig . 8 show the evolution of the spectral signature of the teeth impacts in accordance with the excitation level for two excitation frequencies: 30 Hz and 60 Hz. For both frequencies a threshold of excitation shows a break-even point from which the impact spectrum extended strongly from 500 Hz to more than 5000 Hz. In all the experiments, the signal level strongly decreases around 3200 Hz: this is due to the sensor support plate on which measurements are performed, which plays the rule of dynamic absorber tuned at 3200Hz (Fig . 8).

Parametric Analysis

With the aim of showing the influence of acceleration, velocity and displacement levels, three classes of experiments have been conducted with the excitation frequency between 30 Hz and 60

Hz:

• at a constant displacement magnitude,

• at a constant velocity magnitude,

• at a constant acceleration magnitude,

The excitation is controlled thanks to an accelerometer placed on the electrodynamics shaker head.

These experiments were performed with several levels of excitation.

The figure .5 represents periodgrams according to the excitation frequency for each of the three classes of measurements.

Periodgrams with a constant magnitude of the excitation acceleration is the most invariant with excitation frequency (in term of energetic level of impacts). According to a simple linear approach, forces can be described in terms of acceleration. So the acceleration of the loose gear depends on the impact force and consequently on the excitation level of the main shaft.

The number of impacts on a given side strongly depends on the displacement level induced by the angular oscillations of the main shaft: for a given main shaft displacement magnitude, the energetic level of impacts increases with the excitation frequency, but the number of impacts keep the same.

6 Model

Global Description

The model proposed in the present paper is based on a non linear Single angular Degree Of Freedom (Fig . 8) which can relate the main shaft behaviour to the loose gear in terms of force, torque, acceleration, speed and position in translation and rotation.

The non linear approach is obvious: for an excitation composed of a single frequency, the dynamic response can be broken down into several frequencies over ten times the excitation frequency.

The very first excitation is introduced by the behaviour of the main shaft. The neutral gear dynamic rattle appears with impacts between loose gears and the main shaft. The only possible transmission path for vibrations is the shaft, which is supposed to have a linear behaviour. Moreover, the main non linear behaviour is concentrated between the main shaft and the loose gear.

For this paper several models were considered in order to pro-pose the most appropriate one for the study. The very first tested models are lumped models described by [START_REF] Barthod | étude de l'influence des caractéristiques de l'acyclisme sur le bruit de "graillonnement" dans les boites de vitesses automobiles[END_REF] and [START_REF] Ajmi | model for simulating the quasi-static and dynamic behaviour of solid wide-faced spur and helical gears[END_REF]. These models are made with specific non linearity of elasticity and damping with the objective to performing analytical solutions section after section. The dynamic description of the contact and impacts is performed as a succession of sections (based on periodic solutions). In such models, the dynamic response between two temporal sections (large and discontinuous evolution of the stiffness for example) does not take into account the gradual evolution of the stiffness between the two states. Although, several studies ( [START_REF] Henriot | Quelques réflexions sur la génération du bruit et des vibrations dans les engrenages[END_REF], [START_REF] Robbe-Valloire | Modélisation de la topologie microgéométrique[END_REF]) had established that the dynamic response of solids in contact depends strongly on the progression of the stiffness function and more precisely on the geometric defaults (shape and micro geometry in terms of roughness). Simulations performed with these models overestimate the number and the levels of impacts and underestimate the duration of each impact (still using parametric identification obtained by updating the models with measurements). For these reasons this kind of Lumped Model was abandoned.

Other tested techniques ( [START_REF] Blomqvist | Detection of rattle sources in gearbox using correlation technique[END_REF]) consist in building the solution in the frequency domain with a semi analytic method. This approach needs a specific analytical description of elasticity and damping (such as polynomial decomposition) and a large number of terms if realistic simulations are required. More over, this method could not describe an important phenomenon experimentally observed: several impacts occur on the same side of a tooth before changing sides. For these reasons this kind of method was not retained either.

Models based on Finite Element Method ( [START_REF] Crochiere | Multi-rate signal processing[END_REF]) are very heavy models in terms of computing time and memory. Moreover, it remains difficult to describe the complete geometry with specific defaults as roughness, which are important parameters in the dynamic of the shocks. However, this technique could be successfully used with the aim of identifying the structural stiffness of solids. For dynamic excitations this kind of model could not be used with reasonable computation time except for specific kinds of excitation ( [START_REF] Crochiere | Multi-rate signal processing[END_REF]). In the present study, the roughness had to be taken into account and the excitation is not supposed to be known. This is the reason why this method has not been developed in this paper.

The proposed Lumped Model can be considered as a prolongation of classical Single Degree Of Freedom with non linear stiffness and damping. Models of stiffness and damping were chosen with the aim of taking into account the backlash, the geometric default of tooth (roughness) and the nature of the contact.

The elementary model for rotation on the z axis is defined by:

I L θL + K θ (θ M , θ L ) + C θ θM , θL , θ M , θ L + C θL = B (1)
with:

• B : external torque

• C : viscous coefficient for dissipative behaviour introduced by lubricant between the loose gear and the shaft.

The transmission error ξ on the curvilinear axis for translation displacement (on pitch diameters) is defined by:

ξ = ξ L -ξ M = r L θ L -r M θ M (2)
where r L and r M are respectively the pitch radius of the loose gear and the main gear.

