Antoine Dechezleprêtre

Matthieu Glachant
email: glachant@ensmp.fr

Yann Ménière

Technology transfer by CDM projects: a comparison of Brazil, China, India and Mexico

, we gave a general description of technology transfers by CDM projects and we analyzed their drivers. In this paper, we use the same data and similar econometric models to explain inter-country differences. We focus on 4 countries gathering about 75% of the CDM projects: Brazil, China, India, and Mexico. 68% of Mexican projects include an international transfer of technology. The rates are respectively 12%, 40% and 59% for India, Brazil and China.

Our results show that transfers to Mexico and Brazil are mainly related to the strong involvement of foreign partners and good technological capabilities. Besides a relative advantage with respect to these factors, the higher rate of international transfers in Mexico seems to be due to a sectorcomposition effect. The involvement of foreign partners is less frequent in India and China, where investment opportunities generated by fast growing economies seem to play a more important role in facilitating international technology transfers through the CDM. International transfers are also related to strong technology capabilities in China. By contrast, the lower rate of international transfer (12%) in India may be due to a better capability to diffuse domestic technologies.

Introduction

The success of post-Kyoto climate policies will crucially hinge on the involvement of fast growing emerging countries such as China, India or Brazil. Such involvement however raises difficult policy issues that largely shape the current climate negotiations. To reduce the greenhouse gas (GHG) intensity of their growth paths, emerging countries would have to implement environmentally friendly technologies on a massive scale. Thus far, most of these technologies have been developed and used in developed countries. To catch up, developing countries must either develop the technology by their own means, or acquire it abroad -two costly options. Against this background, enhanced action on technology development and transfer was marked as one of the objectives the December 2007 Bali road map, and discussions have started in the Expert Group on Technology Transfer to find effective and acceptable mechanisms to fulfil this goal.

The Clean Development Mechanism (CDM) of the Kyoto protocol was a first attempt to address these challenges. CDM allows industrialized countries which have accepted emissions reduction targets to develop or finance projects that reduce GHG emissions in non-Annex 1 countries in exchange for emission reduction credits. Since reducing GHG emissions in a less-developed country may be cheaper than doing so domestically, it helps Annex 1 countries to achieve their emission reduction target at a lower cost. Besides saving abatement costs, the goal of the CDM is to promote sustainable development in non-Annex 1 countries (for a review on this aspect of the CDM, see [START_REF] Olsen | The Clean Development Mechanism's Contribution to Sustainable Development: a Review of the Literature[END_REF]. It is also considered by many as a key means to boost technology transfer and diffusion. Projects may in particular lead to international transfer if the technology used in the project is not available in the host country and has to be imported. Although international transfers are not necessarily better than the replication of domestic technology (the latter being in some cases more appropriate to match local conditions), it is of course important to analyze whether the CDM is effective in this respect. We aim to do so in this paper by comparing international technology transfers induced by the CDM in four emerging countries -namely China, India, Brazil and Mexico -which are also the main recipients of CDM projects.

The transfer of GHG mitigation technologies to developing countries is the subject of an extensive general literature (for example, [START_REF] Blackman | The Economics of technology diffusion: implications for climate policy in developing countries[END_REF][START_REF] Yang | Should the north make unilateral technology transfers to the south? North-South cooperation and conflicts in responses to global climate change[END_REF]IPCC, 2000;Yang and Nordhaus 2006).

Numerous case studies of successful technology transfers have also been conducted in order to assess the drivers for and barriers to technology adoption (for instance, OCDE/IEA 2001;[START_REF] Kathuria | Technology for GHG Reduction: A Framework with Application to India[END_REF][START_REF] Ockwell | Key policy considerations for facilitating low carbon technology transfer to developing countries[END_REF]. Ockwell et al. The literature on technology transfers through CDM is more recent but it is growing fast. A good review can be found in Scheider et al. (2008), with a detailed analysis of the CDM contribution to the alleviation of various barriers to technology transfer. Several papers use a quantitative approach. Based on a sample of 63 registered projects, De Coninck, Haake and van der Linden (2007) show that imported technologies originate mostly from the European Union and that the investments from industrialized countries associated with the CDM are small when compared to total foreign direct investments. [START_REF] Seres | Analysis of Technology Transfer in CDM Projects[END_REF] 1 and [START_REF] Dechezleprêtre | The Clean Development Mechanism and the international diffusion of technologies: An empirical study[END_REF] analyze technology transfers respectively in 2293 projects in the CDM pipeline and 644 registered projects. They find transfers in respectively 39% and 43% of these projects (accounting for 64% and 84% of emission reduction claims). Using regression analysis, both papers find that larger projects and projects with foreign participants involve more technology transfer. [START_REF] Dechezleprêtre | The Clean Development Mechanism and the international diffusion of technologies: An empirical study[END_REF] consider other variables such as the technology capabilities of recipient countries, and whether project developers are subsidiaries of Western companies, both of which have significant positive effects on transfers.

As compared to these papers, our originality is to compare different countries and to seek to identify what explains their differences. We follow the econometric approach used in [START_REF] Dechezleprêtre | The Clean Development Mechanism and the international diffusion of technologies: An empirical study[END_REF]. We use the same data and similar econometric models to explain inter-country differences.

The four countries we focus on -Brazil, China, India, and Mexico -gather about 75% of the CDM projects. We seek to highlight and to explain the national specificities of technology diffusion by the CDM, such as differences in the percentage of projects where a technology is imported from abroad.

Although our main focus is on international transfers of technology, we also take into account and discuss country differences as regards the diffusion of purely domestic technology.

The remainder of this article is organized as follows. In section 2 we describe the data. Then we give descriptive statistics by country on the frequency of transfer, on the types of technology involved, etc.

In Section 4, we present an econometric model which is used in Section 5 to explain inter-country differences with respect to technology transfer. We conclude in Section 6.

Data issues

Sources

Our data describe all the 644 projects registered as of May 1 st , 2007. These projects account for an expected 888.5 million tons of CO 2 -equivalent (MtCO 2 eq) emissions reductions by the end of 2012.

We use three main information sources to describe these projects: 1) the UNEP Risoe Center CDM Pipeline database2 , 2) the Project Design Documents, and 3) data from international institutions such as the World Bank and the World Trade Organization for country-level economic and technological variables.

For every CDM project, the UNEP Risoe Center CDM Pipeline database includes the host country, the type of technology, the estimated amount of the annual emissions reductions, the cumulative emissions reductions to the end of the Kyoto period (31 December 2012) and the countries that will buy the carbon credits generated by the project (if already available). We have also collected the registration dates of each project and the name of every country involved, on the UNFCCC website dedicated to CDM projects3 .

The content of the Project Design Documents (PDD) is our main source of information. They are mandatory standardized documents of about 50 pages submitted to the Executive Board by the project developers for registration. In the PDDs, we have collected information about the technology used, whether there is a transfer or not, the type of transfer, the project implementer (name, business sector and name of parent company) and every foreign partner involved (name, location). We have also retrieved information on the role of the project partners: are they credit buyers, consulting companies, PDD consultants or equipment suppliers?

Host country characteristics, including information on GDP, trade or FDI flows have been obtained from the World Bank's World Development Indicators 20064 . We have completed this information with economic performance indicators from the Earth Trends database of the World Resource Institute5 . To proxy the technological capability of a country to import and use advanced technology, we have used the composite index Arco developed by [START_REF] Archibugi | A new indicator of technological capabilities for developed and developing countries (ArCo)[END_REF].

Information on technology transfers

We define technology transfer as the import of a technology from abroad. It is important to keep in mind that this definition does not encompass all forms of technology diffusion. CDM projects may also entail technology transfers within a country, e.g; from an urban to a rural area. Unfortunately such intra-country transfers are difficult to track in PDDs, and therefore they do not lend themselves easily to statistical analysis. By contrast, international transfers can be identified and make it possible to carry out more ambitious analysis. They are also of prime interest for us since they relate directly to international negotiations on technology transfers.

The technology that is transferred may take various forms. Knowledge transfers take place if the local project developer benefits from the transfer of knowledge, know-how, information or technical assistance from a foreign partner. By contrast, an equipment transfer consists in importing equipment, such as wind turbines or gas burners, from a supplier located in a foreign country. Of course, a project can involve both a transfer of equipment and a transfer of knowledge.

We find information on transfers in the PDDs. In principle, the technology to be employed in the project activity is described in section A.4.36 . But this is not a compulsory requirement, and no section is specifically devoted to technology transfer. Indeed, claims of technology transfer can often be found in other [s] sections such as "Description of the project activity" (A.2) or "Barrier analysis" (B.4). Section G ("Stakeholders' comments") sometimes contains interesting information on equipment suppliers.

Further information on the technology employed may also be displayed in the annex. In order to get relevant information, we have read carefully all the PDDs 7 . More details and examples can be found in [START_REF] Dechezleprêtre | The Clean Development Mechanism and the international diffusion of technologies: An empirical study[END_REF].

How reliable is this information? There are several potential problems which we have tried to mitigate.

In some PDDs, a transfer of technology may refer to the simple adoption of a new technology. If the technology provider is clearly located within the country, the project involves no international transfer; consequently our database records no international transfer for that project in that country.

Another difficulty concerns specifically the import of equipment. From a general point of view, the import of goods does not always entail a technology transfer. For instance, importing a DVD player made in China into the U.S. does not. The same is true for CDM projects which might include the import of generic devices. In this regard, we have considered that the import of equipment is associated with a technology transfer as soon as the PDD claims that it is.

It remains that PDD editors have an incentive to overstate the existence of technology transfer as it helps project registration. Accordingly, type I errors are unlikely while type II errors could be frequent even if any claim of technology transfer should be justified in the PDD 8 . Therefore, descriptive statistics regarding technology transfer percentages are probably less reliable than other figures. 9 This is a usual difficulty with this type of study. But one can realistically assume that this bias is randomly distributed over the PDD-writing population. Therefore, this problem probably does not damage our econometric results.

Descriptive statistics by country

In this section, we describe the international technology transfers occurring in CDM projects in Brazil, China, India and Mexico. As shown in Table 1, the share of projects involving such transfers varies greatly across countries. 68% of projects set up in Mexico involve an international technology transfer, but only 12% of projects located in India.

In most cases international transfers are not limited to the import of equipment. The transfer of both equipment and knowledge is observed in 42% of Chinese projects and 46% of Indian projects.

Transfers of knowledge alone are very frequent in Brazil (23%) and in Mexico (68%). This is mainly due to the high number of projects taking place in the agricultural sector in these two countries.

7 For efficiency purposes, we first searched the PDDs for the words "technology", "transfer", "equipment", "supplier", "import", "manufacturer" and "training". If no information on technology transfer could be found through this search, we then read through the entire PDD. 8 A type I error consists of wrongly describing a project as not involving any technology transfer. Conversely, a type II error occurs when a project is wrongly described as involving a technology transfer (when it does not). 9 [START_REF] Haites | Technology Transfer by CDM projects[END_REF] find that 33% of the projects involve transfer, compared to 43% in our data set. One possible reason is that the datasets are slightly different. Another is the procedure used in both papers for encoding technology transfer. We read the entire PDDs whereas [START_REF] Haites | Technology Transfer by CDM projects[END_REF] only searched for the word "technology". We now give more specific information on the types of technology that are transferred in each country.

