LEMNA Pierre-Jean Messe

Nathalie Greenan

Arnaud Chéron

Bruno Decreuse

Dorothée Charlier

Bérangère Legendre

Olivia Ricci

Sylvain Chareyron

Naomie Mahmoudi

Jérémy Tanguy

Laetitia Challe

L ' Yannick

Pascale Horty

Petit

Yannick L'horty

Philomene Mbaye

Pascale Petit

Jean-François Hoarau

Nicolas Lucic

Loïc Emmanuel Duguet

Du Parquet

Julien Albertini

Xavier Fairise

Arthur Poirier

Anthony Terriau

Andri Immigration

Idriss Chassamboulli

Ismael Fontaine

Galvez-Iniesta

Idriss Fontaine

Sabine Garabedian

Maël Jammes

Idriss Fontaine

Helene Veremes

Marie-Noëlle Lefebvre

Eddy Zanoutene

Marnix Amand

Florian Pelgrin

Fabrice Gilles

Sabina Issehnane

Florent Sari

Ferhat Mihoubi

Christopher Ball

Nicolas Groshenny

Özer Karagedikli

Murat Özbilgind

Finn Robinsona

Souleymane Mbaye

Kevin Spiritus

Etienne Lehmann

Sander Renes

Floris T Zoutman

Emilie Dargaud

Armel Jacques

Laurence Jacquet

Anne-Sophie Bruno

Mohamed Ali

Ben Halima

Joseph Lanfranchi

David Gray

Nicolas Moreau

Idriss Fontaine

David Nortes-Martinez

Hélène Vérèmes

Malak Kandoussi

François Langot

Serge Blondel

Radu Vranceanu

Daniel Rakotomalala

Sébastien Ménard

Stéphane Blancard

Maximin Bonnet

Knowledge transmission in the second part of careers: does formal training matter?

Keywords: JEL Classification: J14, J24 Older workers, knowledge transmission, skill obsolescence, technological and organizational change, training

This paper examines to what extent formal training targeted to workers aged 45 and over could enhance their knowledge transmission activities and specifically in changing work environments. This is a key issue for Human Resources practitioners. Allowing older workers to keep on interacting with their colleagues and transmitting their knowledge acquired through experience reduces the risk for firms of losing critical knowledge assets.

Design/methodology approach

We use French matched employer-employee data to estimate the effect of participation in training sessions intended to support change on the probability for workers aged 45-59 of frequently showing work practices to their co-workers.

 show that this is a valid approximation of a random assignment to training.

Originality

This study sheds a new light on the effectiveness of older workers' training. Some contributions argue that training for older worker is not very effective because it has no significant effect on employment duration, earnings or on relative productivity.

We show that specific types of training to update skills after a technological or organizational change allow older workers to keep interacting with their co-workers and pass their knowledge gained through experience, thereby reducing the risk for firms of losing critical knowledge assets.

Research implications

Our findings suggest that two main aspects have to be borne in mind when assessing the effectiveness of training for older workers. First, the reasons for training must be carefully considered, especially if it occurs in response to technological or organizational change in the workplace. Second, the continuation of interactions between older workers and their co-workers must be factored.

If the public debate acknowledges that employee learning and development is critical in times of structural change and crisis, the outcomes of knowledge transmission within workplaces in terms of job satisfaction, turnover intentions, productivity or innovation, which we do not cover in this paper, deserve further investigations. In particular, we believe that studying how the training that supports technological and organizational change influences the relationship between age diversity and firm productivity is a promising avenue for future research.

Practical implications

The implication of this article for human resource managers is that there may be a substantial cost to not updating the skills of older workers after technological or organizational change. Indeed, it is likely that a large proportion of jobs will only be partially automated, which implies that while some tasks will disappear, rendering the corresponding skills obsolete, others will persist and the skills associated with them will remain useful to organisations.

If older workers are excluded from their work collectives after these changes, because their skills have not been updated through training, the knowledge from their accumulated experience that remains valuable will be irrevocably lost when they retire.

Findings

Training with the intention to support change for workers aged 45 and more significantly increases knowledge transmission for training participants. This effect is not strictly related to a supervising role as it is significant for workers without subordinate; it holds when we address the selection bias into training by narrowing down the comparison group.

When training comes as a response to mitigate the potential negative effects of technological or organizational changes in the work environment, it helps workers aged 45-59 maintain their contribution to the knowledge base of the production.

Introduction

The rapid ageing of the workforce is a major challenge for human resource practitioners. They have to maintain older workers' employability to encourage them to stay longer in their jobs. In addition, they have to cultivate greater intergenerational knowledge sharing to deal with increasingly age-diverse teams and organizations. In this setting, European Union employers and trade unions have negotiated in March 2017 a framework agreement on active ageing as well as an inter-generational approach. This agreement has two main goals: improving the ability of workers of all ages to remain healthy and active in work until the legal retirement age and facilitating transfers of knowledge and experience between generations.

However, recent studies that have examined the effective transmission of knowledge from experienced employees point out a strong decline of knowledge transmitters over age 45, i.e. in their second part of careers [START_REF] Masingue | Seniors tuteurs: comment faire mieux?[END_REF][START_REF] Molinié | Avoir un rôle de tuteur[END_REF][START_REF] Greenan | Transmission of vocational skills in the second part of careers: the effect of ICT and management changes[END_REF]. This suggests that the knowledge accumulated by employees approaching retirement age is no longer passed on, which can lead to inefficiencies if younger employees have to relearn on their own instead of drawing on the experience of older employees. To explain this phenomenon, [START_REF] Greenan | Transmission of vocational skills in the second part of careers: the effect of ICT and management changes[END_REF] put forward the role of technological change and specifically the introduction of Information and Communication Technologies (ICTs) in organizations. Some empirical evidence show that these innovations accelerate the obsolescence of specific skills acquired by senior workers [START_REF] De Grip | The economics of skills obsolescence: a review[END_REF] and affect them negatively through adaptability requirements [START_REF] Aubert | New technologies, organisation and age: firm-level evidence[END_REF][START_REF] Bartel | Technological change and retirement decisions of older workers[END_REF]. Consequently, workers aged 45 and over have progressively lost their role of knowledge transmitters in the dynamic work environment of the most technology advanced firms.

Building on these insights, this paper examines to what extent formal training targeted to workers aged 45 and over could enhance their knowledge transmission activities and specifically in changing work environments. We start from the assumption that when an important change in the techniques used in the production process is implemented, part of the skills accumulated by experienced workers become obsolete while others remain valuable for the organization. [START_REF] Backes-Gellner | Skill obsolescence, vintage effects and changing tasks[END_REF] distinguish two kinds of tasks and required skills: those based on technical knowledge (e.g. programming software) and those based on experience (selling or negotiating contracts). From German data, they show that workers performing knowledge-based tasks face greater skill obsolescence than those performing experience-based ones. More generally, [START_REF] Green | Employee involvement, technology and evolution in job skills: A task-based analysis[END_REF] notes that the frontier between tasks and skills affected by the development of ICTs and those that are not remains difficult to draw. This strongly relies on the decisions that companies make to capitalize on the knowledge-based resources held by their employees when the new technology is implemented and its uses defined. In any case, offering training opportunities to workers in the second part of their careers to update their obsolete skills is a way to maintain access to those skills that still contribute to the knowledge base of production. In doing so, these workers remain integrated to the process of knowledge transmission within the organization. Of course, we are not suggesting here that this is the only role of training, nor that it has no potential impact on productivity. We recognize that training is also likely to maintain the employability of workers through the renewal of their skill set. Nevertheless, our empirical strategy does not aim to provide evidence in this respect. We focus on the knowledge transmission effect of training, an indirect effect that the literature has scarcely studied.

The sharp decline with age of participation in training is widely acknowledged [START_REF] Becker | Investment in human capital: A theoretical analysis[END_REF][START_REF] Behaghel | Training and age-biased technical change[END_REF][START_REF] D'addio | Population ageing and labour markets[END_REF]. This can be partly explained by the shorter period when this training investment will be valorized (Cheron and Terriau, 2018), by a lack of motivation of older workers for improving their human capital [START_REF] Warr | Age and personal initiative at work[END_REF] or by employers' perceptions about their lower adaptability [START_REF] Taylor | The ageing workforce: Employers' attitudes towards older people[END_REF][START_REF] Loretto | Employers' attitudes, practices and policies towards older workers[END_REF][START_REF] Van Dalen | Productivity of older workers: Perceptions of employers and employees[END_REF][START_REF] Wang | Mid and late career issues: An integrative perspective[END_REF]. Some specific policies to combat this decline have been pioneered, such as an age-dependent tax deduction on training expenses in the Netherlands [START_REF] Leuven | Evaluating the effect of tax deductions on training[END_REF]. Hence workers aged 45 and over that participate in training are likely to be highly selected. To address this selection bias, we follow the approach suggested by [START_REF] Leuven | An alternative approach to estimate the wage returns to privatesector training[END_REF]. We narrow down the group of untrained workers to those who wanted to attend to a training session but had to cancel their participation on the basis of exogenous reasons. [START_REF] Leuven | An alternative approach to estimate the wage returns to privatesector training[END_REF] show that this is a valid approximation of a random assignment to training. This method allows us to come closer to the causal effect of training for workers aged 45 and over on their probability of passing on their knowledge to other colleagues.