Impacts are conditioned by ξ along the pitch diameter, although the nonlinear dynamic equation for the elementary model is written for the angular variable.

K θ (θ M , θ L ) the angular elasticity of the contact and C θ θ M , θ L , θM , θL the angular dissipation of the contact are simply and respectively linked to K ξ (θ M , θ L ) and C ξ θ M , θ L , θM , θL the same function projected on the axis :

K θ (θ M , θ L ) = r L K ξ (ξ M , ξ L ) (3)
C θ θ M , θ L , θM , θL = r L C ξ ξ M , ξ L , ξM , ξL (4)

Contact Elasticity and Dissipative Functions

Five different cases are considered, depending on transmission error value compared to geometric manufacturing errors and backlash (see figure .11).

The explicit expression of the contact elasticity K ξ (ξ M , ξ L ) is:

• when the gears are in contact on side F+ : ∀ξ ≤ -ε

K ξ = K * LM (ξ L -ξ M ) ξ L -ξ M G α-1 (5)
• when the gears are near the contact on side F+ : ∀-ε ≤ ξ ≤ 0

K ξ = K * LM 2 1 -ε 2 + ξ L -ξ M sin π ε (ξ L -ξ M ) ξ L -ξ M G α-1 (6)
• when the gears are separated :

∀0 ≤ ξ ≤ G K ξ = 0 (7)
• when the gears are near the contact on side F-:

∀G ≤ ξ ≤ G + ε K ξ = K * LM 2 1 -ε 2 -(ξ L -ξ M -G) sin π ε (ξ L -ξ M -G) ξ L -ξ M G -1 α-1 (8)
• when the gears are in contact on side F-:

∀G + ε ≤ ξ K ξ = K * LM (ξ L -ξ M -G) ξ L -ξ M G -1 α-1 (9)
with :

• K * LM the equivalent static stiffness of contact between the teeth L and M ( [START_REF] Henriot | Quelques réflexions sur la génération du bruit et des vibrations dans les engrenages[END_REF])

• α the contact-type coefficient (e.g. sphere to plane, cylinder to cylinder ) ( [START_REF] Hertz | Ueber die beruhung fester elastischer korper[END_REF])(see figure .10

• ǫ the size of manufacturing errors on the tooth ( e.g. shape defects and micro-geometric defaults) ( [START_REF] Robbe-Valloire | Modélisation de la topologie microgéométrique[END_REF])

• G the backlash ( [START_REF] Harris | Rolling bearing analysis[END_REF]) on the pitch diameter.

The equation (6 and 8) ensure the continuity of the function K ξ and its first derivatives.

The dissipative coefficient of the contact is defined with the same existing conditions as for the elasticity function.

This kind of normal contact model is used in several applications:

for particular applications such as the dynamic behaviour of the compressor valve contacts or non lubricated gears, a friction dissipative term in the dissipative function had to be introduced

([29]).
The explicit expression for the dissipative coefficient C ξ ξ M , ξ L , ξL , ξM is:

• ξ ≤ -ε C ξ = C LM ξL -ξM • -ε ≤ ξ ≤ 0 C ξ = C LM 2 1 -ε 2 + ξ L -ξ M sin π ε ξL -ξM • 0 ≤ ξ ≤ G C ξ = 0 • G ≤ ξ ≤ G + ε C ξ = C LM 2 1 -ε 2 -(ξ L -ξ M -G) sin π ε ξL -ξM • G + ε ≤ ξ C ξ = C LM ξL -ξM
Where C LM denotes the associated damping of contact between the teeth L and M ([29])

7 Model Characterization

Aims And Methods

The aim of the proposed model is to simulate the source of dynamic excitation in gear boxes.

The model proposed in the present paper is deterministic and preserves stationary properties. Each parameter is invariant and independent of the excitation level and the frequency.

All parameters, except for dissipative behaviour, were identified without dynamic measurements. Most of them had been produced by Renault SA. Some of them, such as K * LM , K θ and α were determined by using classical engineering gears methods ( [START_REF] Harris | Rolling bearing analysis[END_REF]).

The choice of the model and its parameters had been strongly influenced by the possibility to identify the parameters with a greater degree of accuracy.

Each parameter is determined by a simple analytic model based on the design of gear. But they can also be identified by a single dynamic measure on the studied test bench excited with harmonic signal.

For each degree of freedom, 9 parameters had to be identified for the model:

• geometric parameters: I L , r L , r M • elastic parameters: K LM ,α, ǫ, G • dissipative parameters: C LM , C
These parameters are presented and explained in greater detail • r L and r M : are respectively the pitch radius of the loose gear and the main gear (r L =37 mm and r M =28.25 mm)

Elasticity

• K LM : the equivalent elasticity between the L and M teeth in contact. This can be obtained by a numerical identification with a Finite Element study of teeth in contact or more easily by the retained method defined by Palmgren ( [START_REF] Henriot | Quelques réflexions sur la génération du bruit et des vibrations dans les engrenages[END_REF]). To avoid coupling with α and G parameters and unit incompatibility, the parametric identification is performed on

K * LM = K LM .G (α-1) .
This method reveals a non-dimensional variable ξ * = ξ/G and independent parameters K * LM , α, G. (K * LM =3(10 6 ) N.m -1 ;

K LM =1.03.( 10
8
) N.m -4/3 ).

