The Role of Content and Context in PISA Interest Scales: A Study of the Embedded Interest Items in the PISA 2006 Science Assessment

Keywords: scientific literacy, science interest, large-scale studies international comparisons, multidimensional scaling

.With its innovative approach PISA embeds interest items within the cognitive test unit and its contents and contexts. The main difference from conventional interest measures is that in most questionnaires, a relatively small number of interest items cover broad fields of contents and contexts. The science units represent a number of systematically differentiated scientific contexts and contents. The units' stimulus texts allow for concrete descriptions of relevant content aspects, applications and contexts.

In the analyses, multidimensional item response models are applied in order to disentangle student interest. The results indicate that multidimensional models fit the data. A two-dimensional model separating interest into two different knowledge of science dimensions described in the PISA science framework is further analysed with respect to gender, performance differences and country. The findings give a comprehensive description of students' interest in science. The paper deals with methodological problems and describes requirements of the test construction for further assessments. The results are discussed with regard to their significance for science education.

Only in a few countries, among them Germany, Japan and Austria, do we find an adequate level of interest in terms of the comprehensive definition of scientific literacy. Findings like this require a closer look at interest and its structure.

Interests play an important role in initiating, steering and retaining the engagement in specific domains and learning processes [START_REF] Renninger | The role of interest in learning and development[END_REF][START_REF] Schiefele | Situational and Individual Interest[END_REF].

In educational research, interest has been conceptualised as a phenomenon that emerges from a person's interaction with an object or a certain content [START_REF] Krapp | Structural and dynamic aspects of interest development: Theoretical considerations from an ontogenetic perspective[END_REF]. The personal side of this model has been well elaborated. Research has focused on a variety of the person aspects of the theory, e.g. attention [START_REF] Ainley | Interest, learning, and the psychological processes that mediate their relationship[END_REF][START_REF] Hidi | Interest, a motivational variable that combines affective and cognitive functioning[END_REF], goals [START_REF] Harackiewicz | Predicting success in college: A longitudinal study of achievement--Goals and ability measures as predictors of interest and performance from freshman year through graduation[END_REF][START_REF] Pintrich | The development of academic self-regulation: The role of cognitive and motivational factors[END_REF][START_REF] Sansone | The 'how' of goal pursuit: Interest and self-regulation[END_REF] or levels of learning [START_REF] Alexander | Profiling the differences in students' knowledge, interest, and strategic processing[END_REF][START_REF] Hoffmann | Promoting girls' interest and achievement in physics classes for beginners[END_REF][START_REF] Köller | Does interest matter? The relationship between academic interest and achievement in mathematics[END_REF]. Interests, however, are not sufficiently described when looked at only from the personal side. The objects and domains of interest are sometimes described as opportunity structures for the development of interest. In most approaches, interest is regarded as being domain specific [START_REF] Häussler | Qualitative differences in students' interest in physics and the dependence on gender and age[END_REF]; see also Krapp & Prenzel, this volume). Nevertheless, a much smaller research body can be found on the object aspect (e.g. [START_REF] Prenzel | The selective persistence of interest[END_REF][START_REF] Renninger | Individual interest as context in expository text and mathematical word problems[END_REF][START_REF] Todt | Development of interests[END_REF] of interest.

The topological perspective Häussler and colleagues adopted for the Leibniz-Institute for Science Education (IPN) study on physics interest focuses on the structure of the domain [START_REF] Häussler | Measuring students' interest in physics--Design and results of a crosssectional study in the Federal Republic of Germany[END_REF]Hoffmann, Häussler, & Lehrke, 1998). In their topological model of physics interest, they disentangle specific topics, contexts in which the topics are represented and activities in which the individual can engage. Students are systematically confronted with topics, application areas and activities related to physics. This strategy was stimulating for the development of the embedded interest measures in PISA. While the IPN study used short texts to inform students about the actual topics, PISA 2006 refers to the situated assessment approach used in the cognitive tests. In science education, constructivist principles have been discussed for a long time [START_REF] Duit | Conceptual change--A powerful framework for improving science teaching and learning[END_REF]Fraser & Tobin, 1998). Researchers emphasise the relevance of situations and contexts for understanding and for the construction of meaning, aspects that are also very relevant for valid assessments. PISA takes this into account, and subsequently designs its cognitive items in such a way that they provide a situation or context which puts students into the role of problem solvers in a specific situation. For its innovative interest measure, PISA takes advantage of this structure, even though it has some caveats that will be discussed below.

To construct the embedded interest measure, the structure of the cognitive test was used with its thematically differentiated units. A short description of the topic using a text, a picture and/or a graph, is followed by a number of test items that relate to the stimulus in the beginning. The stimulus introduces the topic and, at the same time, situates and contextualises the test items. In order to find out more about the role a certain context or content could play in the motivational processes, the interest measures are inserted at the end of a number of items of the cognitive test. Instead of giving rather unspecific indicators of being interested in domains like 'physics' or 'cell biology', students are asked to indicate their interest in specific topics. Interest is measured by asking whether or not a student would like to know more about what tooth decay bacteria look like under a microscope or to learn how the body recovers after stopping smoking. Students state their interest in a unipolar Likert-type rating scale.

With this methodological approach, used for the first time in 2006, PISA goes beyond the scope of traditional questionnaires. One major problem of conventional questionnairebased interest measures (and also the PISA interest questionnaire) is that in most cases, a relatively small number of interest items covers broad fields of contents and contexts. The resulting interest measure is rather global and does not give a precise picture of the huge and multifaceted field of science. Students have to give abstracted judgments [START_REF] Schwarz | Self-reports: How the questions shape the answers[END_REF], which are far more likely to be biased than concrete judgments. In order to give information about science interest, students need to summarise their interests in cell biology, astronomy, mechanics and many other scientific areas.

The PISA 2006 science test makes a first attempt to systematically differentiate a number of contexts (i.e. life situations that involve science and technology: health, natural resources, environment, hazards, but also the frontiers of science and technology) and a variety of knowledge areas (living systems, physical systems, technology systems or earth and space systems) which are relevant from a scientific literacy perspective (see Figure 1, OECD, 2006). Therefore, the item pool for the cognitive test offers a comprehensive constellation of science contexts and contents for measuring interests differentiated. As opposed to a traditional questionnaire, the embedded interest items offer a basis of rich information on concrete, situated and contextualised science topics.

In principle the interest structure could be described along the dimensions of the cognitive test. Several constraints, however, have to be taken into account. As a large scale study, PISA is carried out using a rotated booklet design that allows for a high number of test items in a limited span of time. Employing a rotated booklet design in an assessment means that not every student receives all test items. The rotated booklet design provides students with a balanced choice of items which represent only a small fraction of the entire number of items. In PISA 2006, 17% of the science testing time was devoted to attitudinal items; therefore, not all cognitive units were complemented by interest items. Interest items were designed 'for suitable units' (OECD, 2009b); that is, interest items were constructed only for a selection of units. In summary, this means that the data matrix for the embedded interest measures is incomplete and much sparser than the data matrix for the cognitive science assessment. With 52 items available from 13 units rotated over 13 booklets, our analysis models cannot comprise all facets of interest. It is, however, an attempt towards a more comprehensive picture. From a theoretical point of view, a first step towards a differentiation of the global interest in science could take into account the context and the knowledge aspects from the PISA framework, i.e. the knowledge of science dimension. Figure 1 displays the structure of scientific literacy as defined in PISA. The framed aspects were the focus of our attention: the life situations that involve science and technology (contexts) and the systems within which central knowledge aspects are systematised. For the theoretical reasons mentioned above, assuming that science interest is organised along domains and within the structure of a domain, focused on certain contents, contexts and activities (see [START_REF] Häussler | Qualitative differences in students' interest in physics and the dependence on gender and age[END_REF], we expected those aspects to be relevant in identifying dimensions of science interest.

In our study we adopted the argument that interest needs to be studied with regard to contents and contexts, in order to disentangle the object aspect of the person-object relation [START_REF] Krapp | Interest and human development: An educational-psychological perspective[END_REF]. The PISA 2006 science assessment makes a first attempt to measure interests in the respective contexts.

The core of the present study is the multidimensional scaling of the embedded interest items. Instead of a global indicator for science interest, we investigate a more differentiated description of different facets of science interest.

Analysing Different Groups with Regard to Dimensions of Science Interest

Having found dimensions of science interest in the present data, we will pursue a number of questions that help to further clarify the picture by analysing group differences on [START_REF] Gardner | The development of males' and females' interests in science and technology[END_REF][START_REF] Hoffmann | Promoting girls' interest and achievement in physics classes for beginners[END_REF], and consequently we take a look at the distribution of the interest dimensions among boys and girls.