We use data from a French matched employer-employee survey on training and workers' trajectories (DEFIS) conducted in 2015. Respondents declare how frequently they show some work practices to other colleagues, so we can identify the workers who are integrated to knowledge transfer process of their firm. In addition, the DEFIS survey provides a detailed description of employees' training participation and training characteristics and goals. The employee level survey is linked to a firm level that provides information on workplace characteristics and firms' specific Human Resources Management practices. We can control for a large set of reliable company, job and work environment's characteristics that may influence both the access to training at 45 and over and the probability of showing frequently work practices to colleagues. To help us carry out our identification strategy, we rely on specific questions indicating whether there was a training session that respondents wanted to attend but were unable to do so and for what reasons.

We find that participation in training has a positive effect on the probability of transmitting knowledge and skills after age 45. More interestingly, this effect is only significant and substantial when training aims at accompanying a change occurred in the work environment. Other types of training, such as to become more effective at work or to take on responsibilities, do not have a significant impact on the participation of older people in the knowledge transmission process. This suggests that promoting access to training for seniors aged 45 and over to support a change in the working environment is a good human resource practice to keep these employees integrated into the knowledge transmission process. Beyond skills' updating, it favors the retention of valuable knowledge from experienced employees, such as knowledge about subject matter expertise, business relationships and social networks, organizational knowledge and institutional memory or knowledge of governance [START_REF] Joe | Knowledge loss when older experts leave knowledge-intensive organisations[END_REF].

The remainder of our paper is organized as follows. Section 2 presents a short overview of the literature on knowledge transmission. Section 3 describes the data set and some descriptive statistics. In section 4, we explain our empirical strategy and we show our empirical results. Section 5 concludes.

Background

Knowledge transfers between colleagues have been thoroughly studied in the management literature, particularly in terms of mentor-protégé relationships. Starting from the traditional model in which one senior assists with the protégé's personal and professional development (Ragins, 1997), it has been shown that setting mentor-protégé relationships may help professionals learn technical knowledge and organizational ropes as well as improve managerial talent [START_REF] Kram | Mentoring at work: Developmental relationships in organizational life[END_REF]. Furthermore, mentorship is an efficient training and development tool for protégés (Roche, 1979) and it improves mentors' satisfaction [START_REF] Hunt | Mentorship: A career training and development tool[END_REF]. Management literature on mentorship has also introduced social networks theory to go beyond the single dyadic relationship and studied developmental networks [START_REF] Higgins | Reconceptualizing mentoring at work: A developmental network perspective[END_REF].

In economics, while the relevance of learning by doing was recognized in the seminal papers about human capital accumulation (Mincer, 1961;Arrow, 1962;[START_REF] Becker | Investment in human capital: A theoretical analysis[END_REF]Rosen, 1972), early human capital theory has not paid much attention to the role of knowledge transmission on skills' development or productivity and focuses more on education and formal training. Until recently, only a few empirical studies point out the importance of learning from co-workers for human capital development. Bishop (1991) shows that informal training by co-workers or training by watching others have positive and significant effects on new hires' productivity. [START_REF] Loewenstein | Formal and informal training: Evidence from the NLSY[END_REF] report that informal training for new hires, provided by coworkers or supervisors, explains a sizeable amount of within job wage growth. . Liu and Batt (2007) show that returns to informal training depend on whether trainers are supervisors or co-workers.

Recent empirical studies emphasize the importance of knowledge transfers in skills' development of new hires but most of them are controlled laboratory experiments or case studies of one specific firm or occupation. Empirical studies on call-center agents show that new hires benefit from knowledge spillovers from experienced co-workers (De Grip, Sauermann Sieben, 2011;Lindquist et al., 2015). De [START_REF] De Grip | The effects of training on own and co-worker productivity: Evidence from a field experiment[END_REF] extend this result to the whole workforce by showing through a field experiment that call-center agents productivity increase when peers have participated in training, due to likely knowledge spillover from trained to untrained workers. De [START_REF] De Grip | What affects lifelong learning of scientists and engineers?[END_REF] show that scientists or engineers who face a skill gap try to upgrade their skills learning from their colleagues. Among science researchers or teachers, Waldinger (2012) finds little evidence of knowledge spillovers while Jackson and Brugeman (2009) find empirical support of learning from coworkers. Investigating large labor markets, other empirical studies obtain mixed evidence. [START_REF] Muehlemann | Hiring costs and labor market tightness[END_REF] estimate that informal instruction of a new hire by supervisors and co-workers, a prerequisite for him or her to reach full productivity, amounts to about 20% of the total hiring cost. Cornelissen et al. (2017) find small peer effects in wages for high-skilled occupations and large peer effects in wages of lowskilled occupations. They attribute it to peer pressure rather than knowledge spillovers as comparison with co-workers' productivity may encourage an increase in own effort (Falk and Ichino, 2006;Mas and Moretti, 2009;Bandiera et al., 2010;Cornelissen, 2016).

In addition, the interest for knowledge transfers among economists has also been renewed by the development of new internationally comparable data on informal learning at work through the OECD Programme for International Assessment of Adult Competences (PIAAC). Informal learning is defined as the acquisition of skills through learning by doing or learning from supervisors or co-workers (OECD, 2013). A high percentage of workers perceive that informal learning is relevant for them on a daily basis [START_REF] De Grip | The importance of informal learning at work[END_REF]. More precisely, the share of workers who learn new things by doing tasks ranges from 12% in Korea to 53% in Spain and the share of workers who learn new things from supervisors or coworkers ranges from 10% in Korea to 36% in Spain1 . This is consistent with country-specific studies conducted in Netherlands or in France that show that workers learn particularly well from engaging in challenging activities or from cooperating with more experienced colleagues (Borghans et al., 2014;[START_REF] Fournier | Apprentissages informels et «dynamique de travail[END_REF].

Despite the new interest in informal learning, the effect of age is not well documented and has been studied only from the perspective of trainees, i.e. workers who upgrade their skills through learning from others. Using PIAAC, De Grip et al. (2018) exhibit a quadratic relationship between involvement in informal learning and age suggesting that older workers engage in informal learning activities until a turning point at the end of the employee's working life. However, little is known about the characteristics of knowledge transmitters, i.e. workers who transmit their knowledge to their colleagues. Since organizational productivity benefits to a large extent from knowledge transmission, the relationship between age and the ability to transmit knowledge deserves further investigation as well as the mediating role of human resources practices, e.g. provision of formal training.

Data and descriptive statistics

We use a French matched employer-employee survey on training and workers' trajectories (DEFIS) carried out in 2015 by the CEREQ at the request of the National Council for Professional Training Assessments (CNEFP), DEFIS is funded by the Joint Fund for Ensuring Career Paths (FPSPP), currently "France Competences". This survey has been first conducted in March-April 2015 on a representative sample of 3700 firms with 10 employees or more in all market activity sectors excluding agriculture and public administration and on a non-representative one of about 800 companies below this size threshold. In a second stage, around 16000 workers employed in the responding firms in December 2013 were interviewed first in 2015 and then re-interviewed each fall until 2019. This employer-employee matched survey adopts a two-stage sampling design where enterprises are sampled first. The sample of enterprises is randomly drawn from the SIRENE file and the sample of employees from the Déclarations Annuelles de Données Sociales (DADS) of 2013. To take into account the probability that employees that are drawn in the sample respond to the survey, the weighting of the first wave of DEFIS uses a double inference method. A further step is to weight the responding companies in which selected employees provided a response. A final weighting step is carried out using a calibration on the DADS data.

For this study, we restrict our sample to employees aged 30-59. According to the sampling frame, we know that selected employees had been in employment in the responding company for the 18 months before their interview. Since we are interested in knowledge transmission, we exclude from our sample the workers who report having no colleagues. This leads to a working sample of 9684 individuals.

To measure knowledge transmission, we exploit information about the interactions that respondents have with their colleagues. More precisely, the survey asks workers the following question: "Do you often have to show your colleagues how to do part of their work?". answers being "often", "sometimes" or "never". We create a binary indicator equal to one if the respondent answers "Often". Indeed, only 10.9% of respondents answer "never" to this question. In our sample, 35.8%2 of the respondents report that they often show some work practices to their colleagues. Hereafter, we will refer to these individuals as knowledge transmitters.

This question is self-reported, which can be seen as a limitation. In particular, the fact of having participated in a training session could bias the individual perception of knowledge transmission. However, the structure of the questionnaire mitigates this problem since the question on knowledge transmission is located in a section devoted to the organization and working conditions (section C) which precedes the description of the training courses attended (section F). In section C, respondents describe work organization at the date of the survey when in section F they describe the training sessions attended over the past twelve months. Moreover, subjectivity plays a role knowledge transmission as one needs to be confident in one's abilities to effectively transfer. It thus seems difficult not to be aware of showing some work practice to co-workers.