• α : This coefficient is defined according to the type of contact ( [START_REF] Hertz | Ueber die beruhung fester elastischer korper[END_REF]): a simple line contact leads to α=10/9, a single point leads to α=3/2. In fact, for spur or helical gears the actual contact is neither linear nor punctual. This parameter had then to be identified between these two cases and was used to minimize the discrepancy between experimental measures and numerical simulations (Fig . 10). (α=4/3). Generaly speaking gear manufacturing requires that gear teeth are shaped to ensure a contact area limited to the central part of the tooth [START_REF] Harris | Rolling bearing analysis[END_REF].

• ǫ : the size of manufacturing errors on the teeth (shape defects and micro-manufacturing errors) ( [START_REF] Robbe-Valloire | Modélisation de la topologie microgéométrique[END_REF]). This parameter transforms the point discontinuity into a continuous curve over an infinitesimal displacement interval thus the instantaneous stiffness increases from 0 to the equivalent solid stiffness K LM over the deformation interval ǫ of manufacturing errors (ǫ=1.10 -6 m).

• G : the backlash ( [START_REF] Harris | Rolling bearing analysis[END_REF]) on the pitch perimeter (projected onto the ξ axis). This parameter is the main cause of idle gear noise.

As this parameter is well known and statistically verified during the manufacturing process it cannot be excluded from consideration for technological reasons such as assembly or operating constraints. (G=0.025 mm)

The complete elastic function is defined by 6 parameters (r L , r M , K LM , α, ǫ, G) and 2 variables ξ L and ξ M .

K ξ (θ M , θ L ) is exhaustively defined by 4 parameters (K LM , α, ǫ, G). (Fig . 11)

Dissipation

C LM is the associated damping of the L and M teeth in contact ( [START_REF] Henriot | Quelques réflexions sur la génération du bruit et des vibrations dans les engrenages[END_REF]).To avoid coupling with K LM , I L , α, G and rL parameters, the parametric identification is obtained from:

C * LM = C LM 2 K * LM I L r 2 L ( 10
)
This method reveals a non-dimensional and independent parameter C * LM . This parameter cannot easily be predicted before experimentation but it can be accepted as being in the band from 1 to 5 percent.

For several industrial applications this dissipative model can be added or replaced by a friction model with an internal state variable ( [START_REF] Dahl | Solid friction damping of mechanical vibrations[END_REF], [START_REF] De | A new model for control of systems with friction[END_REF], [START_REF] Barabanov | A new model for control of systems with friction[END_REF]). (C * LM =0.04 ; C LM =67 N.s.m -1 ).

The model described in [START_REF] Dahl | Solid friction damping of mechanical vibrations[END_REF] was successfully tested but not retained in the final presented model in order to preserve the simplicity of the global model. More over, the accuracy needed for the dissipation function in this study does not justify the introduction of new parameters.

The complete dissipative function is defined by 5 parameters (r L , r M , C LM , ǫ, G) and 4 variables ξ L , ξ M , ξL and ξM .

C ξ θ M , θ L , θM , θL is completely defined by 3 parameters (C LM , ǫ, G)and produces a dissipative force proportional to the velocity of the impact.

C is the viscous coefficient for dissipative behaviour introduced by a lubricant between the loose gear and the shaft. This coefficient can be identified with an analytic fluid model between two infinite cylinders or with a simple measure of velocity when the loose gear is in free rotation (Fig . 13).

With the intention of constructing independent parameters and to avoid coupling with I L parameter, the parametric identification is led on C * = C/I L This method reveals an independent parameter C * and the time response for an initial velocity S 0 at t 0 is:

S(t) = S 0 e -C * (t-t 0 ) = S 0 e -C I L (t-t 0 ) (11)
S(t) corresponds to computed dash-dot lines in red on Fig. For each comparison, different kinds of kinematic results are shown:

• acceleration on the pitch diameter of the loose gear

• velocity on the pitch diameter of the loose gear

• displacement on the pitch diameter of the loose gear

• transmission error in displacement between the main and loose gears. amplitude peak on the pitch diameter of the main gear)

The comparison based on angular position or displacement along the pitch diameter (ξ axis) (Fig . 12) shows a very strong correlation between experimental measurements and numerical simulations. Often in cases of dynamic systems, observations involving this kind of kinematic variable display a close fit between measurements and models. The comparison of kinematic variables had to be conducted with greater order derivatives to confirm the models.

The simulated velocity describes with precision levels and events during the dynamic simulation: synchronization of the different steps during the excitation, shocks and free motions, are predicted with strong accuracy (Fig . 10).

Acceleration is the more convenient kinematic descriptor for impact identification and vibroacoustic prediction of complete structures. Acceleration levels and the number of impacts are accurately predicted. Simulated peaks decrease in accordance with measurements (Fig . 11).

Another way to identify impacts and dynamic behaviour of gears is to represent the relative displacement between the main gear and the loose gear. This displacement results from the backlash between teeth shown by the gap between the two horizontal dashdotted lines on Fig . 17, .19, .22.

(Fig . 18).

Another way to identify and quantify shocks is obtained in post processing velocities with the intention of measuring the relative velocity when shocks occur (Fig . 19). This graph reinforces the analysis of shocks and can also permit one to quantify the velocity of each shock and thus, the kinematical energy of shocks and its evolutions in Rayleigh (dissipative) and potential (elastic)

Energy.