Findings from PISA analyses on science interests show that the connection between science performance and science interest is complex and difficult to discern (OECD, 2007a, b). The relationship between interest in science and achievement (on an individual as well as on a country level) has been the subject of research for more than 40 years (e.g. [START_REF] Baumert | Interest research in secondary level I: An overview[END_REF][START_REF] Osborne | Attitudes towards science: A review of the literature and its implications[END_REF], and thus we examine the connection between science performance and the dimensions of science interest. Earlier analyses show that high-performing students are overrepresented in the upper interest quartiles [START_REF] Prenzel | Interesse an den Naturwissenschaften[END_REF], even though not all high-performing students are highly interested in science. Interest is an important predictor of high-performing students' choice of a sciencerelated career. In our analysis, we therefore focused on high-performing students (the upper performance quartile within a country), aiming at a more differentiated picture within this group.

On a country level science performance and science interest do not necessarily correspond in an educationally desired way. In some countries, low performance is associated with extremely high interest and vice versa. In order to reduce the complexity of this relationship, we decided to work with a selection of OECD countries. We chose a group of countries that is rather homogeneous with regard to performance and can be compared to one another in terms of the significance of the natural sciences and technology in society. The criterion for the choice of these countries was ranging above the OECD average on the combined science performance scale. Fourteen countries were included in these analyses.

Characteristics of the school systems and aspects of science teaching vary across the OECD countries (OECD, in prep.). Learning environments and their degree of stimulation with respect to the (natural) sciences are an important basis for the development of interests (Krapp & Prenzel, this volume). Fostering scientific literacy means supporting the development of science competencies along with the development of science interest. On a country level, it is helpful to know the general level of science interest and tendencies in the distribution of dimensions of interest and learning preferences among 15 year olds. For example, this information is crucial for purposes like reducing and essentialising curricula.

Most of the countries selected for our analyses already suffer from or at least threaten to suffer from a lack of specialised workforce in the natural sciences (OECD, 2009a). Science interest is an important predictor of students' choice of a science-related career (OECD, 2006). Hence, from a motivational perspective, students with specialised interests are an interesting group to study. This group of students is characterised by a one-sided high interest in one dimension of interest with complementary low interest in the other dimension. Such an orientation might even approach a well-developed interest, which is a very sophisticated stage in the development of interest according to [START_REF] Hidi | The four-phase model of interest development[END_REF]. It is interesting to pay special attention to students belonging to this group. As the selection of countries chosen for our analysis is strongly dependent on young scientists it is vital for those countries' schools to foster interest and engagement in science in order to convince as many highly interested and high-performing students as possible to make their vocational choice accordingly.

Research Questions

As mentioned above, the aim of this study is to explore the contribution of the embedded items for a deeper understanding of facets of science interest. Therefore, the first part of our analyses comprises the questions: Can interest in science be disaggregated with regard to specific components of scientific literacy? Do we find specific dimensions of science interest by which we can describe interest more precisely than in just one global science interest score? We expect that either the contexts PISA provides with the test units or the areas of scientific knowledge could be the basis of our disentangling analysis. Once we have found dimensions of science interest, we analyse group differences on an individual and a country level: We examine the distribution of interest dimensions with regard to gender and to performance groups (high-performing students).

Further, on a country level, we ask: Do we find specific preferences according to the dimensions of interest? Does the distribution of interests vary with respect to different highperforming OECD countries?

Eventually, we ask for specific profiles of interests in the high-performing countries.

Do we find countries where the group of students with a one-sided high interest in one dimension with complementary low interest in the other dimension is larger than in others (and vice versa)?

Methods

Sampling Procedures

The following data analyses and results are based on the international dataset from PISA 2006: a representative sample of 15-year-old students assessed in all 30 participating OECD countries with the instruments of PISA 2006. The total number of students analysed is 251 278 and the population covered by these students is 12 933 228 fifteen year olds from the OECD countries. For the purpose of analysing the multidimensional structure, a random sample of 10% of the cases of the complete dataset was used, and a sample size of 25 476 OECD students was reached. In order to report further findings for OECD countries later in this paper, for each analysis the full sample was used. These analyses also employed the PISA survey weights.

Item Pool

Within the science assessment, interest items were constructed for 18 out of 37 science units. Seventeen percent of the testing time in science was devoted to the attitudinal component. The attitudinal assessment comprises items on 'interest in science', 'support for scientific enquiry' and the 'responsibility towards resources and environment' (OECD, 2006).

Test Format and Administration

After having worked on the cognitive items, students were asked to rate their interest in the topics, objects and activities with which they had just been confronted. Three-item scales were constructed to measure interest within each of the units using four-point Likerttype scales from 'high interest' to 'no interest'. The generic item construction principle focuses the assessment of epistemic interest, e.g. knowing more, understanding better and learning more how/about. The questions were highlighted in grey to ensure that students regarded those questions as an extra part of the assessment which was not part of the cognitive science test. Students were told that there was no right or wrong solution to the questions. Figure 2 gives an example for the items.

Multidimensional Scaling of Interest in Science Items

In order to disentangle science interest we considered the contexts and the knowledge of science categories as potential criteria. Following our theoretical assumptions from scientific literacy and, from an interest perspective, assuming that an object of interest is determined by contents (in the PISA framework the knowledge of science dimension), as well as by contexts, we analysed the interest items with regard to their science content. The specifying either context-or content-specific interests as dimensions measured by the respective interest items from the assessment. Note that science items measuring science performance were not used in these models. A multidimensional generalisation of the partial credit model (Masters, 1982) assuming equal slopes for all items was used to scale the item responses.

To find an adequate multidimensional model, we compared several multidimensional models, starting with one for the contexts, one for the contents and a unidimensional model as a control. The best fitting model was chosen using several selection criteria: (a) The number of interest items available for each context or content dimension must be sufficient. Given the rotated booklet design and the fact that only some science units were complemented with interest items, it is important to achieve a reasonably large proportion of students with observed interest items for each dimension. From a psychometrical perspective, we compared different multidimensional models with regard to several criteria: (b) the likelihood based fit index Bayesian information criterion BIC [START_REF] Bozdogan | Model selection and Akaikes information criterion (AIC): The general theory and its analytical extensions[END_REF]

Results

Disaggregated Science Interest

The first question asked for specific dimensions of science interest by which we can describe interest more precisely than through just one global science interest score. Pursuing this question required a reorganisation of the science units in order to identify meaningful dimensions of science interest. We considered the contexts and the knowledge of science categories as potential criteria (see Figure 1). In a first step, two 4-dimensional models were specified, one on contexts and the other on contents. Due to small item numbers, 'Health' and 'Environment' in the context model (M1, Table 1) were collapsed prior to the analysis. For the contexts model (M2, Table 1), three items that were not allocated to a context originally were considered close enough to be assigned to living systems. Thus, all model comparisons are based on all 52 interest items.

# please insert Table 1 here: Multi-dimensional interest models, scale definitions and results

The dimensions of these models and the respective numbers of items are reported in Table 1, as well as scale means, variances, reliabilities, average error variances of the EAP estimates and correlation matrices. Note that all results in this section were computed on a 10% random sample of the full dataset of N=25 476 students without employing survey weights. The table further displays results from two more models derived from the contents model (M2): To construct alternative models with more items in their dimensions, we reorganised the dimensions of knowledge of science with regard to their content. To eliminate the dimensions consisting of only five items, an alternative model with three content dimensions (M3) is proposed. The former living systems dimension now includes three items on earth and space (from the unit 'dinosaurs') and the former physical systems dimension now includes another two items (from the unit 'moon'). The earth and space units were redistributed: the unit on dinosaurs was allocated to the living systems category, the unit on the moon was assigned to the physical/technology systems category as this is more focused on physical than on living system aspects.

Another model (M4) further collapses physical systems and technology systems into one dimension. The new categories comprise living systems and physical/technology systems in a broader sense and are illustrated in Figure 3. The physical and the technological systems categories are taken together as one dimension. The results of the estimation of these two further models are also reported in Table 1. As a control model, Table 1 also displays the results of a unidimensional model (M5).

Table 2 presents likelihood results which allow for comparing the goodness of fit from these five models. With respect to the BIC, the model on four content dimensions (M2) fits the data best, the model on four context dimensions fits the data second best, model M3 on three contents fits the data third best, followed by model 4 on two content dimensions. As expected, the unidimensional model has the worst fit among all five models. Since models M1, M2 and M3 are rather close in their BIC index, each of these models could potentially be used to describe the structure of interest in science.

please insert table 2 here: Multi-dimensional interest models, goodness of fit M2 shows the highest standard deviations and reliabilities among all five models (see Table 1), indicating that these four dimensions best discriminate students' interests in science. Another indicator for the appropriateness of a model might be the fit of its items.