The main limitation of the question is that it is coarse in the sense that it captures a threshold identifying a higher frequency of knowledge transmission. This may be problematic for measuring a higher level of knowledge transfer when employees have a structural role in the information dissemination network. This is the case for supervisors who are often called upon to show the work because of their hierarchical position. For this reason, we also consider a subsample of 5910 individuals who report having no subordinates. More objective measures of the intensity of knowledge transmission would thus be interesting complements to the subjective measures, but they would probably require another survey framework, such as time use surveys.

Our data allows us to control for a rich set of characteristics, regarding the workers, their work practices and their organizational environment. We have information on the standard socio-demographic characteristics (gender, age, marital status, educational and occupational level), on workers' seniority and daily wage. To control for a potential effect of health, we introduce a dummy for the presence of health limitations. Regarding jobs, we control for full time or part-time work as well as for a set of variables regarding the respondents' self-reported working conditions. [START_REF] Fournier | Apprentissages informels et «dynamique de travail[END_REF] show from the same data set that informal learning is more frequent when the job implies frequent exchanges with colleagues, when the tasks to perform are complex and when the workers are autonomous. We introduce additional controls for teamwork, repetitive tasks, learning from colleagues and frequent meetings.

In addition, DEFIS provides firm level information that captures the organizational environment. In their analysis, [START_REF] Fournier | Apprentissages informels et «dynamique de travail[END_REF] have identified different human resources practices that promote knowledge transmission letting workers autonomous in their daily activities, sharing information about training opportunities or developing workers' ability to transmit their knowledge. In this respect, we construct binary indicators for the presence in the firm of autonomous workgroups, apprentices and a department dedicated to workers training We finally control for the economic situation of the firm, with size dummies, and other binary variables indicating respectively whether the firm is a subsidiary, whether its workforce has been growing and whether it has experienced some recruitment difficulties. This last variable echoes the results of Van Dalen et al. (2015) who show that firms with recruitment difficulties tend to adopt workplace accommodation measures and develop more practices towards their senior workers.

Figure 1 shows the proportion of knowledge transmitters by age group with their confidence intervals at a 5% significance level. We plot the left graph from the whole sample of 9684 individuals while the right graph is built from the subsample of 5910 individuals who have no subordinates. This figure shows that the proportion of knowledge transmitters falls sharply after age 453 . For the remainder of the paper, we restrict our sample to the workers aged 45-59. Table 1 provides descriptive statistics to compare the characteristics of knowledge transmitters with those of workers who do not take up such a role. We do so for the sample of all workers aged 45-59 (N=4731) and for the subsample of workers aged 45-59 who have no management responsibilities (N=2927). These statistics are consistent with the literature related to knowledge transmission and informal learning. Females are under-represented among knowledge transmitters as well as workers with low educational or skill level and seniority, employed in low-paid or part-time jobs. Working conditions also have an influence on workers' participation in knowledge transmission after age 45. Working in teams, learning things from other co-workers and having frequent meetings are positively associated with the probability of showing work practices to other colleagues. In addition, some firms' characteristics and human resources practices favors knowledge transmission after age 45. The probability of showing work practices to other colleagues is higher in large firms, that have set autonomous workgroups and that have encountered recruitment difficulties. Regarding industry-specific correlations, we note an underrepresentation of workers employed in the transport industry among knowledge transmitters.

Our main variable of interest is the participation in training to support change over the last twelve months. Table 1 shows a strong positive correlation between this variable and the probability of frequently showing work practices to colleagues after age 45. In the sample of all workers aged 45-59, the proportion of participants in such training is 8 percentage points (pp) higher among knowledge transmitters than among other workers. When disentangling knowledge transmission from management responsibilities by restricting this sample to workers without any subordinate, the proportion of participants in training is 10.6 pp higher among knowledge transmitters than among other workers. The simple descriptive statistics show that participation in training to support change is positively associated with the probability of being a knowledge transmitter after age 45 especially when the activity of transmission is not related to management responsibilities. However, participation in training could be influenced by a set of observable and unobservable characteristics that could also influence the probability of being a knowledge transmitter. To come closer to the estimation of a causal effect we have to correct for this selection bias. 4 Empirical strategy and results

Correcting for the selection bias

There is evidence that older workers suffer from reduced access to training [START_REF] Behaghel | Training and age-biased technical change[END_REF]. In our sample, the proportion of participants who undertook training sessions to support change is 26.5% for workers aged 30-34 but only 16.6% for workers aged 45 and over. Hence, older workers who participate in these training sessions may have specific characteristics. We need to control them to assess the effect of training on knowledge transmission.

To tackle this issue, we follow the approach suggested by [START_REF] Leuven | An alternative approach to estimate the wage returns to privatesector training[END_REF]. Instead of comparing participants in training to non-participants, we narrow down the comparison group considering only those workers who wanted to participate in training but could not do so because the training session was cancelled for exogenous reasons. In DEFIS, the questions to identify these accidental non-participants to training are: "over the last twelve months, did you intend to participate in training courses to support a change in the work environment?" and "has this training session not started yet, been cancelled or delayed?". Among the sample of 4731 workers aged 45 and over, 3945 did not participate in any training session aiming to support change. Of these non-participants, 933 individuals (23.7%) intended to participate in these training courses and 274 (6.95%) did not do so because the planned training session had not yet started or had finally been cancelled or delayed for reasons external to the worker. 5 . Other reasons for non-participation are less exogenous: "your work schedule was too busy", "the company refused to let you go", "funding for the training was refused", "family or personal constraints", "the training center was too far away"..

If the individual does not directly decide to cancel a training course, we can however discuss its exogeneity. Indeed, the cancellation of a training course may be the result of a decision at the level of the company. If, for example, a company realizes that the change in its production process is not so important after all, it may decide to cancel a planned training action. Training would then be endogenous at the company level. Since our sample consists of firms in which several workers were interviewed, we perform a cross-firm decomposition of the variance of the chosen dummy variable explaining nonparticipation6 . The total variance of this variable is equal to 0.074 and its within-firm component constitutes 66%. This ensures that the variation exploited is mainly within firms. Furthermore, in 70% of the companies where at least one respondent reports that, his or her training session has not yet taken place, has been delayed or cancelled, no other worker reports such a contingency. This confirms the exogenous nature of the chosen non-participation variable used to narrow the comparison group.

To check whether narrowing down the comparison group allows correcting for the selection bias, we compare the characteristics of the participants' group and different comparison groups. The first comparison group includes all the workers who did not participate in any training session to support change (N=3945). The second comparison group retains non-participants who wanted to attend training to support change but did not do so (N=933). The third comparison group is further reduced to those non-participants whose planned training session to support change did not eventually occur for exogenous reasons (N=274). We test whether the differences between the participants' group and the comparison groups are significant for a set of variable likely to influence the participation.

The first two columns of Table 2 show that the participants' group and the first comparison group are significantly different on almost every variable considered. Workers who participated in training to support change are overrepresented among skilled workers who hold full-time jobs, have frequent meetings and learn from other colleagues. Participants' are more likely to be employed in large firms, which have developed autonomous workgroups and in which there is a dedicated training department.

Replacing the first comparison group with the second one eliminates almost all significant differences the two groups. However, one significant difference remains. In the comparison group, the proportion of workers who report having learned work practices from their colleagues is statistically lower than in the participants' group. This difference is no more significant when we move to the third comparison group. Therefore, we make the non-participants' group comparable to the participants' one in terms of observable characteristics when we reduce it to those employees who wanted to attend the training but did not because the training session did not (yet) occur. We then estimate the effect of participation in training to support change on the probability that workers aged 45-59 will frequently transmit their knowledge to their colleagues. We consider different combinations of participants' and comparison groups and report the results from the corresponding Probit models in Table 3 7 . In Panel A, we focus on participants' and comparison groups in the overall sample of workers aged 45-59. Sample n°1 includes all the participants and non-participants (N=4731); Sample n°2 includes all the participants and the non-participants who wanted to attend training but did not do so (N=1719); Sample n°3 include all the participants and the non-participants whose planned training session to support change did not (yet) occur (N=1060).

In Panel B, to disentangle the activity of knowledge transmission from management responsibilities, we narrow down the analysis to the subsample of individuals who report having no subordinates. In the same vein as in Panel A, sample n°4 includes the participants' and the first comparison groups (N=2927), sample n°5 includes the participants' and the second comparison groups (N=949) and sample n°6 is made up of the participants' and the third comparison groups (N=545). We use the full set of variables in Table 2 as control variables.

We report the marginal effect of participation in training to support change on knowledge transmission and the standard error of this effect. In Panel A whatever the sample considered, the marginal effect is positive but small and insignificant even at a 10% level. The fact that workers with hierarchical positions transmit knowledge to their subordinates as part of their functional role may blur the identification of the specific effect of training to support change. This is comforted in Panel B, where we observe an effect that is both positive and statistically significant. When we consider the subsample of workers without any subordinates, whatever the comparison group considered participation in a training session to support change raises the probability for workers aged 45-59 to frequently show work practices to other colleagues by 8.4-13.3 pp. This type of training allows workers aged 45 and over to continue to pass on the part of their expertise that remains valuable and thus to maintain their contribution to the knowledge base of the production.

One can argue that firms facing major changes can either train their workers or lay them off. Consequently, our sample is a selected one in the sense that it includes only workers who have stayed in the firm. In the overall sample of respondents aged 45-59, we observe that about 90% of those aged 45-54 years old are still in employment in the same firm 18 months after having been selected in the sampling frame8 but only 74% of those aged 55-59 years old. The selection bias is likely to be particularly important for the latter group. To test the robustness of our main findings to this selection effect, we performed the same regressions as in Table 3 but restricting our sample to workers aged 45-54 years old 9 . This does not qualitatively change the results. For instance, if in sample n°6 (last column of panel B in Table 3) we retain only workers aged 45 to 54 (N=451), we find a marginal effect of training that is really similar to the one obtained in Table 3 (around 0.12) and significant at the 5% level. Our baseline hypothesis is that participation in training to update skills after a change occurred in the work environment allows workers aged 45 and over to maintain their contribution to the knowledge base of the production by transmitting to their co-workers the part of their expertise that remains valuable. This hypothesis implies that the observed effects are related to the changes occurred in the work environment. If another mechanism explains the relationship between knowledge transmission and training, then other types of training would also be likely to have a positive effect on knowledge transmission.

Apart from "supporting a change in the work environment", DEFIS identifies four other motives for training. In our sample, workers aged 45-59 without subordinates, followed in total over the past twelve months 2437 training sessions. The breakdown of their motives is as follows. They aimed respectively at "supporting change" (15.76%), "being more efficient" (30.57%), "taking on new responsibilities" (12.43%), "having a better knowledge of the work environment" (26.3%) and "reinforcing team building" (14.94%). The median duration is almost the same regardless of the reason for the training and varies from two to three days. A trainer internal to the firm has carried out about 40% of these sessions. This proportion does not really depend on the type of training.

In Table 4, for each alternative type of training, we present marginal effects and standard errors for three combinations of participants' and comparison groups composed of workers without any subordinate. Whatever the comparison group, all the marginal effects are small and non-significant even at a 10% level. These alternative types of training, which are unrelated to technological or organizational change, have no effect on the probability of frequently showing work practices to other colleagues. We conduct this further investigation in Table 5. We consider the subsample of workers aged 45-59 without subordinates and we look at the effect for different combinations of participants' and comparison groups as in Table 3. We observe in This confirms our interpretation that the training/transmission relationship after age 45 is not systematic. As long as older workers are not penalized by a change that prevents them from contributing to the production knowledge base, training and transmission are not significantly related. It is when they face a major change that training becomes effective in mitigating the tangible risk for experienced workers to be excluded from their work group. In that case, it allows workers aged 45-59 to continue to interact with their colleagues and to transmit their knowledge gained through experience, thus reducing the risk of firms losing critical knowledge assets.

Concluding remarks

Starting from the fact that workers aged 45-59 are under-represented among those who frequently transmit their knowledge to other workers, we have investigated the role of training. From data drawn from the matched employer-employee DEFIS survey, we have estimated the marginal effect of participation in training sessions to support change on the probability for workers aged 45-59 of frequently showing work practices to their co-workers. To disentangle knowledge transmission from supervising role, we have also considered a subsample of workers aged 45-59 who report having no subordinates.

To control for the potential selection bias, we have exploited the richness of information provided by the survey, restricting non-participants in training to those workers who wanted to attend training but who did not and reducing further this comparison group to non-participants whose planned training session to support change did not (yet) occur for exogenous reasons. We have checked that this empirical strategy allows to make participants' and restricted comparison groups comparable in terms of characteristics likely to influence the probability of transmitting knowledge.

We find that participation in training to support change for workers aged 45 and more increases significantly the probability of showing frequently work practices to colleagues but only for workers who have no subordinate. This strong positive effect holds when narrowing down the comparison group to correct for selection bias. More interestingly, this training-transmission relationship after age 45 is not systematic. Indeed, training helps workers aged 45-59 to keep on participating to the knowledge base of production only if it intervenes as a response to mitigate potential negative effects of technological or organizational change occurred in the workers' environment. We fail to find any significant positive effect of other types of training on the probability of transmitting knowledge and only find a low and weakly significant positive effect for the narrowest comparison group when respondents have not effectively experienced a change in working environment. These results are consistent with those found by Greenan and Messe (2018) using an earlier edition (2006) of a French matched employer employee survey on computerization and organizational change.

This study sheds a new light on the effectiveness of older workers' training. Some contributions argue that older workers' training is less effective as it has no significant effect on job duration [START_REF] Stenberg | Can adult education delay retirement from the labour market?[END_REF]Behaghel, 2014;Boockman et al., 2018), earnings (Belloni et Villosio, 2015) or on relative older workers' productivity [START_REF] Göbel | Are personnel measures effective in increasing productivity of old workers?[END_REF]. We show that specific types of training to update skills after a technological or organizational change allows older workers to keep on interacting with their colleagues and transmitting their knowledge acquired through experience, which reduces the risk for firms of losing critical knowledge assets. This happens when new techniques or management changes affect a sub-group of tasks within those that are bundled into older workers' jobs, creating a situation where the use of the non-affected skills becomes conditional on mastering a new tool or method.

In terms of research, our findings suggest that two main aspects have to be accounted for when investigating the effectiveness of older workers' training. First we have to carefully check the motives of training and especially if it intervenes as a response to a technological or organizational change occurred in the workplace. Second, we should account for the potential interactions between generations of workers.

If the public debate acknowledges that employee learning and development is critical in times of structural change and crisis [START_REF] De Grip | The importance of informal learning at work[END_REF], the outcomes of knowledge transmission within workplaces in terms of job satisfaction, turnover intentions, productivity or innovation, which we do not address in this paper, deserve further research. In particular, knowledge transfers between older workers and their co-workers is an important area of investigation, likely to be affected by human resource management practices. Indeed, the main studies that examine the effect of training on the productivity of older workers assume perfect substitution between age groups, leaving no room for potential complementarities or interactions between older workers and their colleagues. In particular, we believe that studying how training that supports technological and organizational change influences the relationship between age diversity and firm productivity is a promising avenue for future research.

The implication of this paper for human resources managers is that not updating older workers' skills after a technological or organizational change may imply a cost for firms. As stressed by the recent literature on the consequences of the ongoing technological transformation, a large share of jobs is likely to be only partially automated implying that while some tasks will disappear, making the corresponding skills obsolete, others will persist and their associated skills will remain valuable for organizations [START_REF] Arntz | Revisiting the risk of automation[END_REF][START_REF] Nedelkoska | Automation, skills use and training[END_REF][START_REF] Pouliakas | Determinants of Automation Risk in the EU Labour Market: A Skills-Needs Approach[END_REF]. If older workers are excluded from their workgroup after these changes, because their skills have not been updated through training, valuable organizational knowledge from these experienced employees will be lost. Once these older workers retire, the organization will discard part of its human capital and productivity will be affected negatively. Several papers put forward that the organizational knowledge loss attributable to the departure of experienced workers may bring about a strong fall in productivity [START_REF] Massingham | Knowledge risk management: a framework[END_REF][START_REF] Joe | Knowledge loss when older experts leave knowledge-intensive organisations[END_REF][START_REF] Bartel | Human capital and productivity in a team environment: evidence from the healthcare sector[END_REF][START_REF] Massingham | Measuring the impact of knowledge loss: a longitudinal study[END_REF]. Human resource managers in their tradeoff between exit policies or employee development measures should more carefully account for this point for workers in the second part of their careers.

Figure 1 :Figure 2

 12 Figure 1: Proportion of knowledge transmitters by age group

Table 1

 1 Descriptive statistics of the sample

		All workers aged 45-59	Workers aged 45-59 without any subordinate
		Knowledge transmitter	Knowledge transmitter
		YES	NO	YES	NO
	Characteristics of the worker
	Female	0.263	0.361***	0.311	0.390*
	Aged 45-49	0.461	0.414	0.415	0.398
	Aged 50-54	0.331	0.366	0.360	0.385
	Aged 55-59	0.209	0.221	0.225	0.217
	Single	0.193	0.247*	0.224	0.275
	Primary education	0.102	0.129	0.138	0.136
	Vocational education	0.450	0.531**	0.452	0.549***
	High-school education	0.088	0.103	0.085	0.099
	Undergraduate education	0.200	0.165	0.238	0.159***
	Graduate/Post-graduate education	0.160	0.072***	0.088	0.058
	Blue-collar workers	0.299	0.395***	0.380	0.431
	Clerks	0.320	0.423***	0.377	0.449*
	Engineers/managers	0.381	0.182***	0.242	0.120***
	Health limitations	0.146	0.164	0.200	0.174
	Part-time job	0.046	0.155**	0.071	0.177***
	Log of hourly wage	2.816	2.580***	2.724	2.520***
	Seniority
	< 10 years	0.270	0.348***	0.264	0.368***
	11-20 years	0.297	0.280	0.271	0.276
	21-30 years	0.261	0.237	0.268	0.231
	> 30 years	0.172	0.135*	0.197	0.124**
	Working conditions
	Works in teams	0.715	0.594***	0.659	0.548***
	Has learned from other colleagues	0.477	0.395**	0.516	0.405***
	Has frequent meetings	0.712	0.486***	0.592	0.428***
	Carries out repetitive tasks	0.451	0.569***	0.486	0.420
	Has participated in training to support change	0.226	0.146***	0.221	0.115***
	The employing firm
	Is a subsidiary	0.645	0.559***	0.664	0.565
	Size
	1-19 employees	0.215	0.268	0.202	0.264
	20-49 employees	0.126	0.131	0.087	0.130*
	50-249 employees	0.167	0.203	0.166	0.201
	250-1999 employees	0.200	0.201	0.200	0.21
	2000 employees and more	0.292	0.196***	0.344	0.194***

Table 2

 2 Sample means for the participants' and comparison groups Source : CNEFP-France compétences-Céreq, DEFIS 2015 4.2 The effect of participation in training to support change on knowledge transmission after age 45

	Characteristics	Participants' group	Comparison group n°1	Comparison group n°2	Comparison group n°3
	Female	0.366	0.321	0.322	0.325
	Aged 45-49	0.455	0.423	0.497	0.511
	Aged 50-54	0.331	0.359	0.360	0.377
	Aged 55-59	0.214	0.217	0.243	0.212
	Single	0.225	0.230	0.211	0.262
	Primary education	0.081	0.128	0.036	0.052

Comments: The comparison group n°1 includes all the employees who did not participate in any training session to support change. The comparison group n°2 includes only non-participants who wanted to attend training to support change but did not do so. The comparison group n°3 further reduces the comparison group to nonparticipants whose planned training to support change was eventually cancelled. Statistical significance for differences in means between participants' and comparison groups: *** p<0.01, ** p<0.05, *p<0.1

Table 3 -

 3 The effect of participation in training to support change on the probability of being a knowledge transmitter among workers aged 45-59 The samples n°1, 2 and 3 are constructed considering workers aged 45-59. Sample n°1 includes all the participants and non-participants in training session aiming to support change. Sample n°2 includes participants and non-participants who wanted to attend training to support change but did not do so. Sample n°3 includes participants and non-participants whose planned training session to support change did not (yet) occur. Samples n°4, 5 and 6 are constructed in the same way but considering only the subsample of workers aged 45-59 without any subordinate. The full set of variables in Table2are used as control variables.

	Panel A: Overall sample of workers aged 45-59
		Sample n°1	Sample n°2	Sample n°3
	Marginal effect of participation in training to support change	0.041	0.038	0.017
	Standard errors	(0.033)	(0.039)	(0.049)
	Number of observations	4731	1719	1060
	Panel B: Subsample of workers aged 45-59 without any subordinate
		Sample n°4	Sample n°5	Sample n°6
	Marginal effect of participation in training to support change	0.084**	0.101**	0.133***
	Standard errors	(0.032)	(0.040)	(0.048)
	Number of observations	2927	949	545
	Comments: Statistical significance for marginal effect obtained from Probit models: *** p<0.01, ** p<0.05, *p<0.1
	Source : CNEFP-France compétences-Céreq, DEFIS 2015
	4.3 The influence of other types of training on knowledge transmission after age 45

Table 4 -

 4 The effect of participation in training for other motives than supporting change on the probability of being a knowledge transmitter after age 45 The samples n°4, 5 and 6 are constructed considering workers aged 45-59 without any subordinate. Sample n°4 includes all the participants and non-participants in training session aiming to support change. Sample n°5 includes participants and non-participants who wanted to attend training to support change but did not do so. Sample n°6 includes participants and non-participants whose planned training session to support change was eventually cancelled. The full set of variables in Table2are used as control variables. Knowledge transmission, training and technological or organizational changes It could be that the type of training we consider, i.e. training to support change, takes place when the participants have not encountered any change, either technological or organizational, in their daily activities. This would invalidate our basic assumption that training for workers aged 45-59 is effective if it is used in response to a change that may inhibit them from contributing to knowledge exchanges. To investigate the influence of technological or organizational change on the training-transmission relationship after age 45, we exploit information available in the data. Indeed, DEFIS asks workers the following questions: "has your work been modified by a change in techniques or processes or by an organizational change since January 2014". This information allows computing our marginal effect of interest both for workers who have reported a change in their work and for those who have not reported any.

	Marginal effect of participation in training to…	Sample n°4	Sample n°5	Sample n°6
	…be more efficient at work	0.035	0.055	0.046
		(0.027)	(0.040)	(0.053)
	…take on responsibilities	0.034	0.032	0.010
		(0.039)	(0.048)	(0.053)
	…have a better knowledge of the work environment	0.029	0.026	0.005
		(0.029)	(0.041)	(0.046)
	…reinforce team building	0.005	0.015	-0.048
		(0.037)	(0.044)	(0.049)
	Number of observations	2927	949	545
	Comments: Statistical significance of marginal effects obtained from Probit models: *** p<0.01, ** p<0.05, *p<0.1
	Source : CNEFP-France compétences-Céreq, DEFIS 2015
	4.4

Table 5 -

 5 The effect of participation in training to support change on the probability of being a knowledge transmitter among workers aged 45-59 according to technological or organizational changes occurred in the workers' environment Panel A: Subsample of workers aged 45-59 without any subordinate who did not experience any change in their daily work The comparison group n°1 includes all the non-participants in training session aiming to accompany the change. The comparison group n°2 includes only non-participants who wanted to attend training but did not do so. The comparison group n°3 further reduces the comparison group to untrained workers who wanted to participate in a training session aiming to accompany the change and whose training session has been eventually cancelled. The full set of variables in Table 2 are used as control variables. When considering the subsample of workers aged 45-59 without any subordinate, these shares are respectively 7.22% and 21.32%. Nevertheless, a substantial part of the participants in training sessions to support change had not experienced any change at the time when they were interviewed. A possible reason for this discrepancy is that the change had not yet occurred or it had not affected the daily activity of the respondent.

		Participants +
	Comparison	Comparison	Comparison
	group n°1	group n°2	group n°3

  Table5that marginal effects are positive both in Panel A where workers experienced no change and in Panel B where they did. However, these effects are strong and significant for all comparison groups in panel B only. Thus, without restricting the comparison group training with the intention to support change increases knowledge transmission for training participants by 12.3 pp (significant at the 1% level) if workers have experienced a change in their work environment (Panel B) but has no significant marginal impact if they have not had to face such a situation (Panel A). Reducing the comparison group to those who wished to undergo such training, the effects are +11 pp in the presence of technological or organizational changes, but are not significant in their absence. Reducing the comparison group further to those whose planned training session to support the change did not take place, these effects are +19.2 pp in the former case and +8.6 pp points (significant at the 10% level) in the latter.

For France, these two shares are respectively 45% and

25%.

All the statistics and estimates are weighted to account for unequal sampling rates.

Running a simple Probit model and controlling for age and occupation dummies, we find that for workers without any management responsibilities, being aged

45-49 years old reduces by 9.4% the probability of being a knowledge transmitter. This negative effect is significant at a 5% level.

In the graph, the effect seems to be only significant for individuals aged 45-49 years old. However, running a simple Probit model and controlling for age and occupation dummies, we find that for workers without any management responsibilities, being aged

50-54 years old reduces by 11.9 percentage points the probability of being a knowledge transmitter for non-participants to a training session aiming at supporting a change. In contrast, for workers who participate in such a training session, this negative effect is clearly lower (-6.6 p.p.) and becomes non-significant.

These shares are comparable to the ones described in the Leuven and Oosterbeek's paper. In their sample, among 1145 non-participants, 249 (21.75%) wanted to receive training course and 77 (6.72%) did not do so because of some random event.

We thank our reviewers for this suggestion.

We report in Table6in the Appendix the other coefficients estimated for the baseline Probit regression

As mentioned in section 3, selected individual were working in the responding firms in December 2013, but were interviewed about 18 month later, in 2015.

We thank one of our reviewers for a comment on this issue. The corresponding results are available upon request.

19-1. The interaction between labour force participation of older men and their wife: lessons from France Idriss Fontaine

Appendix


Andrei Zaharescu

Edmond Boyer

Kiran Varanasi

Radu Horaud

Surface Feature Detection and Description with Applications to Mesh Matching

In this paper we revisit local feature detectors/descriptors developed for 2D images and extend them to the more general framework of scalar fields defined on 2D manifolds. We provide methods and tools to detect and describe features on surfaces equiped with scalar functions, such as photometric information. This is motivated by the growing need for matching and tracking photometric surfaces over temporal sequences, due to recent advancements in multiple camera 3D reconstruction. We propose a 3D feature detector (MeshDOG) and a 3D feature descriptor (MeshHOG) for uniformly triangulated meshes, invariant to changes in rotation, translation, and scale. The descriptor is able to capture the local geometric and/or photometric properties in a succinct fashion. Moreover, the method is defined generically for any scalar function, e.g., local curvature. Results with matching rigid and non-rigid meshes demonstrate the interest of the proposed framework.