22

• The experimental study outlines phenomena and parameters responsible for the idle gear noise.

• The model built is a non linear Single Degree Of Freedom in rotation composed of 9 independent and invariant parameters:

The identification of each parameter can be performed with an analytical and numerical approach or by measurements on a single dynamic test. To find the independence of each parameter and for a better accuracy, identifications presented in section 7 are performed with reduced variables and parameters.

In several cases these reduced parameters are dimensionless.

• All simulations were carried out with the same set of parameters and reveal a very close fit with experimental measurements. The kinematic of the impacts is well predictable in terms of magnitude, duration, for number of one-sided impacts as well as the decrease in the magnitude of each impacts.

• Both simulations and experimental measurements reveal the following important points:

• several impacts can often appear on one side of the gear mesh,

• the number of impacts on one side strongly depends on the angular displacement of the main shaft,

• the energy level of impacts mainly depends on the acceleration level of the main shaft oscillations,

• The excitation level at which impacts appear is identified, predicted and closely connected with the acoustic phenomenon called idle gear noise.

• The next step of this study is to undertake a vibroacoustic model of the gear box based on this study of a single couple of gears. The aim of this work is to show the predictable vibroacoustic behaviour of a complete gear box with a small number of (non linear) Degrees of Freedom.

1 List of symbols 1

 1 Definition Variable Angular position of the main shaft θ M Angular position of the loose gear θ L Definition Parameters Viscous coefficient introduced by lubricant between shaft and gear C Strutural damping of the L and M teeth in contact C LM Angular damping of the contact C θ Translational damping of the contact (along ξ-axis) C ξ Momentum inertia of the loose gear I k zz(k) (P ) = I L Strutural elasticity of the L and M teeth in contact K * LM Angular elasticity of the contact K θ Translational elasticity of the contact (along ξ-axis) K ξ Radius of the loose gear r L Radius of the main shaft r M Geometric defaults on the tooth ε Exponent defined by the nature of the contact α Backlash G Pitch diameter on the loose gear ξ L Pitch diameter on the main shaft ξ M

in paragraphs 7 .2 to 7 . 4 . 7 . 2

 77472 Numerical values for each parameter are given in the case of the studied gear box. Corresponding simulations are presented in paragraph 8, and compared with the experiments. Geometric parameters • I L : Moment of Inertia for the loose gear. Normally, this parameter is obtained numerically from the geometrical model and the density of the material. (I L =0.32(10 -3 ) kg.m 2 )

9 (C

 9 * =78 s -1 ; C=0.025 N.s.m). When these dash-dot lines are compared with free motion episodes of the loose gear, they represent a good method to estimate the quality of the numerical identification of C.8 Numerical and Experimental ComparisonAll simulations shown were performed with numerical values defined in section 7 for each parameter of the model built.In an effort to improve the graphic legibility and kinematical analysis, measured and simulated signals are studied between 0 Hz and 1000 Hz using classical filtering techniques (10 th order Cauer filter with phase compensations). Three comparisons between the numerical simulations and the experimental measures are shown. Each comparison is defined by the excitation level of the main shaft and its frequency:• 30 Hz, 300 Rad.s -2 RMS (12 m.s -2 amplitude peak on the pitch diameter of the main gear)• 60 Hz, 300 Rad.s -2 RMS (12 m.s -2 amplitude peak on the pitch diameter of the main gear)• 30 Hz, 500 Rad.s -2 RMS (20 m.s -2 amplitude peak on the pitch diameter of the main gear) Other comparisons were made for frequencies between 30 Hz and 60 Hz (by 2 Hz steps) and for several angular accelerations from 100 Rad.s -2 RMS to 700 Rad.s -2 . The presented comparisons cover a large field of significant cases with a high level of idle gear noise.

8. 1

 1 Initial Comparison: The Lower Limit Frequency Of Idle Gear Noise The first comparison is undertaken at an excitation frequency of 30 Hz and an acceleration of 300 Rad.s -2 RMS (12 m.s -2

Fig . 1 .Fig . 2 .Fig . 3 .Fig . 4 .Fig . 5 .Fig . 6 .

 123456 Fig .1. Schematic diagram of the experimental setup

Fig . 8 . 34 Fig . 9 .Fig . 11 .Fig . 13 .Fig . 24 . 2 . 1 27 . 2 Pictures of the experimental setup 28 . 3 Schematic diagram of gear configuration 29 . 4 30 . 5 31 . 6 . 7 2 excitation 33 . 8 . 9

 834911132421272283294305316723389 Fig .8. Evolution of loose gear acceleration spectral shape versus main shaft acceleration level

Fig .7. Kinetical gear behaviour for a 30 Hz -500rad/s 2 excitation

Fig .17. Relative displacement (instantaneous backlash) for 30 Hz and 300 Rad s -2

Acknowledgment

The authors express their gratitude to Renault SA for the financial support of this study.

Second Comparison: The Upper Limit Frequency Of Idle Gear Noise

The second comparison is performed at a frequency of 60 Hz and an acceleration of 300 Rad.s -2 RMS (12 m.s -2 amplitude peak on the pitch perimeter of the main gear). The model's parameters are numerically the same as those used for the previous comparison and those identified in section 7.

The evolution of the frequency from 30 Hz to 60 Hz reduces the number of shocks from 4 or 5 to only 2 or 3 shocks each time on the same side. The level of acceleration does not change significantly as the frequency increases (Fig . 13).