Table 3 presents the number of items from each model that do not fit well according to the root mean square error of approximation (RMSEA) between the expected and observed item characteristic curves. A value greater than 0.05 is regarded as an indication of the misfit of a particular item. The numbers of non-fitting items range from 9 items in M4 to 17 items from the total of 52 items in M1.

please insert table 3 here: Multi-dimensional interest models, number of non fitting items In order to check whether these misfits might be explained by a bad fit of the items into the assumed multidimensional structures expressed by the models, we reestimated the five models, specifying different slopes for each item. Thus, the reestimated models are multidimensional generalisations of a generalised partial credit model (GPCM), with two parameters for each item (labelled 2PL). The numbers of non-fitting items are given in Table 3; Table 4 shows the likelihood results for the reestimated 'slopes'-models. With respect to the BIC, the reestimated slopes models show a better fit than the respective multidimensional Rasch models, as expected (Table 4). With estimated slopes, the three contents model M3 seems to be the best fitting model. Allowing each item to fit into the multidimensional structure to its own degree obviously reduces the number of dimensions needed to explain the observed responses. The descriptive statistics, variances, correlations and reliabilities of these models are quite similar to the values of the corresponding multidimensional Rasch models (see Table 1) and are not reported here.

The number of non-fitting items from the slopes models range from zero to nine items (see Table 3). For the unidimensional model M5 and the two contents model M4, all item misfits seem to be due to an unfitting (too restrictive) multidimensional structure, since these items fit when estimating their slopes. For the other three models, the results are somewhat more complex. Assuming higher dimensional models fitting the data (M1, M2 and M3), the number of non-fitting items should become smaller. However, with higher dimensional models, the discriminations of the dimensions increase as well and the item fit statistic shows a higher sensibility to item misfit. By allowing slopes for these models, however, the number of non-fitting items is reduced, even if some items with bad fit remain.

An inspection of the slope parameters reveals which items do not fit well into the dimensional structure. By definition, in our slopes models, the mean of the slopes is set to one. If a generalised Rasch model holds as well, most slope parameters will be close to one.

With the unidimensional slopes model M5-2PL, most items show slopes around 1, ranging from about 0.8 to 1.2; only some single items and the items from units S519 'airbags' and S527 'extinction of the dinosaurs' show lower slopes, down to 0.6. The items of unit S519 fit well within the three-and four-dimensional slope models M3-2PL and M2-2PL. The items in unit S527 only show slopes sufficiently close to one in the four contents model M2-2PL. Given the results presented above, choosing a multidimensional model according to different criteria will result in different choices. Discriminative power (i.e. standard deviation and reliabilities) would allow the use of both four-dimensional models, supported by the global model fit. With a high priority for measurement precision and for not having too few items in a dimension, one could decide to base further reporting on the three content model M3. If the numbers of items for each dimension should also be fairly well balanced, the two content models M4 would be optimal.

Results for the Dimensions of Interest in the OECD Countries

In the remainder of this paper, we present interest results for OECD countries. Even though other models are more sophisticated and fine-grained, the distinction between living systems on the one hand and physical and technological systems on the other resembles the content structure of science as a domain very well, as seen from a student perspective. The distinction provides the strongest contrast, both psychologically and regarding the content aspect of interest. Vocational choices are made along these categories, too, as they reveal a typical area of tension. We therefore further analyse student interests with regard to the two content subscales of science interest of model M4, physical/technology systems, based on 22 items and living systems, based on 30 items. This model's dimensions are illustrated in Figure 3. remarkable overlap between the two dimensions which, however, tends to be somewhat lower than the correlation between the performance in the different PISA domains (OECD, 2009b, p. 215). This finding once again supports the result that the two dimensions of science interest can be seen as two distinct categories.

We analysed gender differences at a country level, i.e. for the entire sample of students in the OECD countries. Figure 4 displays the results: The means for the interest dimensions with regard to gender range from 0.21 to 0.45. Girls show a more pronounced interest in topics in the field of living systems compared with the topics of Physical/Technology Systems, and for boys it is the opposite, even though the differences between the dimensions are slightly smaller for boys than for girls. For the living systems dimension, we find a small difference between girls and boys (M girls = 0.33, S.E.=0.017 vs. M boys = 0.35, S.E.=0.014), while it is larger (and statistically significant) for the Physical/Technology Systems dimension (M girls = 0.21, S.E.=0.016 vs. M boys = 0.45, S.E.=0.014).

please insert Figure 4 here: Gender differences for two dimensions of interest in the OECD sample Using a subsample of students consisting of the students in the highest performance quartile, we explored the question: Do high-performing students differ in the dimensions of interest from the complete sample of students in the OECD countries? Figure 5 Compared to the other students in the OECD sample, the students in the subsample of highperforming students are significantly more interested in both dimensions of interest. This applies for boys (53.6%) as well as for girls (46.4%). The gender differences among the high performers go in the same direction as for the whole sample. The picture of gender differences for the high-performing students resembles the situation of the entire OECD sample, only on a higher level of interest. Boys and girls in the highest-performing quartile of the sample are equally interested in living systems (M girls =0.56, S.E.=0.022 vs. M boys = 0.55, S.E.=0.022). For Physical/Technology Systems there is a clear and statistically significant difference in favour of the boys: M girls = 0.30, S.E.=0.022 vs. M boys = 0.56, S.E.=0.024. In fact, the level of boys' interest in Physical/Technology Systems for the whole sample (M=0.41, S.E.=0.016) is considerably higher than the high-performing girls' interest in this dimension. The already small interest difference for males for the two dimensions dissolves completely in the high-performing subpopulation: Boys in the highest quartile are equally interested in Living and Physical/Technology Systems.

In the next step we analysed whether or not the distribution of interests varies for the OECD countries ranging significantly above the OECD mean in science performance. For this analysis we compared 14 high performing OECD countries with the OECD average. On first glance, Figure 6 S.E.=0.042), the interest scores are quite low for both dimensions of interest, compared to the OECD average (M living = 0.34, S.E.=0.013, M phys./tech. = 0.33, S.E.=0.012). Regarding the dimensions of interest we also find considerable differences within the countries: As opposed to the OECD means which indicates slightly higher interest for the living systems dimension than for the physical/technology systems dimension, we find an interest preference for physical/technology systems in a number of high-performing countries: Ireland, Austria, Germany, the Netherlands, Japan and Finland. It needs to be mentioned that in some countries this difference is on a rather high-interest level, in others on a low level, each compared to the OECD average. In order to have a realistic impression of the differences we translated the displayed scores back to the original values on the interest scale. The four-point Likert scale comprises the categories 0 = no interest, 1 = low interest, 2 = medium interest and 3 = high interest. On average for the entire OECD sample, the mean for the two interest dimensions is M rawscore-living = 1.66 and M rawscore-phys/tech. = 1.63. For the Netherlands, for example, one of the countries with the lowest interests, the means are M living = 1.32 and M phys/tech. = 1.32.

Finally we asked for specific profiles of interest dimensions in the high performing countries. We asked whether, within a country, we could identify groups of students scoring high on one dimension of interest while the other dimension is of minor interest for them. In order to answer this question we calculated interest ratios for the high-performing countries. within the data of the OECD countries differentiating higher versus lower interest in each of the interest dimensions. The OECD acts as the reference group for this analysis. We then analysed profiles of students indicating higher or lower interest in living systems and physical/technology systems. In the OECD, 9% of the students were found to indicate a profiled interest either way rather than indicating a higher or lower interest in both dimensions.

The analysis was carried out by applying a median split of interest scores in both dimensions

please insert Table 5 here: Interest Profiles for the high performing OECD-countries Table 5 displays the percentages of students with (nonprofiled and) profiled interests.

On the OECD level a total of 82% of the students are 'nonprofiled', i.e. they either score high or low in both dimensions of interest (Column 2 and 5). The interesting groups are 9% of the students who are either not interested in living systems while they are at the same time highly interested in physical/technology systems (Column 3), and another 9% of the students who are highly interested in living systems and score low in the physical/technology dimension (Column 4). If we look at Finland, the highest performing OECD country in science, we find that 63.7% of the students are not interested in either dimension of science interest. Column 3 describes the percentages of students with profiled interests: In Finland, 9.9% of the students are strongly interested in physical/technology systems and not interested in living systems, slightly more than on the OECD level. The interests for the group of students in Column 4 are profiled vice versa. Those students are highly interested in living systems and hardly interested in physical/technology systems. For Finland as an example country, such students represent 6.7% of the sample, less than for the OECD sample. Column 5 displays the students who score high in both dimensions of science interest, which applies for 19.8% of the Finnish students. The last column gives a ratio that allows an estimation in comparison to the OECD mean: A ratio below 1 implies that the group of students who are not interested in living than the group where it is vice versa. Finland is an example country that shows that, compared to the OECD average, there is a larger group with one-sided interest in Physical/Technology Systems and a smaller group of students whose interest profile is the other way around. This also applies for Japan, the Netherlands, Germany, the UK, Austria and Ireland. Canada, New Zealand, Australia, Korea, the Czech Republic, Switzerland and Belgium are the highperforming OECD countries where it is the other way around: In those countries the group of students which is unidirectionally interested in living systems is larger than in the OECD average.

Discussion and Conclusion

This study aimed at exploring new methodological approaches to optimise the analyses of the insights from PISA data, i.e. the embedded science interest assessment. Our paper provides an approach to generate additional information on student interest in science by disentangling dimensions of the PISA interest assessment.