Introduction

The detection, characterization, and matching of various 2D or 3D features from visual observations is of great importance for a large variety of applications such as modeling, tracking, recognition or indexing, among others. The vast majority of existing methods detect features using either photometric information available with 2D images or geometric information available with 3D surfaces. However, recent progress in image based 3D modeling and rendering allows to recover both photometric and geometric information from multiple images [START_REF] Seitz | A comparison and evaluation of multi-view stereo reconstruction algorithms[END_REF]. Whenever such models are available, photometric 2D features or geometric 3D features, if taken separately, have limited informative capabilities with respect to the potential richness of the data. This is the case, for example, with deformable and/or articulated objects, since image appearance is only partially robust to motions and geometric properties alone are not always robust, e.g., the topology of the model can change considerably with varying object poses. Therefore, we believe that photometric and geometric information need to be handled in a consistent and simultaneous manner. To this purpose, we observe that photometric information available with 3D models can be viewed as scalar functions defined over 2D manifolds and, as such, represent a generalization of planar image domains to non-planar domains. We can thus build on the existing image feature extraction theories and investigate their extensions to 2D manifolds.

The contribution of this paper is twofold: first we develop a methodology for feature-based characterization using operators acting on scalar functions defined over 2D manifolds; second, we derive a novel family of interest point detectors and descriptors that take into account both the surface geometry and the photometric information. To this aim, operators such as the discrete convolution and the discrete gradient, are defined for scalar functions on discrete surfaces, i.e., meshes, thus taking into account both the functions' differential properties as well as the surfaces' intrinsic geometry. Based on these operators, a new inter-est point detector and a new local descriptor are introduced, namely MeshDOG and MeshHOG. MeshDOG is a generalization of the DOG operator [START_REF] Marr | Theory of edge detection[END_REF][START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF] and it seeks the extrema of the Laplacian of a scale-space representation of any scalar function defined on a discrete manifold. Mesh-HOG is a generalization of the histogram of oriented gradients (HOG) descriptor recently introduced for describing 2D images [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF]. The new descriptor is defined with respect to the measurements available at each of the discrete surface's vertices and it can work with features photometric features, as well as with geometric feature, such as curvature, geodesic integral, etc.

As it is the case with the more classical image operators, detectors and descriptors are not uniquely defined over surfaces and MeshDOG and MeshHOG were chosen in light of their quasi-invariance to transformations such as rotation and scale. In addition, they exhibit a number of attractive properties: (i) there are no perspective distortions, since computations are achieved in 3D; (ii) there are no false detections due to occlusions; (iii) the descriptor captures both the local 3D geometry and the local gradient information of the scalar function; (iv) no planar mesh embedding is necessary; (v) within a multiple-camera setting, the descriptor can fuse the photometric information coming from different images in order to provide more robust image-invariant photometric information.

The organization of the paper is as follows. Section 2 discusses related works. Section 3 describes the mathematical formulation used to to build a number of operators on discrete manifolds. Section 4 and 5 introduce the local feature detector and descriptor, respectively. Section 6 presents and discusses the results, before concluding in section 7.

Related Work

Photometric functions over planar domains (local image features): Developing robust 2D features, invariant under changes in illumination, viewpoint, scale and orientation has been one of the long term research goals in the area. Currently, SIFT [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF] and HOG (histogram of oriented gradients) [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] are among the most widely used descriptors for their robustness to the transformations just cited. Interest points may coincide to the extrema of the Laplacian of the photometric function, and they are detected at various resolution scales using the difference of Gaussians (DOG) approximation of the Laplacian, see [START_REF] Mikolajczyk | A performance evaluation of local descriptors[END_REF] for a detailed review. Alternatively, spatio-temporal descriptors have also been proposed [START_REF] Wong | Extracting Spatiotemporal Interest Points using Global Information[END_REF][START_REF] Kläser | A spatio-temporal descriptor based on 3D-gradients[END_REF], by considering the 3D spatio-temporal volume defined by a short image sequence over time. Such space-time features can be seen as local features defined over 3D grids. We extend the DOG operator to non-planar surfaces instead of dealing with volumetric grids.

Geometric functions over surfaces (local geometric features): 3D spin images [START_REF] Johnson | Using spin images for efficient object recognition in cluttered 3D scenes[END_REF] and 3D shape contexts [START_REF] Körtgen | 3D shape matching with 3D shape contexts[END_REF][START_REF] Frome | Recognizing objects in range data using regional point descriptors[END_REF] are among the most successful surface descriptors. These are descriptors that rely solely on the surface geometry. See [START_REF] Tangelder | A survey of content based 3D shape retrieval methods[END_REF][START_REF] Bustos | Feature-based similarity search in 3D object databases[END_REF] for a detailed survey. Typically these descriptors characterize the neighbourhood of a specified surface region. A number of methods have been proposed for automatic identification of interest regions on surfaces, taking into account geometrical features. Scale-space extrema based on the averaged mean curvature flow are proposed in [START_REF] Schlattmann | Scale space based feature point detection on surfaces[END_REF]. Alternatively, [START_REF] Novatnack | Scale-dependent 3D geometric features[END_REF] defines the scale space in a planar parametrization of the surface using the normal map and searches for the extrema. Gradient operators are defined over a planar vector field. While this formulation could be used as an alternative mathematical framework in current work, the required planar parameterization introduces an additional level of complexity that the currently proposed method avoids. [START_REF] Lee | Mesh saliency[END_REF] proposes a mesh saliency method, based on the center-surround operator, adapted from the visual attention literature. Photometric information is not taken into account by these methods.

Photometric functions over surfaces (local augmented surface features):

In [START_REF] Wu | 3D model matching with viewpoint invariant patches (vips)[END_REF] a SIFT-based descriptor on 3D oriented patches is proposed, i.e., VIP (Viewpoint Invariant Patches), which was used for 3D model matching. It constitutes a first attempt to devise a descriptor that includes both geometry (normal orientation) and photometric information. In [START_REF] Starck | Correspondence labelling for widetime free-form surface matching[END_REF] the authors propose a concatenated surface descriptor taking into account both geometry (a region descriptor based on geodesic-intensity histograms), and photometric information (edge and corner descriptors that take into account the local isometric mapping to R 2 ). The approach proposed in this paper is similar in spirit to [START_REF] Wu | 3D model matching with viewpoint invariant patches (vips)[END_REF], but, instead, considers full 3D gradients and histograms.

Many applications make use of local features, in particular in the context of surfaces: surface registration, nonrigid shape matching and object recognition. For instance [START_REF] Novatnack | Scale-dependent/invariant local 3D shape descriptors for fully automatic registration of multiple sets of range images[END_REF] proposes an image-based descriptor using the local R 2 embedding of the normal information on the mesh in order to perform surface registration. Also, a recent number of works, e.g. [START_REF] Furukawa | Dense 3D motion capture from synchronized video streams[END_REF][START_REF] Ahmed | Dense correspondence finding for parametrization-free animation reconstruction from video[END_REF][START_REF] De Aguilar | Marker-less 3D feature tracking for mesh-based human motion capture[END_REF][START_REF] Varanasi | Temporal surface tracking using mesh evolution[END_REF], address the non-rigid mesh matching problem using observations from multiple views. The vast majority of the proposed methods (the only notable exception being [START_REF] Furukawa | Dense 3D motion capture from synchronized video streams[END_REF]) uses both geometric information extracted from surfaces and photometric data available with images. The latter is first extracted using 2D image descriptors (such as SIFT [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF]), and subsequently backprojected onto the mesh. This sparse description is generally used to bootstrap dense matching. Surface descriptors may well be used for 3D object recognition, as it has been already done in [START_REF] Shilane | The princeton shape benchmark[END_REF] using the Princeton shape benchmarking database1 .

Our work contributes to these efforts by taking a different, yet complimentary approach, namely image-feature detection and description methodologies are extended to features defined onto 2D manifolds.

Problem formulation

Let S denote the set of all possible discrete parametrizations of the admissible 2D manifolds in R 3 . We will consider in particular uniformly sampled triangulated meshes S ∈ S, namely meshes whose facets are triangles of approximately the same area and whose vertices' valence is close to 6. We notice that such an uniform mesh can be obtained from a non-uniform mesh through simple mesh operations, as proposed in [START_REF] Kobbelt | Multiresolution shape deformations for meshes with dynamic vertex connectivity[END_REF]. This absolves us of the necessity of complex techniques that ensure proper samplings of scalar fields over S, while keeping generality. It is interesting to notice that an image can be viewed as a "flat" uniformly sampled mesh, i.e., a grid of vertices with valence 4 and whose facets are squares or rectangles.

S can also be viewed as a graph S(V, E), where V = {v i } 1≤i≤N is the set of mesh vertices and E = {e ij } is the set of mesh edges between adjacent vertices. We denote by e avg the average edge length. We associate a 3D point v ∈ R 3 with each vertex v. The ring of a vertex rg(v, n) is the set of vertices that are at distance n from v on S, where the distance n is the minimum number of edges between two vertices. Thus rg(v, 0) is v itself and rg(v, 1) is the set of direct neighbours of v (see Figure 2). The neighbourhood N n (v) is then the set of rings {rg(v, i)} 0≤i≤n . We further denote -→ n v the unit vector normal to the surface S at vertex v, computed as the average direction of the normals of the triangles incident to v.

We consider a scalar function f : S → R. In order to be able to estimate discrete gradient information, we first recall the definition of the directional derivative of a scalar function on a manifold [START_REF] Hirani | Discrete Exterior Calculus[END_REF]:

Definition 1 (Directional Derivative) Let ∇ S f denote the gradient operator of f on S, the directional derivative of f at v ∈ S is defined as: D-→ u f (v) = ∇ S f (v) • - → u , (1)
for any direction -→ u in the tangent plane of S at v.

Using the fact that up to first order:

f (v j ) -f (v i ) = ∇ S f (v i ) • (v j -v i ) around v i ,

we have the following definition:

Definition 2 (Discrete Directional Derivative) The discrete directional derivative of f is defined as:

D-→ eij f (v i ) = 1 || --→ v i v j || (f (v j ) -f (v i )), (2)
∀e ij ∈ E and where || --→ v i v j || = ||v j -v i ||. ∇ S f (v i
) is by definition a vector in the tangent plane at v i and the above definition allows us to estimate its directional values around v i . Hence, two such non-null local directional gradients are, in principle, sufficient to estimate the gradient ∇ S f (v i ) at v i . This is a generalization of the classical way of computing gradients in the image using two orthogonal directions. In practice however, we prefer to use all the directional gradients provided by the first ring of a vertex: indeed, this redundancy guarantees a more robust operator:

Definition 3 (Discrete Gradient) the gradient operator ∇ S f (v i ) of f at v i ∈ S is defined as: ∇ S f (v i ) = vj ∈rg(vi,1) (w ij D-→ eij f (v i )) -→ u ij , (3)
where w ij weighs the contribution of D-→ eij and -→ u ij is the normalized projected direction of --→ v i v j in the tangent plane at v i .

The weights w ij should be chosen in order to balance the contributions of the local directional derivatives with respect to their associated directions in the tangent plane. The gradient is defined as a weighted mean of directional derivatives, since directional derivatives are projections of the gradient onto given directions. Assuming that S is uniformly sampled and thus that neighbours around v i are equally spaced we get:

w ij = 1 val(vi)
where val(v i ) is the valence of v i . For non uniformly sampled meshes, the weights are a function of the angles between the directions -→ u ij around v i in the tangent plane at v i .

Finally, we define the discrete convolution operator on a mesh: Definition 4 (Discrete Convolution). The convolution of the function f with a kernel k is:

(f * k)(v i ) = 1 K vj ∈Nn(vi) k(|| --→ v i v j ||)f (v j ), (4)
where the kernel weighs the participation of neighbouring vertices v j as a function of their distances from vertex v i and

K = vj ∈Nn(vi) k(|| --→ v i v j ||
) is a normalization factor. Notice that, as for the discrete gradient, we assume a uniformly sampled mesh and thus that contributions of neighbouring vertices v j in the above expression are equally weighted with respect to their spatial arrangements. Another remark is that, generally, we use the above definition with the first ring only, i.e., n = 1.

Feature Detection (MeshDOG)

Feature detection is comprised of three steps, as illustrated in Figure 3. First, the extrema of the function's Laplacian (DOG) are found across scales using a one-ring neighbourhood. Second, the extrema thus detected are thresholded. Third, the unstable extrema are eliminated, thus retaining those mesh locations exhibiting some degree of cornerness.

Scale-space extrema. We propose a scale-space representation of scalar function f defined on a mesh. We consider the convolution operation on meshes (see Definition 4) using a Gaussian kernel, defined as:

g σ (x) = exp (-x 2 /2σ 2 ) σ √ 2π .
The scale space of f is built progressively:

f 0 = f , f 1 = f 0 * g σ , f 2 = f 1 * g σ , etc
. Convolved functions are subtracted, e.g., DOG 1 = f 1 -f 0 , DOG 2 = f 2 -f 1 , etc., in order to obtain the difference of Gaussian operator. An example can be observed in Figure 4, where the model used is frame 30 from pop2lock sequence from the University of Surrey, and the features being shown are colour and mean curvature. An important observation is that, when building the scale space, the mesh geometry does not change, but the different scalar functions defined on the mesh, i.e. f 1 , f 2 , DOG 1 , DOG 2 . We have chosen σ = 2 The feature points are selected as the maxima of the scale space across scales, followed by non-maximumsuppression, using the one ring neighbourhood, in the current and in the adjacent scales.

(c) f 2 (d) f 64 (e) DOG 2 (f) DOG 64 (g) f 2 (h) f 64 (i) DOG 2 (j) DOG 64
Thresholding. From the extrema of the scale space, only the top β = 5% of the maximum number of vertices are being considered, sorted by magnitude. We have chosen a percentage value versus a hard value threshold in order to keep the detector flexible, no matter which feature is being considered, without the need for normalization.

Corner Detection. Additionally, in order to eliminate more non-stable responses, we retain the features that exhibit corner characteristics. As proposed in [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF] this can be done using the Hessian operator: : represent here a local coordinate system in the tangent plane of v, typically the gradient direction for -→ x and its orthogonal direction for -→ y . The ratio between the largest λ max and the lowest λ min eigenvalues of the Hessian matrix is a good indication of a corner response, which is independent of the local coordinate frame. We typically use λ max /λ min = 10 as a minimum value to threshold responses.

H(v) = d xx (v) d xy (v) d yx (v) d yy (v) , (5)

Feature Descriptor (MeshHOG)

The descriptor t v for vertex v is computed using a support region, defined using a neighbourhood ring size r, as depicted in Figure 2. For each vertex from the neighbourhood v i ∈ N r (v), the gradient information ∇ S f (v i ) is computed using [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF]. As a first step, a local coordinate system is chosen, in order to make the descriptor invariant to rotation. Then, a histogram of gradient is computed, both spatially, at a coarse level, in order to maintain a certain high-level spatial ordering, and using orientations, at a finer level. Since the gradient vectors are 3 dimensional, the histograms are computed in 3D.

Neighborhood size. The number of rings r for the support region is chosen adaptively based on a more global measure, such that the descriptor is robust to different spatial samplings and to scaling. The value of r is chosen such that it covers a proportion α r from the the total mesh surface, where α r ∈ (0, 1). By denoting A S as the total surface area of the mesh S, which can be computed as the sum of all triangle areas, the ring size r is:

r = round 1 e avg α r A S Π , (6)
assuming that the surface covering the ring neighbourhood can be approximated with a circle and that the mesh S is equally sampled, with the average edge size e avg . In practice, we use an r corresponding to α r = 1%.

Local Coordinate System. A local coordinate system can be devised using the normal -→ n v and two other unit vectors, residing in tangent plane P v of v. Given a unit vector -→ a v ∈ P v , the local coordinate system is given by

{ - → a v , - → n v , - → a v × - → n v }.
Vector -→ a v is computed as the direction associated to the dominant bin in a polar histogram, with b a = 36 bins. The histogram is computed by considering the projected vertices v i in P v and taking into account their gradient magnitudes. We weigh ||∇ S f (v i )|| by a Gaussian with σ = e avg r/2, based on the geodesic distance from v. In order to reduce aliasing and boundary effects of binning, votes are interpolated bilinearly between neighbouring bins when computing the histograms. We use the same weighting and interpolation technique for any further binning.

Histograms. Instead of computing full 3D orientation histograms, as proposed in [START_REF] Kläser | A spatio-temporal descriptor based on 3D-gradients[END_REF], we project the gradient vectors to the 3 orthonormal planes, describing the local coordinate system. This provides us with a more compact representation of the descriptor. For each of the three planes, we compute a 2 level histogram. Firstly, the plane is divided in b s = 4 polar slices, starting with an origin and continuing in the direction dictated by the right hand rule with respect to the other orthonormal axis vector. When projected onto the plane, each vertex v i will fall within one of the spatial slices. For each spatial slice, we compute orientation histograms with b o = 8 bins for each of the projected gradient vectors ∇ S f (v i ) of the vertices v i that projected onto that spatial slice, as shown in Figure 5(d).

Descriptor. The final descriptor is obtained by concatenating b s × b o histogram values for each of the three planes, followed by L-2 normalization.

Mesh Matching

We are validating the proposed detector and descriptor using a mesh matching approach. Let us consider two meshes S 1 and S 2 of the same object. The two meshes do not necessarily have the same number of vertices. Using the proposed approach, n 1 interest points are detected on S 1 , which are characterised by descriptors t 1 i , with i ∈ [1..n 1 ]. Similarly, n 2 interest points are detected on S 2 , characterised by descriptors t 2 j , with j ∈ [1..n 2 ]. Matching. We use an intuitive greedy heuristic in order to select the a set of best matches. For each descriptor t 1 i from surface S 1 , we find the best matching descriptor t 2 j from surface S 2 in terms of the Euclidean distance

d ij = ||t 1 i -t 2 j || .
We perform cross validation, by checking that t 2 j 's best match is indeed t 1 i . Finally, we only accept the candidate match is the second best match is significantly worse (γ = 0.7 or less from the best match score). This is not meant to fully solve the matching problem, as would a global approach [START_REF] Starck | Correspondence labelling for widetime free-form surface matching[END_REF]. It is merely intended for validation and for evaluation of our detector and descriptor.

Datasets. In our evaluation we consider the following scenarios: (i) the two meshes are representations of the same rigid object, which can thus be aligned using a rotation, translation and scale; (ii) the two shapes are representations of the same non-rigid object, i.e. a moving person. In this context, we are introducing the datasets.

• Matching rigid objects: we are considering reconstructions of the same object using different camera sets.