Transmission error clearly shows only two shocks on the same side of the tooth before changing sides (Fig . 14). When the frequency increases the number of shocks on the same side is reduced and the length of free motion phases of the loose gear significantly shortens.

Velocity (Fig . 15) is also correctly predicted in terms of magnitude and events by the numerical simulation.

Third Comparison: Effect of an excitation level increase

The third comparison is performed at a frequency of 30 Hz and an acceleration of 500 Rad/s 2 RMS (20 m.s -2 amplitude peak on the pitch perimeter of the main gear). The model's parameters are numerically the same as those used for the previous comparisons and those identified in section 7.

When the excitation magnitude increases, the number of shocks increases as does their acceleration magnitude (Fig . 16).

Shock period and number can also be predicted from the transmission error diagram (Fig. .17). Shocks seem to happen over the interval G between teeth (red dash-dotted lines). In fact, this effect is due to the coupling between dynamic behaviour of the loose gear and the non linear stiffness of its teeth (progression depending on manufacturing errors ǫ).

When the excitation magnitude increases, the model maintains a


O Lézoray

A Elmoataz

V.-T Ta

NONLOCAL PdES ON GRAPHS FOR ACTIVE CONTOURS MODELS WITH APPLICATIONS TO IMAGE SEGMENTATION AND DATA CLUSTERING

Keywords: Partial difference Equations (PdEs), Graphs, Global active contours, Nonlocal, Clustering

We propose a transcription on graphs of recent continuous global active contours proposed for image segmentation to address the problem of binary partitioning of data represented by graphs. To do so, using the framework of Partial difference Equations (PdEs), we propose a family of nonlocal regularization functionals that verify the co-area formula on graphs. The gradients of a sub-graph are introduced and their properties studied. Relations, for the case of a sub-graph, between the introduced nonlocal regularization functionals and nonlocal discrete perimeters are exhibited and the co-area formula on graphs is introduced. Finally, nonlocal global minimizers can be considered on graphs with the associated energies. Experiments show the benefits of the approach for nonlocal image segmentation and high dimensional data clustering.

INTRODUCTION

With the advent of our digital world, much different kind of data are now available. On the contrary to classical images and videos, these data do not necessarily lie on a Cartesian grid and can be irregularly distributed. To represent a large number of data domains (images, meshes, social networks, etc.), the most natural and flexible representation consists in using weighted graphs by modeling neighborhood relationships. Therefore, there is much interest in the transposition of signal and image processing tools for the processing of functions on graphs, e.g., spectral graph wavelets [START_REF] Hammond | Wavelets on graphs via spectral graph theory[END_REF] or PDE on graphs [START_REF] Grady | The piecewise smooth Mumford-Shah functional on an arbitrary graph[END_REF][START_REF] Elmoataz | Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing[END_REF]. In this paper, we consider active contour models. They can be casted in two models: geometric models with level sets and variational models with Total Variation. We consider the second model and extend it to the processing of arbitrary data on graphs as well as unify their local and nonlocal processing for images. To do this transcription, we use the framework of PdEs [START_REF] Elmoataz | Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing[END_REF] along with a family of discrete gradients. We introduce nonlocal perimeters as well as the coarea on graphs. We then consider nonlocal global minimizers for segmentation and clustering and provide an application example with nonlocal Chan-Vese on graphs.

PARTIAL DIFFERENCE EQUATIONS ON GRAPHS

In this section, we recall definitions and operators on graphs.

All the operators presented in this section are naturally local and nonlocal [START_REF] Elmoataz | Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing[END_REF]. Indeed, local and nonlocal configurations are expressed by graph topology and edges' weights.

Preliminary notations

A weighted graph G = (V, E, w) consists in a finite set V of vertices and a finite set E ⊂ V × V of weighted edges. We assume G to be undirected, with no self-loops and no multiple edges. Let e i j = (v i , v j ) be the edge of E that connects vertices v i and v j of V. Its weight, denoted by w i j = w(v i , v j ), represents the similarity between its vertices and is computed using a positive symmetric function w. Notation v i ∼ v j is also used to denote two adjacent vertices. The volume of a set of vertices A can be computed by vol

(A) = ∑ v i ∈A d A (v i ) with d A (v i ) = ∑ v j ∼v i w(v i , v j
), ∀v i ∈ A the degree of a vertex relatively to a set of vertices A. Let H (V) be the Hilbert space of real-valued functions defined on the vertices of a graph. A

function f : V → R of H (V) assigns a real value x i = f (v i )
to each vertex v i ∈ V. By analogy with functional analysis on continuous spaces, the integral of a function f ∈ H (V), over the set of vertices V, is defined as V f = ∑ V f . Similarly, let H (E) be the space of real-valued functions defined on the edges of G. Both Hilbert spaces are endowed with the usual inner products.