Brazil

CDM projects in Brazil belong to two main types: renewable energy production and biogas recovery in breeding farms and landfills (see table 3). Renewable energy projects mostly consist of hydro power and biomass energy production. The latter are usually set up in sugar mills where bagasse -a residue from sugarcane processing -is used as a feedstock for cogeneration of heat and electricity. These power plants rely on direct-fired systems that are very similar to usual fossil-fuel fired power plants.

India

India is the main host country for CDM projects but as mentioned above, international technology transfer is very limited. However this does not imply that there is no technology diffusion. As in China, biomass energy and hydro power projects rely on local technologies (see Table 5). But, contrary to China, most wind power projects use equipment produced by local manufacturers (mainly Suzlon and Enercon India).

Energy efficiency measures in industry -power generation from waste heat recovery or reduction of steam consumption -are usually designed locally. However, technology partnerships have been set

up in a few projects. For example, Technovacuum Russia has supplied a technology aiming at reducing steam consumption in a petroleum refinery and Giammarco-Vetrcoke Italy has implemented a solution to reduce energy consumption at an ammonia plant. The technology used in the three HFC destruction projects also comes from Europe (Ineos UK, SGL Acotec and Caloric Anlagenbau Germany).

Interestingly, the unique solar power project in India has been developed through a partnership between a German physicist Wolfgang Scheffler -who has invented the so-called Scheffler reflectors for solar cooking -and Indian institutions. Among the other Mexican projects with technology transfer, there is one large HFC project, which yields more annual emission reductions than the 69 biogas recovery projects altogether, and three wind power projects using turbines supplied by Gamesa Eolica. Two landfill gas projects have been developed through a partnership between EcoMethane and technology providers from UK, Biogas Technology Ltd and ENER*G.

Econometric model

In the previous section, we have presented statistics describing inter-country differences in international technology transfers by CDM. These statistics do not help us to understand what drives these differences. For instance, 59% of the Chinese projects involve an international transfer while the percentage is only 12% in India. Why is it so? Is it because the technological capability of India is less than that of China or, by contrast, because India can rely on local technology? Is it due to sector composition effect -Indian projects may take place in economic sectors where a transfer is less likely? Is it due to project characteristics? In this section, we present an econometric model which we will use in the next section to answer these questions. Econometric analysis allows us to determine the specific effect of each variable on the likelihood that a project involves international technology transfer, all other factors being held constant.The model is very close to the models presented in [START_REF] Dechezleprêtre | The Clean Development Mechanism and the international diffusion of technologies: An empirical study[END_REF].

Model specification

Let TECH_TRANSFER denote a binary variable equal to 1 if a project involves a technology transfer (regardless of the nature of this transfer), and to 0 otherwise. To examine the relationship between TECH_TRANSFER and a set of explanatory variables, the following logit equation is estimated:

Pr( _ 1
) 1

e TECH TRANSFER e

Ω Ω = = +

(1) with:

( ) ( )

0 1 2 3 4 5 6 6 7 8 9 1 0 ( ) ( _ ) ( ) ( _ ) ( ) ( _ ) ( _ ) ( _ ) ( _ ) ( _ ) _ n n o LOGSIZE CREDIT BUYER SUBSIDIARY SIMILAR PROJECTS TRADE FDI INFLOWS GDP GROWTH LOG POPULATION GDP PERCAPITA CARBON INTENSITY TECH CAPACITY SECTOR CO α α α α α α α α α α α α α α Ω = + + + + + + + + + + + + + ( ) o UNTRY ε + α i is
a vector of coefficients to be estimated and ε is a random term identically independently distributed following a Gumbel extreme distribution. We use a set of regression variables at the project and country levels that are likely to influence the probability that a CDM project involves some international transfer of technology.

According to [START_REF] Schneider | Understanding the CDM's Contribution to Technology Transfer[END_REF], technology transfers through CDM projects are hindered by four types of barriers pertaining respectively to their commercial viability; the lack of information on the existence and functioning of the CDM, or on available technologies; a lack of access to capital; and the institutional framework in the host country. Following their analysis, we identify three variables at the project level that may help to alleviate the first three barriers.

We use the log of the project size (LOGSIZE), as measured by its annual emissions reduction, as an indicator of the commercial viability of CDM projects. 10 As a general rule, the CDM registration process entails large transaction costs that are fixed and therefore represent a strong impediment to small scale projects [START_REF] Michaelowa | Transaction Costs of the Kyoto Mechanisms[END_REF]. Similarly, upfront investment costs are higher when technology is imported from industrialized countries [START_REF] Schneider | Understanding the CDM's Contribution to Technology Transfer[END_REF]. This is especially true when the technology is at an early commercialization stage, which it is often the case with environmentally sound technologies [START_REF] Wilkins | Technology Transfer for Renewable Energy Overcoming Barriers in Developing Countries[END_REF]. Consequently, we can expect projects involving technology transfer to be more viable if they are large.

The two other project variables relate to the access to information and capital. SUBSIDIARY is a dummy variable indicating whether the project is implemented in the subsidiary of a company located in an Annex 1 country. The involvement of a parent company can facilitate technology transfers in many ways. It may help manage the CDM registration, provide expertise at the technology level, or provide an easier access to capital.

Financial barriers can also be alleviated thanks to the participation of one or more credit buyers that are not parent companies but rather carbon founds. Before the project developer can sell the credits, the UNFCCC must first certify, issue and register the emission reduction and this administrative process takes time. Selling credits through a forward contract can be of great help. It reduces the risk surrounding the investments by adding a guaranteed revenue stream. One can assume that credit buyers also give advice and bring expertise that may ease technology transfer. We therefore define CREDIT_BUYER as a dummy variable indicating the participation of one or more credit buyers in the project, and expect a positive effect of this variable on the probability of international transfer.

The remaining variables characterize the capability of the host country to attract international technology transfers. We include the country size (LOG_POPULATION), the per capita GDP (GDP_PERCAPITA) and the carbon intensity of the economy (CO2_INTENSITY) as usual control variables 11 . Although they are likely to affect positively the number of opportunities to undertake CDM projects, it is not obvious how they could influence the probability that those projects involve international technology transfers. By contrast, we can expect the variable GDP_GROWTH to have a positive impact on such transfers. Indeed a fast growth hinges on sustained investments which offer more opportunities for implementing new technologies through CDM projects.

Empirical evidence indicates that the adoption of a new technology is strongly associated with human capital, supporting infrastructure and research and development activities (Blackman 1997). In order to measure this technological capability (TECH_CAPABILITY), we use the ArCo technology index 10 Using the logarithm of the size ensures that the few very large HFC projects do not have a disproportionate influence on the results. 11 Per capita GDP and population are similarly used as control variables in previous works (see for instance [START_REF] Haites | Technology Transfer by CDM projects[END_REF][START_REF] Seres | Analysis of Technology Transfer in CDM Projects[END_REF]. We added the carbon intensity of the economy as a control variable following several requests to do so by readers of previous versions of this work. As could be expected, we find no significant effect of this control variable.

developed by [START_REF] Archibugi | A new indicator of technological capabilities for developed and developing countries (ArCo)[END_REF]. This composite indicator captures three aspects determining technological capabilities: the creation of technology (number of patents and number of scientific articles), the technological infrastructures (internet penetration, telephone penetration and electricity consumption) and the development of human skills (percentage of tertiary science and engineering enrolment, mean years of schooling and literacy rate). It must be noticed that the technological capability, although favoring international technology transfers at a macroeconomic level, may also imply that the technology required for CDM projects are available locally. To take this possibility into account, we add as a country variable the number of other CDM projects using the same technology within the host country (SIMILAR_PROJECTS). Of course we can expect that international technology transfers are less likely when similar projects are carried out in the same country.

There is also strong empirical evidence that international trade and Foreign Direct Investments (FDI)

promote the transfer of technology across countries [START_REF] Coe | North-South R&D spillovers[END_REF]. A country openness to global trade can indeed alleviate barriers pertaining to access to information and to technology. It may also denote a favorable institutional environment. Accordingly, we use the variable TRADE which is the ratio of the sum of exports and imports of merchandise on GDP and FDI_INFLOWS which is the level of incoming FDI in the host country.

Finally, SECTORi and COUNTRYi are vectors of sector dummies and country dummies, respectively.

They control for sector-and country-specific characteristics that are not captured by the other variables.

Estimation results

Results are displayed in Table 7. The overall quality of the estimation is reasonably good. The

McFadden pseudo R-squared is 0.36 and the model correctly predicts 80% of the observed outcomes.

The coefficients exhibit the expected signs.

We will be very quick on the comments of these results as this was the prime goal of the companion paper (Dechezlepretre et al., 2008). Technology transfer increases with the size of the project (LOGSIZE). The participation of one or more credit buyers in the project (CREDIT_BUYER variable) also increases the likelihood that the project involves technology transfer. Marginal calculations show that a project with a credit buyer has a 16% higher probability of involving a technology transfer.

Being the subsidiary of a company from an Annex 1 country (as indicated by the dummy variable SUBSIDIARY) clearly favors the transfer of technology. The coefficient is highly significant and much larger than that of CREDIT_BUYER. In marginal terms, the transfer likeliness of a project located in the subsidiary of an Annex 1 company is 50% higher.

Turning next to country variables, it is worth noting that the average annual rate of GDP growth from 2000 to 2004 (GDP_GROWTH) has a very high impact on the likeliness of technology transfer: one additional percentage point of average GDP growth raises transfer likeliness by 19%. The variables LOG_POPULATION , GDP_PERCAPITA and CO2_INTENSITY have no significant effects in the regression.

As expected, trade openness (TRADE) reinforces the likelihood of technology transfer. In contrast, the share of FDI inflows in GDP has a negative impact on transfer. This may be due to the fact that capital links are already captured by the variable SUBSIDIARY. National technological capabilities (TECH_CAPABILITY) have a positive and significant impact on transfer likeliness, while the number of other CDM projects using the same technology within the host country (SIMILAR_PROJECTS) lowers the probability of transfer.

Country comparison

In this section, we use the econometric model presented in section 2 in order to analyze the impact of the explanatory variables on the overall rate of technology transfer in the different host countries. The discussion about the sign of the coefficients does not yield information about the size of the effects of the explanatory variables. In order to compare these effects across countries, we draw Figure 1 using the model's results. Figure 1 is based on the following calculation. Let i x be the average value of the variable i

x in a sample of projects and let i β denote the value of its coefficient. Then, the product The two other variables, namely CREDIT_BUYER and SUBSIDIARY, denote important differences in countries' capacities to attract foreign partnerships. China and Mexico have clearly benefited from the involvement of foreign credit buyers. The advantage of Mexico is even stronger as regards foreign subsidiaries, for which Brazil is also well positioned. In contrast, India performs poorly with respect to both variables.

Turning next to country variables, the strong effect of GDP_GROWTH clearly indicates that international technology transfers are more likely in fast growing economies. Although all countries have substantial growth rates, the very fast economic growth in India and in China seem to be decisive factors in their abilities to generate projects involving technology transfers.

International technology transfers are also strongly correlated to national technology capabilities (TECH_CAPACITY). Beside a small lag in the case of India, all countries benefit in equal proportions from attractive technological capabilities. One must however balance this effect with the impact of the variable SIMILAR_PROJECTS which denotes the number of other CDM projects using the same technology within the host country. Local availability of technologies has comparable negative impacts on the likelihood of technology transfers in each country. It mitigates the positive effect of TECH-CAPACITY, without suppressing it entirely. Again, the net impact is the lowest in India, which suggest that India has been particularly successful in relying on domestic technology capabilities to diffuse carbon mitigation technology through the CDM.

Sector dummies are interesting in that they reflect the sector-composition effect. Figure 1 suggests that inter-country differences are not that much influenced by this. The exception is Mexico. One possible explanation is that this country gets very specialized in biogas recovery in breeding farms which frequently entail technology transfer.

Finally, the country dummies -BRAZIL, CHINA, INDIA and MEXICO -capture factors that are not taken into account by the other country-level variables (TRADE, FDI, GDP_GROWTH and TECH_CAPACITY). They may reflect administrative peculiarities -difference in intellectual property regimes, etc.-which are not described in the database. Figure 1 shows that these unobserved factors play a strong role in explaining country differences. Although, by nature, these effects are difficult to interpret, it is likely that the national policies with respect to CDM play an important role. China has for instance been slow in setting up a Designated National Authority (DNA) to help setting up CDM projects. In contrast, Mexico and Brazil seem to benefit of more proactive policies vis-à-vis CDM projects 12 .

We can now complete the discussion by relating these results with each country's performance in terms of technology transfers. Comparing the countries in Figure 1 suggests two different types of country profiles, namely Mexico and Brazil on the one hand, and China and India on the other hand.

12 Remember that every host country must give its approval to CDM projects through its DNA. Interestingly, the Brazilian Designated National Authority (DNA) is hosted by the Ministry of Science & Technology, while in the great majority of cases, the DNA is hosted by the Ministry of Environment or by some national environmental protection agency.

The relative success of Mexico (where the transfer rate is 68%) in attracting foreign technology when compared to other countries is mainly due a sector-composition effect (in particular, there are many projects of biogas recovery in breeding farms, a sector where transfers prevails) combined with good technological capabilities and a strong involvement of parent companies in Mexican subsidiaries.

Brazil has a similar profile but in lesser proportions. The effect of GDP_GROWTH is slightly stronger than in Mexico, while the positive impact of sector composition, foreign subsidiaries and technological capabilities is weaker.

The profiles of India and China are quite different. Indeed neither of them has experienced a strong involvement of foreign partners. The transfer rate of 59% in China is mostly explained by the dynamism of its economy (GDP_GROWTH), combined with good technological capabilities. In comparison with China, the lower rate of international technology transfers (12%) in India can be explained by a (relative) smaller advantage in terms of growth rates and technological capabilities, but also by a stronger propensity to rely on domestic capabilities to diffuse technology through the CDM.

Conclusion

We have described the international transfers of GHG mitigation technologies induced by the Clean Development Mechanism in Brazil, China, India and Mexico using a dataset including 644 CDM projects registered until May 2007.

Our analysis shows very large differences across countries. The percentage of projects where an international technology transfer takes place ranges from 12% in India to 68% in Mexico. Moreover, very different technologies are concerned. In Brazil and Mexico, projects recovering biogas in breeding farms represent an important share of the overall transfer. In China, Mexico and Brazil, the import of wind turbines is widespread whereas India mainly relies on local suppliers. Nevertheless, some technologies are imported whatever the country. This is true for HFC or N 2 O destruction technologies used in very large projects in the chemical industry. This is also the case of landfill gas capture and flaring.

Note that a high transfer rate does not mean that the country performs better than others. Consider the example of Indian wind power projects. India would seem to perform badly in this area since transfer frequency is low (23%) as compared to others (between 75% and 100%). But it is so because India is in fact more advanced in this area and has leading domestic producers like Suzlon.

We also develop an econometric analysis to investigate what drives these transfers. Our results highlight various patterns of technology diffusion. Transfers to Mexico (68% of CDM project) and Brazil (40%) are related to the same factors, namely the strong involvement of foreign partners and good technological capabilities. The high Mexican rate seems to be due to a relative advantage against Brazil with respect to these factors. Mexico moreover benefits from a sector-composition effect: many Mexican projects concern biogas recovery in breeding farms, a sector where transfers prevail.

The pattern of technology diffusion is quite different in China (59%) and India (12%). The involvement of foreign partners is less frequent, and international transfers seem rather related to the investment opportunities generated by fast growing economies. Our results suggest that technological capabilities may play different roles in both countries. Strong technology capabilities are positively correlated with international transfers in China. By contrast, the technology capabilities of India seem to be rather geared towards the replication of CDM projects involving domestic technologies only.

What are the policy lessons of this analysis? Excluding macro variables like GDP growth, the results stress the importance of project partnerships: promoting projects in subsidiaries of Annex 1 countries' companies and involving a credit buyer in the project clearly alleviate barriers to international transfers.

Our results also highlight the importance of capacity building as a means to accelerate technology diffusion. A strong technology capability facilitates the import of foreign technology, but it is also a source of domestic technologies to be diffused locally. Depending on which aspect is emphasized, it may thus be leveraged for very different patterns of technology diffusion.

β

  represents the average impact of i x on the linear predictor Ω of Equation (1). Calculating the value of i i x β for every variable allows setting the average weight of each variable against the decision to transfer technology. Figure1represents these weights for the different countries. Using the same metric, each bar measures the impact of the variable on an average CDM project in each country. Finally, we only represent statistically significant variables.

Figure 1 -

 1 Figure 1 -Comparative impacts of the explanatory variables for the different countries

Table 1 -International technology transfer by host country

 1

	Country	Total number of [N] projects	Number of projects involving technology transfer only only Knowledge Equipment Knowledge Equipment +	Percentage of technology transfer
			[E]	[K]	[B]	[(E+K+B)/N]
	India	225	10	5	13	12%
	Brazil	99	8	23	9	40%
	Mexico	78	4	45	4	68%
	China	71	11	1	30	59%
	Total	473	33	74	56	34%

Table 2

 2

gives additional information on the projects. In average, Chinese projects are much larger. This is essentially due to the presence of 7 huge projects of HFC-23 destruction. The percentage of projects which are located in the subsidiary of Annex 1 countries' companies is interesting as one might expect more transfers in these projects. In this regard, China and India sharply differ from Brazil and Mexico, where such projects are much more frequent. Finally, the presence of a foreign credit buyer may also facilitate transfer. They are involved in most projects in China and Mexico, but only in 36% of the Indian projects.

Table 2 -Project characteristics by host country

 2

	Variables	China	India	Brazil	Mexico
	Average size (ktCO 2 eq/year)	816.7	85.2	160.0	76.5
	Median size (ktCO 2 eq/year)	110	26	42	17
	Projects implemented in a subsidiary of annex I company	0%	3%	28%	56%
	Projects with a foreign credit buyer	89%	36%	52%	97%

  Thus there is no need to import technologies. Hydropower is also common in Brazil as it supplies more than 80% of electricity in this country. A few wind energy projects use turbines supplied by Enercon, Germany.The second most popular type of CDM projects in Brazil is biogas recovery. They generally entail technology transfer. In particular, projects in breeding farms mitigating biogas resulting from the decomposition process of animal effluents present interesting channels of technology diffusion. 85% of these projects benefit from technology transfers from AgCert. This Irish consulting company provides farmers with turnkey solutions, including training sessions on how to operate the technology. It also operates in Mexico as will see below.However, in terms of emission reductions, the most important projects concern landfill gas capture and N 2 O destruction. Projects in landfills mainly use foreign technology. In particular, several projects set up in subsidiaries of French companies Veolia Environnement and Suez benefited from internal transfers of know-how.As for the N 2 O destruction project, there is only one huge project in a chemical facility producing adipic acid. It amounts for nearly 6 million tons of annual CO 2 eq reductions, i.e. 38% of the annual reductions in Brazil by CDM projects. The plant is owned by Rhodia and the Brazilian facility benefits from transfers of know-how from the facility of Chalampé located in France.

Table 3 -Main project types and international technology transfers in Brazil

 3 China is the leading country for HFC-23 destruction projects. These 7 projects represent 80% of the annual reductions in China and they always entail a technology transfer. The French company Vichem provides the HFC destruction technology of 4 out of 7 projects. The rest is supplied by Japanese corporations.As landfill gas capture and flaring is new in China, local CDM developers have frequently cooperated with foreign suppliers such as Waste Management New Zealand or Energi Gruppend Jylland Denmark. This leads to an 85% rate of technology transfer in this area.

			Percentage of	Average	Total annual
	Type of technology	Number of projects	projects involving intern. technology.	project size (annual	reductions (ktCO 2 eq)
			transfer	ktCO 2 eq)
	Biomass energy	34	9%	51	1747
	Biogas recovery in agriculture (breeding farms)	20	90%	74	1477
	Hydro power	19	11%	45	852
	Landfill gas recovery	13	85%	402	5225
	N 2 O destruction	1	100%	5961	5961
	Wind power	4	75%	42	169
	Energy efficiency (industry)	2	0%	47	93
	Fossil fuel switch	5	20%	20	99
	Fugitive gas recovery	1	100%	220	220
	3.2 China
	China also implements many renewable energy projects as shown in Table 4. The country can rely on
	local technologies for hydro power and biomass energy projects but depends upon imported turbines
	for wind power projects. The main suppliers of wind turbines are Gamesa Eolica (Spain) with 12

projects and Vestas (Denmark) with 8 projects. Notably, 55% of the wind projects registered in April 2007 use turbines manufactured by the local firm Goldwind. Imported turbines have higher capacities on average than locally produced turbines (1.11 MW against 750 kW).

Table 4 -Main project types and international technology transfers in China

 4

			Percentage	of	Average	Total annua
	Type of technology	Number of projects	projects involving intern. technology	project size (annual	reductions (ktCO 2 eq)
			transfer		ktCO 2 eq)
	Wind power	34	74%		112	3807
	Hydro power	13	0%		104	1349
	HFC decomposition	7	100%		6743	47200
	Biomass energy	5	20%		160	802
	Methane destruction	3	66%		462	1387
	Energy efficiency (industry)	3	66%		804	2413
	Landfill gas recovery	4	100%		163	652
	N 2 O destruction	1	100%		350	350
	Reforestation	1	0%		26	26

Table 5 -Main project types and international technology transfers in India

 5

			Percentage of	Average	Total annual
	Type of technology	Number of projects	projects involving intern. technology	project size (annual	reductions (ktCO 2 eq)
			transfer	ktCO 2 eq)
	Biomass energy	78	8%	38	2926
	Energy efficiency (industry)	54	17%	85	4595
	Hydro power	30	0%	34	1030
	Wind power	26	23%	29	763
	Reduction of the share of clinker in cement production	13	0%	119	1544
	Biogas (other)	7	0%	32	224
	HFC decomposition	3	100%	2589	7766
	Fossil fuel switch	4	25%	43	171
	Energy efficiency (services)	1	100%	3	3
	Energy efficiency (supply side)	6	0%	6	38
	Solar power	1	100%	1	1
	3.5 Mexico

Mexico is very specific: almost 90% of CDM projects concern biogas recovery in breeding farms (Table

6

). AgCert -the Irish company previously evoked for Brazil -has initiated 41 projects involving technology transfers through training of local staff. Granjas Carroll Mexico -the largest commercial pig producer in Mexico -has developed 24 projects with the help of the EcoSecurities (though no technology transfer is claimed in this case). The CDM has clearly enhanced the diffusion of biogas mitigation among Mexican pork producers.