In particular, we are using meshes obtained employing the method described in [START_REF] Zaharescu | Transformesh: a topology-adaptive mesh-based approach to surface evolution[END_REF], using the publicly available datasets from the Middleburry Multi-View Stereo site [START_REF] Seitz | A comparison and evaluation of multi-view stereo reconstruction algorithms[END_REF]. The Dino datasets contains two meshes, one with 27,240 vertices obtained from 16 cameras and the other of 31,268 vertices generated from 47 cameras. Similarly, the Temple datasets contains two meshes, one with 78,019 vertices obtained from 16 cameras and the other of 80,981 vertices generated from 47 cameras.

• Matching non-rigid objects from synthetic data: we consider a synthetically generated dataset entitled Synth-Dance of a human mesh with 7,061 vertices moving across 200 frames.

• Matching non-rigid objects from real data: additionally, we use frames 515-550 from the INRIA Dance-1 sequence 2 , where the same reconstruction method [START_REF] Zaharescu | Transformesh: a topology-adaptive mesh-based approach to surface evolution[END_REF] was employed to recover models using 32 cameras. The models have vertices ranging between 16,212 and 18,332.

Photometric information. The colour of each vertex of the surface is computed by considering the median colour in the visible images. We assume that the colours of a vertex follow a non-Gaussian distribution, due to errors that can occur around occluding contours. In the Synth-Dance dataset the vertices are randomly coloured.

Examples of Matching Rigid Objects

We present our results on the Dino and Temple datasets in Figure 6, where we have run tests where the colour and the mean curvature were used as features, as well as cases in which we have created a new descriptor by concatenating the MeshHOG descriptors for colour and mean curvature. The results are interesting. Even when just curvature is used for the descriptor, there seems to be enough discriminability to account for a number of correct matches varying between 10-30, depending on the detector and the dataset. Both the Dino and the Temple datasets are rather challenging, due to the fact that, at a first glance, they do not have a large number of distinguishing non-repetitive features in terms of their visual aspect. Additionally, it seems that using just the colour as a feature provides the best results in terms of the number of matches. This is so, we can argue, because the descriptor inherently incorporates certain mesh geometry information by design of the operators.

2 https://charibdis.inrialpes.fr/ These are the only results presented in the paper where different features were used for the descriptor. All the other results are generated using colour information.

Examples of Matching Non-Rigid Objects

Comparison with back-projected 2D features. We present a comparison between the proposed mesh matching framework using MeshHOG descriptor with another framework, currently employed in a number of mesh matching methods (see Section 2), that uses back-projected image descriptors. In the image based framework, the matching is performed in the images and only then is back-projected onto the surface. In our comparisons, we used the SIFT image descriptor. When matching the two surfaces, only matches from the same cameras are considered. In order to be able to carry such a comparison for the Synth-Dance dataset, we have generated images for 16 virtual cameras, distributed in a circular pattern around the object.

Synthetic comparative results are presented in Figure 7. The mesh in the first frame was matched with the mesh at any of the other 199 frames across the sequence. As it can be observed, the MeshHOG descriptor generates very few false positives in comparison with the SIFT equivalent, clearly demonstrating the advantages of the proposed approach.

In addition, we present empirical results in Figure 8 for for the INRIA Dance-1 sequence. As it can be observed, the second best match ratio threshold γ = 0.7 tends to be more aggressive for SIFT. There are only 54 matched found using the SIFT back-projected method between frame 525 and 526, whereas MeshHOG finds 119 matches. Even when matching across distant frames (530 and 550), our proposed method finds 13 correct matches, versus the SIFT descriptor, that fails. It is to be expected, since most of the inter-frame matches are due to local creases formed by the clothes.The head is the only unique feature that can be robustly matched across time.

Resilience to Noise

There are two kinds of uniformly distributed noise being applied: geometry noise (changing the vertices v) and colour noise (changing values f (v) held in each vertex). The colour noise relates to % of the total amount of a maximum 255 RGB value noise, whereas the geometry noise relates to the % of the total amount of a maximum e avg noise level. As it can be observed in Figure 9, the method does not generate more false positives when the amount of noise increases. The Dino dataset has a larger number of false positives, since the two meshes are not perfectly identical, being the result of a 3D reconstruction method from multiple images, which introduces some errors. In the Synth-dance dataset, the colour noise influences the descriptor accuracy more than the geometry noise, whereas in the Dino dataset the situation is reversed. This stems from the fact that the meshes in the two datasets have a relatively different number of vertices, which will in turn directly influence the ring neighbourhood size r (r = 7 for Synth-dance, and r = 15 and r = 16 for Dino), always chosen to represent α r of the total mesh area.

Integration with mesh tracking. We have integrated the MeshHOG descriptor within an existing mesh tracking ap-proach, described in [START_REF] Varanasi | Temporal surface tracking using mesh evolution[END_REF], by replacing the sparse matching step based on back projected SURF descriptors with the currently introduced descriptor. For more details, see [START_REF] Zaharescu | Constributions to Spatial and Temporal 3-D Reconstruction from Multiple Cameras[END_REF].

The running time of computing such a descriptor depends on the descriptor neighbourhood size. For example, in the synth-dance dataset, computing 706 descriptors using a neighbourhood size r = 7 took under 1 second, while computing 2724 descriptors using a ring neighbourhood size r = 15 took 35 seconds. The machine used for the test was a Core2Duo 2.4GHz Intel with 2 Gigs of RAM running Mac OS.X. The code has been developed in C++ and it is available for download from 3 .

Conclusion

We have introduced MeshDOG and MeshHOG, a new 3D interest point detector and a new 3D descriptor, defined on uniformly sampled triangular meshes. The descriptor is able to capture the local geometric and/or photometric properties in a succinct fashion. It is robust to changes in orientation, rotation, translation and scale. We have presented results of matching various rigid and non rigid datasets, both on real sequences and on synthetically generated data. They demonstrate that local features detected on meshes using both photometric and geometric information are more robust than traditional purely photometric features detected in images.

Figure 1 .

 1 Figure 1. The feature detection method described in this paper can be applied to any scalar function defined over a 2D manifold such as the meshed surface shown here: photometric data (a) and associated points of interest (b); mean surface curvature (c) and the detected features (d).

Figure 2 .

 2 Figure 2. A vertex v and its rings (left) and the first ring of v (right).

1 3

 1 e avg and have performed 93 convolutions.

Figure 3 .

 3 Figure 3. Feature detection shown with photometric data. (a) Original mesh (27240 vertices); (b) Scale-space extrema (5760 vertices left); (c) Thresholding (1360 vertices left); (d) Corner detection (650 vertices left).

Figure 4 .

 4 Figure 4. An example of processing (a) photometry and (b) mean curvature. Scale space photometric representation (c)-(f) and scale space representation of mean curvature (e)-(j).

  where d xx , d xy and d yy are second partial derivatives. We estimate them by applying the definition of directional derivatives (1) twice, e.g. d xy = ∇ S D-→ x f (v) • -→ y , where the gradient is computed using (3). The directions -→ x and -→ y

Figure 5

 5 Figure 5. a) 3D Histogram -polar mapping used for creating histograms via binning of 3D vectors; b) Choosing 3 orthogonal planes onto which to project the 3D Histogram. c) Polar Coordinate system used for creating histograms via binning of 2D vectors, shown in this example with 8 polar slices. d) Example of a typical spatial and orientation histograms, using 4 spatial polar slices and 8 orientation slices.

Figure 6 .

 6 Figure 6. "Rigid" matching results -Dino and Temple datasets. (a) (c) Matching results when using the colour both as a detector and as a feature; (b) (d) Error distribution when using different combinations of features for both detection and matching.

  Figure 7. "Non-rigid" matching using synthetic data -dancer-synth dataset. Comparison between MeshHOG and SIFT matching results. Matches between frames 1 and 50 are visually depicted in (a),(d). There are 364 matches for MeshHOG and 343 matches for SIFT. They are also quantified in the error histograms (b),(e). The histogram bins are of size equal to eavg. The last bin groups all the errors greater than 20 * eavg. Additionally, the average histogram errors are shown in (c),(f) for matching frame 1 with x, where x ∈ [2..200].

Figure 8 .

 8 Figure 8. "Non-rigid" matching using real data -Dance-1 sequence. (a) Matches between frames 525 and 526 using Mesh-HOG (119 matches); (b) Matches between frames 530 and 550 using MeshHOG (13 matches); (c) Matches between frames 525 and 526 using SIFT (54 matches); (d) Matches between frames 530 and 550 using SIFT (0 matches).

3

  http://perception.inrialpes.fr/

Figure 9 .

 9 Figure 9. Resilience to noise. Two kinds of noise are being applied: geometry noise (changing the vertices v) and colour noise (changing values f (v) held in each vertex). a) Synth-dance dataset (frame 1 and 50) -True Positive (TP); b) Synth-dance dataset (frame 1 and 50) -False Positive (FP) Ratio; c) Dino dataset -True Positives (TP); d) Dino dataset -False Positive (FP) Ratio.

http://shape.cs.princeton.edu/benchmark/

Acknowledgements. This research was supported by the VISIONTRAIN RTN-CT-2004-005439 Marie Curie Action within the European Community's Sixth Framework Programme.