The empirical results showed that all multidimensional models have the potential to discriminate students along all dimensions of science interest analysed here. With regard to global model fit, the four-dimensional models are well suited to describe the interest structure; however, the three-and two-dimensional content models are applicable as well. The fit of single items was analysed by an item fit statistic and by estimating slopes in alternative slope models. An inspection of the estimated slopes revealed that with a few exceptions, all items fit into the multidimensional structure of the four-dimensional content model. Two units were found to fit within the four-dimensional content model M2 but not within any of the other models.

In summary, the multidimensional analyses seem to be best suited to reflecting the dimensionality of the interest items model M2. Thus, the assessment of a multidimensional structure of interest aimed at with the embedded items was successful. However, defining a From a methodological perspective, however, some remaining challenges should be pointed out. The main disadvantage of implementing the innovative approach in PISA 2006 lies in the small number of interest items distributed over a complex rotated booklet design:

This does not necessarily support a fine-grained analysis of interest dimensions and their interrelations. In order to examine interest dimension structures more comprehensively, a higher number of embedded items in the respective dimensions would be helpful. Ideally, if more testing time could have been allocated for the embedded attitudinal items, a minimum of 3 units comprising 9 interest items for each designated dimension within the 29 newly constructed science units in PISA 2006 would have been appropriate. Our analyses contribute to a better estimation of the required number of items. Even if the test construction is a very complicated procedure and the required number of items is high, we find the results at hand already quite instructive and we conclude that eventually, it would be very interesting to pursue this approach. If it is too complicated to do so in a large scale format that employs a rotated booklet design, it could instead be carried out in other studies, even in other-large scale studies like in standard-based assessments where all students work on all test items. living systems, while males' interest in physical/technology systems is higher than that of females. This also applies for students in the upper quartile of the performance scale. The interest level in the group of high-performing students is higher than in the entire sample.

Boys and girls show considerable interest on a comparable level. High-performing girls are less interested in Physical/Technology Systems than the boys of the same group.

From a teaching perspective it is important to understand the students' motivational situation and to know their learning preferences in order to adaptively foster interests and to design instruction in a way that meets specific learning preferences (Darling-Hammond & Bransford, 2005). This is supported by the results presented. Knowing the preferences and special interests of students, boys as well as girls, can help to improve instruction in the natural sciences.

Studying the high-performing countries resulted in interesting findings: Firstly, for high performing countries in the OECD sample we find considerable differences in the level of interest. Figure 6 shows that high performance in a country is not necessarily complemented with high interest levels. The ideal stated in the definition of scientific literacy in PISA is not reached in a number of countries. Taking a closer look at the dimensions of interest reveals further considerable differences. In some countries students prefer living systems, while in other countries physical/technology systems are of higher interest to students.

So-called profiled interests were also studied: For the high-performing countries, the group of students with high interest in one dimension complemented with a low level of interest in the other dimension and vice versa was studied. We found countries with profiles in favour of the living systems dimension with a higher percentage of students in this group and countries where the profiled group with a preference for the Physical/Technology Systems is larger, as shown in the ratio built in comparison to the reference ratio of 1 on the OECD level. From an educational systems perspective, these findings on the country level are very instructive with regard to fostering interests. For high-performing countries it is interesting to know how successful they are in fostering interest in science and at the same time enabling the development of high performance in science.

Students with specialised interests are also very interesting from a teaching perspective. They indicate high interest in one dimension while simultaneously disregarding the other dimension. In societies which are in need of specialised scientists, these groups can be very important, as interests play an important role in vocational choices. Another perspective is the educational goal of supporting the development of competence along with the development of interests. Fostering interests could be a differentiated and systematic process, starting from an area of interest and transferring to aspects in the wider context of one topic. Hence, the interest profiles can be helpful in finding out about learning preferences in relation to the need to foster interests.

The present findings leave some questions unanswered; therefore, analyses will be conducted in order to further describe students' interest in more detail, in terms of their connection to students' views on a science-related career or background variables like socioeconomic factors or other parental variables.

F

  At first glance, the incomplete data structure is sufficient for analyses on an aggregated item level. A general indicator for interest in science can be successfully established on the data structure at hand. Using this aggregation, however, does not provide much extra information compared to the very global interest indicator based on the usual questionnaire approach in large-scale assessment. Thus, in order to disentangle dimensions of science interest in the PISA 2006 assessment, we refer to the interest item pool PISA produced.

  on the interest aspect of the attitudinal component of scientific literacy, which was measured with 52 interest items embedded in 18 units.

  please insert Figure 2 here: Example Items for interest in science. (Source: OECD, 2009b, p.36, for the cognitive unit, see p.104f.)

  contexts are operationalised as areas of application in the PISA 2006 framework (Figure 1), labelled as Frontiers of Science, Hazards, Environment, Health and Others. The contents distinguished in the PISA 2006 framework are living systems, technology systems, physical systems, earth and space systems (Figure 3).

  , (c) the estimated latent person variances, and (d) reliabilities of the interest dimensions based on Expected A Posteriori estimates (EAP) were employed in order to test to what degree the dimensions discriminate interest between students. Additionally, (e) we compared average variances of the measurement errors as indicators for the precision of the interest estimates. All model results are reported in the logit metric defined by a multidimensional Rasch model with slopes fixed to either zero or one. The interest parameters refer to the mean item difficulty being zero within each scale. In order to investigate item fit, generalised partial credit models with slope parameters were employed, fixing a slope either to zero or estimating it. All multidimensional models were specified as (restricted) general diagnostic models (von Davier, 2007) and estimated via maximising the marginal likelihood using the egpcm software by von Davier (2009). Page 12 of 40 URL: http://mc.manuscriptcentral.com/tsed Email: editor_ijse@hotmail.co.

  However, the standard deviations and reliabilities of model M1 are similar. The three content dimensions of model M3 are virtually the same as the respective content dimensions in model M2. The standard deviations and reliabilities from the two-dimensional content model are close to the respective content dimensions of M3 and M2, quite close for living systems and somewhat smaller for the collapsed dimension physical/technology systems. The the smallest standard deviation and a rather high reliability.Note that the reliabilities given are derived from EAP estimates, and thus are based on the items of the respective dimensions and on the covariance among all dimensions of a model.The average error variance (of the EAPs) of each dimension provides a closer look into the measurement precision of the estimates. This error variance obviously correlates with the number of items: M5 shows the smallest error, while the dimension defined on five items for earth and space systems in model M2 shows the greatest error. In terms of measurement error the unidimensional model and model M4 are considered best; the other three models are less precise in measuring their dimensions.

  please insert Figure3here: The Reorganisation of the Knowledge of Science-Categories In a first step, the two dimensions of interest were analysed for the OECD countries' sample. The results are presented on a logit scale defined by the item response model; all results reported below were computed on the complete country samples using the PISA survey weights. The mean (M) for the living systems dimension is M=0.34, with Standard Deviation SD=1.45 and Standard Error S.E.=0.013. For the physical/technology systems, we 33 with SD=1.47 and S.E.=0.012. The latent correlation between the two dimensions is r living systems, physical/technology systems = 0.80. The latent correlation shows a

  presents the results, analysed separately for males and females. # please insert Figure 5 here: Differences for high performing students (upper quartile) for the two dimensions of interest versus the other students in the OECD sample

  displays remarkable differences in the interest structure of highperforming countries. please insert Figure 6 here: Interest in living vs. physical/technology systems: all students The scores for both interest dimensions in countries like Japan (M living = 0.41, S.E.=0.041, M phys./tech. = 0.44, S.E.=0.043), Austria (M living = 0.31, S.E.=0.051, M phys./tech. = 0.37, S.E.=0.046) and Germany (M living = 0.39, S.E.=0.038 M phys./tech. = 0.44, S.E.=0.043) are which means that they are similar to the OECD average. The fact that on the OECD level the average interest is approximately similar to the countries with highly interested students is due to the selection of (high-performing) countries: It is a common finding in PISA that low-performing countries show considerable interest compared to high performers. In other, likewise high-performing countries like the Netherlands (M living = -0.38, S.E.=0.045 M phys./tech. = -0.34, S.E.=0.043), Finland (M living = -0.43, S.E.=0.045, M phys./tech. = -0.34, S.E.=0.044) or New Zealand (M living = -0.18, S.E.=0.042, M phys./tech. = -0.24,

  the same time they are interested in physical/technology systems is larger

  five items does not warrant stable results from further analyses. Therefore, we constructed alternative models on three and two dimensions. With further analyzing M3 instead of M2 or M4 instead of M3, reporting may lose a bit of the potential to obtain differential insights into science interest structure but gains stability in the results. With regard to the numbers of items in each dimension, it might be indicated to report interest along the content dimensions based on the three-or two-dimensional models M3 or M4.

For

  further reporting in this paper we choose model M4 differentiating science interest along the prominent distinction in life sciences, living systems in terms of PISA and rather hard, physical and technological sciences, called physical systems and technology systems in PISA. According to these two categories, we found gender differences that support findings from a number of studies and indicate that males and females are about equally interested in

  please insert table 4 here: Multi-dimensional interest models with slopes, goodness of fit

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	ACID RAIN				Knowledge • About the natural world
					(knowledge of science)
	Requires people to: How much interest do you have in the following information? How they do so is influenced by: Tick only one box in each row.	• Living Systems • Physical Systems • Technology Systems • Earth and Space
	Context Life situations that involve science and technology	Competencies Interest High	Systems Low • About science itself Medium (knowledge about science) No Interest Interest Interest
	• Identify scientific issues a) Knowing which human activities • Health
	o r F o r F o r F o r F o r • Natural resources contribute most to acid rain • Environment • Hazards • Frontiers of science and technology b) Learning about technologies that minimise • Explain phenomena scientifically • Use scientific evidence F the emission of gases that cause acid rain o c) Understanding the methods used to repair r buildings damaged by acid rain	1 1 1		Attitudes Response to science issues • Interest • Support for scientific enquiry • Responsibility 2 3 2 3 2 3	4 4 4
	P P P P P P
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	Categories of
	knowledge of science	N of Items
	Living Systems	24
	Physical Systems	11	Interest Contents	N of Items
	Technology Systems	9	Living Systems	30
	Earth and Space		Physical/Technology
	Systems	5	Systems	22
	Not assigned to a
	category	3
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	0.50
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	0.10 0.00	F o OECD sample: living r systems	OECD sample: phys./tech.systems
		P female	male
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			Number of		Standard		Error
	Model	Scale	items	Mean	Deviation	Reliability	Variance		Correlation Matrix
	M1 "contexts 4D"							health_env	frontiers	hazards	resources
		health_env	12	0.50	1,70	0.72	0.52	1
		frontiers	17	0.29	1,52	0.66	0.42	0.70	1
		hazards	12	0.60	1,65	0.72	0.59	0.64	0.68	1
		resources	11	0.29	1,71	0.73	0.60	0.74	0.73	0.65	1
	M2 "contents 4D"							earth & space	living	physical	technology
		earth & space	5	0.85	1,73	0.70	0.72	1
		living	27	0.36	1,56	0.78	0.34	0.61	1
		physical	11	0.61	1,65	0.64	0.62	0.57	0.65	1
		technology	9	0.13	1,79	0.78	0.64	0.55	0.72	0.65	1
	M3 "contents 3D"							living	physical	technology
		living	30	0.40	1,51	0.79	0.31	1
		physical	13	0.61	1,61	0.62	0.63	0.67	1
		technology	9	0.13	1,79	0.78	0.65	0.71	0.64	1
	M4 "contents 2D"							living	physical/technology
		living	30	0.39	1,49	0.79	0.31	1
		physical/technology	22	0.38	1,50	0.63	0.46	0.74	1
	M5 "all items 1D"
		total	52	0.38	1,40	0.77	0.26	1

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	Model	log Likelihood # parameters	N	BIC
	M1 "contexts 4D"	-434849	776	25476	877572
	M2 "contents 4D"	-434263	776	25476	876400
	M3 "contents 3D"	-437975	277	25476	878761
	M4 "contents 2D"	-442709	178	25476	887225
	M5 "all items 1D"	-450253	160	25476	902129
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			Slopes model:
	Model	Items with low fit	Items with low fit
	M1 "contexts 4D"	17	9
	M2 "contents 4D"	11	6
	M3 "contents 3D"	11	3
	M4 "contents 2D"	9	0
	M5 "all items 1D"	10	0
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			Interest profiles in %
		living	living		living	living
		phys./tech.	phys./tech.	phys./tech.	phys./tech.	Ratio
	Finland	63.7	9.9	6.7	19.8	0.67
	Canada	53.8	8.8	9.1	28.3	1.03
	Japan	36.1	10.4	8.4	45.1	0.81
	New Zealand	57.8	7.2	9.6	25.4	1.33
	Australia	56.3	8.3	8.9	26.5	1.07
	The Netherlands	64.4	8.5	8.4	18.6	0.98
	Korea	47.7	9.3	10.3	32.7	1.10
	Germany	36.3	10.7	10.0	43.0	0.94
	F o r United Kingdom 56.8 Czech Republic 46.5 F Switzerland 40.1 Austria 38.3 o Belgium 41.8 r Ireland 49.7	8.6 8.6 10.3 12.3 8.6 10.7	8.3 11.1 11.0 9.9 11.5 9.3	26.3 33.9 38.7 39.5 38.1 30.3	0.96 1.29 1.07 0.80 1.33 0.87
	OECD countries	P P 41	9	9	41	1
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uk International Journal of Science Education


Romain Arcila

Cédric Cagniart

Franck Hétroy

Edmond Boyer

Florent Dupont

Temporally coherent mesh sequence segmentations

Keywords: mesh sequence, segmentation, mesh tracking, spectral clustering séquence de maillages, segmentation, suivi de maillage, regroupement spectral

In this report, we consider the problem of fully automatic segmentation of mesh sequences, with or without temporal coherence. More precisely, our goal is to identify model parts that consistently move rigidly over time. We propose a novel framework that incrementally adapts segments along a sequence based on the coherence of motion information within each segment.

In contrast to existing approaches, this framework handles meshes independently reconstructed at each time instant, provided that motion cues are available. It allows therefore for meshes with varying connectivity as well as varying topology. Experiments on various data sets in addition to a quantitative evaluation demonstrate the eectiveness and robustness of the approach.

Segmentations temporellement cohérentes de séquences de maillages Résumé : Nous considérons dans ce rapport le problème de la segmentation entièrement automatique de séquences de maillages, avec ou sans cohérence temporelle. Plus précisément, notre but est d'identier les parties d'un modèle qui se déplacent rigidement de manière cohérente au cours du temps. Nous proposons un canevas nouveau pour adapter ces régions de manière incrémentale le long de la séquence, en se basant sur la cohérence de l'information de mouvement dans chaque région. Contrairement aux approches existantes, ce canevas permet de traiter les séquences de maillages reconstruits indépendamment à chaque pas de temps, pourvu que des indicateurs de mouvement soient disponibles. Il permet donc de segmenter des maillages avec changement de connectivité et/ou changement de topologie. Des expériences sur plusieurs jeux de données ainsi qu'une évaluation quantitative démontrent l'ecacité ainsi que la robustesse de cette approche.

Introduction

Shape segmentation is the process of partitioning shapes into segments or regions that are semantically consistent with respect to a given criterion. Segmentation interest appears in several application domains where semantic information on shapes, e.g. body parts, are required for interpretation purposes, for instance motion analysis and recognition applications. When considering static 3D models, such process is faced with the issue of nding appropriate criteria to dene segments [START_REF] Shamir | A survey on mesh segmentation techniques[END_REF]. On the other hand, dynamic or temporal sequences of evolving 3D models overcome this limitation by providing motion information that naturally yield criteria, including rigidity, enabling the identication of segments. This is particularly relevant with multi-view acquisition systems, e.g. [START_REF] Starck | Surface capture for performance based animation[END_REF], or animation software, e.g. Blender [ble] for example, that can build temporal mesh sequences of dynamic scenes usually composed of rigid, sometimes articulated, moving parts.

Segmenting mesh sequences is anyway a challenging task. The main reason lies in the inconsistency of the produced meshes that may result from various factors. First, multi-view reconstructions are usually performed independently over time, hence providing meshes with dierent connectivities and even, occasionally, dierent topologies. Second, meshes are perturbed by the noise accumulated in the acquisition pipeline. Approaches have been proposed to recover temporally consistent meshes from inconsistent ones, e.g. [START_REF] Vlasic | Articulated mesh animation from multi-view silhouettes[END_REF]. While providing rich information for segmentation over time sequences, they usually require a reference model that introduces an additional step in the acquisition pipeline, hence increasing the noise level. Moreover, the reference model strongly constrains shape evolution to a limited domain.

In light of this, we propose an iterative scheme that clusters vertices into rigid segments along sequences using motion information between successive frames.

In each frame, rigid segments can be rened by separating parts that present inconsistent motions or, on the contrary, merged when neighboring segments present similar motion. Motion information are estimated by matching meshes at successive instants. The contribution of the approach is threefold: it is fully automatic and does not require prior knowledge on the observed shape;

it handles arbitrary shape evolutions, including changes in topology;

segmentation can be computed on the y.

This method can provide two kinds of segmentations: either one segmentation for the whole sequence, or a segmentation which evolves during the sequence, representing the currently displayed motion.

The remainder of the paper is as follows. Section 2 discusses related work.

Section 3 outlines the proposed scheme. Sections 4 and 5 presents the matching between two successive time instants and the segmentation algorithm respectively. Section 6 shows comparisons with previous works and an evaluation proposal. Finally, we conclude in section 7.

Related Work

Spatio-temporal mesh sequences can be split into two categories: dynamic meshes, or temporally coherent mesh sequences: all meshes of the sequence have the same number of vertices and connectivity. This means that only Euclidean coordinates of the mesh vertices change over time;

unconstrained mesh sequences : each mesh of the sequence has its own numbers of vertices and faces and its own connectivity. Moreover, global topology changes can happen. Such sequences are sometimes called 3D videos, or time-varying meshes. Figure 1 shows some frames of the Balloon sequence which is a multi-camera reconstructed sequence.