Difference operators

Let f : V → R be a function of H (V). The difference opera- tor [3] of f , noted d w : H (V) → H (E)
, is defined on an edge

(v i , v j ) ∈ E by: (d w f )(v i , v j ) = w(v i , v j ) 1/2 ( f (v j ) -f (v i )).
The directional derivative of f , at a vertex v i ∈ V, along an edge e i j = (v i , v j ), is defined as ∂ f

∂e i j v i = ∂ v j f (v i ) = (d w f )(v i , v j ).
Based on the difference operator, the weighted morphological external and internal difference operators are respectively [START_REF] Ta | Nonlocal PDEs-based morphology on weighted graphs for image and data processing[END_REF]:

(d + w f )(v i , v j )= max 0, (d w f )(v i , v j ) and (d - w f )(v i , v j )= -min 0, (d w f )(v i , v j
) . The correspond-ing external and internal partial derivatives are

∂ + v f (v i ) = (d + w f )(v i , v j ) and ∂ - v f (v i ) = (d - w f )(v i , v j
(d * w H)(v i ) = -div w (H)(v i ) = ∑ v j ∼v i w(v i , v j ) 1/2 (H(v j , v i ) -H(v i , v j )).

Gradients and norms

The weighted gradient operator of a function f ∈ H (V), at a vertex v i ∈ V, is the vector operator defined by

(∇ w f)(v i ) = [∂ v j f (v i ) : v j ∼ v i ] T .
The L p norm of this vector is defined by [START_REF] Elmoataz | Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing[END_REF]:

(∇ w f)(v i ) p p = ∑ v j ∼v i w(v i , v j ) p/2 f (v j )-f (v i ) p .
Similarly, we have

(∇ ± w f)(v i )=[∂ ± v j f (v i ) : v j ∼ v i ] T and (∇ ± w f)(v i ) p p = ∑ v j ∼v i w(v i , v j ) p/2 M ± 0, f (v j )-f (v i ) p
with M + = max and M -= min. These norms exhibit the following property:

(∇ w f)(v i ) p p = (∇ + w f)(v i ) p p + (∇ - w f)(v i ) p p .

Anisotropic p-Laplace operator

The weighted p-Laplace anisotropic operator of a func-

tion f ∈ H (V), noted ∆ a w,p : H (V) → H (V), is defined by: (∆ a w,p f )(v i ) = 1 2 d * w (|(d w f )(v i , v j )| p-2 (d w f )(v i , v j )).
Using the definitions of d w and d * w , we obtain this expression [START_REF] Lezoray | Partial differences as tools for filtering data on graphs[END_REF]:

(∆ a w,p f )(v i ) = ∑ v j ∼v i (γ a w,p f )(v i , v j )( f (v i ) -f (v j )). with (γ a w,p f )(v i , v j ) = w p/2 i j | f (v i ) -f (v j )| p-2
. This operator is nonlinear if p = 2. In this latter case, it corresponds to the combinatorial graph Laplacian.

VARIATIONAL FRAMEWORK OF ACTIVE CONTOURS ON GRAPHS

In this section, we introduce a family of nonlocal regularization functionals that verify the co-area formula for p = 1. The gradients of a sub-graph are introduced and their properties studied. Then, we show that there are strong relations, for the case of a sub-graph, between the introduced nonlocal regularization functionals and nonlocal discrete perimeters. Finally nonlocal global minimizers can be considered on arbitrary graphs with the associated nonlocal energies.

Nonlocal regularization functionals

Let us consider nonlocal regularization functionals based on weighted total variation (TV w ) on graphs R w,p :

H (V ) → R of a function f ∈ H (V ): R w,p ( f ) = ∑ v i ∈V D w f(v i ) p
p with 0 < p < +∞. D w is a discrete gradient operator that can be among ∇ w and ∇ ± w . By replacing D w with the previous previous discrete gradients one obtains several functionals: R w,p ( f

) = ∑ v i ∈V ∑ v j ∼v i w(v i , v j ) p/2 f (v j )-f (v i )
p , and

R ± w,p ( f ) = ∑ v j ∈V ∑ v j ∼v i w(v i , v j ) p/2 M ± 0, f (v j )-f (v i ) p with property R w,p ( f ) = R + w,p ( f ) + R - w,p ( f ).
Finally, using the chain rule, we can show [START_REF] Lezoray | Partial differences as tools for filtering data on graphs[END_REF] that

∂R a w,p ( f ) ∂ f (v i ) = 2p(∆ a w,p f )(v i ).
3.2. Nonlocal perimeters and co-area on graphs

Gradients of a sub-graph

Let A be a set of connected vertices with A⊂V. We denote by ∂ + A and ∂ -A, the external and internal boundary sets of A, respectively. A -=V \ A is the complement of A. For a given vertex v i ∈V,

∂ + A = {v i ∈A -: ∃v j ∈A with (v i , v j )∈E} and ∂ -A = {v i ∈A : ∃v j ∈A -with (v i , v j )∈E}.
We now consider the gradients of a sub-graph and study their properties. We denote by χ A the indicator function defined by χ : V → {0, 1}.

Proposition 1

For a given vertex, if v i ∈ A, then χ A (v i ) = 1 and χ A (v i ) = 0 otherwise. The L p -norm of the directional gradient of the indicator function is defined as:

∇ ± w χ A (v i ) p p = χ ∂ ± A (v i ) ∑ v j ∼v i ,v j ∈A ± w(v i , v j ) p/2 (1)
with the notation A + =A and A -=V \ A.

Proof. By studying cases where v i ∈ A, v i / ∈ A and similarly with the neighbors of v i , we can prove that only the inner and outer set of vertices ∂ ± A are concerned. This leads to the following property:

∇ w χ A (v i ) p p = ∇ + w χ A (v i ) p p + ∇ - w χ A (v i ) p p 3.2.