Table 6 -Main project types and international technology transfers in Mexico

 6

	Type of technology	Number of projects	Percentage of projects involving transfer intern. technology	Average project size ktCO 2 eq) (annual	Total annual reductions (ktCO 2 eq)
	Biogas recovery in agriculture (breeding farms)	69	65%	31	2146
	HFC decomposition	1	100%	2155	2155
	Hydro power	2	50%	43	87
	Landfill gas	2	100%	186	373
	Wind power	3	100%	400	1201
	Biogas (other)	1	100%	4	4

Table 7 -Regression results of model explaining TECH_TRANSFER

 7

	Dependant variables	Coefficients
	LOGSIZE	0.2806*** (0.0843)
	CREDIT_BUYER	0.5050** (0.2509)
	SUBSIDIARY	2.3511*** (0.3579)
	SIMILAR_PROJECTS	-0.4103*** (0.1206)
	TRADE	0.0090* (0.0057)
	FDI_INFLOWS	-0.2674* (0.1363)
	GDP_GROWTH	0.6882*** (0.2225)
	GDP_PERCAPITA	-0.0001 (0.0001)
	LOG_POPULATION	-0.2566 (0.2641)
	CARBON_INTENSITY	0.0002 (0.0003)
	TECH_CAPABILITY	0.0722* (0.0400)
	SECTOR i	─
	COUNTRY i	─
	Nb of observations	643
	Pseudo-R2	0.36
	Percentage of correct predictions	79.8 %
	Notes: Standard errors in parentheses; * denotes significance at 10% level, ** denotes significance at 5% level,
	and *** denotes significance at 1% level.

In an extension of[START_REF] Haites | Technology Transfer by CDM projects[END_REF].

The database is available at http://cdmpipeline.org/

http://cdm.unfccc.int/Projects/index.html

Available online at http://devdata.worldbank.org/wdi2006

http://earthtrends.wri.org/

The Guidelines for completing the PDD available from UNFCCC indicate that "this section should include a description of how environmentally safe and sound technology, and know-how to be used, is transferred to the host Party(ies)."

Appendix

Projects and technology transfers by type of technology


J.-Y Brua
email: brua@math.u-strasbg.fr

Adaptive estimators for nonparametric heteroscedastic regression models

Keywords: Adaptive estimation, Heteroscedastic regression, Kernel estimator, Minimax, Nonparametric regression MSC: primary 62G08, secondary 62G05, 62G20

come

Adaptive estimators for nonparametric heteroscedastic regression models

Introduction

A lot of research studies are devoted to the problem of estimating a regression function and specially a function belonging to a Hölder class. In this direction it is known from [START_REF] Sacks | Asymptotically optimum kernels for density estimation at a point[END_REF] that the linear minimax estimator is a kernel estimator in the case of a quasi-Hölder regression function estimated at a single point with squared error loss. This estimator is within 17 percent of asymptotically minimax over all procedures (see [START_REF] Donoho | Geometrizing rates of convergence[END_REF]). Furthermore it is proved in [START_REF] Donoho | Asymptotic minimax risk for sup-norm loss: solution via optimal recovery[END_REF][START_REF] Korostelev | Exact asymptotically minimax estimator for nonparametric regression in uniform norm[END_REF] that a kernel estimator is asymptotically efficient when the Hölder regression function or its k th derivative is estimated with the sup-norm global loss. The reader is referred for instance to [START_REF]Efromovich Nonparametric Curve Estimation[END_REF][START_REF] Golubev | Asymptotically minimax estimation of a regression function in an additive model[END_REF][START_REF] Donoho | Wavelet shrinkage: asymptopia?[END_REF] for other regression works.

Our regression problem is the following. Suppose we observe data from:

y k = S(x k ) + g(x k , S)ξ k , k ∈ {1, . . . , n}, (1)
where x k = k/n, (ξ k ) k∈{1,...,n} are independent identically distributed standard Gaussian random variables. We are interested in the estimation of the regression function S at a given point z 0 ∈]0; 1[. We point out the fact that in this heteroscedastic regression model the variance of the noises depends on the unknown function S and on the regressors x k . This kind of model is used in financial analysis or in medical research (see, e.g., [START_REF] Goldfeld | Nonlinear Methods in Econometrics[END_REF]). More recently we can find such a model in [START_REF] Cai | Adaptive variance function estimation in heteroscedastic nonparametric regression[END_REF] where the authors reduce their classical regression model with variance V depending only on the regressors to a specific regression model where the regression function is nearly V and the noise term depends on V . We assume that the regression function belongs to a Hölder class but its smoothness parameter β remains unknown. We consider the absolute error loss and for the corresponding risk for which we aim at constructing an adaptive estimator which attains the minimax rate. Because of the adaptation, this rate differs from the one in the case where β is known. Many papers deal with adaptive problems, see for instance [START_REF] Barron | Risk bounds for model selection via penalization[END_REF][START_REF] Galtchouk | Efficient adaptive nonparametric estimation in heteroscedastic regression models[END_REF][START_REF] Lepskiȋ | A problem of adaptive estimation in Gaussian white noise[END_REF][START_REF] Lepskiȋ | Asymptotically minimax adaptive estimation. I. Upper bounds. Optimally adaptive estimates[END_REF][START_REF] Lepskiȋ | Asymptotically minimax adaptive estimation. II. Schemes without optimal adaptation. Adaptive estimates[END_REF]. Our construction is based on those one can find in [START_REF] Lepskiȋ | A problem of adaptive estimation in Gaussian white noise[END_REF][START_REF] Galtchouk | Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes[END_REF] for adaptive estimation of the drift coefficient in diffusion processes. We propose an adaptive kernel estimator which attains the minimax rate, that is to say that its risk has an asymptotic finite upper bound whereas the minimax risk is bounded away from zero. If these two bounds coincide we say that the estimator is asymptotically efficient.

We proceed with the method developed in [START_REF] Galtchouk | Asymptotically efficient estimates for nonparametric regression models[END_REF] in the homoscedastic and non adaptive case. In this paper the risk of an estimator is defined as the supremum of the absolute error loss taken over a neighborhood (called weak Hölder class) of functions that allows an arbitrary large derivative but has an additional weak Hölder condition (see [START_REF] Donoho | Geometrizing rates of convergence[END_REF]). It has led to the heteroscedastic case studied in [START_REF] Brua | Asymptotically efficient estimators for nonparametric heteroscedastic regression models[END_REF]. In these cases it is shown that a kernel estimator is asymptotically efficient, with the minimax rate n β/(2β+1) . Here we found the same asymptotic lower bound for the minimax risk as in the non adaptive case with the minimax rate n ln n β/(2β+1) , but unfortunately not the same asymptotic upper bound for the risk of the adaptive kernel estimator.

The paper is laid out as follows. Section 2 gives the description of the problem with all assumptions needed and all definitions of necessary mathematical objects. In section 3 we construct an adaptive estimator for which an upper bound of the risk is found. The lower bound of the minimax risk is given in section 4. Appendix A contains technical results for our proofs.

Statement of the problem

Consider model [START_REF] Sacks | Asymptotically optimum kernels for density estimation at a point[END_REF] where S ∈ C 1 ([0; 1], R) and g : [0; 1] × C 1 ([0; 1], R) -→ R * + are unknown functions. We want to estimate the regression function S at a fixed point z 0 ∈]0; 1[. The problem assuming that

S ∈ M,K>0 H(M, K, β), where β ∈ [β ; β ] ⊂]1; 2[, H(M, K, β) = S ∈ C 1 [0; 1] : ||S || ≤ M, sup x,y∈[0;1] |S (y) -S (x)| |x -y| α ≤ K , with β = 1 + α and ||f || = sup x∈[0;1] |f (x)|, remains open.
As we can find in [START_REF] Brua | Asymptotically efficient estimators for nonparametric heteroscedastic regression models[END_REF] and [START_REF] Galtchouk | Asymptotically efficient estimates for nonparametric regression models[END_REF] where non adaptive problems of asymptotically efficient estimation for nonparametric regression models are solved, we define the local weak Hölder class U z 0 ,δ at the point z 0 for the true value of the parameter β as:

S ∈ C 1 ([0; 1], R) : S ≤ δ -1 ; ∀h ≥ 0, 1 -1 S(z 0 + hu) -S(z 0 ) du ≤ δh β ,
(2) where δ ∈]0; 1[. The smoothness parameter β is supposed to be unknown whereas the interval [β ; β ] is considered as known. Notice that

1 -1 S(z 0 + hu) -S(z 0 ) du = 1 -1 z 0 +uh z 0 (S (t) -S (z 0 ))dt du, (3)
so we have for all S ∈ H(M, K, β),

1 -1 S(z 0 + hu) -S(z 0 ) du ≤ 2K β(β + 1) h β .
That is why the class U z 0 ,δ is called a weak Hölder class. The risk of an estimator Ŝ of S(z 0 ) is defined over the neighborhood U z 0 ,δ by

R z 0 ,δ,n ( Ŝ) = sup β∈[β ;β ] sup S∈Uz 0 ,δ N (β) g(z 0 , S) E S | Ŝ-S(z 0 )|, where N (β) = n ln n β/(2β+1)
. We assume that there exists two known constants g > 0 and g < ∞ such that

g ≤ inf 0≤x≤1 inf S∈C 1 ([0;1]) g(x, S) ≤ sup 0≤x≤1 sup S∈C 1 ([0;1]) g(x, S) ≤ g .
Moreover we suppose that the function g is uniformly continuous with respect to both variables x and S.

Upper bound

The paper [START_REF] Galtchouk | Asymptotically efficient estimates for nonparametric regression models[END_REF] handles the homoscedastic non adaptive case, considering the kernel estimator

S * h (z 0 ) = 1 q n (h) n k=1 Q x k -z 0 h y k ,
where

Q = I [-1,1] , h = n -1/(2β+1) and q n (h) = n k=1 Q x k -z 0 h . Taking into account
the fact that β is unknown we can not use such an estimator because the bandwidth h depends on β. That is the reason why we create a partition of the interval [β ; β ] in the following way:

β l = β + l β -β ln n , l = 0, . . . , [ln n],
where [a] denotes the integral part of a number a, and we define the corresponding bandwidths

h l = h(β l ) = n ln n -1/(2β l +1) . Then we set l = max 0 ≤ l ≤ [ln n] : max 0≤j≤l S * h l (z 0 ) -S * h j (z 0 ) - λ N j ≤ 0 ,
where

N j = N (β j ) and λ = 2 + 2 √ 2g β 2β + 1 - β 2β + 1 1/2 .
Notice that l really exists because the set above contains the index 0. The adaptive estimator is now defined as Ŝn = S * hl (z 0 ). Furthermore we associate with the unknown parameter β the unique integer l(β) ∈ {0, . . . ,

[ln n] -1} such that β l(β) ≤ β < β l(β)+1 .
The following result gives an upper bound for the risk of the adaptive estimator.

Theorem 3.1 One has lim sup δ→0 lim sup n→∞ R z 0 ,δ,n ( Ŝn ) ≤ λ e β -β g . Proof : Fix δ ∈]0; 1[ and write | Ŝn -S(z 0 )| = | Ŝn -S(z 0 )|I { l≥l(β)} + | Ŝn -S(z 0 )|I { l<l(β)} =: I 1 + I 2 ,
where

I 1 = [ln n] j=l(β) | Ŝn -S(z 0 )|I { l=j} = [ln n] j=l(β) |S * hj (z 0 ) -S(z 0 )|I { l=j} .
For all j = 0, . . . , [ln n], we note

S * hj (z 0 ) -S(z 0 ) = ζ n (β j ) q n (β j ) + B n (β j ), with q n (β j ) = q n (h(β j )), ζ n (β j ) = 1 q n (β j ) n k=1 Q x k -z 0 h j g(x k , S)ξ k , B n (β j ) = 1 q n (β j ) n k=1 Q x k -z 0 h j (S(x k ) -S(z 0 )). The variance of ζ n (β j ) is then σ 2 n (β j , S) := 1 q n (β j ) n k=1 Q x k -z 0 h j g 2 (x k , S).
One has

ζ n (β j ) q n (β j ) = |Z n (β j )| σ n (β j , S) q n (β j ) , with Z n (β j ) ∼ N (0, 1).
In addition, taking the decomposition used in [16, §4]:

B n (β j ) = nh j q n (β j ) 1 -1 (S(z 0 + h j u) -S(z 0 )) du + nh j q n (β j ) R n (β j ),
where

|R n (β j )| ≤ 6δ -1 n .
Using the weak Hölder condition we obtain

|B n (β j )| ≤ δ nh 1+β j q n (β j ) + 6δ -1 h j q n (β j )
, and then

|S * h j (z 0 ) -S(z 0 )| ≤ δ nh 1+β j q n (β j ) + 6δ -1 h j q n (β j ) + |Z n (β j )| σ n (β j , S) q n (β j ) . ( 4
)
Hence

I 1 ≤ [ln n] j=l(β) |S * h j (z 0 ) -S * h l(β) (z 0 )|I { l=j} + |S * h l(β) (z 0 ) -S(z 0 )|I { l≥l(β)} ≤ [ln n] j=l(β) λ N j I { l=j} + |S * hl(β) (z 0 ) -S(z 0 )| ≤ λ N l(β) + |S * hl(β) (z 0 ) -S(z 0 )| ≤ λ N (β) e β -β + δ nh(β l(β) ) 1+β q n (β l(β) ) + 6δ -1 h(β l(β) ) q n (β l(β) ) + |Z n (β l ( β) )| σ n (β l(β) , S) q n (β l(β) )
.

Let us show that lim sup

δ→0 lim sup n→∞ sup β∈[β ;β ] sup S∈U z 0 ,δ N (β) g(z 0 , S) E S I 1 ≤ λ e β -β g . ( 5
)
As

N (β)h(β l(β) ) β ≤ 1 and q n (β l(β) ) ∼ 2nh(β l(β) ), we have lim δ→0 lim sup n→∞ sup β∈[β ;β ] sup S∈U z 0 ,δ δ N (β) g(z 0 , S) nh(β l(β) ) 1+β q n (β l(β) ) = 0. ( 6
)
Moreover it is easy to see that lim sup

n→∞ sup β∈[β ;β ] sup S∈Uz 0 ,δ N (β) g(z 0 , S) 6δ -1 h(β l(β) ) q n (β l(β) ) = 0. ( 7
)
Since σ n (β l(β) , S) is bounded by g and

N (β) 2 q n (β l(β) ) = nh(β l(β) ) q n (β l(β) ) n ln n 2(β-βl(β))/(2β+1)(2βl(β)+1) 1 ln n ≤ nh(β l(β) ) q n (β l(β) ) e 2(β -β ) 1 ln n , one has lim sup n→∞ sup β∈[β ;β ] sup S∈Uz 0 ,δ N (β) g(z 0 , S) E S |Z n (β l(β) )| σ n (β l(β) , S) q n (β l(β) ) = 0. ( 8
)
Finally from ( 6) -( 8), we get [START_REF]Efromovich Nonparametric Curve Estimation[END_REF]. Now recall that

I 2 = | Ŝn -S(z 0 )|I { l<l(β)} and let us prove that lim n→∞ sup β∈[β ;β ] sup S∈Uz 0 ,δ N (β) g(z 0 , S) E S I 2 = 0. (9) We have { l < l(β)} = l(β)-1 j=0
{ l = j} and by definition of l,

{ l = j} ⊂ j+1 i=0 S * h i (z 0 ) -S * h j+1 (z 0 ) > λ N i . Remark that for 0 ≤ i ≤ l ≤ l(β), one has h i ≤ h l ≤ h(β) and q n (β i ) ≤ q n (β l ).
Then denoting Z * n = max l |Z n (β l )| and using (4) and lemma (A.1), we obtain for

0 ≤ i ≤ l ≤ l(β): S * h l (z 0 ) -S * h i (z 0 ) > λ N i ⊂ S * h l (z 0 ) -S(z 0 ) + S(z 0 ) -S * h i (z 0 ) > λ N i ⊂ 2δh(β) β + 12δ -1 nh i h(β) + 2g q n (β i ) Z * n > λ N i ⊂ 2g q n (β i ) Z * n > λ -2δ -12δ -1 h(β ) N i , because nh i > N i and N i < h(β) -β .
The precedent inclusions are true for a sufficiently large n and we will consider this case from now to the end of the proof.

As

q n (β i ) N 2 i = q n (β i ) nh i ln n, lemma A.1 brings us √ ln n ≤ q n (β i ) N i ≤ √ 3 ln n.
Setting Λ n := λ -2δ -12δ -1 h(β ), we can write

( l = j) ⊂ j+1 i=0 S * h i (z 0 ) -S * h j+1 (z 0 ) > λ N i ⊂ Z * n > Λ n 2g √ ln n =: A n and I 2 ≤ l(β)-1 j=0 |S * h j (z 0 ) -S(z 0 )|I {A n } .
To get (9) we write

N (β) g(z 0 , S) E S I 2 ≤ N (β) g(z 0 , S) l(β)-1 j=0 E S δnh 1+β j q n (β j ) + 6δ -1 h j q n (β j ) + |Z n (β j )| σ(β j , S) q n (β j ) I {A n } ≤ l(β)-1 j=0 N (β) g(z 0 , S) δnh 1+β j q n (β j ) P S (A n ) + l(β)-1 j=0 N (β) g(z 0 , S) 6δ -1 h j q n (β j ) P S (A n ) + l(β)-1 j=0 N (β) g(z 0 , S) σ n (β j , S) q n (β j ) E S |Z n (β j )|I (A n )
and study the asymptotic behavior of each term. Let Z ∼ N (0, 1) and

A n := |Z| > Λ n 2g √ ln n .
For the first term one has

l(β)-1 j=0 N (β) g(z 0 , S) δnh 1+β j q n (β j ) P S (A n ) ≤ δ g l(β)-1 j=0 N (β)h β j P S (A n ) ≤ δ g l(β)-1 j=0 n ln n 2β(β j -β)/(2β j +1)(2β+1) P S (A n ) ≤ [ln n] 2 δ g n ln n 2β(βl(β)-1-β)/(2βl(β)-1+1)(2β+1) P S (A n ) ≤ [ln n] 2 δ g n ln n 2β(β l(β)-1 -β)/(2β+1) 2 2 2g Λ n √ ln n 1 √ 2π e -(Λ n ) 2 4g 2 ln n ≤ 4δg √ 2πg Λ n e -2β /(2β +1) 2 e 2β /(2β +1) 2 [ln n] 2 √ ln n n -(Λ n ) 2 4g 2 .
Since λ > 2 the last term tends to zero as n goes to infinity. We handle the second term as well.

Using the Cauchy-Schwarz inequality for the third term, we obtain

N (β) g(z 0 , S) l(β)-1 j=0 E S |Z n (β j )| σ n (β j , S)
q n (β j )

I {A n } ≤ g g l(β)-1 j=0 N (β) q n (β j ) (P S (A n )) 1/2 ≤ g g l(β)-1 j=0 N (β) q n (β j ) [ln n] 1/2 (P S (A n )) 1/2 ≤ 2(g ) 3/2 g Λ n √ 2π [ln n] 3/2 √ ln n q n (β ) N (β )n -(Λ n ) 2 /8g 2 .
By definition of λ this term tends to zero as n goes to infinity. Eventually we have proved ( 9) which, connected to (5), completes the proof.

Lower bound

In this section we give the lower bound for the minimax risk. We will consider a family of functions in U z 0 ,δ defined with another bandwidth h(β) = n -1/2β+1 (ln n) -2β/2β+1 such that n h(β) = N 2 (β) and h is a increasing function of β.

Theorem 4.1 For all δ ∈]0; 1[ the following inequality holds

lim inf n→∞ inf S R z 0 ,δ,n ( S) ≥ E|ξ| √ 2 , ξ ∼ N (0, 1),
where the infimum is taken over all estimators S of S(z 0 ).

Proof : For ν ∈ 0; 1 4 , denote S ν (x) = 1 N (β) V ν x-z 0 h(β)
, where the function V ν is defined by

V ν (x) = 1 ν +∞ -∞ Qν (u)m u -x ν du, Qν (u) = I {|u|≤1-2ν} + 2I {1-2ν≤|u|≤1-ν}
and m is a nonnegative function, infinitely differentiable on R, such that m(z) = 0 for any |z| ≥ 1 and

1 -1 m(z)dz = 1.
We can easily show that for all 0 < ν < 1 4 , one has V ν (0) = 1 and

1 -1 V ν (x)dx = 2. Now fix b > 0, ν ∈]0; 1/4[, δ ∈]0; 1[ and denote S ν,u (x) = u N (β) V ν x -z 0 h(β) for
x, u ∈ R. Thanks to lemma A.2 we can say that if |u| ≤ b then there exists an integer n b,δ > 0 such that S ν,u ∈ U z 0 ,δ for all n ≥ n b,δ . Hence for n ≥ n b,δ , we have

R z0,δ,n ( S) ≥ sup β∈[β ;β ] sup |u|≤b N (β) g(z 0 , S ν,u ) E Sν,u | S(z 0 ) -S ν,u (z 0 )| ≥ sup β∈[β ;β ] 1 2b b -b N (β) g(z 0 , S ν,u ) E S ν,u v a S(z 0 ) -S ν,u (z 0 ) du =: sup β∈[β ;β ] I n (a, b, β)
where v a (x) = |x| ∧ a, a > 0.

Let P Sν,u be the law of (y (1) k ) k=1,...,n , where y (1) k = S ν,u (x k )+g(x k , S ν,u )ξ k , and P the law of (y (0) k ) k=1,...,n , where y (0) k = g(x k , S ν,u )ξ k . These two measures are equivalent and the corresponding Radon-Nikodym derivative at the point (y 1 , . . . , y n ) is

ρ n (u, β) = dP S ν,u dP (y 1 , . . . , y n ) = exp uς n (β)η n (β) - u 2 2 ς 2 n (β)
where

ς 2 n (β) = 1 N 2 (β) n k=1 V 2 ν x k -z 0 h(β) g 2 (x k , S ν,u ) and η n (β) = 1 N (β)ς n (β) n k=1 V ν x k -z 0 h(β) g 2 (x k , S ν,u ) y k .
Under the law P, η n (β) is a standard Gaussian random variable.

Setting

σ 2 ν = 1 g 2 (z 0 ,0) 1 -1 V 2 ν (z)dz, lemma A.3 implies sup β∈[β ;β ] |ς n (β) -σ ν | ---→ n→∞ 0. (10)
Rewrite

ρ n (u, β) = exp uσ ν η n (β) -u 2 σ 2 ν 2 + r n (β) , where r n (β) is a Gaussian random variable with expectation u 2 2 (σ 2 ν -ς 2 n (β)
) and variance u 2 (ς n (β) -σ ν ) 2 . Then using [START_REF] Barron | Risk bounds for model selection via penalization[END_REF] and Tchebychev inequality we get for ε > 0

lim n→∞ sup β∈[β ;β ] P (|r n (β)| > ε) = 0. (11)
As a consequence we can show that ρ n (u, β)

L ---→ n→∞ ρ ∞ (u) := exp uσ ν η -u 2 σ 2 ν 2 uniformly in β and lim n→∞ sup β∈[β ,β ] P |e rn(β) -1| > ε = 0. Denoting ψ a,n ( S, S ν,u ) = v a (N (β)( S(z 0 ) -S ν,u (z 0 ))
) and E the expectation for the probability measure P, one has

I n (a, b, β) ≥ 1 2b b -b EI B d (β) ψ a,n ( S, S ν,u ) g(z 0 , S ν,u ) n (u, β)du + δ n (a, b, β) (12) =: J n (a, b, β) + δ n (a, b, β), (13)
where

B d (β) = {|η n (β)| ≤ d} and d = σ ν (b - √ b), b > 1, n (u, β) = exp uσ ν η n (β) - u 2 σ 2 ν 2 , δ n (a, b, β) = 1 2b b -b EI B d (β) ψ a,n ( S, S ν,u ) g(z 0 , S ν,u ) θ n (u, β)du, θ n (u, β) = ρ n (u, β) -n (u, β) = exp uσ ν η n (β) - u 2 σ 2 ν 2 (e r n (β) -1). Let us show that inf S sup β∈[β ;β ] δ n (a, b, β) ---→ n→∞ 0. ( 14
)
We can easily prove that Eρ ∞ (u) = 1 and Eρ n (u, β) = 1. Then the sequence {ρ n (u, β), n ≥ 1} is uniformly integrable (see [18, p.32]). As n (u, β) is bounded on

B d (β)
, we obtain the uniform integrability of I Bd(β) ψa,n( S,Sν,u)

g(z 0 ,S ν,u ) θ n (u, β), n ≥ 1 . We remark that exp uσ ν η n (β) -u 2 σ 2
E I B d g(z 0 , S ν,u ) ψ a,n ( S, S ν,u )θ n (u, β) ≤ K sup β∈[β ;β ] E|e r n (β) -1| ≤ Ke u 2 σ n ν 2 .
By bounded convergence we obtain finally sup

β∈[β ;β ] δ n (a, b, β) ≤ 1 2b b -b sup β∈[β ;β ] E I B d g(z 0 , S ν,u ) ψ a,n ( S, S ν,u )θ n (u) du ---→ n→∞ 0,
and then [START_REF] Lepskiȋ | Asymptotically minimax adaptive estimation. II. Schemes without optimal adaptation. Adaptive estimates[END_REF]. Now we are interested in the term J n (a, b, β) in [START_REF] Lepskiȋ | Asymptotically minimax adaptive estimation. I. Upper bounds. Optimally adaptive estimates[END_REF].

Firstly rewrite n (u, β) = ζ n (β)e -σ 2 ν (u-η n (β)) 2 /2 where ζ n (β) = e η 2 n (β)/2 and ηn (β) = η n (β) σ ν , then J n (a, b, β) = 1 2b b -b EI Bd(β) ζ n (β) v a (u -(t n (β)) g(z 0 , S ν,u ) exp - σ 2 ν 2 (u -ηn (β)) 2 du, with t n (β) = N (β) S(z 0 ).
Further if ξ ∼ N (0, 1) and if we set ξ = ξ σ ν , ζ = e ξ 2 /2 , Bd = {|ξ| ≤ d} and Ẽ the expectation for the probability measure of ξ, we have

J n (a, b, β) = ẼI Bd ζ 1 2b b -b v a (u -t n (β)) g(z 0 , S ν,u ) exp - σ 2 ν 2 (u -ξ) 2 du.
Since g is uniformly continuous and bounded away from zero, the following term

sup β∈[β ;β ] ẼI Bd ζ 2b b -b v a (u -t n (β)) exp - σ 2 ν 2 (u -ξ) 2 1 g(z 0 , S ν,u ) - 1 g(z 0 ,
ẼI Bd ζ 1 2b b -b v a (u -t n (β)) g(z 0 , 0) exp - σ 2 ν 2 (u -ξ) 2 du.
Moreover we can write

Ẽ I Bd ζ 2b b -b v a (u -t n (β)) g(z 0 , 0) e -σ 2 ν 2 (u-ξ) 2 du ≥ Ẽ I Bd ζ 2b √ b - √ b v a (t -t n (β) + ξ) g(z 0 , 0) e -σ 2 ν 2 t 2 dt ≥ Ẽ I Bd ζ 2b √ b - √ b v a (t) g(z 0 , 0) exp - σ 2 ν 2 t 2 dt
J n (a, b, β) ≥ σ ν √ 2π ∞ -∞ |t| g(z 0 , 0) exp - σ 2 ν 2 t 2 dt.
Eventually using the fact that σ

2 ν ---→ ν→0 2 g 2 (z 0 , 0)
and combining the result with ( 14) finish the proof.

Appendix A.

Lemma A.1 There exists an integer N such that if n ≥ N , then

1 ≤ q n (β i ) nh i ≤ 3,
for all i = 0, . . . , [ln n].

Proof : Writing

q n (β i ) nh i = 1 nh i n k=1 Q x k -z 0 h i = [n(z 0 + h i )] -[n(z 0 -h i )] nh i , the following inequalities [n(z 0 + h i )] ≤ n(z 0 + h i ) ≤ [n(z 0 + h i )] + 1, -1 -[n(z 0 -h i )] ≤ -n(z 0 -h i ) ≤ -[n(z 0 -h i )],
show that 2 -

1 nh i ≤ q n (β i ) nh i ≤ 2 + 1 nh i .
But there exists an integer N such that if n ≥ N , then nh(β ) ≥ 1. As h is an increasing function of β, we get for n ≥ N :

2 - 1 nh(β ) ≤ q n (β i ) nh i ≤ 2 + 1 nh(β ) ,
and then the desired result.

Lemma A.2 For any ν ∈]0; 1 4 [ and δ ∈]0; 1[ there exists an integer n ν,δ > 0 such that S ν ∈ U z0,δ for all n ≥ n ν,δ .

Proof : Firstly for all h ≥ 0 one has

ν 2 isI

 2 bounded by a constant independent of β on B d (β) and that ψ a,n ( S, S ν,u ) is bounded by a. Then lim n→∞ sup β∈[β ;β ]P I B d (β) ψ a,n ( S, S ν,u ) g(z 0 , S ν,u ) |θ n (u, β)| > ε = 0. B d (β)ψa,n( S,Sν,u)g(z 0 ,S ν,u ) θ n (u, β), n ≥ 1 that lim n→∞ sup β∈[β ;β ] E I B d (β) ψ a,n ( S, S ν,u ) g(z 0 , S ν,u ) θ n (u, β) = 0.Furthermore there exists a constant K > 0 such that sup β∈[β ;β ]

1 - 1 1 - 1 |m 1 - 1 |mg 2 g 2

 11111122 (S ν (z 0 + uh) -S ν (z 0 )) du = 0. Moreover we can write|S ν (x)| ≤ 2 νN (β) h(β) (u)|du ≤ 2 ν n (1-β )/(2β +1) (ln n) 3β /2β +1Hence for any fixed δ ∈]0; 1[, there exists n ν,δ ∈ N * such that if n ≥ n ν,δ , then|S ν (x)| ≤ δ -1 and S ν ∈ U z 0 ,δ .Lemma A.[START_REF] Donoho | Asymptotic minimax risk for sup-norm loss: solution via optimal recovery[END_REF] The following limit holds sup β∈[β ;β ] There exists an integer n 0 such that for all n ≥ n 0 and all β ∈ [β ; β ], we have [z 0 -h(β); z 0 + h(β)] ⊂ [0; 1]. As a consequence for n ≥ n 0 we getς 2 n (β) = 1 h(β) z0+ h(β) z0-h(β) (x, S ν,u )µ n (dx) = (x, S ν,u ) ν n (dx)

with µ n = 1 n n k=1 δ k/n and ν n =

As g is uniformly continuous, we have:

We easily prove that for a real function f defined on R and continuous at the point

and applying (A3) to the function x → 1 g 2 (x, 0) and using (A2), we obtain

Now we want to show that

Let F be a function such that F = V 2 ν and denote u k = x k -z 0 h(β) again. For all n ≥ n 0 , one has

Finally, as we have

g 2 (z 0 , 0) ν n (dx) -

V 2 ν (z) g 2 (z 0 , 0) dz , (A4) and (A5) implies lemma A.3.


B Kerautret
email: kerautre@loria.fr

J.-O Lachaud

B Naegel

International Journal of Shape Modeling

Keywords: curvature estimators, corner detection, multi-scale analysis, dominant point extraction

 , which come from the discrete geometry community, can now process damaged data and integrate the amount of noise in their analysis. In this paper, we propose a comparative evaluation of these estimators, testing their accuracy, efficiency, and robustness with respect to several type of degradations. We further compare the best one with the visual curvature proposed by Liu et al. 14 , a recently published method from the computer vision community. We finally propose a novel corner detector, which is based on curvature estimation, and we provide a comprehensive set of experiments to compare it with many other classical corner detectors. Our study shows that this corner detector has most of the time a better behavior than the others, while requiring only one parameter to take into account the

Curvature estimation is an important task for shape analysis, matching or segmentation. This geometric characteristic has indeed many nice properties for these applications: it is invariant under isometries; it allows to discriminate between straight, smoothly curved, and corner parts; all feature or dominant points are defined from curvature. Its main default is to be sensitive to noise and small perturbations. The correct, precise, and robust estimation of curvature along a shape boundary description is thus a prerequisite for many applications in image analysis.

As noted by several authors (see for instance Utcke [START_REF] Utcke | Error-bounds on curvature estimation[END_REF] ), it is already difficult to get a precise estimation of curvature along digital contours that are perfect digitizations of regular shapes. Furthermore, real applications tend to add perturbations to the original data: low resolution digitization, pepper and salt noise, quantification noise, several scales of analysis. It is for instance very difficult to compute the correct curvature for the perturbated images (b,c,d) of Fig. 1. This is why the range of applications of curvature estimators is for now rather limited in the image analysis and computer vision community.

Recently, four new methods were proposed to compute curvature (or pseudo curvature) along the boundary of noisy discrete data. First Nguyen et al. [START_REF] Nguyen | Curvature estimation in noisy curves[END_REF] define a new curvature estimator (called NDC estimator) by extending the estimator of osculating circles (called CC estimator) proposed by Coeurjolly et al. [START_REF] Coeurjolly | Discrete curvature based on osculating circle estimation[END_REF] by using blurred segments. By this way, the estimation of osculating circles had a better behavior on noisy contours and was also meaningful for a non connected (but ordered) set of points. Then a second approach (called GMC estimator), proposed by Kerautret and Lachaud [START_REF] Kerautret | Robust estimation of curvature along digital contours with global optimization[END_REF] , suggests to minimize curvature while satisfying geometric constraints derived from tangent directions computed on the discrete contour. The robustness to noise of this approach was given also by blurred segments but used in a different manner. The third curvature estimator (called BCC estimator), proposed at the same time by Malgouyres et al. [START_REF] Malgouyres | Binomial convolutions and derivatives estimations from noisy discretizations[END_REF] , suggests to use discrete binomial convolution to obtain a convergent estimator adapted to noisy data. From these three new estimators we can add the new notion of multi-scale visual curvature (called VC estimator) introduced by Liu et al. [START_REF] Liu | A unified curvature definition for regular, polygonal, and digital planar curves[END_REF] . This new definition takes into account both digitization artefacts and noise in data with a scale parameter.

In this work, we propose to evaluate experimentally the accuracy, efficiency, and robustness of the three first estimators by using different test contours and by measuring the precision of the estimation with several grid sizes. The visual curvature is also compared with the GMC estimation and with the expected curvature. After comparing the accuracy of curvature fields obtained with low, intermediate, and fine resolutions, we measure the robustness of these estimators on noisy contours. The GMC estimator [START_REF] Kerautret | Robust estimation of curvature along digital contours with global optimization[END_REF] appears the most stable and robust. Considering its qualities, we then define a new corner detector based on the GMC curvature estimation. This detector is compared with other standard corner detectors and with the recently-published corner detector based on digital multi-scale visual curvature.

The paper is organized as follows. In the following section, after introducing briefly some notions on digital blurred segments, we give a short description of the four curvature estimators. Then the experimental comparison of these estimators is given in Section 3. In Section 4, we propose a new corner detector based on GMC curvature estimation. We then present a comparative evaluation with several other corner detectors, and eventually show its potential for multi-scale contour representation.

Discrete Curvature Estimators

In this section, we recall briefly the definitions of five curvature estimators, which have all been recently published and which are all able to analyze digital and damaged shape data. For the sake of completeness, we begin by describing the notion of digital blurred segments, which is used by the two first estimators.

Blurred segments were introduced by Debled-Rennesson et al. [START_REF] Debled-Rennesson | Optimal blurred segments decomposition of noisy shapes in linear times[END_REF] . Note that a comparable concept and algorithm was proposed independently by Buzer 2 . A blurred segment is a piece of thick digital straight line, whose width is the vertical diameter of its convex hull. From this definition the authors proposed an algorithm for the recognition of blurred segments of width ν. Note that the two following curvature estimators use a variant of the original definition of blurred segment, which is not restricted to the hypothesis that points are added with increasing x coordinate (or y coordinate). Blurred segments are illustrated in Fig. 2 and Fig. 3(b).

R BS(l, k) BS(k, r)
Fig. 2. Illustration of curvature estimation using NDC estimator. The blue lines represent the two blurred segments BS(l, k) and BS(k, r) of width 5.

Estimator based on osculating circles (CC and NDC)

The curvature estimator proposed by Nguyen and Debled-Rennesson 7 follows the same concept as the estimator proposed by Coeurjolly et al. [START_REF] Coeurjolly | Discrete curvature based on osculating circle estimation[END_REF] (called CC estimator). The latter is based on the estimation of osculating circles. More precisely, by denoting C a discrete curve, C i,j the sequence of points going from C i to C j and BS ν (i, j) the predicate "C i,j is a blurred segment of width ν". They consider the points C r and C l defined such that:

BS ν (l, k) ∧ ¬BS ν (l -1, k) ∧ BS ν (k, r) ∧ ¬BS ν (k, r + 1).
With this definition, they determine the curvature of width ν at the point C k from the radius of the circle passing through C k , C l and C r . By noting

s 1 = || ---→ C k C r ||, s 2 = || ---→ C k C l || and s 3 = || ---→ C l C r ||
, the authors give the following expression of radius R ν of the circle associated to the point C k :

R ν (C k ) = s 1 s 2 s 3 (s 1 + s 2 + s 3 )(s 1 -s 2 + s 3 )(s 1 + s 2 -s 3 )(s 2 + s 3 -s 1 )
If the vectors ---→ C k C r and ---→ C k C l are not collinear then the curvature of width ν can be determined with sign(det( --------→

C k Cr,C k C l )) Rν (C k )
. Otherwise the curvature is set to 0.

Global Minimization Curvature estimator (GMC)

The estimator proposed by Kerautret and Lachaud is based on two main ideas [START_REF] Kerautret | Robust estimation of curvature along digital contours with global optimization[END_REF] . The first one is to take into account all the possible shapes that have the same digitization defined by the initial contour and to select the most probable contour. This selection is done with a constrained global optimization which minimizes the squared curvature among admissable shapes. From this idea, we can expect a precise result even with a low contour resolution. The second idea is to obtain an estimator that is not sensitive to noise or to non perfect digitization process by determining the constraints with blurred segments.

For the purpose of minimizing curvature, the tangential cover 8 is computed on the discrete contour and the minimal and maximal possible tangent values are defined for each pixel based on the directions of surrounding maximal segments. Fig. 3(a) illustrates tangential cover and the bounds on the tangent directions is shown in Fig. 4(a). In the space of tangent directions (as in Fig. 4 geometry is represented by a piecewise linear curve which must stay within the constraint boxes. Minimizing the curvature consists in moving each point (x i , y i ) of this curve along the y axis and between the bounds defined by the boxes, such that (x i , y i ) is as close as possible to the line joining (x i-1 , y i-1 ) to (x i+1 , y i+1 ).

The global minimization method is a standard relaxation process. The estimated curvature is then simply the slope of this curve at each point.

For the robustness to noise and non perfect digitization process, the discrete maximal segments were replaced by the discrete maximal blurred segments previously described. Note that the definition of the minimal and maximal slopes of the blurred segments is notably different from the non blurred case (see [START_REF] Kerautret | Robust estimation of curvature along digital contours with global optimization[END_REF] for more details).

Binomial Convolution Curvature Estimator (BCC)

Malgouyres et al. proposed an algorithm to estimate derivatives with binomial convolutions [START_REF] Malgouyres | Binomial convolutions and derivatives estimations from noisy discretizations[END_REF] which is claimed to be multigrid convergent. They define the operator Ψ K which modifies the function F : Z → Z by the convolution with kernel K : Z → Z. For instance the backward finite difference is given by Ψ δ F where the kernel δ is defined by:

δ(a) =    1 if a = 0 -1 if a = 1 0 otherwise
Then the authors give the smoothing kernel defined as:

H n (a) =            n a + n 2 if n is even and a ∈ {-n 2 , ..., n 2 } n a + n+1 2 if n is odd and a ∈ {-n+1 2 , ..., n-1 2 } 0 otherwise.
Finally the derivative kernel D n is given by D n = δ * H n and the derivative estimator is [START_REF] Beus | An improved corner detection algorithm based on chaincoded plane curves[END_REF] 2 n Ψ Dn F . An interesting point of the method is that higher order derivatives have similar expressions:

D 2 n = δ * δ * D n
And for the curvature we compute:

D 2 n (x) * D n (y) -D 2 n (y) * D n (x) D n (x) 2 + D n (y) 2
The value of n is defined by n = h 2(α-3)/3 , where h represents the grid size and α ∈]0, 1] is an additional parameter which can be associated to the amount of noise. More precisely, a value close to 0 allows more noise than a value close to 1.

Visual Curvature (VC)

Liu et al. [START_REF] Liu | A unified curvature definition for regular, polygonal, and digital planar curves[END_REF] have recently proposed a definition of curvature which is suited to regular, polygonal, and digital planar curves. Robustness to noise is achieved through a scale parameter. The pseudo-curvature estimator is then thresholded to detect corners. As we will show on experiments, their corner detector seem to be the most reliable at the moment.

A number N of directions is given by the user (the authors suggest 128): the bigger is N , the more precise can be the estimator. The contour is seen as a parametric curve (x(s), y(s)). Then N different height functions H αi are generated by rotating the parametric curve (x(s), y(s)) along N directions between 0 and π. The profile along the y-axis gives an height function H αi for each direction. For a given neighborhood S(v) of size ∆S around the point of curvilinear abscissa v, the visual curvature at v is defined as

K N,∆S (v) = π N -1 i=0 #[H αi (S(v))] N ∆S ,
where #[H αi (S(v))] represents the number of local extreme points of the height function H αi in the neighborhood S(v).

This definition of curvature was shown to tend toward the standard absolute curvature for regular curves, when N tends toward infinity and ∆S toward 0. Unfortunately, no convergence speed is given. Furthermore, this definition is useless both for digital contours and for noisy shapes, since they have plenty of extreme points (the steps) in every direction.

The authors therefore add a scale measure to any extreme point v along a direction: it is the minimum height of the two peaks surrounding the point v divided by the diameter of the shape along this direction. A peak is the maximal height variation of the function before it reaches again the same height as v. A strong extreme point has then a scale measure close to 1 (such as any point on the convex hull) while weak extreme points have low scale measure (close to 0). The multiscale visual curvature of scale λ is then

K λ N,∆S (v) = π N -1 i=0 #[H λ αi (S(v))] N ∆S ,
where #[H λ αi (S(v))] represents the number of local extreme points of the height function H αi in the neighborhood S(v), such that their scale measure is no smaller than λ.

This definition is more suited to digital or noisy contours, where small extreme points can be removed by the scale parameter. Although the given value is a qualitative estimation of curvature, it is not a correct nor precise quantitative estimator of curvature. In order to better detect and localize corners, Liu et al. have further proposed to cut the input contour into pieces of T points, and then to gather all the visual curvature of each piece onto its point with maximal visual curvature. This allows that several significant curvature points, which are close to each other, are considered as one corner. This latter definition is called the digital visual curvature.

To sum up, the visual curvature requires four parameters: the number N of directions, the size ∆S of the neighborhood, the scale λ, the number of points T for gathering curvature.

Experimental Comparisons

In this section, we first compare experimentally the three estimators GMC, CC-NDC and BCC, both on perfectly digitized data and on noisy shapes. Since the notion of visual curvature does not represent rigorously the same type of information and is not directly suited to digital contours, we present it in the last part of this section and we give only a comparative evaluation on noisy contours.

The objective is to measure the accuracy of the previously described estimators and to give execution times. For this purpose several data sets were generated with different grid sizes (h), for coarse (h = 1), medium (h = 0.1), and fine resolution (h = 0.01). Fig. 6 illustrates the two shapes defined with a digitization grid step equal to 1 (image "flower" (a) and image "polygon" (b)). We choose these test shapes to analyze the performances of the estimators both on smooth shapes with varying curvature and quick turns, and on polygonal shapes with corners. The three estimators have been applied on the previous data. For the BCC estimator, the parameter α was set to 1 and the value of n was thus set to h -4/3 . For the other estimators, no parameters were used. From the resulting curvature graphs of Fig. 6, it can be seen that, for both the flower and the polygon, the BCC estimator shows numerous oscillations with coarse grid sizes (h=1 and h=0.1). But the finer the grid size is, the more the values of curvature of the BCC estimator are stable. It appears that it is not the case with the CC estimators. Indeed the oscillations appear to increase as the grid size gets finer. For the GMC estimator we can see that it is very stable since no oscillations are visible. Other error measures and execution timings were performed on these shapes and they may be found in a previous work [START_REF] Kerautret | Comparison of discrete curvature estimators and application to corner detection[END_REF] .