Dynamic meshes are generally created by deforming a reference mesh, using skeleton driven animation system. While dierent in nature from meshes generated from real images, a signicant eort has been devoted to their temporal segmentation. Unconstrained mesh sequences are generally captured from videos and have been a subject of interest over the last years in the computer vision domain. This kind of sequences now tends to be used by both communities [START_REF] Starck | Surface capture for performance based animation[END_REF][START_REF] Vlasic | Articulated mesh animation from multi-view silhouettes[END_REF].

Segmentation of a static shape is a common problem in several areas, i.e.

texture mapping among others and many solutions have been proposed, see Shamir [START_REF] Shamir | A survey on mesh segmentation techniques[END_REF] for a survey. Conversely, segmentation of temporal mesh sequences has received less attention. Several methods have been proposed to compute motion based segmentation of dynamic meshes. Among them, [START_REF] Wang | Segmenting a deforming mesh into near-rigid components[END_REF][START_REF] De Aguiar | Automatic conversion of mesh animations into skeleton-based animations[END_REF] segment a dynamic mesh into rigid components. In particular, de Aguiar [START_REF] De Aguiar | Automatic conversion of mesh animations into skeleton-based animations[END_REF] proposes a spectral approach which relies on the fact that the distance between two points is invariant under rigid transformation. In this paper, we consider unconstrained mesh sequences and hence cannot use the invariant proposed by de Aguiar et al.

In contrast to [START_REF] Vlasic | Articulated mesh animation from multi-view silhouettes[END_REF][START_REF] Cagniart | Iterative deformable surface tracking in multi-view setups[END_REF] which transform an unconstrained mesh sequence into a consistent mesh sequence, we only use the motion information available between two frames.

A main issue of unconstrained mesh sequences is that there is no direct matching between vertices of consecutive frames. This matching needs to be explicitly computed. This can be done using one of the numerous methods proposed in the literature [START_REF] Knossow | Inexact matching of large and sparse graphs using laplacian eigenvectors[END_REF] for example. [START_REF] Franco | Learning Temporally Consistent Rigidities[END_REF] propose such a method for mesh sequences, but it requires as input the desired number of clusters.

In this paper, we propose a fully automatic approach to segment meshes with arbitrary evolution, including topology changes. To the best of our knowledge, no similar method exist in 3d. Interestingly, our approach shares similarities with the recent work of Brox and Malik [START_REF] Brox | Object segmentation by long term analysis of point trajectories[END_REF] in the case of image segmentation. It should be noted that even if our method aims at segmenting mesh

Method Outline

Our method takes as input a mesh sequence, where each mesh can have its own connectivity, and with potentially global topology changes. It uses an iterative approach and is designed to work on a small time window, containing a few (typically, 5) meshes. Working on only a few meshes at the same time allows to handle long sequences composed of meshes with a high number of vertices.

Our algorithm alternates between two steps at iteration k:

Matching between 2 consecutive frames and displacement vectors computation within a time window from frame k.

Segmentation of frame k and mapping to frame k + 1.

This method relies on ecient matching and segmentation algorithms, that we describe in section 4 and 5.

We create a motion-based segmentation by clustering vertices into rigid components. Using a simple boolean parameter, we can generate two dierent types of segmentations:

global segmentations : a single segmentation is sought over the full sequence;

time-varying segmentations : the segmentation of a given frame represents the current motion around this frame. Clusters can appear and disappear along the sequence.

The notations used throughout the article are the following:

N : number of frames;

F k : k th frame that can be composed of several meshes;

F k : k th frame F k registered to F k+1 ; nv(F k ): number of vertices in F k ; v (k) i : vertex with index i in F k ; Ng(v (k) i ): 1-ring neighbors of vertex v (k) i .
Note that we always use k as the index for a frame, and i, j as the indices for vertices in a frame. Vertex Matching

The objective of this step is, given meshes at frames

F k , k ∈ [0..N -1], to pro- vide a mapping from vertices v (k) i to vertices v (k+1) j
for 1 ≤ k < N , and a possibly dierent mapping from vertices v

(k+1) j to vertices v (k) i .
This mapping is further used to propagate segment labels over the sequence. We proceed iteratively according to the following successive stages: rst, meshes at frames k and k + 1 are registered, then displacement vectors and correspondences are estimated. The following subsections detail these stages. Please note that the segmentation algorithm presented further is independent of the matching performed and mainly requires displacement vectors. Thus, an approach based on 3D scene ow (e.g., [SAL * 08]) could also be considered.

Mesh Registration

The matching stage of our approach aims at establishing a dense cross parametrization between pairs of adjacent meshes of the sequence. Among the many available algorithms for this task, we chose to favor generality by casting the problem as the registration of two sets of points and normals. This means that we exclusively use geometric cues to align the two meshes, even when photometric information is available like in the case of meshes reconstructed from multi-camera systems. Thus, our approach also handles the case of software generated mesh sequences.

We implemented the method of Cagniart et al. [START_REF] Cagniart | Iterative deformable surface tracking in multi-view setups[END_REF] that iteratively deforms the mesh F k to t the mesh F k+1 . This approach decouples the dimensionality of the deformation from the complexity of the input geometry by arbitrarily dividing the surface into elements called patches. Each of these patches is associated to a rigid frame that encodes for a local deformation with respect to the reference pose F k . The optimization procedure is inspired by ICP as it iteratively re-estimates point correspondences between the deformed mesh and the target point set and then minimizes the distance between the two point sets while penalizing non rigid deformations of a patch with respect to its neighbors.

Running this algorithm in a coarse-to-ne manner by varying the radii of the patches has proven in our experiments to robustly converge, and to be faster than using a single patch-subdivision level.

Mapping and Displacement Vectors Extraction

By using the previous step, we get the registered mesh F k of the frame F k on frame F k+1 . To create a mapping from F k to F k+1 , we nd for each vertex in F k the closest vertex in F k+1 using Euclidean distance. We also create a mapping from F k+1 to F k by nding for each vertex in F k+1 the closest vertex in F k . Both mappings are necessary for the subsequent stage of our algorithm (see Section 5.3).

The displacement vector of each vertex v

(k) i in F k such that 0 i < nv(F k ) is given as --→ DV i (k) = v (k) i -v (k) i , with v (k) i the corresponding vertex in F k . Inria Figure 2: Matching process.
At the end of this stage, we get the mappings and displacement vectors between each pair of successive frames (see Figure 2).

Motion-based Segmentation

Our algorithm proceeds iteratively over a time sequence. First, in the case of a time-varying segmentation, neighboring clusters, in the current segmentation at time k, that present similar motions are merged (Section 5.2). Then, the current segmentation is rened using the displacement vectors at time k (Section 5.1). Finally, the segmentation is remapped onto the mesh at k + 1 (Section 5.3).

Features of the approach are:

it can process a sequence with numerous frames, where each frame can have a huge number of vertices. It only requires few frames in memory at a time. For instance, results shown on Figure 3 correspond to a sequence of 300 frames with approximately 15,000 vertices per frame; it can control cluster creation and keep track of the transformations.

In the case of a global segmentation (Section 5.4), for which no cluster merging is allowed, the segmentation computed on the last frame is the nest one.

This segmentation is then mapped back onto the whole sequence.

Note that we use a time window, 5 frames from k to k +4 in our experiments, to compute displacements at each frame k. The segmentation of a frame is performed using a spectral clustering approach applied on the graph of vertices of the frame. Edges in this graph correspond to edges on the mesh. An edge is weighted by the distance between the transformations of its 2 vertices. This is in contrast to [START_REF] De Aguiar | Automatic conversion of mesh animations into skeleton-based animations[END_REF] where Euclidean distances between vertices are considered. Intuitively transformation distances better constrain rigidity than Euclidean distances and our experiments conrm this fact.

Our segmentation algorithm produces, by construction, connected clusters since the atomic operations over clusters are: splitting a cluster in connected clusters through spectral clustering, and merging neighboring clusters.

Spectral Clustering

Using mappings and displacement vectors computed as previously explained, we build a graph whose edges are weighted according to rigid transformation similitude between their two endpoints. The normalized Laplacian matrix associated with this graph is computed and the clustering is performed on this matrix.

Graph Construction

To compute the spectral clustering, the weighted adjacency matrix of the graph associated to the vertex transformations between two consecutive frames is build. The nodes of this graph are the vertices of the frame at time k. To construct this graph, we rst compute the rigid transformation which maps

each vertex v (k) i
of the frame F k , and its one-ring neighborhood Ng(v

(k)
i ), onto the frame F k . This transformation is computed using Horn's method that estimates a 4 × 4 matrix representing the best rigid transformation between 2 point clouds. A transformation matrix T (k) i is therefore associated to each vertex v (k) i . To compute the weights of the graph W (k) edges, we use the following expression [START_REF] Murray | A Mathematical Introduction to Robotic Manipulation[END_REF]:

w (k) ij = 1 log(T (k) i -1 T (k) j ) 2 if i = j, 0 if i = j. (1)
As demonstrated in [START_REF] Murray | A Mathematical Introduction to Robotic Manipulation[END_REF], this distance is the most meaningful one between rigid transformations in the sense that it corresponds to distances on the man-Inria ifold of rigid transformations SE(3). We comment on the consequences of this choice in section 6.3.2.

Laplacian Matrix and Clustering

Using the weighted adjacency matrix W (k) , we build the normalized Laplacian matrix L

(k) rw as follows and use the well-known Shi-Malik normalized spectral clustering algorithm [START_REF] Shi | Normalized cuts and image segmentation[END_REF] to segment the graph.

D (k) ii = j∈Ng(v (k) i )
w ij (k) .

(2)

L (k) = D (k) -W (k) .
(3)

L (k) rw = D (k)-1 L (k) = I (k) -D (k)-1 W (k) . ( 4
)
This method assumes the number K of clusters to be known. We compute K using the eigen gap method (see section 6.3.1 for a discussion on the eigen gap method).

Merging

In the case of a time-varying segmentation, we merge at each time step neighboring clusters with similar motions before applying the spectral clustering. To this aim, the distance between the transformations of neighboring clusters is thresholded using T merge , where the transformation of a cluster is estimated over all its vertex displacements using the Horn's method.

The merging starts with the cluster with the minimal residual error, and stops when this process cannot be applied anymore. The residual error for a cluster corresponds to the average distance, for all points v (k) i of the cluster, between the point v (k+1) i and the location of v (k) i after the rigid transformation computed for the cluster is applied. An issue that arises from having two different thresholds for the cluster merging and splitting operations is that two clusters which are merged can then be split right afterwards. To avoid this situation, we apply spectral clustering on the set of the two clusters before merging them in practice. If they are split during clustering, then we do not eectively merge them.

Mapping onto the next frame

The segmentation is computed at each time step on the current frame F k . Labels are then mapped onto the frame F k+1 using the bi-directional mapping dened in Section 4.2. Clusters are rst transferred using the mapping from frame F k to F k+1 . Then for all unmatched vertices in F k+1 , we use the mapping from F k+1 to F k . Clusters which are mapped on 2 dierent meshes are split.

Global Segmentation

In order to get a time-coherent global segmentation over the whole sequence, the segmentation obtained for the last frame need to be transferred to all previous frames. To this purpose, we apply the mapping process described in the previous section in reverse order, from last to rst frame. For each pair of successive frames (F k , F k+1 ) we rst remap using the vertex matching from F k+1 to F k , then for all remaining points, we use the vertex matching from F k to F k+1 .

6

Experiments and Discussion

6.1

Results and Comparisons with Previous Works

Vertex Matching

The vertex matching computation is an important step since our segmentation algorithm relies on it. Figure 4 shows the result of vertex matching between two successive frames of an unconstrained mesh sequence. Computation time is about 30 seconds for two frames with approximately 7000 vertices each.

[AKH * 10] used another matching method which provides similar results, but the proposed method outperforms their matching algorithm which takes about 13 minutes to complete computation with the same data. Note that outliers in the matching are not explicitly taken into account in the segmentation, however their inuence is limited by the threshold on the cluster size that tends to force their merges with neighboring clusters.

Segmentation of Dynamic Meshes

As shown in Figure 5, our method yields visually similar results to previous methods dedicated to dynamic meshes, i.e. temporally coherent sequences. Note that each cluster is connected: even if some clusters in Figure 5 share the same color, they represent dierent clusters.

Global Segmentation of Unconstrained Mesh Sequences

To the best of our knowledge, [AKH * 10] is the only previous method which computes global segmentation of unconstrained mesh sequences. Our method shares some similarities with their framework; however, our segmentation algorithm allows for better cluster evolutions by enabling both merge and split operations and by using a spectral clustering method for that purpose. On dynamic meshes such as the horse sequence (see Figure 5), our segmentation results are closer to the presented results in [START_REF] De Aguiar | Automatic conversion of mesh animations into skeleton-based animations[END_REF] and in [START_REF] Wang | Segmenting a deforming mesh into near-rigid components[END_REF]. The improvement over [AKH * 10] illustrates the benet of the spectral clustering.

Time-Varying vs. Global Segmentations

Quantitative Evaluation

A quantitative and objective comparison of segmentation methods is an ill-posed problem since there is no common denition of what an optimal segmentation should be in the general case. The segmentation evaluation has been recently addressed in the static case using ground truth (i.e. segmentations dened by humans) [START_REF] Benhabiles | A framework for the objective evaluation of segmentation algorithms using a ground-truth of human segmented 3D-models[END_REF]. In the dynamic mesh case, none of the previously cited articles proposes an evaluation of the obtained segmentations. We propose the following framework to evaluate a mesh sequence segmentation method.

Optimal Segmentation

The optimal segmentation of a mesh animation (dynamic mesh or unconstrained mesh sequence) into rigid components can be guessed when the motion is known. This is, for instance, the case with skeleton-based mesh animations, as created in the computer graphics industry. In this case, each mesh vertex of the sequence is attached to at least one (usually, no more than 4) joints of the animation skeleton, with given weights called skinning weights. These joints are organized in a hierarchy, which is represented by the bones of the skeleton that are, therefore, directed. For our evaluation, we attach each vertex to only one joint among the related joints, the furthest in the hierarchy from the root joint. If this joint is not unique, we keep the one with the greatest skinning weight. Each joint has its own motion, but several joints can move together in a rigid manner.

For a given frame, we can therefore cluster joints of the animation skeleton into joint sets, each joint set representing a dierent motion. We now dene as an optimal cluster the set of vertices related to joints in the same joint set. Since we know the motion of each joint, we exactly know, for each frame, what are the optimal clusters. This denition can be applied in the general case of unconstrained mesh sequences, provided that each vertex of each frame can be attached to a joint.

However, we only tested it in the more convenient case of a dynamic mesh.

Error Criteria

We propose the following criteria in order to evaluate a given segmentation with respect to the previously dened optimal segmentation:

Frame Assignment Error (FAE): for a given frame, the ratio of vertices which are not assigned to the correct cluster. This includes the case of clusters which are not created, or which are wrongly created;

Vertex Assignment Error (VAE, in the case of dynamic meshes): for a given vertex, the ratio of frames in which the vertex is not assigned to the correct cluster.

Results

We tested our algorithm on a walking cat skeleton-based animation (see Figure 7 and the accompanying video). We get a time-varying segmentation with a FAE up to 17%, for the worst frame. Wrongly assigned vertices correspond to the cat skin around joints and to a wrong subdivision in cat paw, i.e. in the less rigid areas.

In the case of global segmentations, and if we do not take into account matching issues, the FAE is the same for all frames. For the cat sequence, the FAE is also 17%. The VAE varies between 0% and 100%, and is only relevant as a relative criterion to compare vertices. On the cat sequence vertices in highly rigid areas (paws, tail, body) are always assigned to the correct cluster, while some vertices around joints can be assigned to the same neighboring cluster for all frames. 6.3 Discussion

User Dened Parameters

We use three parameters to drive the segmentation: the number of clusters, the minimum cluster size and the merge threshold. The eigen gap method is used for each frame to determine the number of clusters to create, but is not always reliable. To overcome this problem and to avoid the creation of many subclusters, we use the minimum cluster size parameter. In all of our experiments, we have arbitrarily set this parameter to 5%. This value works well enough for reconstructed mesh sequence as meshes in these sequences are uniform. This threshold is used for instance on the balloon sequence to ensure that the ball is not split into many subclusters as it is hard to nd a rigid transformation on the ball. It improves computation time but also avoids small cluster creation around articulations, which are usually not rigid. Finally, the merge threshold is used to decide if two clusters represent the same motion (see section 5.2). It is closely related to the eigen gap value, since if these thresholds are not set correctly, the algorithm can merge two clusters which are split by the spectral clustering right afterwards.

Distance Function

We have tried our algorithm with three distance functions to compute distances between rigid transformations:

1. Residual errors between clusters; 2. Frobenius norm;

3. Logarithm of matrices (see Equation 1).

In our experiments, the residual errors between clusters are really dicult to set correctly. The logarithm of matrices provides the best result but if the computation time is a hard constraint, then the Frobenius norm yields also quite good results. Actually on dynamic meshes, both yield the same results. However, on unconstrained mesh sequences logarithm based distance yields better results, i.e. more regular clusters, than the Frobenius distance.

Current Limitations

We are currently aware of three limitations in the proposed algorithm:

our method clearly depends on the the quality of the matching process.

Important errors in matching computation can lead to wrong results; clusters which are wrongly subdivided are transferred to the following frames, meaning that errors on an early frame in the sequence can aect the whole segmentation. This issue is less important on time-varying segmentation since clusters will be merged later. Nevertheless, on global segmentations, it can lead to wrong results. Such errors are generally due to error in the matching process; segmentation can slightly drift: this is due to the fact that we only consider 2 frames when matching.