Nonlocal functionals and nonlocal perimeters

Now, we show that there are strong relations, for the case of a sub-graph, between the previous discrete regularization functionals and discrete perimeters.

Proposition 2

By replacing f by the indicator function χ A in the regularization functionals, one has: R ± w,p (χ A ) = 1 2 vol(∂A).

Proof.

Using equation (1) and the nonlocal functionals

, R + w,p (χ A ) = ∑ v i ∈V ∑ v j ∼v i ,v j ∈A w(v i , v j ) p/2 χ ∂ + A (v i ) = ∑ v i ∈∂ + A ∑ v j ∈∂ -A w(v i , v j ) p/2 = ∑ v i ∈∂ + A d ∂ -A (v i ) = 1
2 vol(∂A). Similarly, one has R - w,p (χ A ) = 1 2 vol(∂A), and R w,p (χ A ) = vol(∂A).

All the previous functionals can be then seen as the inner (R - w,p (χ A )), outer (R + w,p (χ A )) or global (R w,p (χ A )) p-cost of a cut between two set of vertices A and A -. As a consequence, all these previous expressions can be seen as nonlocal discrete perimeters of a sub-graph:

2Per ± w,p (A) = Per w,p (A) = 2R ± w,p (χ A ) = R w,p (χ A )
This shows explicitly the relations between the functionals we defined with Total Variation and nonlocal discrete perimeters of a sub-graph. Interestingly, we can recover on grid graphs with specific weights, a discrete analogue of the continuous nonlocal perimeters introduced in [6].

Co-area formula on weighted graphs

Now we show that for p = 1, the introduced nonlocal functionals do verify the co-area formula.

Proposition 3

For any function f ∈ H (V) and for p = 1, one has the following property:

R * w,1 ( f ) = +∞ -∞ R * w,1 (χ { f >t} )dt (2)
where

R * w,1 ( f ) is among R w,1 ( f ) and R ± w,1 ( f ) with χ { f >t} (v i ) = 0 if f (v i ) < t and χ { f >t} (v i ) = 1 if f (v i ) ≥ t.

Proof.

All the functionals R * w,1 appearing in (2) are of the form R w,1 ( f

) = ∑ v i ∈V ∑ v j ∼v i w(v i , v j )| f (v j ) -f (v i )| or R ± w,1 ( f ) = ∑ v i ∈V ∑ v j ∼v i w(v i , v j )( f (v j ) -f (v i )) ± with x + = max(0, x) is the nonnegative part of the real number x (x -= (-x) + ). Since |a -b| = +∞ -∞ |χ {a>t} -χ {b>t} |dt and (a -b) ± = +∞ -∞ (χ {a>t} -χ {b>t} ) ± dt, R * w,1 clearly satisfies (2).
This result is different from [START_REF] Bresson | Non-local unsupervised variational image segmentation models[END_REF] that assumes that the set of vertices of the graph belong to a submanifold and then make use of the co-area formula on manifolds. Our proposal does not need this assumption and is much simpler since have the co-area formula for any weighted graph with p = 1.

Nonlocal active contours models as global minimizers for segmentation and clustering

The usual drawback of active contours methods is the existence of local minimizers with the solution highly dependent of the initial condition. A recent method, introduced by Bresson and Chan [START_REF] Bresson | Fast global minimization of the active contour/snake model[END_REF][START_REF] Bresson | Non-local unsupervised variational image segmentation models[END_REF], proposes to redefine the active contour model into a model which gives global minimizers. Using our framework we show how we can introduce nonlocal global minimizers of active contours models on graphs. Let us consider this very general minimization problem on graphs:

min f :V→{0,1} TV w ( f ) + λH( f , f 0 ) (3)
where TV w ( f ) is a weighted total variation of f . Both TV w ( f ) and H( f , f 0 ) are considered as satisfying the co-area formula. When λ = 0, this energy can be considered as the nonlocal discrete analogue on graphs of [START_REF] Bresson | Non-local unsupervised variational image segmentation models[END_REF]. When λ = 0, it is the discrete analogue to the continuous min-cut of [START_REF] Houhou | Semisupervised segmentation based on non-local continuous min-cut[END_REF] since this reduces to minimize Per * w,p (A) = vol(∂A) = cut(A, A -). Now, we show how such a minimization can be solved but we restrict ourselves to the following minimization problem:

min f :V→{0,1} E w,1 ( f ) = R * w,1 ( f ) + λ ∑ v i ∈V r( f 0 (v i )) f (v i ) (4)
where

R * w,1 ( f ) is among R w,1 ( f ) and R ± w,1 ( f ).
As previously shown, the first part R * w,1 ( f ) of the energy E w,p ( f ) satisfies the co-area formula. It can also be easily shown that it is also the case for the second part of the energy. Minimization problem ( 4) is non-convex and, as shown in [START_REF] Nikolova | Algorithms for finding global minimizers of image segmentation and denoising models[END_REF] for the continuous analogue, can be reformulated to consider a convex relaxation problem to determine a global minimizing solution. Therefore, a new minimization problem is considered:

min f :V→[0,1] Êw,1 ( f ) = R * w,1 ( f ) + λ ∑ v i ∈V r( f 0 (v i )) f (v i ) (5)
Following the approach in [START_REF] Nikolova | Algorithms for finding global minimizers of image segmentation and denoising models[END_REF], one can show, that every level set of a minimizer of ( 5) is a solution of the original optimization problem (4). As a consequence, to obtain a global solution f : V → {0, 1} to the problem (4), one thresholds any function f : V → [0, 1] that is a solution of ( 5) and f = χ A ( f ).