In order to measure the stability of the curvature estimators on noisy shapes, noise was added according to the model proposed by Kanungo [START_REF] Kanungo | Document Degradation Models and a Methodology for Degradation Model Validation[END_REF] . Fig. 7 shows such noise addition (images (a) and (b)). From these results, we can see that the GMC estimator is very stable compared to the NDC estimator. Even with width 2 the results are stable if we ignore a small part with negative curvature (image (f)). In the same way, the results of the BCC estimators appear satisfying although a large size n was needed to obtain stable values. The fact that there are no oscillation in the case of the GMC constitutes a real advantage to design a simple and efficient curvature based-algorithm.

Curvature estimations with the multiscale visual curvature estimator were performed on the noisy versions of the flower and polygon shapes (see Fig. 8) at two different scales (λ = 0.5 and λ = 0.05). The result on the flower shape shows clearly that the visual curvature cannot be used to extract accurate measures on the parts outside corners. It appears that even near corners and even with larger scale val- (2) detect all the maximal curvature regions defined by sets of consecutive points:

R k = {(p i ) i∈[a,b] | ∀i, κ(p i ) = κ(p a ) ∧ κ(p a-1 ) < κ(p a ) ∧ κ(p b+1 ) < κ(p b ) ∧κ(p a ) > 0}
(3) for each region R k mark the point p (a+b)/2 as a corner.

(4) (optional) select only corner points with curvature greater than a threshold value κ min .

Note that a quantification of the curvature field was applied in order to simplify the detection of the local minima/maxima (set to 1e -3). In the rest of this paper we have not applied the optional step (4) since it adds a new parameter and it could be fixed for each particular application. The concave corner detection is deduced by replacing the maximal by the minimal curvature regions.

The previous algorithm was applied on the standard image collection available on-line 5 (see Fig. 9). All the corner detections were performed using the GMCCD algorithm with parameter ν = 2, which is a commonly used value for real data presenting a weak amount of noise. We can see on Fig. 9 that some parts, which are considered as corners, are more local maxima. Note that these points could be easily removed by applying a small threshold on the curvature values of the local maxima. For this standard dataset our results are comparable to previous works, but we can already note that only one parameter is required and it remains unchanged for all data considered as non noisy. This is not the case for example for the corner detectors described below.

The images of Fig. 10 were generated from the previous test images and with the same noise model as mentioned in the previous section. The parameter ν was set to 4 according to the noise level. Note also that we are currently working on an automated determination of this parameter. Despite the important amount of noise, the corners were well detected. The only defect can be seen on the left wing of the plane (h), but it disappears when using a width equals to 5. Even with a lot of noise, corners can be detected as illustrated on Fig. 20d (ν = 25). Note that the total execution time was approximately 3 s for the test images of Fig. 9 and Fig. 10.

Comparison with other corner detectors

We have compared our method with other corner detectors. Moreover, in order to assess the versatility of our corner detector, experiments were conducted on noisy versions of the classical reference shapes [START_REF] Chetverikov | [END_REF] . In a first experiment, our method was compared with the five methods implemented on the internet 5 (RJ73 17 , RW75 18 , FD77 9 , BT87 1 , IPAN03 4 ). A set of parameters, left constant for all the 8 images, was chosen for each of the five methods in order to obtain the best possible results. The results as well as the parameters used in the experiments are described respectively in Fig. 11, Fig. 12, Fig. 13, Fig. 14, and Fig. 15.

The IPAN03, FD77 and BT87 methods are very sensitive to noise and therefore give poor and instable results (see Fig. 11, Fig. 14, Fig. 15). A lot of true corners are missing while false detections are frequent.

The RJ73 and RW75 methods give the best results and appear to be rather robust to noise. Using parameter κ = 0.05, RJ73 enable to detect all true corners, while some false corners are detected. The RW75 method with the parameter κ = 0.05 give the best result, with only a few false detections (see Fig. 13b) and a few corners missing (see Fig. 13 (c),(f),(g),(h)). However the RW75 method does not achieve the degree of robustness of the GMCCD method.

In a second experiment, the corner detector based on the visual curvature 14 (which we will refer to as VCCD) was experimented. Similarly to the GMCCD algorithm, a point is detected as corner each time the curvature exceeds a particular threshold DK 0 . In this experiment we have generated a large number of tests in order to obtain the best results. We have used the following list of parameters: ∆S = {2, 4, 8, 10, 15}, λ = {0.05, 0.1, 0.2, 0.3, 0.4}, T = {10, 20, 30, 50} with N = 300 and applied a list of threshold DK 0 from 0.3 to 2 by step of 0.2 (1300 tests for each shape). Fig. 16 and Fig. 17 show two sets of results obtained with the same set of parameters for all the test shapes. We can see that the corners are well detected but on the round parts of the shape we can detect some wrong corners (on shapes (b) and (c)) and some corners are often missing (shapes (g) and (c)).

(a) (b) (c) (d) (e) (f) (g) (h)
Other comparisons of GMCCD with a recent morphological corner detector 3 can be found in previous work [START_REF] Kerautret | Comparison of discrete curvature estimators and application to corner detection[END_REF] , but the results were not interesting on noisy shapes compared to GMCCD and VCCD.

Multiscale comparisons between GMC and VC curvature estimators

Since the two curvature estimators GMC and VC define a multi-scale analysis of the contour, we can compare the contour simplifications obtained from these two estimators. We propose a simple way to simplify the contour by using GMC curvature estimated with a width ν and by using only the frontier points associated to the constant curvature areas. Such a contour is denoted by G ν . Figure 18 illustrates the results obtained with our approach. In the same way Liu et al. proposed a definition of λ-scale approximation of a contour by removing all points for which curvature vanishes (denoted by V λ ). This representation is illustrated with different values of λ on Fig. 19. We can remark that the number of points selected by GMC is smaller and more representative than the ones selected by the visual curvature.

To conclude this part on experimentations and comparisons, we applied the two detectors GMCCD and VCCD on the shapes of figure Fig. 1 (see Fig. 20). The corner detection was successful with the GMCCD on all damaged data: coarse resolution, weak pepper and salt noise, and multiscale interpretation. For the VCCD only convex corners were detected and with the noisy shape (f) there appear some false corners on the large round part of the shape. Increasing the scale to λ = 0.3 does not remove them. For the execution times, GMCCD appears most of the time faster if we exclude the result with large scale (d,h). Note that the parameter N was set to 128 as recommended by the authors of VCCD.

(a) (b) (c) (d) (e) (f) (g) (h)
(e) (f) (g) (a) (b) (c) (d)

Conclusion

In this paper we have compared several recent discrete curvature estimators suited to the analysis of noisy digital data. From these experimentations the GMC estimator appears the most stable and was selected to define a new curvature-based corner detector. The results obtained demonstrate a great robustness to noise and they outperform other classical and recent corner detectors. Furthermore our new estimator requires only one parameter to set, which has a simple geometric interpretation related to the noise level along the contour and the scale for which corners are considered.

Fig. 1 .

 1 Fig. 1. Example of damaged data: (a) original shape, (b) low digitization artefacts, (c) pepper and salt noise or quantification effects, (d) damaged data due to imperfect segmentation and noise removing image processing.

Fig. 3 .

 3 Fig. 3. Illustration of tangential cover (a) and blurred segments with different widths (b).

Fig. 4 .

 4 Fig.4. Illustration of bounds defined from maximal segments (a) (extracted from Fig.3(a)). The plot (b) shows the constraints defined on each variable.

Fig. 5 .

 5 Fig. 5. Illustration of function Hα i

mainFig. 6 .

 6 Fig.6. Comparisons of the CC, GMC and BCC estimators on the flower (a,d,f,h) and on the polygon (b,e,g,i). The graph (c) shows the real curvature of the flower shape.

Fig. 7 .

 7 Fig. 7. Curvature estimations on the noisy version of the shapes of Fig. 6. Each row shows the result obtained for each estimator with different parameter values related to the amount of noise (width ν for GMC and NDC and mask size for the BCC estimator) .

Fig. 9 .

 9 Fig. 9. Results of corner detection using GMCCD (with parameter ν = 2). Convex and concave corners are respectively represented by crosses and disks.

mainFig. 10 .

 10 Fig. 10. Results obtained on noisy version of the original image database. The parameter ν for the GMCCD algorithm was set to 4 for all images.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Result of IPAN (IPAN03) using parameters: d min = 7, αmax = 30.

Fig. 13 .Fig. 14 .

 1314 Fig. 13. Result of Rosenfeld and Weszka (RW75) using parameter: κ = 0.10.

Fig. 15 .

 15 Fig. 15. Result of Beus and Tei (BT87) using parameters: k 1 = 3, k 2 = 7, τ = 0.05, S = 600.

Fig. 16 .Fig. 17 .

 1617 Fig. 16. Results obtained with the visual curvature (VC) with the following parameters: ∆S = 2, λ = 0.1, T = 30, N = 300, DK 0 = 0.6

Fig. 19 .

 19 Fig. 19. Shape simplification based on Visual Curvature.

Fig. 20 .

 20 Fig. 20. Comparison between GMC and VC based corner detectors on different damaged versions of the shape of Fig. 1. The others parameters for VC, ∆S, T , and N , were respectively set to 4, 50 and 128.

  G 1 : 656 pts (84.98%) G 2 : 350 pts (92.98%) G 3 : 276 pts (93.68%) G 6 : 172 pts (96.06%) G 10 : 130 pts (97.02%) G 50 : 68 pts (98.44%) Fig. 18. Multiscale shape representation by using GMC and all changing curvature points. V 0.01 : 2061 pts(52.81%) V 0.02 : 1608 pts(63.18%) V 0.03 : 1410 pts(67.71%)

	V 0.06 : 1072 pts(75.45%)	V 0.1 : 774 pts(82.28%)	V 0.5 : 225 pts(94.84%)

B. Kerautret, J.-O Lachaud, B. Naegel

B. Kerautret, J.-O Lachaud, B. Naegel

B. Kerautret, J.-O Lachaud, B. Naegel

B. Kerautret, J.-O Lachaud, B. Naegel

Acknowledgments

A preliminary version of this work was presented at the International Symposium on Visual Computing, Las Vegas, NV, 2008, Special Track on Discrete and Computational Geometry and their Applications in Visual Computing 12 .

* This work was partially funded by the ANR project GeoDIB, n • ANR-06-BLAN-0225. † Bertrand Kerautret was partially funded by a BQR project of Nancy University 1 May 1, 2009 1:39 WSPC/INSTRUCTION FILE main

ues, there exists some oscillations which are not present with the GMC estimator (Fig. 8).

In the following, we exploit the stability of the GMC estimator in order to implement a robust and fast corner detector.

Application to Corner Detection

We introduce an original algorithm to detect corner points, which is based on the GMC estimator introduced in Section 2. In the sequel we refer to this algorithm as GMCCD. This algorithm requires a parameter ν, which is the estimated thickness of the noise along the shape boundary, expressed in pixels. The main steps of the algorithm for the detection of convex corner points are:

(1) for all contour points (p i ) i∈I compute curvature κ(p i ) with the GMC estimator, setting the width parameter to ν.