As shown in our quantitative evaluations, vertices which are wrongly assigned to a cluster are located near articulations. Vertices in rigid regions are generally correctly clustered.

Conclusion

We have presented a segmentation method which takes as input a mesh sequence, either reconstructed from multi-view videos or created using an animation software. It produces a global segmentation or a time-varying segmentation into rigid components and handles topological changes. The method is fast and yields visually consistent results. It uses a few parameters which can be easily set. We have also proposed a framework for quantitative evaluations of rigid segmentation methods, and we have shown that our method behaves favorably with respect to existing approaches. Future works include the use of a bayesian framework to track and predict clusters. We also plan to improve the vertex assignments around articulations to clusters and optimize our implementation. 8

Figure 1 :

 1 Figure 1: Frame examples from a multi-cam unconstrained mesh sequence (balloon sequence [Bal11].

Figure 3 :

 3 Figure 3: First row: time-varying segmentation generated by our algorithm on the dancer sequence [SH07]. First frames are decomposed into 6 segments, then the right arm and right hand clusters are merged since they move the same way. Finally, this cluster is split again. Note that we can handle topology changes (in the last frames, the left arm is connected to the body). Second row : time-varying segmentation of a sequence with 15,000 vertices per mesh.

Figure 4 :

 4 Figure 4: Result of vertex matching. (a) Full display. (b) Partial display.

Figure 5 :

 5 Figure 5: Segmentation results on a dynamic mesh.

Figure 6 :

 6 Figure 6: Global segmentation result on the balloon sequence.

Figures 3

 3 Figures 3 shows time-varying segmentations computed on the Balloon and Dancer sequences. Figure 6 shows a global segmentation computed on the Dancer sequence. By construction, global segmentations contains more clusters than time-varying segmentations since no merging process is applied. Thresholds for both time varying and global segmentation of the Dancer sequence share the same value, except for the eigen gap threshold which is slightly lower in the time varying segmentation case. In general, threshold values for a sequence are easily found in a few trials. The computation time of the segmentation between 2 frames of the Dancer sequence is approximately 3 minutes with a (not optimized) Matlab implementation. Additional results appear in the accompanying video.

Figure 7 :

 7 Figure 7: Result on a skeleton-based synthetic animation. (a) Computed timevarying segmentation. (b) Optimal time-varying segmentation, for the same frame.

  Distance Function . . . . . . . . . . . . . . . . . . . . . . 13 6.3.3 Current Limitations . . . . . . . . . . . . . . . . . . . . .

	6.3.1	User Dened Parameters	. . . . . . . . . . . . . . . . . .	13
	6.3.2
				14
	7 Conclusion		14
	8 Acknowledgements		14
	RR n°7856

RESEARCH CENTRE GRENOBLE -RHÔNE-ALPES

Inovallée 655 avenue de l'Europe Montbonnot 38334 Saint Ismier Cedex Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr

RR n°7856

Acknowledgements

The Dancer sequence is courtesy of University of Surrey [SH07]. The Balloon sequence is courtesy of Inria Grenoble [Bal11]. The Cat sequence is courtesy of Inria Grenoble [AHLD07].

Inria


M L Hattali
email: hattali.lamine@gmail.com

J Barés

L Ponson
email: cponson@caltech.edu

D Bonamy

LOW VELOCITY SURFACE FRACTURE PATTERNS IN BRITTLE MATERIAL: A NEWLY EVIDENCED MECHANICAL INSTABILITY

Keywords: Fracture; mechanical instability, fragmentation, brittle polymer glass

The occurrence of various instabilities at very high speed is well known to occur in brittle fracture and significant advances have recently been obtained in the understanding of their origin. On the other hand, low speed brittle crack propagation under pure tension loading (mode I) is usually thought to yield smooth crack surfaces. The experimental investigation reported here questions this statement. Steady cracks were driven in brittle glassy polymers (PolyMethyl Methacrylate -PMMA) using a wedge-splitting geometry over a wide range of low velocities (10 -9 -10 -1 m/s). Three distinct patterns can be observed on the post-mortem fracture surfaces as crack velocity decreases: perfectly smooth at the highest speed, regularly fragmented at intermediate speed and macroscopically rough at the lowest speed. The transition between the two latter is reminiscent of chaotic transition.

Introduction

Rupture problems are encountered in a variety of situations: Earthquakes, Damage in biological tissues caused by disease, and aircracft or space structure, civil engineering…. In brittle materials such as ceramics, rocks and glass, crack propagation is the fundamental mechanism leading to material failure. As such, its understanding and modelling have attracted much attention from engineers and physicists (see. e.g. [START_REF] Lawn | Fracture of Brittle Solids[END_REF][START_REF] Freund | Dynamic Fracture Mechanics[END_REF][START_REF] Ravi-Chandar | Dynamic Fracture[END_REF] for reviews in this topic).

In this respect, quantitative fractography, i.e. examination and analysis of the fracture is a powerful tool to provide information on the fracture processes that have occurred during the fracture of the tested materials (see [START_REF] Hull D Fractography | Observing, Measuring and Interpreting Fracture Surface Topography[END_REF] for a review): Statistical analysis of the small scale roughness can e.g. be used to infer fracture process process zone or crack growth direction (see [START_REF] Bonamy | [END_REF] for review); Wallner lines can be used to infer crack velocity [START_REF] Richter | Stress Wave Fractography, In Fractography of Glass[END_REF][START_REF] Bonamy | [END_REF], conic markings observed in Polymethyl methacrylate (PMMA) fracture surfaces point out the presence of dynamic damaging through microcracking [8,9], etc.

The occurrence of various instabilities in tensile crack propagation at very high speeds is well known [10,11,12], on the other hand, it is usually though that slow crack growth in nominally brittle materials are well described by Linear Elastic Fracture Mechanisms, without any instabilities [13]. The experimental investigation reported here questions this statement. In this paper, we report steady state pure mode I fracture experiments in PMMA, the archetype of brittle amorphous materials and observe on the post-mortem fracture surfaces experimental evidence of fragmentation instabilities at low velocities.

Experimental procedure

Wedge Splitting gemeotry [14] was chosen to drive the cracks in PMMA since this geometry allows (i) stable crack growth; (ii) pure tension loading and (iii) access to a wide range of velocities. A schematic representation of the system studied is shown in figure 1a. The idea consists in pushing a steel wedge in a groove (25 x 25 x 15 mm 3 ) cut of the specimen (140 x 125 x 15 mm 3 ). To minimize friction between wedge and lateral support, two steel jaws equipped with roller bearings are placed between the wedge and the groove. Constant velocity (10 -9 to 10 -4 m/s) is then applied to the wedge and hence allow to load the specimen in tension. A stable crack is then induced and growth stably and slowly (at a velocity proportional to the wedge velocity) in the specimen, up to the end of this one.

The location of the crack tip as a function of time is determined using a camera Ueye. Image processing gives the position of the crack tip and consequently the instantaneous velocity ν and its variation with crack length. Postmortem analysis of crack surfaces are scanned via a Bruker mechanical profilometer (stylus tip radius 2 µm, out-of-plane resolution 0.1 nm) as shown in figure 1b. Our topographical maps h (x; z) correspond to 14x15 mm 2 scans. Each one consists of a series of 999 lines parallel to the front direction Oy, each line containing 324 data points. Three distinct patterns can be observed on the post-mortem fracture surfaces: Perfectly smooth, regularly fragmented (Fig. 3a) and macroscopically rough (Fig. 3b). The observation of one or the other of these patterns depends essentially on the crack velocity. The usual surface fracture pattern of PMMA at crack tip velocity is featureless and smooth to the naked eye, tough sometimes vividly hued. Below a critical velocity V c ~ 3.5-4 cm/s, we observe a transition from this mirror-smooth pattern to one with almost regularly spaced fragments (Fig. 2e). The two patterns are separated on the fracture surface by an arrest line that suggest important acceleration/deceleration in the crack velocity. As the crack tip velocities continue to decrease and become smaller than 4 µm/s, a second transition is noted and the regularly fragmented fracture pattern becomes a rough one (Fig. 2a). As for the first transition, this fractographic change is accompanied by a sudden drop in the crack velocity, from 1.5 10 -5 m/s to 4.10 -6 m/s, although the wedge speed was maintained constant over the whole fracture test (Fig. 4).

Figure 1 :

 1 Figure 1: Schematic representation of the experimental apparatus. (a) The wedge-splitting geometry; Principe of applying splitting load and velocity measurements. The dashed line indicates the path of the crack. (b) Surface profile measurements.

Figure 2 Figure 2 :

 22 Figure2shows the postmortem fracture surface of PMMA for various wedge speeds. Light gold coatings were employed to provide better reflection from the otherwise transparent surfaces.

Figure 3 :Figure 4 :

 34 Figure 3: Surface fracture pattern realized with the help of a mechanical profilometer 3D (stylus tip radius 2 µm, out-of-plane resolution 0.1 nm) a) Fragmented surface pattern b) Rough surface pattern.