To establish the solution of the minimization problem (4), we can solve the following system of equations:

∂ Êw,1 ( f ) ∂ f (v i ) = ∂R * w,1 ( f ) ∂ f (v i ) + λr( f 0 (v i )) = 0, ∀v i ∈ V, (6)
First it is easy to show that one has the following relation

∂R w,1 ( f ) ∂ f (v i ) = 2 ∂R + w,1 ( f ) ∂ f (v i ) = 2 ∂R - w,1 ( f ) ∂ f (v i ) = 2∆ a w,1 f (v i ).
To approximate the solution of the minimization problem (5), the algorithms exposed in [START_REF] Bresson | Non-local unsupervised variational image segmentation models[END_REF] can be used (dual/projection, graph cuts). We choose to use, for the sake of simplicity, the linearized Gauss-Jacobi method and obtain the following iterative algorithm, ∀v i ∈ V :

f (t+1) (v i ) = -λr( f 0 (v i )) + α ∑ v j ∼v i γ f (t) w (v i , v j ) f (t) (v j ) α ∑ v j ∼v i γ f (t) w (v i , v j ) (7)
where γ

f (t) w (v i , v j ) = w(v i , v j ) 1/2 | f t (v i ) -f t (v j )| -1 ε . With α = 1 for R ±
w,1 ( f ) and α = 2 for R w,1 ( f ), and the initial condition f (0) = f 0 . The problem being relaxed on R, at each iteration t, the solution is projected on [0, 1] by max(min( f t , 1), 0).

APPLICATIONS

In the following, we discuss the application of our proposed approach to Chan-Vese (CV) model. Since graphs of the arbitrary topologies can be considered, our approach enables to consider any type of discrete data that can be represented by a graph. Moreover, it can be used to segment image with nonlocal configurations and to cluster high dimensional data. We can directly express the discrete analogue on graphs of the CV model as the following minimization, ∀v i ∈ V :

min f :V→{0,1} R * w,1 ( f ) + λ ∑ v i ∈V f (v i )m(c 1 , v i ) + (1 -f (v i ))m(c 2 , v i ) (8)
where m(a, v i ) = (af 0 (v i )) 2 . c 1 and c 2 are the means inside and outside the object. For a fixed f , the minimizer (c 1 ( f ), c 2 ( f )) can be explicitly computed. For fixed (c 1 , c 2 ), the minimization over f can be solved with algorithm [START_REF] Bresson | Non-local unsupervised variational image segmentation models[END_REF] with

r( f 0 (v i )) = (c 1 -f 0 (v i )) 2 -(c 2 -f 0 (v i )) 2 .
Therefore, the minimization of CV model on graphs is performed as an alternate minimization with respect to (c 1 , c 2 ) and with respect to f . Figure 1 shows the application of our CV model on many different graphs with R w,1 ( f ). First column presents a noisy textured image with an initial contour, the segmentation result on local (4-adjacency graph with Gaussian weights computed on pixel values) and nonlocal (4-adjacency graph coupled with a 4-Nearest Neighbor graph selected in a 9 × 9 window and Gaussian weights computed on 3 × 3 patches) graphs. The benefits of nonlocal patch-based configurations for image segmentation is evident for segmenting textured objects. Second column presents an image with an initial contour, the considered graph (a super-pixel graph obtained from an over-segmentation with Gaussian weights on region mean values), and the obtained partition of the super-pixel graph. This shows the interest of our formulation that can be used on a different representation of the image. Third column presents a mesh with an initial contour (on pig's back), the considered mesh (a triangular mesh with Gaussian weights on vertices grey values) with an image of text projected onto the surface, and the obtained partition of the mesh (all the text is extracted). Again, the algorithm remains the same, only graph changes. Our approach on meshes is therefore much more general than the recent framework proposed in [START_REF] Lai | A framework for intrinsic image processing on surfaces[END_REF]. Last column presents an example of high-dimensional data classification: an image database with an initial random partition, the considered graph (a 10 nearest neighbors graph weighted with Gaussian weights on 16 × 16 vectors associated to the image of each vertex). This shows how our CV models can be used in the space of images for classification; our CV model also corresponding a TV-regularized 2-means on graphs.

CONCLUSION

In this paper we proposed nonlocal PdEs on graphs for active contours models with applications to image segmentation and data clustering. Starting from nonlocal regularization functionals on graphs, we showed they verify the coarea formula on graphs and introduced nonlocal perimeters on graphs. These ingredients are then used for nonlocal active contours models as global minimizers for segmentation and clustering. Experimental results with the Chan-Vese models showed the benefits of the approach that enables to perform the binary partitioning of any graph.

Fig. 1 .

 1 Fig. 1. Chan-Vese on various graphs (local and nonlocal, super-pixel, mesh, k-nearest neighbors) for image segmentation and data clustering.

  ). The adjoint of the difference operator, noted d * w : H (E) → H (V), is a linear operator defined, for all f ∈ H (V) and all H ∈ H (E), byd w f , H H (E) = f , d * w H H (V). This adjoint operator d *

	w , of
	a function H ∈ H (E), can by expressed at a vertex v i ∈ V
	by the following expression: