Virginie Zampa
email: virginie.zampa@u-grenoble3.fr

Philippe Dessus
email: philippe.dessus@umpf-grenoble.fr

Validating a Computer-based Tutor that Promotes Self-Regulated Writing-to-Learn

Keywords: Latent Semantic Analysis, Self-regulated Learning, Computer-based Feedback

In this chapter, we introduce in this chapter Apex 2.1 (i.e., Assistant for Preparing EXams), a prototype system that provides automated feedback to e-learning students on the summaries they wrote about the courses they attended. At first we present some theoretical underpinnings on how the system has been designed then we detail Apex 2.1 architecture. Eventually, the first results of a validation study involving three groups of stakeholders (students, teacher, administrator) are presented. The utility, acceptability and usability of the system are examined.

INTRODUCTION

This chapter is devoted to the presentation of Apex 2.1 (an Assistant for Preparing EXams), a prototype system that delivers automated pieces of feedback to e-learning students on the summaries they wrote about the courses they attended. Apex 2.1 first enables students to choose a subject content they want to study, from a query composed of one or more key-word(s). Then the system delivers a set of documents semantically close to this query. At any time the students can write out a summary of the document they have read and understood and can get an automated feedback on this summary. If they do so, students can revise their summaries and enter in a self-regulated workflow:

reading-writing-feedback-revision, which could lead to a better understanding of the chosen subject content.

The remainder of this chapter is as follows. We first introduce the theoretical underpinnings on which this system is based, as well as its architecture and the semantic analysis method (Latent Semantic Analysis) on top of which it is built. We eventually detail its first usability evaluation study upon three different categories of users.

WRITING TO LEARN

In a nutshell, the design of Apex 2.1 starts with the assumption that students should write to learn and that the better their self-reflection (and self-assessment) on their learning is, the better their learning will be. Elaborating more on this assumption will lead to a better explanation on the possible use of Apex 2.1 in an e-learning situation.

According to the so-called "writing-to-learn" approach, the activity of writing yields to learning, in exploring and emphasizing the relations between ideas ("strong text" view, [START_REF] Emig | Writing as a mode of learning[END_REF], due to very close similarities between writing and thinking. Further research [START_REF] Klein | Reopening inquiry into cognitive processes in writing-to-learn[END_REF] showed that the writing-to-learn approach actually shares four different views and hypotheses:

-the spontaneous writing hypothesis, in which writers generate knowledge de facto, without any planning or revision processes (close to the knowledge telling process, [START_REF] Bereiter | The psychology of written composition[END_REF];

-the forward search hypothesis, in which ideas are externalized in texts and new ones are then inferred through re-reading;

-the backward search hypothesis, claiming that writers first have rhetorical goals from which ideas and arguments derive, and thereby writers learn;

-the genre writing hypothesis, suggesting that writers use their knowledge of genre structure to create and analyse relationships between elements of texts, which in turn leads to learning.

These different hypotheses highlight the idea that writing is a self-regulated activity with multiple goals, whose account cannot be constrained, and that e-learning settings make this assumption more crucial. In such settings, feedback to students' learning cannot be as frequent as in presence, and students have to cope with their isolation from each other and to engage in self-regulated activities. Research has been developed on self-regulation of learning in a distance learning context, and Vovides et al. (2007) devised a model on this activity. First, students work on the object level, in preparing their activity according to the ongoing task. They can also apply some cognitive strategies for carrying out these activities. Then, students perform a first draft assessment of their production (its adequacy, its relation to their knowledge, etc.).

Thirdly, a reflection on a meta level allows them to perform a comparison between the latter comparison and the object level, often offered by artifacts (computer-based services, prompts, etc.), so as to compare their perceived level of learning with proposed one by the artifacts. Eventually, the students can perform some adaptations to their work, which in turn fuels the possible update of the object level and can be reacted in a further loop.

Figure 1 -A Model of Self-Regulated Learning in e-learning (after Vovides, et al., 2007, p. 68).

We claim that Apex 2.1 enables students to have a deep understanding of the course they learn without (at least immediate) support from their teachers or tutors. This software belongs to a new line of e-learning services allowing their users to be freely engaged in workflows from which they can easily move to another one and which fosters their self-regulation. It is worth noting that both teachers and tutors benefit from Apex's use as well, by providing an overall view of students' activity and understanding. Thus they can focus on higher-level or more individualized student support. Since the use of this kind of services is new, our aim is to collect fine-grained evidence of their usage, and views on the way students get acquainted with them.

SYSTEM DESCRIPTION

Apex 2.1 has already been subject to a first implementation [START_REF] Dessus | Using production to assess learning: An ILE that fosters Self-Regulated Learning[END_REF], but no evaluation in real-world settings has been undertaken so far. Apex 2.1 is written in PHP and C, and built on top of Latent Semantic Analysis [START_REF] Landauer | A solution to Plato's problem: the Latent Semantic Analysis theory of acquisition, induction and representation of knowledge[END_REF], a statistical method introduced in the next section. A running version of this service is available at http://augur.wu-wien.ac.at/apex2/Recovery/progPhp/Apex2.php.

SYSTEM ARCHITECTURE

The way Apex 2.1 works is depicted in Figure 2. Two main loops (in yellow or grey) are successively carried out: the first one is a reading loop, allowing students to read a set of texts semantically related to a query, the second one is a writing loop, in which students write out a summary from the source texts previously meant as understood, and can get feedback on this understanding. Let us explicate the reading loop. A session typically starts when a student types a query, i.e., some key-words related to the subject domain or the theme he or she wants to be acquainted to (see Figure 3). Then Apex 2.1 displays the student a first text to read, Once the text has been read, the student indicates whether it is understood or not, and whether he or she is able to summarize it (see Figure 4). During this step, the learner model is updated, indicating whether the text is summarizable. Apex 2.1 gives texts to read as long as there is no summarizable text yet and texts sufficiently close to the query but not delivered yet. On the contrary, Apex 2.1 suggests the student to perform a new query. Once the student has understood at least one text, he or she can either read more texts on the given topic or start a writing loop. In the latter case, a table with the list of all the summarizable texts is displayed (Figure 5). The first column indicates the reading order of each text, the second column its ID number. The third column displays the first lines of each text read so far, which can be displayed as a whole by clicking on the + button.

The "summarize" (résumer) button of the fourth column gives access to the writing zone (displaying the latest version of the summary, if it exists, or an empty field if not).

Eventually, the last two columns respectively display the semantic proximity between the summary and the corresponding course text, as computed by LSA, and the comparison of this value and the student's opinion about his or her own comprehension of the text (i.e., "You said you understood the course text and your summary shows it is actually the case", « Vous pensez avoir compris le texte et votre résumé le prouve »). ID, the "already-presented" identifier (0 or 1), the "understood" identifier (0 or 1), the proximity value (i.e., the LSA-based semantic comparison between the source-text and the corresponding student's summary). This text file is updated for every student's move and is re-initialized for every new query.

The student, after any feedback request, can perform one of the following actions:

-revise the summary and ask for a new feedback, since it is available for modification;

-go back to the table with so far read texts (see figure 5) and start with a new summary, or modify one already existing;

-read a new course text;

-perform a new query;

-quit Apex 2.1.

LATENT SEMANTIC ANALYSIS

Latent Semantic Analysis (LSA, [START_REF] Landauer | A solution to Plato's problem: the Latent Semantic Analysis theory of acquisition, induction and representation of knowledge[END_REF]) is a well-known technique that captures semantic information in texts by uncovering word usage regularities.

Extensive research has proven its efficiency in the domain of natural language processing, and more specifically for computer-based instruction-tutoring systems, interactive learning environments [START_REF] Dessus | An overview of LSA-based systems for supporting learning and teaching[END_REF]. LSA represents the pieces of text to be analysed (e.g., text courses, students' writings) in a multidimensional space. The processing steps are the following. LSA takes as input a large set of texts and a wordparagraph matrix of co-occurences is firstly built, then its dimension is reduced (to about 300 dimensions) by means of a statistical procedure. This reduction enables the emergence of semantic relations between words, paragraphs or texts. Thanks to this method two words can be considered as semantically close to each other, if they appear in similar contexts (i.e., sentences, paragraphs, texts), even if hey actually never appeared in the same one. However, this method works best only if a sufficiently large corpus of words is processed (i.e., multi-million word large corpus).

Apex 2.1 is built on top of LSA, which is used for providing the most adequate texts to the students (reading loop), and also to assess to what extent students' summaries are semantically close to the course texts they refer to (writing loop). Initially, we determined an arbitrary threshold value (0.6), above which the selected texts are sufficiently close to the query (reading loop). With regard to the writing loop, the same threshold has been used, and the following prompts can be displayed: "You said you understood this text and obviously you did" when the semantic proximity between the source text and its summary is above the threshold, and "You said you understood this text, but obviously you didn't" on the contrary.

SYSTEM VALIDATION

The validation of Apex 2.1 is based on a corpus of sixty-six course texts (i.e. course notes or research articles) on natural language processing (NLP) and/or computer assisted language learning (CALL). These texts are homogenous regarding their length in words (an average of 650 words each). The texts provided in the reading loop were taken from this corpus. We added a larger and more general corpus to this base, so as to integrate general knowledge about language as well. This "general" corpus contains 101,123 different forms. It is formed from various texts representing several types, styles and vocabularies (articles from the daily newspaper Le Monde, texts written by children, texts written for children) and has been subject to validations [START_REF] Denhière | A semantic space for modeling children's semantic memory[END_REF].

In order to test Apex 2.1 in conditions as close as possible to e-learning settings, we defined three groups of participants.

-a "user" group, consisting of 11 masters degree or PhD students in NLP and/or CALL at our university. This group used Apex 2.1 within the same conditions as e-learning students.

-a "demo" group, consisting of 28 bachelor degree students in educational sciences at our university. This group was given a demonstration of a standard learning session using Apex 2.1 and answered a questionnaire afterwards.

-a "teacher/administrator" group, consisting of two persons: the manager of the pedagogical ICT department at the IUFM (i.e., "Institut universitaire de formation des maîtres", or Teacher Training Institute) of Grenoble university, and the manager of digital workspaces in the same institution. This group followed the same tasks as the previous one (demo).

The reason why the last two groups have been enrolled (those not testing Apex 2.1 practically) was to evaluate the degree to which such a tool is acceptable to potential groups of users and prescribers. In brief, the participants of the first two groups, as learners, performed an empirical "evaluation" task whereas the participants of the latest group performed an "inspection" task, i.e., a prescriptive analysis of the possible uses of the system.

PARTICIPANTS TASK DESCRIPTION

The "user" group participants were of an average age of 24.6 years old (standard deviation 4.26) and were distributed as follows: five first year master students, two second year master students and four PhD students. Before starting the experiment, they were asked to acknowledge the protocol and to sign it, and then they had to fill in a questionnaire about their course revision methods. Afterwards, they used the software by entering keywords of their own choice. They had to summarize five texts of their choice (that they had read and that they considered they understood), without any time limit. Their usage trace on the software was recorded (e.g., chosen keywords, number of searches, texts read, time taken to read each text, texts understood or not, duration of each loop, etc.). At the end of this task, they were asked to fill in a second questionnaire regarding their opinion on the software. The whole set of documents given to this group is available at http://augur.wu-wien.ac.at/apex2/Expe/groupe-utilisateur.pdf.

The "demo" group participants (average age: 23.4 years old, SD: 4.5) had to fill in the same questionnaire as the user group regarding their revision methods. Then they were shown a screencast demonstration of Apex 2.1 and answered a second questionnaire stating their opinion about Apex 2.1. The questionnaire is available at http://augur.wu-wien.ac.at/apex2/Expe/groupe-demo.pdf.

The "teacher/administrator" group consisted of two persons, one aged 42, the other one aged 60. They followed a demonstration of Apex 2.1 and answered a questionnaire available at http://augur.wu-wien.ac.at/apex2/Expe/groupe-admin.pdf.

HYPOTHESES

We defined the following hypotheses, based on criteria of utility, usability and acceptability for CALL [START_REF] Bétrancourt | L'ergonomie des TICE : quelles recherches pour quels usages sur le terrain[END_REF][START_REF] Tricot | Utilité, utilisabilité, acceptabilité : interpréter les relations entre trois dimensions de[END_REF]:

a) The utility hypothesis assumes that the use of the system induces a benefit for the user (time-wise, interest-wise, etc.). It can be split in two sub-hypotheses:

-Query-reading hypothesis: Apex 2.1 can be used as a search engine to find texts relevant to the theme chosen by the user. The texts have been provided by teachers; they are thus appropriate and from reliable sources. Hence the system allows students to read texts both relevant and understandable to them.

-Self-regulation/evaluation hypothesis: Students can freely enter each loop (queryreading and writing-evaluation), which allows them to regulate their learning better; furthermore, Apex 2.1 provides them with feedback on their validated summaries.

b) The usability hypothesis, which corresponds to Apex 2.1's handiness, refers to browsing and the interface. The aim is to evaluate, from an ergonomic point of view, the ease with which the system can be used and with which it can help the users achieve the goals they have determined.

c) The acceptability hypothesis corresponds to the "value of the intellectual representation (attitudes, opinions, etc. whether positive or not) of a system, its utility and its usability1 " (Tricot, et al., 2003, p. 396). This corresponds to a general point of view about usage demands induced by the system itself; in our case, this corresponds to knowing whether students find Apex 2.1 easy to use and whether they would use it for their own work if they were given the opportunity.

RESULTS

In this chapter we present the results for all three participant groups. The previous hypotheses have only been studied in depth for the "users" group.

THE "USERS" GROUP

As noted previously, we gathered two types of data from the "users" group: their traces (users' behaviour in terms of queries, readings and writings) and their answers to the questionnaires. The analysis of the traces has shown the following points.

Utility/Query-reading hypothesis. 7 users out of 11 only carried out only one query (see Table 1), and thus worked on the same theme for all texts, two carried out two queries, one carried out three and the last one four (average number of queries: 1.64, SD 1,03).

Regarding reading, all 11 participants read between 5 and 13 texts (average 8, SD 3.46).

However, we should note that most texts judged as "not understood" and thus not summarized were only skimmed through. Out of 33 texts, 5 were read in more than 2 minutes and the average reading time for the 28 others is 30 seconds. It also seems that, when participants indicate whether they have understood the text and are capable of summarizing it, they give an appreciation of interest rather than an evaluation of their understanding. If the text is of some interest to them, they read it and summarize it, otherwise they skim through the text and turn to the next one. Apex 2.1 would then be used to select texts corresponding to the user's expectations, which validates our first hypothesis.

Utility/Self-regulation-evaluation hypothesis. The following data only refer to read and summarized texts. On average, the inter-participant time taken to read a text varies between 3'24" and 14'19" (average 8'2"; SD 2'50"). This inter-participant variation is thus very high. Regarding the intra-participant variation, the standard deviation on reading time ranges from 41" to 5'52", with an average of 2'40". All participants followed the instructions and wrote at least five summaries, taking from 1'49" to 27'40 to write them down (average writing time for each text, 7'5", SD 4'16"). The average summaries evaluation was 0.73; only 6 out of the 55 summaries had a grade below the threshold (0.6) and 3 out of the 6 corresponded to a single participant. Only one summary was rewritten. The students thus did not modify their summary after obtaining their evaluation. They did not take the feedback into account and did not try to improve them. This can probably be attributed to the fact that most feedback indicated that the summaries were correct, which was enough for them. More detailed feedback, rather than simply a correct/incorrect evaluation, would have certainly had more impact.

Table 1 -General data on reading activities using Apex Regarding the analysis of both questionnaires, the first questionnaire referred mainly to using computers for course revision. Most participants mentioned they supplemented their courses with documents found on the Internet (7 out of 11 on a "weekly" or "nearly daily" basis). Furthermore, they simultaneously used pen-and-paper and computer to revise (8 out of 11 distributed between "nearly every day" and "several times a day"). Hence, the "users" participants are used to revising with their computer.

This first questionnaire allowed us to verify that they are used to working on a computer and thus any problems of usability would be directly attributable to Apex 2.1 and not to lack of expertise in browsing, note taking or computer searches. The second questionnaire referred to Apex 2.1 usability and was divided in two parts.

Usability hypothesis. In the first part, named "usage difficulties" and elaborated from the NASA-TLX test [START_REF] Hart | Development of NASA-TLX (Task Load Index): Results of empirical and theoretical research[END_REF], we note that for the majority of users, Apex 2.1:

-does not lead to any physical pressure, -leads to a pressure related to the experiment itself, -is easy to use.

In the second part, named "functions of Apex 2.1", we can note that, for the majority of the participants:

-the texts provided corresponded to the query, -the texts provided were suitable, precise, etc.

This confirms our second hypothesis. Finally, most participants would use Apex 2.1 from time to time if accessible, which confirms our acceptability hypothesis.

THE "DEMO" GROUP

The "demo" group answered a two-part questionnaire. Regarding their use of computers to revise courses, participants supplemented their course notes with documents found on the Internet less frequently (13 out of 28 "around once a week" and 9 out of 28 "around once a month") and used more the paper-and-pen method on its own to revise (15 out of 28 "nearly every day"). Their answers on computer and paper-and-pen simultaneous usage were: 1 "never", 9 "once a month", 10 "once a week", 6 "nearly every day", 2 "several times a day". Regarding Apex 2.1's functions, most participants think that Apex 2.1 could help them acquire knowledge as easily as, or even more easily than, revision methods (19 answers, 11 "more easily", 8 "as easily"). Thus, the acquired knowledge is considered to be accurate (for 15 participants out of 28) and wide (14 out of 28) with their usual method. Finally, if available, 20 out of them would use Apex 2.1 from time to time for course revision purposes.

THE "TEACHER/ADMINISTRATOR" GROUP

Both participants in this group judged that using Apex 2.1 could be an easier way to acquire new knowledge, compared to traditional methods (paper-pen), and furthermore that the new knowledge would be more precise. Both would suggest a frequent use of the tool to their students. They considered that Apex 2.1 would provide students with better focus on the information required to learn course content. The administrator pointed out that the software gives students access to a wider range of information but remains easy to use.

SUMMARY OF RESULTS

Carrying out this study with three groups has given us a twofold evaluation: an "inspection evaluation" with the "teacher/administrator" group and a more "empirical" evaluation with the other two, with criteria of utility, usability and acceptability.

Regarding the inspection evaluation, both teachers/administrators consider that Apex 2.1 fulfils all three criteria. Regarding the empirical evaluation, the "demo" group validates all three criteria. On the other hand, concerning the "users" group, the utility hypothesis regarding texts evaluation is not completely validated. When summaries are judged correct, participants do not try to improve them, and when judged incorrect, participants did not reassess them. However, the second part of the hypothesis has been validated, as well as both other hypotheses.

CONCLUSION

The purpose of this chapter was to describe the underlying principles and a first validation of Apex 2.1. The software, dedicated to exam preparation, provides different uses to the learner. Firstly, Apex 2.1 can be used as a text database with an integrated search engine. The advantage of this use, compared to an Internet search engine, lies in the fact that the text base has been built by the teacher and therefore only contains suitable texts (which avoids wasting time in Internet searches), and furthermore only contains reliable texts (which is not necessarily the case on the Internet). The search engine, based on LSA, allows the partial reduction of problems related to polysemy, synonymy and inflections [START_REF] Landauer | A solution to Plato's problem: the Latent Semantic Analysis theory of acquisition, induction and representation of knowledge[END_REF].

Apex 2.1 has been designed to help learners acquire knowledge through reading and writing (the "write-to-learn" approach). The advantage is to bring a flexible approach where the user evolves according to his or her wishes. Furthermore, he or she brings feedbacks to "real" contents, rather than answers to Multiple Choice Question papers or other closed exercises.

During the validation process, we have noticed that participants select the texts they wish to read and summarize. Most texts that they indicate that they are unable to summarize are texts that they have not read thoroughly. We also noticed that most participants indicate that Apex 2.1 is easy to use and that they would use it from time to time if available. This therefore supports the continuing development of this software.

This version contains some limitations. The main one is the lack of feedback refinement after the evaluation. Indicating to the learners that they seem to have understood the text does not encourage them to try and improve their summaries. On the other hand, indicating that the text has not been understood without giving any suggestions is not constructive. For these reasons, in the next version currently under development, we aim at remedying these problems and allowing different kinds of feedback such as:

coherence within the summary (so as to detect breaks in coherence); -links between summary and source-text (so as to indicate the user sentences that might be off the point, or even those that can be reused in the summary).

This will then allow users to know on which points to work again in order to improve their summary or synthesis. The feedback would then improve both summary writing techniques (e.g., concentrating on inter-paragraph coherence), but also on the content (thanks to the emphasis on semantic links between source-texts and syntheses).

Furthermore, in order to improve their appropriation of texts they have read, the learners will be able to highlight important sentences and take notes in the dedicated notepad. Finally, they will be able to keep a history of the different versions of their summaries and the associated notes and comments.

This article shows that it is possible to provide e-learning students with a tool that automatically assesses some semantic properties of their written production, and thus their understanding. This tool, though imperfect yet, has been positively appreciated by three categories of potential users.

l'évaluation des EIAH. In C. Desmoulins, P. Marquet & D. Bouhineau (Eds.), Actes de la conférence EIAH 2003 (pp. 391-402). Paris: INRP/ATIEF.

Vovides, Y., Sanchez-Alonso, S., Mitropoulou, V., & Nickmans, G. (2007). The use of e-learning course management systems to support learning strategies and to improve self-regulated learning. Educational Research Review, 2(1), 64-74.

Figure 2 -

 2 Figure 2 -The way Apex 2.1 functions. User-machine interactions.

  which is the closest text to the query. All the texts are presented to the student successively and are the closest to the query possible, as indicated by LSA. The learner model is updated accordingly, indicating that a given text has been actually presented and read.

Figure 3 -

 3 Figure 3 -A message starting the reading loop, asking the student to type some keywords: "What is the course topic you want to read?"

Figure 4 -

 4 Figure 4 -The interface for reading course texts. Once the text has been read, the student can assess his or her own comprehension: "I can summarize the text" or "I cannot summarize the text".

Figure 5 -

 5 Figure 5 -Overall view of the texts read so far, displayed at the beginning of the writing loop.

Free translation of « la valeur de la représentation mentale (attitudes, opinions, etc. plus ou moins positives) à propos d'un système, de son utilité et de son utilisabilité »(Tricot, et al.,

2003, p. p. 396)

Note from the authors

This work is partly supported by the LTfLL (Language Technologies for Lifelong Learning) research project, FP 7, ICT-STREP. We wish to thank Thomas Lebarbé and Lucy Garnier for translating a part of the paper, the participants of the validation study, as well as the MSH-Alpes, Grenoble for providing material support.


Peijiang Liu
email: pjliu@cse.buaa.edu.cn

Yunhong Wang
email: yhwang@buaa.edu.cn

Member, IEEE Di Huang
email: dhuang@buaa.edu.cn

Member, IEEE Zhaoxiang Zhang
email: zxzhang@buaa.edu.cn

Member, IEEE Liming Chen
email: liming.chen@ec-lyon.fr

Learning the Spherical Harmonic Features for 3D Face Recognition

Keywords: 3D face recognition, Spherical Depth Map, spherical harmonics, feature selection

In this paper, a competitive method for 3D face recognition (FR) using Spherical Harmonic Features (SHF) is proposed. With this solution 3D face models are characterized by the energies contained in spherical harmonics with different frequencies, thereby enabling the capture of both gross shape and fine surface details of a 3D facial surface. This is in clear contrast to most 3D FR techniques which are either holistic or feature-based using local features extracted from distinctive points. First, 3D face models are represented in a canonical representation, namely Spherical Depth Map (SDM), by which SHF can be calculated. Then, considering the predictive contribution of each SHF feature, especially in the presence of facial expression and occlusion, feature selection methods are used to improve the predictive performance and provide faster and more cost-effective predictors. Experiments have been carried out on three public 3D face datasets, namely SHREC2007, FRGC v2.0 and Bosphorus, having increasing difficulties in terms of facial expression, pose and occlusion, and which demonstrate the effectiveness of the proposed method.

I. INTRODUCTION

The face is a natural and non-intrusive biometrics which can be easily acquired contactlessly.

Unfortunately, despite the great strides made over the last thirty years, face recognition (FR) in 2D is still a great challenge due to factors as diverse as illumination variations, pose changes and facial expressions [START_REF] Phillips | Overview of the face recognition grand challenge[END_REF]. With the advent of readily available 3D scanners, there has been an increasing interest in using 3D data instead of, or in addition to 2D images to overcome the limitations in 2D FR, in particular viewpoint and lighting variations [START_REF] Pears | From 3D point clouds to pose-normalised depth maps[END_REF], and to achieve better recognition accuracy than 2D FR [START_REF] Akarun | 3D face recognition for biometric applications[END_REF], [START_REF] Bowyer | A survey of approaches and challenges in 3D and multi-modal 3D + 2D face recognition[END_REF], [START_REF] Gokberk | 3D face recognition: Technology and applications[END_REF]. As 3D FR operates on 3D face scans delivered by 3D sensors and they are supposed to convey the accurate shape information of facial surfaces, 3D FR needs to address two kinds of challenges in order to meet real-world application requirements, namely 3D data sensing in unconstrained settings and improved algorithms with high recognition accuracy while displaying strong robustness to various variations and factors of disturbance such as facial expressions, partial face data and pose changes [START_REF] Bowyer | A survey of approaches and challenges in 3D and multi-modal 3D + 2D face recognition[END_REF]. In this paper, we concentrate on the second challenge and propose a competitive method for 3D face recognition using Spherical Harmonic Features.

A. Related work

Face recognition is a pattern recognition problem which requires a proper representation of face data highlighting interclass differences while decreasing intra-class variations in order to facilitate the later stage of matching and recognition. In this subsection, we overview major 3D facial representations and the corresponding matching algorithms. State of the art on feature selection can be found in [START_REF] Guyon | An introduction to variable and feature selection[END_REF], [START_REF] Blum | Selection of relevant features and examples in machine learning[END_REF], [START_REF] Kohavi | Wrappers for feature subset selection[END_REF], [START_REF] Liu | Computational methods of feature selection[END_REF].

1) 3D Face Representations and registration: 3D face data are the scans delivered by 3D sensors which accurately capture the shape information of facial surfaces. As such, 3D FR can theoretically make use of any existing shape representation used in 3D shape retrieval, e.g., Extended Gaussian Image [START_REF] Horn | Extended gaussian images[END_REF], Shape Index [START_REF] Dorai | Cosmos-a representation scheme for 3D free-form objects[END_REF], Spin Image [START_REF] Johnson | Spin-images: A representation for 3-D surface matching[END_REF], Shape Histogram [START_REF] Ankerst | 3D shape histograms for similarity search and classification in spatial databases[END_REF], Shape Distribution [START_REF] Osada | Shape distributions[END_REF], Spherical Harmonic Representation [START_REF] Kazhdan | Rotation invariant spherical harmonic representation of 3D shape descriptors[END_REF], Representation based on surface reconstruction [START_REF] Wendland | Scattered data approximation[END_REF]. Nevertheless, 3D face data has also its own particularities which need to be taken into account. First of all, all 3D face scans are similar in their configuration. As compared to the general issue of 3D shape-based retrieval which performs retrieval of fundamentally different object classes, e.g., car, table, glass, etc., 3D FR performs the recognition within face models of a [START_REF] Jebara | 3D pose estimation and normalization for face recognition[END_REF] DRAFT same class. As a result, a proper 3D face representation needs to highlight the distinctiveness of 3D facial surfaces for a better discriminating power. Moreover, some prior anatomical knowledge of human faces can be utilized to enhance the performance of representation. Besides, 3D face scans as delivered by current 3D sensors are generally partial face models having pose variations and facial expressions.

3D FR algorithms can directly operate on the raw 3D point clouds of 3D face scans or depth maps, also called range images, extracted from the latter in projecting the 3D depth data to a 2D image according to the z-depth of the 3D points. However, a growing number of works in literature have investigated alternative 3D facial representations robust to facial expressions and/or highlighting the distinctiveness of human faces, including the isometric deformations of a facial surface [START_REF] Bronstein | Expression-invariant representations of faces[END_REF], geometry image [START_REF] Kakadiaris | Three-dimensional face recognition in the presence of facial expressions: An annotated deformable model approach[END_REF], indexed facial level curves [START_REF] Samir | An intrinsic framework for analysis of facial surfaces[END_REF], ridge image [START_REF] Mahoor | Face recognition based on 3D ridge images obtained from range data[END_REF],

Signed Shape Difference Map (SSDM) [START_REF] Wang | Robust 3D face recognition by local shape difference boosting[END_REF], Multi-Scale Extended LBP (MS-eLBP) [START_REF] Huang | A novel geometric facial representation based on multi-scale extended local binary patterns[END_REF], to name a few.

Most 3D FR techniques require that 3D face scans be aligned before the generation of a appropriate facial representation and the extraction of facial features for matching [START_REF] Akarun | 3D face recognition for biometric applications[END_REF], [START_REF] Bowyer | A survey of approaches and challenges in 3D and multi-modal 3D + 2D face recognition[END_REF].

Existing techniques on registration still need to be improved regarding their robustness and accuracy. Gokberk et al. [START_REF] Gokberk | 3D shape-based face representation and feature extraction for face recognition[END_REF] reviewed popular algorithms of 3D face registration and recognition based solely on shape information. In the thesis of Jebara [START_REF] Jebara | 3D pose estimation and normalization for face recognition[END_REF], symmetric blob detection, limb extraction and signature analysis were used to locate eyes, mouth and nose respectively. But its success largely depends on anthropometry knowledge. Gordon [START_REF] Gordon | Face recognition based on depth maps and surface curvature[END_REF] used a cylindrical coordinate system to represent depth information under the assumption that the head was vertical. Colbry et al. [START_REF] Colbry | Detection of anchor points for 3D face verification[END_REF] used shape index to detect the anchor points in 3D face scans and achieved 99% success rate in finding the anchor points in frontal images and 86% success rate in scans with large variations in pose and expression. Conde et al. [START_REF] Conde | 3D facial normalization with spin images and influence of range data calculation over face verification[END_REF] successfully applied Spin Image to facial normalization and face verification. Tang et al. [START_REF] Tang | Towards more accurate 3D face registration under the guidance of prior anatomical knowledge on human faces[END_REF] found the reference point and nose line for face registration by computing the symmetry plane of the nose region using prior anatomical knowledge. Pears et al. [START_REF] Pears | From 3D point clouds to pose-normalised depth maps[END_REF] gave an elaborate review of related work and applied radial basis function (RBF) to creating a pose-normalized depth map.

2) Recognition algorithms: 3D FR algorithms can be categorized according to the nature of their matching strategy, either holistic [START_REF] Haar | A 3D face matching framework[END_REF], [START_REF] Samir | An intrinsic framework for analysis of facial surfaces[END_REF], [START_REF] Al-Osaimi | An expression deformation approach to non-rigid 3D face recognition[END_REF], [START_REF] Boehnen | 3D signatures for fast 3D face recognition[END_REF], [START_REF] Xu | Automatic 3D face recognition from depth and intensity gabor features[END_REF], [START_REF] Mahoor | Face recognition based on 3D ridge images obtained from range data[END_REF], [START_REF] Llonch | 3D face recognition with sparse spherical representations[END_REF], [START_REF] Wang | Robust 3D face recognition by local shape difference boosting[END_REF], [START_REF] Berretti | 3D face recognition using isogeodesic stripes[END_REF] or feature-based [START_REF] Huang | Textured 3D face recognition using biological vision-based facial representation and optimized weighted sum function[END_REF] , even though this is highly dependent on the facial features extracted from June 29, 2012 DRAFT a given facial representation. In this section, we only discuss some new state-of-the art 3D FR techniques as a complement to the existing elaborate reviews [START_REF] Akarun | 3D face recognition for biometric applications[END_REF], [START_REF] Bowyer | A survey of approaches and challenges in 3D and multi-modal 3D + 2D face recognition[END_REF], [START_REF] Gokberk | 3D face recognition: Technology and applications[END_REF].

Haar and Veltkamp [START_REF] Haar | A 3D face matching framework[END_REF] propose a 3D face matching framework by combining three curves of 45 face samples in SHREC2007 [START_REF]Shape retrieval contest of 3D face models[END_REF] database and achieve the highest mean average precision (MAP) of 0.78.

Samir et al. [START_REF] Samir | An intrinsic framework for analysis of facial surfaces[END_REF] propose an intrinsic Riemannian framework for the analysis of facial surfaces.

Firstly, they represent facial surface as indexed facial curves derived from a surface distance function. Then, a Riemannian metric denoted the differential geometry of face shapes is achieved for the following purposes: computing the distances between faces, finding optimal deformations between faces, defining and computing the average of a given set of faces.

Al-Osaimi et al. [START_REF] Al-Osaimi | An expression deformation approach to non-rigid 3D face recognition[END_REF] present an automatic approach which can accurately discriminate between expression deformations and interpersonal disparities. An expression deformations pattern is first learnt from training data in PCA eigenvectors, by which they compute the similarity measures between 3D faces and hence recognize faces under any facial expression. The verification rates achieved on the FRGC2.0 database at 0.1% FAR are 98.35% and 97.73% for scans under neutral and non-neutral expressions respectively.

Boehnen et al. [START_REF] Boehnen | 3D signatures for fast 3D face recognition[END_REF] propose a fast face recognition method based on a vector representation named 3D signature whose elements correspond to the fixed surface points in a face-centered coordinates system. The proposed method runs more than three orders of magnitude faster than a traditional ICP-based distance implementation, without sacrificing accuracy. Finally, they achieve an improvement of 0.2% to 0.5% more than the previous highest performing on FRGC2.0 by a new Match Sum Thresholding (MST) fusion approach. Huang et al. [START_REF] Huang | Textured 3D face recognition using biological vision-based facial representation and optimized weighted sum function[END_REF] propose a biological vision-based facial description, namely Perceived Facial Images (PFIs), along with a local feature-based SIFT matching. They achieve 98% rank-one recognition rate on FRGC v2.0 and competitive results on the challenging 3D TEC dataset consisting of 3D scans of 107 pairs of twins [START_REF] Vijayan | Twins 3D face recognition challenge[END_REF].

Holistic matching schemes are based on holistic facial representations which make use of all the facial shape data for discrimination. They hence generally require an accurate normalization step with respect to pose and scale changes. The disadvantage is also that they are vulnerable to occlusions and facial shape deformations, e.g., facial expressions. Now feature-based approaches achieve the matching procedure through local features which are extracted typically from the prominent points on facial surfaces [START_REF] Mian | Keypoint detection and local feature matching for textured 3d face recognition[END_REF], [START_REF] Veltkamp | Shrec '11 track: 3d face models retrieval[END_REF], e.g., curvature extrema. Feature-based matching has the potential advantage of being robust to facial expression, pose and lighting changes and even to partial occlusions. The downside of this scheme is the difficulty in extracting sufficient informative feature points from similar or smooth 3D facial surfaces.

B. Overview of the proposed approach

In this paper, we propose Shape-based Spherical Harmonic Features (SHF) for 3D FR. These SHFs capture both gross shape and fine surface details of facial surfaces through the energies contained in spherical harmonics at different frequencies. They are computed on a canonical facial representation, namely Spherical Depth Map (SDM). Because the predictive contribution of each feature in SHF is different, especially in the presence of facial expressions and possible partial occlusions, we further investigate feature selection methods to improve the predictive performance and provide faster and cost-effective predictors.

Shape-based Spherical Harmonic Features were successfully applied to 3D shape-based interclass retrieval [START_REF] Funkhouser | Search engine for 3D models[END_REF], [START_REF] Kazhdan | Rotation invariant spherical harmonic representation of 3D shape descriptors[END_REF]. When applying SHF to 3D FR which is an intra-class problem, it is necessary to properly address the following two issues. The first is how to reliably calculate SHF for a 3D face scan. Funkhouser et al. [START_REF] Funkhouser | Search engine for 3D models[END_REF] achieves a translation and pose normalized 3D shape descriptor by moving the model so that the center of mass lies at the origin and by discarding phase information. However, such an approach is only suited to the context of wholeobject to whole-object matching [START_REF] Kazhdan | Rotation invariant spherical harmonic representation of 3D shape descriptors[END_REF]. In the context of 3D FR which has to perform partial 3D face matching, the center of mass translation can not be expected to provide optimal results.

Therefore, we propose a novel representation of 3D face scans, subsequently named Spherical Depth Map (SDM), to provide a canonical representation for arbitrary 3D scans represented in unstructured point clouds. The second issue is how to improve the discirminating ability of SHF for 3D FR. Some researchers calculate the L2-norm of each frequency component at each radius in order to obtain a rotation invariant shape descriptor, however, the information lost will seriously deteriorate the performance of face recognition [START_REF] Funkhouser | Search engine for 3D models[END_REF], [START_REF] Kazhdan | Rotation invariant spherical harmonic representation of 3D shape descriptors[END_REF]. Therefore, original SHF is used in our method after an exact face alignment based on Spherical Depth Map [START_REF] Liu | Automatic and robust 3D face registration using multiresolution spherical depth map[END_REF].

Feature selection aims to find faster and more effective predictor while improving or maintaining the prediction performance [START_REF] Guyon | An introduction to variable and feature selection[END_REF]. In contrast with other subspace-based dimensionality reduction techniques, e.g., Principal Component Analysis (PCA) or Independent Component Analysis (ICA), feature selection techniques do not change the original representation of the features, but merely select a subset of them. As a result, feature selection techniques preserve the original semantics of the features, which offers the advantage of interpretability [START_REF] Saeys | A review of feature selection techniques in bioinformatics[END_REF]. According to the relations with predictor, feature selection algorithms broadly fall into two categories:

filter and embedded methods. In this study, the representative methods of two categories are investigated in order to select the most relevant SHF to a given predictor.

The proposed method consists of two stages: an off-line training stage for the selection of relevant SHF features and an online 3D FR stage as Fig. 1 illustrates. In both off-line and online stage, 3D face data are first preprocessed, in particular to remove spikes and fill holes. Then face-aligned Spherical Depth Map is generated by the proposed pipeline in our early work [START_REF] Liu | Representing 3D face from point cloud to face-aligned spherical depth map[END_REF].

In the training stage (off-line), we calculate a large set of SHF on training face data for the purpose of relevant feature selection. In the online 3D FR stage, only the selected subset of SHF is involved in classification, thereby significantly reducing the computational burden so that the process can work in real time.

The contributions of this work can be summarized as follows:

II. SPHERICAL DEPTH MAP

Before the calculation of Spherical Harmonic Features, 3D faces must be formalized as a spherical function with the same translation scale. The center of mass translation as proposed in [START_REF] Funkhouser | Search engine for 3D models[END_REF], [START_REF] Kazhdan | Rotation invariant spherical harmonic representation of 3D shape descriptors[END_REF] only worked well in the context of whole-object to whole-object matching. In the context of 3D FR where we have only frontal facial surfaces, the center of mass translation cannot be the optimal solution. To overcome this defect, a novel method for face shape representation, namely Spherical Depth Map (SDM), is proposed in this paper, which translates the face shape so that the center of the fitting sphere rather than the center of mass is at the origin.

The basic assumption in the generation of a SDM associated with a 3D face scan is that the shape of human head can be viewed as a sphere. Although the human head has a shape rather close to an ellipsoid, which is also homeomorphic Riemannian manifold as the sphere, a spherical fitting has similar performance but less computational complexity.

The generation of a SDM is achieved through the following steps as Fig. 2 shows. More details can be found in our earlier work [START_REF] Liu | Automatic and robust 3D face registration using multiresolution spherical depth map[END_REF], [START_REF] Liu | Representing 3D face from point cloud to face-aligned spherical depth map[END_REF].

(1) A sphere is fitted to the point clouds of a 3D face scan as shown in Fig. 3.

June 29, 2012 DRAFT (2) 3D face shape is translated so that the center of the fitted sphere is at the origin.

(3) The scale of Cartesian coordinates is normalized into an uniform interval.

(4) The Cartesian coordinates of the input face point clouds is transformed into spherical coordinates (r, θ, ϕ).

(5) To calculate the SHF, the face pose is automatically normalized into a uniform orientation [START_REF] Liu | Representing 3D face from point cloud to face-aligned spherical depth map[END_REF].

By interpolation on a grid of θ and ϕ, the SDM of a 3D face scan can be viewed in a rectangle of [θ(0 ∼ π), ϕ(0 ∼ 2π)] as shown the last image in Fig. 2.

Fig. 3 shows the three-view drawings of a head shape after spherical fitting. As we can see, the face shape can be now described using a coordinate system attached to the face shape. The Iterative Closest Point (ICP) algorithm has proved its effectiveness for 3D face alignment by iteratively minimizing the Mean Square Error (MSE) of all point pairs between two face scans [START_REF] Besl | A method for registration of 3D shapes[END_REF]. However, the original ICP algorithm cannot handle the case of 3D face scans with different scales. Fig. 4 shows an example of face scale variations in which all faces are from the June 29, 2012 DRAFT same subject in the FRGC2.0 database. Therefore, we achieve the generation of face-aligned SDM of 3D face using the method that we proposed in an earlier work [START_REF] Liu | Representing 3D face from point cloud to face-aligned spherical depth map[END_REF]. To improve the accuracy of face alignment for the generation of robust SDMs, we made use of another landmark on 3D face scans in addition to the nose-tip. Although many other landmarks can be used which can automatically be accurately located [46] [47], the lowest point between the nose-tip and the center of forehead is employed to control the scale of 3D face scans in this work. Based on the previous work, the accuracy of the face alignment is improved by the following process:

(1) Find the symmetry plane of the input 3D face scan.

(2) The point having the lowest spherical depth on the intersection of the symmetry plane and the input 3D face scan is located as the required additional landmark as Fig. 6 shows. (3) The scale of the input 3D face scan is normalized by scaling the face so that the angle between the nose-tip and the detected landmark is equal to that of a reference model.

Although the asymmetry of human faces can be seen as a biometric trait for person identification [START_REF] Liu | Facial asymmetry quantification for expression invariant human identification[END_REF], the preprocessing step of finding 3D face symmetry plane in this work is quite different from that work. We aim to locate a plane with which the input face can be divided into two relative symmetric parts for alignment. In fact, such information is also required in [START_REF] Liu | Facial asymmetry quantification for expression invariant human identification[END_REF] in terms of "midline" to calculate the face asymmetry.

All of the above precessing is conducted automatically except a few failures caused by 3D face scans with an occlusion in the Bosphorus database. Finally, we obtain the face-aligned SDM of each 3D face model. Fig. 7 shows several examples of face-aligned SDMs from the three databases.

The advantages of using SDMs can be summarized as follows:

• The generation of SDMs is a Mesh free method. The process of generating SDMs can simply operate on scattered point clouds of a 3D face scan, so no mesh or triangulation is required.

• Spherical scale invariance. Constrained by the configuration of the human head, the face area on the fitted sphere is invariant (about π/2×π/2). Therefore, the face scale is naturally June 29, 2012 DRAFT normalized on the sphere.

• Dealing with face pose variation is as simple as a rigid rotation around the center of the fitted sphere.

• Object-centered shape representation. The face shape is described by a coordinate system attached to the 3D face. The origin of the coordinate system is the center of the fitted sphere which is derived from the face model itself.

• Complete shape representation. Only coordinate transformation (one-to-one mapping) is conducted on the point clouds during the generation process. There is no loss of information.

As a result, the original face model can be completely reconstructed.

• Robustness to noise, small occlusion and expression variations. The center of the fitted sphere is calculated based on the global configuration of the head shape, which is insensitive to local variations caused by point-level noise, small occlusion or facial expression.

The original SDM associated with a 3D face scan is still a discrete representation, i.e. the points on a SDM bear no regular structure. This makes it difficult to be used directly. Nevertheless, by interpolation, an implicit function radius = f (θ, φ) can be reconstructed from a given discrete SDM. Similar to range images, SDMs can be used as preprocessing step in many methods and applications, including Principal Components Analysis (PCA), Linear Discriminant Analysis (LDA), Gabor filter, Support Vector Machines (SVM), boosting methods, SIFT and so on.

June 29, 2012 DRAFT

III. SPHERICAL HARMONIC FEATURES

Based on the SDM representation, with the same spherical coordinate system, it is natural to make use of spherical harmonics for the description of 3D facial surfaces. Although this technique has been introduced to characterize general 3D objects, using spherical harmonics for 3D facial surface description has not been investigated yet in the literature. Indeed, 3D FR is quite a different problem from that of general 3D object retrieval which is an inter-class retrieval problem in the sense that it needs to retrieve fundamentally different object classes, e.g., glass, chair, car. In the case of 3D FR, one needs to recognize different instances of the same class, i.e. the human face, thus requiring features or descriptors having enough discriminating power necessary for a fine-grained accuracy.

In mathematics, spherical harmonics [START_REF] Schonefeld | Spherical Harmonics[END_REF], [START_REF] Freeden | Constructive approximation on the sphere with applications to geomathematics[END_REF] are a set of solutions to Laplace's equation in spherical coordinates, that form a set of basis functions of an orthogonal space:

Y l m (θ, ϕ) = K m l p l m (cos θ)e imϕ (1)
where the symbol p m l denotes the associated Legendre polynomials that are real-valued and defined over the range [-1, 1]. K m l is simply a scaling factor to normalize the functions:

K l m = (2l + 1) 4π (l -m)! (l + m)! (2)
3D face in spherical depth map is a spherical function f (θ, φ) which can be expanded into a linear combination of all spherical harmonics scaled by their associated coefficients h l m:

f (θ, ϕ) = ∞ l=0 l m=-l h lm Y l m (θ, ϕ) (3)
Considering that 3D FR only requires real valued spherical harmonics, the equation of real Spherical Harmonics can be set forth as follows:

y l m (θ, ϕ) =          √ 2K l m cos(mϕ)p l m (cos θ), m > 0 √ 2K l m sin(-mϕ)p l -m (cos θ), m < 0 K l 0 p l 0 (cos θ), m = 0 (4)
where p m l and K m l are the same as Equation (1).

June 29, 2012 DRAFT Finally, the associated coefficients h l m, i.e. the spherical harmonic features (SHF), are obtained by the following equation:

h lm = 2π 0 π 0 f (θ, ϕ)y l m (θ, ϕ) sin θdθdϕ ( 5
)
where sinθ is a regulatory factor for the integral in spherical coordinates. This equation is suitable for symbolic integration. The integral has to be evaluated numerically using Monte Carlo integration [START_REF] Caflisch | Monte carlo and quasi-monte carlo methods[END_REF]. The numerical solution for the above integral problem can then be rewritten as:

c lm = 1 n n j=1 f (x j )y l m (x j )4π = 4π n n j=1 f (x j )y l m (x j ) ( 6
)
where n is the number of sample points on f (θ, φ), i.e. the input 3D face model. 4π is a weight in the Monte Carlo estimator for every sample with equal probability.

In the end, a 3D face model is thus characterized by the energies at different frequencies of spherical harmonics. The energies at low frequencies capture the gross shape of the facial surface whereas those at high frequencies describe the fine facial surface details. As a result, these SHFs make it possible to capture both global and local facial shape information. This is in clear contrast to most 3D FR techniques known so far in the literature, which are either holistic, thus based on a global facial representation, or feature-based making use of local features extracted from a number of distinctive points.

IV. FEATURE SELECTION

As SHFs at different frequencies capture different geometric properties of a given facial surface, with the energies at low frequencies capturing the gross shape while those at high frequencies describe the fine details, and given the fact that 3D face scans can convey variations due to facial expression and partial occlusion, we proceed to learning the spherical harmonic features (SHF) which best discriminate the identities of 3D face scans through a feature selection method.

Feature selection, also known as variable selection or attribute selection, aims at selecting an optimal subset of discriminative features for building a robust classifier. The purpose of feature selection is three-fold: improving the prediction performance of the predictors, providing faster and more cost-effective predictors, and providing a better understanding of the underlying process

A. Relief-F

Relief-F is a type of supervised feature weighting algorithm which belongs to the category of filter methods. The key idea of Relief-F is to estimate the contribution of features according to their ability to distinguish between the nearest instances. The scheme of Relief-F is summarized as follows:

(0) Initialize W (a) = 0 for each feature a. m is a user-defined integer.

(1) Randomly select an instance R i .

(2) Find k nearest neighbors H j from the same class with R i .

(3) Find k nearest neighbors M j (C) from each class C = class(R i ).

(4) For each feature a, calculate the corresponding weight W (a) through the following equation:

W (a) = W (a) - k j=1 d(a, R i , H j )/(m • k) + C [K k j=1 d(a, R i , M j (C))]/(m • k) (7)
K = P (C) 1 -P (class(R i ))
where d(a, I 1 , I 2 ) calculates the difference between two instances I 1 and I 2 with respect to feature a. P (C) is the prior probability of class C. 1 -P (class(R i )) denotes the sum of probabilities for class C = class(R i ).

June 29, 2012 DRAFT [START_REF] Gokberk | 3D face recognition: Technology and applications[END_REF] Repeat the above process from (1) for m times. Finally return the weight vector W of all features.

B. minimal-Redundancy-maximal-Relevance mRmR selects a subset of best features according to the criterion of minimal-redundancymaximal-relevance based on mutual information. Given two variables x and y, their mutual information can be defined as:

I(x, y) = p(x, y) log p(x, y) p(x)p(y) dxdy (8)
where p(•) is the probability density function. The criterion of maximal-relevance is to select features S satisfying the following equation:

max D(S, c), D(S, c) = 1 |S| x i ∈S I(x i , c) (9)
where c is a class label. However, it is well known that the combination of individually good features does not necessarily lead to good classification performance because the features selected according to maximal-relevance could have rich redundancy. To remedy this drawback, minimalredundancy criterion is added to reduce the redundancy between features:

min R(S), R(S) = 1 |S| 2 x i ,x j ∈S I(x i , x j ) (10)
The combination of the above two criteria is max(D-R). In practice, incremental search method is normally used to select the near optimal subset of features. Given the selected subset S m-1 , the mth feature can be selected from the set {X -S m-1 } by the following equation:

max[I(x j , c) - 1 m x i ∈S m-1 I(x j , x i )], x j ∈ X -S m-1 (11)

C. Random Forests

Breiman [START_REF] Breiman | Random forests[END_REF] gives the definition of Random Forests as a classifier consisting of a collection of

decision trees (h(x, Θ k ), k = 1, 2, • • •)
, where Θ k are independent identically distributed random vectors. Each decision tree casts a unit vote for the final output with respect to input x. The core task of Random Forests is the construction of each decision tree. Further details can be found in [START_REF] Breiman | Random forests[END_REF].

June 29, 2012 DRAFT

V. EXPERIMENTAL RESULTS AND ANALYSIS

The effectiveness of the proposed method is evaluated in this section. First, we shortly describe the three datasets on which the experiments were carried out. We then evaluate the discriminating power of the proposed spherical harmonic features (SHF). This is followed by experiments highlighting the usefulness of feature selection methods. Finally, 3D FR experiments using selected SHF are performed and the results compared with the state of the art.

A. Face Databases

Our experiments are conducted on three popular face datasets, namely SHREC2007 [START_REF]Shape retrieval contest of 3D face models[END_REF],

FRGC2.0 [START_REF] Phillips | Overview of the face recognition grand challenge[END_REF] and Bosphorus [START_REF] Savran | Bosphorus database for 3D face analysis[END_REF]. Table I gives a comparative overview of these three datasets. Because the computation of Spherical Harmonic Feature (SHF) at different frequencies through Equation ( 5) and ( 6) requires the global description of a 3D shape, 3D face scans having only the half face in Bosphorus are excluded from our experiments. Therefore, a total of 3954 faces from 105 subjects in Bosphorus are used in the experiments.

B. Spherical Harmonic Features

As described in section III, there exists an infinity of spherical harmonics at different frequencies. Therefore, the computation of all SHFs is infeasible and one needs to find out a tradeoff between the descriptive power of SHFs and the associated computational cost. In this work, we In order to test the discriminating power of SHF which is now approximated by a 676 dimensional feature vector, we carried out face verification experiment on the three aforementioned datasets, that is to say, the query set and the target set are all the faces in each dataset. Fig. 8 gives the ROC curves of the proposed SHF. The verification rate on the three datasets at FAR = 0.001 is 92%, 90% and 81.4%, for SHREC2007, FRGC2.0 and Bosphorus respectively. We compared SHF with some state-of-the-art methods on the FRGC2.0 dataset. All experiments were performed under the same conditions, i.e. under "All versus All" partition. Table II shows the comparison of verification rate at FAR = 0.001. Our approach achieves the highest verification rate when using a single feature.

C. Learning the Discriminating Spherical Harmonic Features

The next step is to test the usefulness of feature selection methods. Given that learning is required for these feature section methods, it is important to check their skill of generalization on unseen subjects [START_REF] Llonch | 3D face recognition with sparse spherical representations[END_REF]. To this end, we divided the subjects used for training and testing into two disjoint sets in the experiments as shown in Table III. Given that the three datasets are organized by subject and each subject has a similar distribution, our divisions were based on subject. For example, the 64 subjects with 640 face scans of SHREC2007 were split into a Since Relief-F is a feature weighting algorithm [START_REF] Robnik-Sikonja | Theoretical and empirical analysis of ReliefF and RReliefF[END_REF], a feature list ranked by weight, representing the discriminating ability of the corresponding feature, is returned by Relief-F. To evaluate the effectiveness of Relief-F, KNN classifier was conducted on the selected feature set by incrementally increasing the number of features used on the test set until the full list of features was reached. For each dataset, 10-fold cross validation was used to guarantee the reliability of the results. Fig. 9 shows the accuracy curves on the three datasets. For the purpose of comparison in order to highlight the generalization skill of the chosen features, accuracy on the training sets was also calculated in the same manner.

Minimum-Redundancy-Maximum-Relevance (mRmR) selects a subset of features that have higher correlation to the prediction performance [START_REF] Peng | Feature selection based on mutual information: Criteria of Max-Dependency, Max-Relevance, and Min-Redundancy[END_REF]. In this study, 300 variables ranked by the correlation were selected from the 676 Spherical Harmonic Features. The effectiveness of these features was then evaluated in the same way as Relief-F, as shown in Fig. 10. By observing the curves in figure 9 and 10, we can draw the following conclusions:

• By comparing the results between the training set and testing set, it can be seen that the features selected by Relief-F and mRmR have a very good generalization skill on unseen subjects.

• The features selected by Relief-F and mRmR perform the best on SHREC2007, then on FRGC v2.0 and lastly on Bosphorus. This accords with the increasing difficulties of these three datasets in terms of face data, face expression variations, pose changes and even occlusion.

• Relief-F performs consistently better than mRmR on the three datasets.

As we can see from Fig. 9 and 10, the recognition accuracy increases quite quickly and attains a point of inflection when the number of features selected by both Relief-F and mRmR is augmented. This recognition accuracy then remains stable with very slight improvement in the case of FRGC v2.0 and Bosphorus or even decreases in the case of SHREC2007 when the number of selected features is further increased. The selection of relevant features in our case thus requires a tradeoff between computational complexity and recognition accuracy. While the selection of more features may lead to better recognition accuracy, it also leads to a higher computational complexity of the predictor. Many solutions to this question exist, e.g., second derivative or Receiver Operating Characteristic analysis, which automatically determine the number of selected features to be used based on the requirements of the application in hand. In this work, based on the performance of the features selected by Relief-F and mRmR on the testing set as shown on Fig. 9 and 10, we simply manually selected the subset of features being used in the three datasets. These feature subsets were then used in the task of face recognition.

D. Experiments on 3D FR

To validate the improvements of feature selection on 3D FR, both verification and identification experiments were carried out. There are a series of well-known protocols in FRGC2.0 for face recognition experiments. Unfortunately they are only suitable for FRGC2.0 data. As a result, we designed some more general experiment protocols in order to enable the comparison of the performance of the proposed methods on the three datasets. These experiment protocols are first explained and are then followed by the experimental figures.

For face verification, the query set and the target set were all faces from the three testing sets.

Therefore, the size of the similarity matrixes on the three datasets is 320 × 320, 2034 × 2034 and 1977 × 1977, respectively. Fig. 11 gives the ROC curves of the feature subsets by Relief-F and mRmR in comparison with all features on the three test sets. As we can see from these curves, Relief-F once again outperforms mRmR and the use of all features.

June 29, 2012 DRAFT For face identification, 10-fold cross validation using KNN classifier was employed to calculate the identification rate on the three datasets. Furthermore, this accuracy was compared with 10fold cross validation using SVM classifier. Table V shows the comparison of recognition rates.

The results of Random Forests are listed in the last line of the table.

Fig. 1 .

 1 Fig. 1. The flowchart of our methods.

Fig. 2 .

 2 Fig. 2. The generation of face-aligned SDM.

Fig. 3 .

 3 Fig. 3. Three-view drawings of face shape with fitting sphere.

Fig. 4 .

 4 Fig. 4. Face scale variations in FRGC2.0 database. Faces are showed in same coordinates system.

Fig. 5 .

 5 Fig. 5. The influence of the foreign matters on SDM. As the second column shows, the spherical scale of 3D face in SDM is variant because of the foreign matters.

Fig. 6 .

 6 Fig. 6. The control of the scale of 3D face in SDM.

Fig. 7 .

 7 Fig. 7. SDM examples of 3D face models from SHREC2007 (left), FRGC v2.0 (middle) and Bosphorus (right). Face scans in FRGC2.0 have varied facial expressions whereas those in Bosphorus have varied occulisions. The faces in SDM are cropped by a square having the size of π/2 × π/2.

June 29 ,

 29 2012 DRAFThave chosen l = (1 ∼ 25), which yields (25 + 1) 2 = 676 coefficients, along with the Simple euclidian distance to compute the similarity of two SHFs.

Fig. 8 .

 8 Fig. 8. ROC curves of SHF on three datasets.

Fig. 9 .

 9 Fig. 9. Performance evaluation of Relief-F on the three datasets by 10-fold cross validation using the KNN classifier.

Fig. 10 .

 10 Fig. 10. Performance evaluation of mRmR on the three datasets by 10-fold cross validation using the KNN classifier.

Fig. 11 .

 11 Fig. 11. The comparison of ROC curves between Relief-F, mRmR and All Features.

  of the number of subjects, facial expression, pose and occlusion. Under the combined effect of the discriminating power as provided by SHF and the feature selection method, the recognition rate by Relief+SVM only decreases slightly on FRGC v2.0 with facial expression variations and June 29, 2012 DRAFT Bosphorus with pose changes, facial expression variations and occlusions. These experimental results demonstrate the effectiveness of the proposed method. In addition, the computation cost is significantly reduced by using feature selection methods. The average time for the computation of l = (1..25) band SHF (676 features) of a face scan in FRGC2.0 decreases from 2.72 seconds to 0.8 seconds, thereby cutting the computational cost by a factor of 3. The experiments were implemented using Matlab on a PC with Intel(R) Core(TM)2 Quad CPU 2.40 GHz CPU and 4G RAM. Therefore, there is still room to further acceleration of the computation process. VI. CONCLUSIONS In this paper a novel competitive method for 3D FR has been proposed, which is based on Spherical Harmonic Features (SHF) reliably computed on a canonical 3D face representation, namely Spherical Depth Map (SDM). Feature selection methods are further used to improve the predictive performance and provide faster and more cost-effective predictors. The experimental results on three public 3D face datasets, namely SHREC2007, FRGC v2.0 and Bosphorus, highlight the discriminating power of the proposed Spherical Harmonic Features. Although there is a slight decrease in performance when expression and occlusion are taken into account, proper feature selection method can compensate for this defect. Finding or designing better feature selection method for SHF will be our future objective.

  Based on the relations with respect to predictors, feature selection algorithms broadly fall into two categories: filter and embedded methods. The filter methods take into account the general properties of training data and evaluate the relevance of features without involving any predictor algorithm, whilst the embedded methods incorporate feature selection as part of the training process, and the feature relevance is evaluated analytically according to the associated predictor. In this work, three representative methods for feature selection, namely Relief-F[START_REF] Robnik-Sikonja | Theoretical and empirical analysis of ReliefF and RReliefF[END_REF], mRmR[START_REF] Peng | Feature selection based on mutual information: Criteria of Max-Dependency, Max-Relevance, and Min-Redundancy[END_REF] (filter model) and Random Forests[START_REF] Breiman | Random forests[END_REF] (embedded model), are investigated. These three methods are different from each other as they achieve the selection of discriminating features through different properties from the training data. Our aim is to discover which kind of technique is most suitable for the selection of the most discriminating SHFs.

	that generated the data [6].
	June 29, 2012	DRAFT

TABLE I A

 I COMPARATIVE OVERVIEW OF THE THREE PUBLIC 3D FACE DATASETS.

	Databases	SHREC2007 FRGC2.0	Bosphorus
	Subject	64	466	105
	Total	640	4007	4666
			smile
	Expression	neutral	surprise	35 expressions
			angry
	Pose	NA	NA	13 yaw, pitch,
				cross rotations
	Occlusion	NA	NA	4 occlusions

TABLE II THE

 II COMPARISON OF VERIFICATION RATE AT FAR = 0.001 USING THE FRGC2.0 DATABASE.

		Single feature-mode	Fusion
	Maurer et al. [56]	77.6%	86.9%
	Cook et al. [57]	-	92.31%
	Mian et al. [58]	72.55%,74.31%	86.6%
	Faltemier et al. [59]	<84.8%	93.2%
	Queirolo et al. [37]	-	96.5%
	Wang et al. [21]	-	98.13%
	Berretti et al. [34]	81.2%	-
	Our method	90%	-
	training and testing set of equal size, each having 320 face scans of 32 subjects. The other two
	databases were divided in the same way.

TABLE III DISJOINT

 III SETS FOR TRAINING AND TESTING. THE NUMBERS IN THE TABLE ARE LISTED IN TERMS OF FACE

		SCANS/SUBJECTS.
	Databases	SHREC2007 FRGC2.0 Bosphorus
	Training set	320/32	1973/170	1977/58
	Test set	320/32	2034/296	1977/47

  Table IV summarizes the number of selected features in each subset, per dataset and per feature selection method.

TABLE IV THE

 IV NUMBER OF SELECTED FEATURES IN EACH FEATURE SUBSET.

	Databases SHREC2007 FRGC2.0 Bosphorus
	Relief-F	148	188	171
	mRmR	135	168	153

TABLE V THE

 V COMPARISON OF IDENTIFICATION RATES. THE IDENTIFICATION RATES USING ALL SHF ARE ALSO CALCULATED ASAs we can see from TableV, the best results, respectively 97.86%, 96.94% and 95.63% on the three datasets, were achieved by Relief-F+SVM. Remember that SHREC2007, FRGC v2.0 and Bosphorus, as summarized in table I, are three datasets of increasing difficulties in terms

		THE BASELINE.
		SHREC2007 FRGC2.0 Bosphorus
	KNN(all)	89.75%	88.24%	75.87%
	SVM(all)	97.24%	96.85%	94.42%
	Relief-F+KNN	96.53%	88.25%	76.61%
	Relief-F+SVM	97.86%	96.94%	95.63%
	mRmR+KNN	94.99%	86.27%	73.92%
	mRmR+SVM	96.47%	89.52%	87.79%
	RF	96.83%	92.97%	89.22%

June 29, 2012 DRAFT

ACKNOWLEDGMENT This work is in part funded by the National Basic Research Program of China (No. 2010CB327902), the National Natural Science Foundation of China (No. 60873158, No. 61005016, No. 61061130560), the Sino-French project 3D Face Analyzer supported by the French research agency, Agence Nationale de Recherche (ANR), through the grant ANR 2010 INTB 0301 01, and the 3D Face Interpreter project supported by the LIA 2MCSI lab between the group of Ecoles Centrale and Beihang University.