Sebastian Geisler
email: sebastian.geisler@rub.de

Katrin Rolka

AFFECTIVE VARIABLES IN THE TRANSITION FROM SCHOOL TO UNIVERSITY MATHEMATICS

Keywords: transition to and across university mathematics, teaching and learning of analysis and calculus, dropout, study success, affect

The transition from school to university mathematics is a difficult step for students, which many of them do not succeed to manage immediately. In this contribution we use questionnaires, which measure mathematics-related affective variables as well as subject-unspecific affective variables and students' achievement during the semester to predict the outcome of the exam at the end of the first semester (as a first indicator for success in their studies) and the students' attendance in this exam (as an indicator for early dropout). We are interested in whether the mathematics-related or the "general" affective variables are more suitable to predict the students' exam attendance and the exam outcome. The students' achievements during the semester turned out to be the best predictor for the exam outcome, whereas the students' attendance was best predicted by their interest in mathematics.

INTRODUCTION

Dropout is a big problem for German universities, especially in mathematics. Nearly 80% of all mathematics students drop out or change their subjects -most of them during their first year at university [START_REF] Dieter | Vier von fünf geben auf (Four out of five are giving up)[END_REF]. However, this is not a typical German phenomenon: Chen (2013) reports similar figures of dropout and subject change for the United States. In this paper, we do not distinguish between students who drop out and leave the university system without examination and those who "just" leave mathematics and change to another subject.

At Ruhr-University Bochum, where our study takes place, students begin their studies of mathematics with two lecture courses in the first semester -calculus I and linear algebra I. Two so called "mini-tests" are written during the semester to prepare the students for their first "real" exam at university. These "mini-tests", designed by the lecturer, cover conceptual and procedural knowledge about definitions and proof which have been discussed in the lecture before. 73% of the students at Ruhr-University (who attended the exams) failed their calculus exam in 2017. Furthermore, 25% of the students did not even attend the exam. Following the goal to improve the support of students who are at risk to fail their exam and/or drop out, we are interested in which way students at risk differ from those who succeed.

Due to data protection regulations, it was not possible for us to identify which students really dropped out from mathematics at Ruhr-University in 2017. Instead, we could match the results from our questionnaires with the results of the exam at the end of the first semester. We therefore could identify which students were successful in their exam, which failed and which did not attend the exam. Students who did not attend the exams might have dropped out before or may be at risk to drop out soon. [START_REF] Baars | Early Identification and Characterization of Students Who Drop Out in the First Year at University[END_REF] found that students who do not attend their exams in the first semester have a high risk to drop out.

THEORETICAL BACKGROUND

Both, dropout and study success, are considered to be influenced by multiple factors, which are often called predictors, such as the socio-economic and school background, personal psychological prerequisites, learning behaviour and study conditions [START_REF] Tinto | Dropout from Higher Education: A Theoretical Synthesis of Recent Research[END_REF][START_REF] Heublein | Ursachen des Studienabbruchs in Bachelor-und in herkömmlichen Studiengängen (Reasons for Dropout in Bachelor-and conventional Studyprogramms)[END_REF][START_REF] Thiel | Ergebnisse der Befragung der Studierenden in den Bachelorstudiengängen an der Freien Universität Berlin -Sommersemester 2008[END_REF].

The predictors, which are listed by the students for their decision to drop out, are called dropout reasons. In Germany, most dropped out mathematics students name the course requirements (e.g. failed exams, work-overload) (33%) and low motivation (25%) as their main reasons to quit their studies of mathematics. Other reasons such as study-conditions (13%) and reasons related to health-or financial problems (12%) and personal reasons like family problems (10%), were less important -specially for early dropout [START_REF] Heublein | Ursachen des Studienabbruchs in Bachelor-und in herkömmlichen Studiengängen (Reasons for Dropout in Bachelor-and conventional Studyprogramms)[END_REF].

Given the fact, that most dropped out mathematics students in Germany name the requirements at university and their lack of motivation as crucial for their decision to drop out, we want to shed light on the following affective variables, which are considered to influence the students' motivation and academic achievements: mathematical self-concept, interest in mathematics, beliefs concerning the nature of mathematics, basic needs and general self-efficacy. These variables are briefly discussed in the following.

Mathematical self-concept

The self-concept can be seen as the mental model of one's personal competences, abilities and properties, or "in very broad terms, self-concept is a person's perception of himself" [START_REF] Shavelson | Self-concept: Validation of construct interpretations[END_REF] as cited in [START_REF] Bong | Academic Self-Concept and Self-Efficacy: How Different Are They Really[END_REF]. The self-concept is influenced by the students' former experiences and achievements and can itself influence students' motivation [START_REF] Bong | Academic Self-Concept and Self-Efficacy: How Different Are They Really[END_REF]. The self-concept is considered to be domain specific. [START_REF] Rach | The Transition from School to University in Mathematics: Which Influence Do School-Related Variables Have[END_REF] found that the mathematical self-concept is a significant predictor for dropout but not for students' success in the first semester.

Interest in mathematics

The interest in mathematics is considered to have a positive impact on the learning of mathematics. [START_REF] Schiefele | Der Fragebogen zum Studieninteresse (FSI) (The Studyinterest-Questionnaire)[END_REF] define interest as a specific relation between a person and an object. The interest in the subject that one is studying is rather stable, since it has been developed over a longer time through different experiences.

Due to contradictory results in various studies the impact of interest in mathematics on students' performance and success in their studies of mathematics remains uncertain. [START_REF] Rach | The Transition from School to University in Mathematics: Which Influence Do School-Related Variables Have[END_REF] found no significant influence of the interest in mathematics on students' success during the first semester or on their risk to drop out. However, [START_REF] Blömeke | Ausbildungs-und Berufserfolg im Lehramtsstudium im Vergleich zum Diplomstudium -Zur prognostischen Validität kognitiver und psycho-motivationaler Auswahlkriterien (Success in pre service Teachers Studyprograms in contrast to Diplom Studyprograms -Regarding the prognostic validity of cognictive and psycho-cognitive criteria)[END_REF] found significant correlations between the interest in mathematics and the students' intention to drop out.

Beliefs concerning the nature of mathematics

It has been widely discussed that the nature of mathematics changes with the transition from school to university (e.g. [START_REF] Rach | The Transition from School to University in Mathematics: Which Influence Do School-Related Variables Have[END_REF]. Mathematics in German schools is often focused on applying mathematical techniques to solve real world problems (modelling, problem solving). New mathematical contents are regularly presented more intuitively with examples and illustrations and yield on an intuitive or practical understanding of the concepts. Mathematics at university is more theoretically and proof oriented. New concepts are presented in a rather formal and abstract way and therefore less illustrated than in school. The focus often lies on encouraging logical and abstract thinking. The students have to develop understanding for deductive argumentations and proof -applying the theory is less important than at school. This change from a practical to a theoretical approach is not easy for most students. Many of them feel a big gap between mathematics at school and university [START_REF] Geisler | Dropout & Persistence in University Mathematics[END_REF]. This feeling might be a result of unfulfilled expectations and incongruences between the mathematical "reality" at university and their established beliefs concerning the nature of mathematics, which are based on their school experiences. [START_REF] Daskalogianni | Beliefs Overhang: The Transition from School to University[END_REF] call this phenomenon "belief overhang". [START_REF] Andrà | Drop-out undergraduate students in mathematics: an exploratory study[END_REF] found hints that students' beliefs concerning the nature of mathematics can influence their decision to drop out or to stay. Traditionally we distinguish between a static view, where mathematics is viewed as a summary of (unconnected) rules, facts and techniques, and a dynamic view, where mathematics is considered as a process and a creative field of research [START_REF] Grigutsch | World views of mathematics held by university teachers of mathematics science[END_REF]. However, it is yet unclear which beliefs are beneficial for a successful transition from school to university.

Basic Needs

Following the framework of self-determination theory [START_REF] Ryan | Self-Determination Theory and the facilitation of Intrinsic Motivation, Social Development, and Well-Being[END_REF], there are three basic psychological needs that are important for the well-being of humans and to generate motivation: social relatedness, competence and autonomy. In the special situation of the transition from school to university mathematics, many students do not experience autonomy and competence [START_REF] Liebendörfer | Interest in Mathematics and the first Steps at University[END_REF]. This is problematic since Faye and Sharp (2008) found that especially the feeling of competence is strongly associated with motivation in university. In an explorative case-study, we found hints for the impact of social relatedness and competence on the decision to drop out [START_REF] Geisler | Dropout & Persistence in University Mathematics[END_REF].

General self-efficacy

The general self-efficacy is the strength of a persons' belief to be able to reach certain goals and to solve problems by his or her own competences and abilities [START_REF] Luszczynska | General self-efficacy in various domains of human functioning: Evidence from five countries[END_REF]. This general belief is not limited to a special domain like mathematics or special academic settings. In contrast to the self-concept, self-efficacy is more focused on the consequences of one's own competences and abilities than on the competences and abilities themselves. That's why self-concept is rather past oriented whereas the self-efficacy focuses on the future [START_REF] Bong | Academic Self-Concept and Self-Efficacy: How Different Are They Really[END_REF]. Besides, self-concept is considered to be the more stable variable. Self-efficacy can influence the students' motivation in the sense that students who believe that they are able to succeed in their studies of mathematics are more motivated to put effort in their learning than those who believe that they have no chance in the exams. Selfefficacy is therefore associated with academic achievement [START_REF] Luszczynska | General self-efficacy in various domains of human functioning: Evidence from five countries[END_REF][START_REF] Bong | Academic Self-Concept and Self-Efficacy: How Different Are They Really[END_REF]. Students with lower self-efficacy have a higher risk to drop out than those with higher self-efficacy [START_REF] Krieger | Determinanten des Studienabbruchs in naturwissenschaftlich orientierten Studiengängen (Determinants of Dropout in Naturesciences[END_REF].

Achievement during the semester

The students' achievement is an important factor for success and dropout. In [START_REF] Tinto | Dropout from Higher Education: A Theoretical Synthesis of Recent Research[END_REF] framework, achievement, as a part of the academic integration, is important for the decision to drop out or to stay. In an explorative case-study, [START_REF] Geisler | Dropout & Persistence in University Mathematics[END_REF] found hints that students who are not satisfied with their achievement during the first semester sometimes drop out, even if they are successful in their exams at the end of the semester. Though achievement is closely connected with the perceived feeling of competence.

RESEARCH QUESTIONS

In order to support students who are at risk to fail their exam or even to drop out we want to know in which way these students differ from those who succeed. Following the theoretical background described above, we decided to focus on the students' achievements during the first semester and on affective variables which are likely to influence students' motivation. Since the dropout rate in mathematics is high compared to other subjects, it seems plausible that mathematics related variables have an important impact on dropout and success. We therefore distinguish between mathematics-related affective variables (mathematical self-concept, interest in mathematics, beliefs concerning the nature of mathematics) and more "general" affective variables (basic needs, self-efficacy). We are interested in whether the mathematics-related or the "general" affective variables are more suitable to predict students' exam attendance and their exam outcome. This leads to the following research questions:

Differences between the three groups of students 1) Which differences in the affective variables and the achievements can be found between students who do not attend the exam, students who fail in the exam and those who succeed?

Prediction of the students' exam attendance 2.1) In which way can the mathematics-related affective variables predict the students' attendance for the exam at the end of the first semester? 2.2) In which way can the "general" affective variables and the students' achievements approve this prediction?

Prediction of the exam outcome 3.1) In which way can the mathematics-related affective variables predict the outcome of the exam at the end of the first semester? 3.2) In which way can the "general" affective variables and the students' achievements approve this prediction? METHODOLOGY 209 students in the calculus lecture in wintersemester 2016/17 voluntarily participated in our study. Undergraduate mathematics students as well as pre-service teachers in mathematics usually attend this lecture during their first year at university. The questionnaires were filled out during the lecture in the mid of the first semester, taking into account that students cannot rate their satisfaction of the basic needs at the begin of the semester. Due to incomplete datasets, only N=193 cases could be included in our analysis. The instruments used in our questionnaire can be found in Table 1. All reliabilities (Cronbachs α) were at least sufficient -except for the reliability of the autonomy-subscale. All items were answered on a five-point Likert scale (1=totally disagree; 5=totally agree). To measure the students' achievement during the semester, we used their results in the first "mini-test" (1 to 12 points).

The results of the questionnaires were analysed using a MANOVA (to prevent the accumulation of the α-error compared with t-tests) to answer research question 1. To answer the other research questions, we used linear and binary logistic regressions.

RESULTS

Students who do not attend the exam at the end of the first semester differ significantly from those who fail the exam and those who succeed in almost all affective variables (research question 1), except for the static beliefs (Table 2). Focussing on the mathematics-related affective variables, the biggest difference between the three groups of students can be found in the interest in mathematics, which can explain 13% of the variance (η 2 =0.13***). Regarding the "general" affective variables, the self-efficacy turned out to explain the most variance between the three groups of students (η 2 =0.1***). Taking into account all measured variables, the biggest difference between the three groups of students can be found in their achievements in the "mini-test" (η 2 =0.19***). To answer research questions 2.1 and 2.2, three different logistic regression models were tested (Table 3). Model 1 only contains the mathematics-related affective variables. The only significant predictor for the students' attendance in this model was interest in mathematics. Model 1 can explain 19% of the variance and is able to classify 71% of the students correctly as attending or not attending. Model 2 additionally contains the affective variables basic needs and self-efficacy. None of these variables has a significant influence on the students' attendance and they do not improve the students' classification. In contrast, the students' achievements are a (weak) significant predictor for the students' attendance (Model 3). The inclusion of the students' achievement can improve the classification of the students (75.1% correct) and increases the explained variance (Nagelkerke's R 2 =0.26). Note that the interest in mathematics is still the most significant predictor in Model 3. We used three linear regression models to answer research questions 3.1 and 3.2 (Table 4). In Model 1 only the mathematics-related affective variables were included. None of them can predict the exam outcome significantly. All these variables together only explain 5% of the variance in the exam outcome. The "general" affective variables, specially the self-efficacy, seem to have a bigger impact on the exam outcome (Model 2). The inclusion of these variables improves the variance that can be explained (R 2 =0.13). The self-efficacy is a highly significant predictor. The most important (highly significant) predictor for the exam outcome is the students' achievement in the "mini-test" (Model 3). The inclusion of the achievement increases the explained variance to 37%. Multicollinearity of the variables in our models is at least tolerable (tolerance>0.48, VIF<2) and should not have a big impact on results.

CONCLUSION

The interest in mathematics is the most important predictor for exam attendance in our study, whereas the mathematical self-concept has no significant influence. This is contradictory to Rach's and Heinze's (2016) findings, where only the mathematical self-concept was able to predict the attendance. However, both studies have in common that a mathematics-related affective variable is the most important predictor for the attendance, whereas the "general" affective variables have no influence.

The mathematics-related affective variables do not predict the exam outcome. The only affective variable that significantly predicts the exam outcome is the general self-efficacy. This is rather surprising, taking into account that self-efficacy and selfconcept are (at least theoretically) closely connected variables and it seemed plausible that the mathematics-related variable provides more insights. However, in contrast to the mathematical self-concept, the general self-efficacy is not only focussed on one's competences and achievements in mathematics but also takes into account subject unspecific competences which could be beneficial at university, too.

All in all, it turned out that the students' achievements can predict both, exam outcome and exam attendance. Interest in mathematics is suitable to predict the exam attendance, whereas self-efficacy can predict the exam outcome. Firstly, this finding shows that, mathematics related as well as general affective variables play an important role in the transition from school to university. Secondly, it suggests that success and dropout should not necessarily be viewed as two sides of a coin.

Our study has some limitations. We conducted data from only one university, which could lead to cohort specialities. Furthermore, questionnaires were filled out in the mid of the semester during the lecture. Students who do not (regularly) attend lectures or have dropped out before have not been captured by our study. Some results might be different if we could capture those students, too.

Our on-going research will now focus on a more detailed characterisation of dropped out students, taking into account cognitive and metacognitive variables (e.g. students learning behaviour) as well. In addition, it seems to be useful to identify different types of dropped out students in mathematics. This might help to design and evaluate more individualized supporting programs for students in the transition.

Table 1 : Instruments with numbers of items, reliability and item-example

 1

	construct	source	No. of items /	Item-example
			Crobach's α
	Interest	Schiefele et al.	12 / 0.82	"It is personally important for me
		2007		that I can study mathematics."
	Self-Concept	Kauper et al.	4 / 0.82	"I am very good in mathematics."
		2012
	Beliefs: static	Laschke &	6 / 0.67	"Mathematics means learning,
		Blömeke 2013		remembering and applying."
	Beliefs: dynamic	Laschke &	6 / 0.73	"Mathematics involves creativity
		Blömeke 2013		and new ideas."
	Social Relatedness Kauper et al.	6 / 0.78	"I feel comfortable with the other
		2012		students."
	Competence	Kauper et al.	3 / 0.66	"I get clear and detailed feedback
		2012		on my achievements."
	Autonomy	Kauper et al.	3 / 0.58	"I can do tasks in my way."
		2012
	Self-Efficacy	Beierlein et al.	3 / 0.88	"I can solve most problems on my
		2012		own"

16 10.89 2.75 22.74 0.19 *** Table 2: Means, standard deviations and results of the variance analysis *p<0.05 **p<0.01 ***p<0.001

		No Attendance		Failed	Succeeded
			N = 54	N = 101		N = 38
		M	SD	M	SD	M	SD	F	η 2
	Interest	2.99	0.66	3.43	0.7	3.7	0.56	13.89	0.13 ***
	Self-Concept	2.64	0.83	2.93	0.67	3.11	0.58	5.45	0.05 **
	Beliefs: static	3.77	0.66	3.79	0.58	3.68	0.5	0.46
	Beliefs: dynamic 3.07	0.68	3.46	0.65	3.56	0.62	8.22	0.08 ***
	Self-Efficacy	2.46	0.96	2.78	0.78	3.25	0.69	10.18	0.1 ***
	Social Related.	3.77	0.68	4.07	0.55	4.23	0.57	7.69	0.08 **
	Competence	2.94	0.87	3.23	0.86	3.43	0.81	3.91	0.04 *
	Autonomy	2.89	0.72	3.28	0.79	3.42	0.66	6.83	0.07 **
	Achievement	6.19	3.9	7.81	3.

Table 3 : Results (coefficients Exp(B)) of the logistic regression to predict the exam attendance -*p<0.05 **p<0.01

 3

		Model 1	Model 2	Model 3
	Interest	2.43 **	2.5 **	2.42 **
	Self-Concept	1.22	0.89	0.8
	Beliefs: static	1.14	1.3	1.29
	Beliefs: dynamic	1.66	1.3	1.24
	Self-Efficacy		1.33	1.18
	Social Relatedness		1.3	1.14
	Competence		0.93	1.04
	Autonomy		1.4	1.39
	Achievement			1.13 *
	Nagelkerke's R 2	0.19	0.22	0.26
	Correct classification	71 %	71 %	75.1 %

Table 4 : Results (standardized beta coefficients) of the linear regression to predict the exam outcome -*p<0.05 ***p<0.001

 4

		Model 1	Model 2	Model 3
	Interest	0.08	0.06	0.01
	Self-Concept	0.17	-0.03	0.04
	Beliefs: static	-0.09	-0.01	0.02
	Beliefs: dynamic	0.05	-0.02	-0.05
	Self-Efficacy		0.41 ***	0.2 *
	Social Relatedness		0.11	0.07
	Competence		-0.08	0.02
	Autonomy		-0.03	-0.07
	Achievement			0.54 ***
	R 2	0.05	0.13	0.37


Sławomir B Ąk

Marco San Biagio

Ratnesh Kumar

Vittorio Murino

François Brémond

Exploiting Feature Correlations by Brownian Statistics for People Detection and Recognition

Keywords: brownian descriptor, covariance descriptor, pedestrian detection, re-identification

Characterizing an image region by its feature inter-correlations is a modern trend in computer vision. In this paper, we introduce a new image descriptor that can be seen as a natural extension of a covariance descriptor with the advantage of capturing nonlinear and non-monotone dependencies. Inspired from the recent advances in mathematical statistics of Brownian motion, we can express highly complex structural information in a compact and computationally efficient manner. We show that our Brownian covariance descriptor can capture richer image characteristics than the covariance descriptor. Additionally, a detailed analysis of the Brownian manifold reveals that in opposite to the classical covariance descriptor, the proposed descriptor lies in a relatively flat manifold, which can be treated as a Euclidean. This brings significant boost in the efficiency of the descriptor. The effectiveness and the generality of our approach is validated on two challenging vision tasks, pedestrian classification and person re-identification. The experiments are carried out on multiple datasets achieving promising results.

I. INTRODUCTION

D ESIGNING proper image descriptors is a crucial step in computer vision applications, including scene detection, target tracking and object recognition. A good descriptor should be invariant to illumination, scale and viewpoint changes. This usually involves a high-dimensional floatingpoint vector encoding a robust representation of an image region [START_REF] Bay | Speeded-up robust features (surf)[END_REF], [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF]. Typically, descriptors employ simple statistics (i.e. histograms) of features extracted by different kinds of image filters (gradients [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF], binary patterns [START_REF] Wang | An hog-lbp human detector with partial occlusion handling[END_REF]). In recent studies, a trend has emerged that consists in discarding the intrinsic value of the features, encoding instead their intercorrelations. The most well-known image descriptor following this idea is the covariance descriptor [START_REF] Tuzel | Region covariance: A fast descriptor for detection and classification[END_REF]. This descriptor encodes information on feature covariances inside an image region, their inter-feature linear correlations and their spatial layout. The correlation-based descriptors show a consistent invariance to many aspects (scale, illumination, rotation), making them a good choice for representing object classes with high intra-class variability (e.g. pedestrians [START_REF] Tuzel | Pedestrian detection via classification on riemannian manifolds[END_REF]). Moreover, correlation-based descriptors are superior to other methods for absorbing inter-camera changes (e.g. for matching objects registered by different cameras [START_REF] Hirzer | Person re-identification by descriptive and discriminative classification[END_REF], [START_REF] Ma | Bicov: a novel image representation for person re-identification and face verification[END_REF]).

In this paper, we focus on correlation-based descriptors, revisiting fundamentals of covariance. We highlight that the covariance descriptor measures only linear dependence between features, which might not be enough to capture the complex structure of many objects. As an example see Fig. 1 which illustrates the correlation between two features extracted from the patch of a pedestrian image. Intensity values and the corresponding gradient magnitudes are plotted together to show the dependency. Most pixels of the patch have high intensity and low gradient (homogeneous regions). This produces the dense distribution in the lower-right corner of the plot. The most informative pixels are captured by the strap structure that show a non-monotone (nonlinear) dependency. Interestingly, the classical covariance will not capture this information as it measures only the linear correlation between features. In the result the covariance descriptor may produce a diagonal matrix, which is not a sufficient condition for statistical independence; actually, a non-monotone relation exists. This indicates information loss when using the covariance descriptor.

We overcome this issue by employing a novel descriptor based on Brownian covariance [START_REF] Bak | Brownian descriptor: a rich metafeature for appearance matching[END_REF], [START_REF] Székely | Brownian distance covariance[END_REF]. The classical covariance measures the degree of linear relationship between features, whereas Brownian covariance measures the degree of all kinds of possible relationships between features. We show that our novel descriptor is computationally efficient and more effective than the covariance descriptor (Sec. II).

This paper makes the following contributions:

• We discuss the covariance descriptor and highlight its constraints and limitations as a dependency measure (Sec. III-A).

• We propose a new image region descriptor that is a natural extension of covariance (Sec. III-B): the proposed descriptor is referred to as Brownian descriptor due to its analogy to the Brownian covariance.

• We illustrate advantages of the new descriptor over the classical covariance descriptor using synthetic data and theoretical analysis (Sec. III-D) and we provide an efficient algorithm for extracting the descriptor employing integral images (Sec. III-C).

• We show the generality of the descriptor, validating it on different vision tasks (Sec. IV). We show that this descriptor can handle both inter-and intra-class variations, e.g. pedestrian classification and person re-identification.

The results bear out that this descriptor reaches sufficient trade off between discriminative power and invariance. Finally, we demonstrate that the Brownian descriptor outperforms the classical covariance descriptor in terms of both efficiency and accuracy. The paper draws conclusions in Sec. V by discussing future perspectives.

II. RELATED WORK

One of the most common problems in object detection and recognition is to find a suitable object representation. For historical and computational reasons, vector descriptors that encode the local appearance of the data have been broadly applied. In this sense, many different techniques have been developed in the literature. As shown in [START_REF] San Biagio | Heterogeneous auto-similarities of characteristics (hasc): Exploiting relational information for classification[END_REF], many of these techniques follow two complementary paradigms: "featurebased" and "relation-based". The former takes into account measurable intrinsic characteristics of an object, such as color or shape information. Most of well-known descriptors included in this sub-group are: Scale-Invariant Feature Transform (SIFT) [START_REF] Lowe | Object recognition from local scale-invariant features[END_REF], Histogram of Oriented Gradients (HOG) [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF], Local Binary Pattern (LBP) [START_REF] Ojala | A comparative study of texture measures with classification based on featured distributions[END_REF], [START_REF] Wang | An hog-lbp human detector with partial occlusion handling[END_REF]. The latter paradigm consists of considering the intrinsic value of these cues, encoding their inter-relations: the most known descriptor following this line is the covariance of features (COV) [START_REF] Tuzel | Region covariance: A fast descriptor for detection and classification[END_REF], in which linear relations between features are exploited as elementary patterns.

A. Feature-based descriptors

SIFT descriptor, originally proposed in [START_REF] Lowe | Object recognition from local scale-invariant features[END_REF], is used for a large number of purposes in computer vision related to point matching between different views of a 3-D scene and view-based object recognition. SIFT descriptor is invariant to translations, rotations and scaling transformations in an image domain and robust to moderate perspective transformations and illumination variations. Experimentally, SIFT descriptor has been proven to be very useful for image matching and object recognition under real-world conditions [START_REF] Sivic | Video Google: A text retrieval approach to object matching in videos[END_REF], [START_REF] Li | A bayesian hierarchical model for learning natural scene categories[END_REF], [START_REF] Lazebnik | Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories[END_REF], [START_REF] Nowak | Sampling strategies for bag-of-features image classification[END_REF].

However image descriptors must not only be accurate but also highly efficient. SIFT unfortunately is represented by high-dimensional floating point vector bringing significant computational burden, while employed to tasks that require real-time performance. Consequently, HOG descriptor has been revealed. This descriptor is of particular interest in object detection and recognition as it is fast to compute and provides high performance [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF], [START_REF] Dalal | Human detection using oriented histograms of flow and appearance[END_REF], [START_REF] Zhu | Fast human detection using a cascade of histograms of oriented gradients[END_REF], [START_REF] Sabzmeydani | Detecting pedestrians by learning shapelet features[END_REF], [START_REF] Felzenszwalb | A discriminatively trained, multiscale, deformable part model[END_REF], [START_REF] Weinrich | Appearance-based 3d upperbody pose estimation and person re-identification on mobile robots[END_REF]. HOG is considered as the most popular feature used for pedestrian detection.

In [START_REF] Maji | Classification using intersection kernel support vector machines is efficient[END_REF], we can find PHOG, an extension of classical HOG descriptor for pedestrian detection. The authors showed that PHOG can yield better classification accuracy than the conventional HOG and it is much computationally lighter while having smaller dimension. However, these HOG-like features that capture edge and local shape information might perform poorly when the background is cluttered with noisy edges [START_REF] Wang | An hog-lbp human detector with partial occlusion handling[END_REF].

Originally proposed by [START_REF] Ojala | A comparative study of texture measures with classification based on featured distributions[END_REF], Local Binary Pattern (LBP) is a simple but very efficient texture operator which labels pixels of an image according to the differences between values of the pixel itself and the surrounding ones. It has been widely used in various applications and has achieved very good accuracy in face recognition [START_REF] Ahonen | Face description with local binary patterns: Application to face recognition[END_REF]. LBP is highly discriminative and its key advantages, namely its invariance to monotonic gray level changes and computational efficiency, make it suitable for demanding image analysis tasks such as human detection [START_REF] Enzweiler | Multi-cue pedestrian classification with partial occlusion handling[END_REF].

B. Relation-based descriptors

Recently, in contrast to the classical feature descriptors discussed above, a novel trend has emerged that consists of considering the intrinsic value of image features, encoding their inter-relations. The most popular descriptors exploiting feature correlations in images is the covariance descriptor [START_REF] Tuzel | Region covariance: A fast descriptor for detection and classification[END_REF]. This descriptor represents an image patch by the covariance of its features such as spatial location, intensity, higher order derivatives, etc.

Covariance descriptor was first introduced for object matching and texture classification [START_REF] Tuzel | Region covariance: A fast descriptor for detection and classification[END_REF]. Since then it has also been intensively employed in many other computer vision applications, such as pedestrian detection [START_REF] Tuzel | Pedestrian detection via classification on riemannian manifolds[END_REF], [START_REF] Jayasumana | Kernel methods on the riemannian manifold of symmetric positive definite matrices[END_REF], [START_REF] Yao | Fast human detection from joint appearance and foreground feature subset covariances[END_REF], person reidentification [START_REF] Hirzer | Person re-identification by descriptive and discriminative classification[END_REF], [START_REF] Bak | Multiple-shot human reidentification by mean riemannian covariance grid[END_REF], [START_REF] Bak | Boosted human reidentification using riemannian manifolds[END_REF], [START_REF] Bak | Learning to match appearances by correlations in a covariance metric space[END_REF], [START_REF] Cai | Matching groups of people by covariance descriptor[END_REF], object tracking [START_REF] Porikli | Covariance tracking using model update based on lie algebra[END_REF], action recognition [START_REF] Harandi | Spatio-temporal covariance descriptors for action and gesture recognition[END_REF] and head orientation classification [START_REF] Tosato | Characterizing humans on riemannian manifolds[END_REF].

As covariance matrices do not lie on a Euclidean space, each of these studies addresses the problem of using the covariance descriptor in a non-trivial machine learning framework. Several optimization algorithms on manifolds have been proposed for the space of positive semi-definite matrices Sym + d [START_REF] Tuzel | Region covariance: A fast descriptor for detection and classification[END_REF], [START_REF] Tuzel | Pedestrian detection via classification on riemannian manifolds[END_REF], [START_REF] Yao | Fast human detection from joint appearance and foreground feature subset covariances[END_REF], [START_REF] Tosato | Characterizing humans on riemannian manifolds[END_REF], [START_REF] Goh | Clustering and dimensionality reduction on riemannian manifolds[END_REF]. The most common approach consists in mapping covariance matrices to the tangent space that can be treated as an approximation of a Euclidean space [START_REF] Tuzel | Pedestrian detection via classification on riemannian manifolds[END_REF], [START_REF] Yao | Fast human detection from joint appearance and foreground feature subset covariances[END_REF]. Performing mapping operations involves choosing the tangent point on the manifold, which usually is determined either by the mean of the training data points -Karcher mean [START_REF] Karcher | Riemannian center of mass and mollifier smoothing[END_REF], or by the identity matrix [START_REF] Tosato | Characterizing humans on riemannian manifolds[END_REF]. The logarithmic and exponential maps are iteratively used to map points from the manifold to the tangent space, and vice-versa. Unfortunately, the resulting algorithms suffer from two drawbacks: first, the iterative use of the logarithmic and exponential maps makes them computationally expensive, and, second, they only approximate true distances on the manifold by Euclidean distances on the tangent space. Another possibility is to compute a metric directly on Sym + d , which estimates the geodesic distance [START_REF] Tuzel | Region covariance: A fast descriptor for detection and classification[END_REF], [START_REF] Tosato | Characterizing humans on riemannian manifolds[END_REF]. Using this approach, we preserve real distances between each pair of samples. Unfortunately, both solutions involve a high computational cost.

For the above mentioned reasons, in contrast to the previous approaches, we present new insights into the covariance descriptor, raising fundamental limitations of covariance as a dependence measure. In this paper, we design a novel descriptor driven by recent achievements in mathematical statistics related to Brownian motion [START_REF] Bakirov | Brownian covariance and central limit theorem for stationary sequences[END_REF], [START_REF] Székely | Brownian distance covariance[END_REF]. The new descriptor can be treated as a point in a Euclidean space, making the descriptor computationally efficient and useful for real-time applications. This novel descriptor not only brings tremendous matching speed-up in comparison to the classical covariance, but also keeps more information on feature correlations inside an image region.

In the following sections we raise fundamental constraints of covariance as a dependency measure and we define the Brownian descriptor.

III. BROWNIAN DESCRIPTOR

This section introduces the Brownian descriptor, discussing its advantages over the classical covariance. Before elaborating the Brownian descriptor, we discuss the classical covariance descriptor proposed in [START_REF] Tuzel | Region covariance: A fast descriptor for detection and classification[END_REF], highlighting its limitations.

A. Limitations of the classical covariance

Image feature inter-relation are often captured by the covariance matrix. This descriptor encodes information on feature variances inside the image region, their covariance with each other and their spatial layout. It enables to fuse different types of features, while producing a compact representation.

Let I be an image and F be a n-dimensional feature image extracted from I F = φ(I),

where function φ can be any mapping. Usually, the most applied mappings contain intensity values, color, gradients, filter responses, etc. . Recently, we can also find other types of mappings, e.g. based on infrared images, depth or motion flow.

In the result, each pixel can be expressed by a n-dimensional feature point determined by mapping φ.

For a given region R ⊂ F containing Z pixels, let {f z } z=1...Z be the n-dimensional feature points inside R. Each feature point f z is characterized by function φ. We represent region R by the n × n covariance matrix (C R ) with (i, j)-th element expressed as

C R (k, l) = 1 Z -1 Z z=1 f z (k) -µ(k) f z (l) -µ(l) , ( 2
)
where µ is the mean of f z points. The diagonal entries are variances of each feature, whereas the off-diagonal entries are the covariances between pairs of features.

Standardization. Covariance values are very often normalized by the product of corresponding standard deviations

ρ kl = C R (k, l) C R (k, k)C R (l, l) , (3)
and are referred to as the Pearson Product-Moment Correlation Coefficients.

1) Limitations due to linear dependency measure ρ measures a linear correlation between two variables (the strength of the linear dependence). However, as it is computed with respect to the mean of the feature (note µ components in Eq. ( 2)), it is not able to measure nonlinear or non-monotone dependence (see Sec. III-D and Fig. 2 for elaboration).

2) Limitations due to choice of metric

As we have already mentioned in Sec. II, covariance matrices do not lie on a Euclidean space. Computing distance between two covariance descriptors, we need to either assume a Riemannian manifold employing geodesic distance or map covariance to a tangent space approximating distances. Both solutions are computationally intensive and unfavorable in practice. Moreover, well known machine learning techniques are not adequate for learning on complex manifolds, often producing over-fitted classifiers.

B. Brownian covariance

Brownian descriptor inherits the theory from recent advances in mathematical statistics related to Brownian covariance [START_REF] Székely | Brownian distance covariance[END_REF]. In particular, it is based on the sample distance covariance statistics that measures dependence between random vectors in arbitrary dimension. In the following sections we introduce distance covariance V 2 , sample distance covariance V 2 n , and their relations to Brownian covariance W. The mathematical notations and formulas are in accordance with [START_REF] Székely | Brownian distance covariance[END_REF].

1) Distance covariance V 2 Let X ∈ R p and Y ∈ R q be random vectors, where p and q are positive integers. f X and f Y denote the characteristic functions of X and Y , respectively, and their joint characteristic function is denoted f X,Y . In terms of characteristic functions, X and Y are independent if and only if f X,Y = f X f Y . A natural way of measuring the dependence between X and Y is to find a suitable norm to measure the distance between f X,Y and f X f Y .

The Distance covariance V 2 [START_REF] Bakirov | Brownian covariance and central limit theorem for stationary sequences[END_REF] is a new measure of dependence between random vectors and can be defined by

V 2 (X, Y ) = ||f X,Y (t, s) -f X (t)f Y (s)|| 2 (4) = 1 cpcq R p+q |f X,Y (t,s)-f X (t)f Y (s)| 2 |t| 1+p p |s| 1+q q dtds, (5)
where c p and c q are constants determining norm function in R p ×R q , t ∈ X, s ∈ Y . This measure is analogous to classical covariance, but with the important property that V 2 (X, Y ) = 0 if and only if X and Y are independent. In [START_REF] Bakirov | Brownian covariance and central limit theorem for stationary sequences[END_REF] distance covariance is seen as a natural extension and a generalization of the classical covariance measure. It extends the ability to measure linear association to all types of dependence relations. Further, distance covariance can be computed between any random vectors in arbitrary dimension. For more theoretical and practical advantages of this new dependency measure the interested reader is refereed to [START_REF] Bakirov | Brownian covariance and central limit theorem for stationary sequences[END_REF].

2) Sample distance covariance V 2 n
Designing a new image descriptor, we are interested in finding relations between finite distributions (limited amount of pixels). Thus, we can employ a sample counterpart of distance covariance [START_REF] Székely | Brownian distance covariance[END_REF]. The sample distance covariance V 2 n between random vectors X and Y is defined as

V 2 n (X, Y ) = 1 n 2 n k,l=1 A kl B kl , (6)
where A kl and B kl are simple linear functions of the pairwise distances between n sample elements. These functions are defined in the following. For a random sample

(X, Y) = {(X k , Y k ) : k = 1 . . . n} of n i.i.d random vectors (X, Y ) from their joint distribution, compute the Euclidean distance matrices (a kl ) = (|X k -X l | p ) and (b kl ) = (|Y k -Y l | q ). Define A kl = a kl -āk• -ā•l + ā•• , k, l = 1, . . . , n, (7)
where

āk• = 1 n n l=1 a kl , ā•l = 1 n n k=1 a kl , ā•• = 1 n 2 n k,l=1 a kl . (8)
Similarly, we define

B kl = b kl -bk• -b•l + b•• .
Standardization. Similarly to covariance which has its standardized counterpart ρ, V 2 n has its standardized version referred to as distance correlation R 2 n and defined by

R 2 n (X, Y ) =    V 2 n (X,Y ) √ V 2 n (X)V 2 n (Y ) , V 2 n (X)V 2 n (Y ) > 0; 0, V 2 n (X)V 2 n (Y ) = 0, (9)
where

V 2 n (X) = V 2 n (X, X) = 1 n 2 n k,l=1 A 2 kl . (10)
3) Brownian covariance W Brownian motion is a stochastic process invented for modeling random movements of particles suspended in a fluid. It describes their trajectories and interactions. These interactions can be expressed by Brownian covariance. Let W be a Brownian covariance. According to [START_REF] Székely | Brownian distance covariance[END_REF] and [START_REF] Bakirov | Brownian covariance and central limit theorem for stationary sequences[END_REF], W measures all kinds of possible relationships between random particles (variables). This means that W(X, Y ) = 0 if and only if X and Y are independent.

The surprising coincidence is that for arbitrary X ∈ R p , Y ∈ R q with finite second moments

W(X, Y ) = V(X, Y ). ( 11
)
For the proof, the interested reader is pointed out to THEOREM 8 in [START_REF] Székely | Brownian distance covariance[END_REF]. Further, we see THEOREM 2 from [START_REF] Székely | Brownian distance covariance[END_REF] that says:

If E|X| α p < ∞ and E|Y | α q < ∞, then almost surely lim n→∞ V n (X, Y ) = V(X, Y ), ( 12
)
where α is a positive exponent on Euclidean distance. This equality holds only if the α moments are finite and 0 < α < 2. Although V can be defined for α = 2, it does not characterize independence. Indeed, the case α = 2 (squared Euclidean distance) leads to the classical covariance measure. In the results in algorithm 1, we assume α = 1 that leads to employing 1 metric, while computing distance matrix (a kl ).

From equations ( 11) and ( 12) we can see that

W(X, Y ) = lim n→∞ V n (X, Y ) ∝ R 2 n (X, Y ). (13)
In the result if R 2 n (X, Y ) = 0, we expect no dependence between variables. This is the main advantages of R 2 n (X, Y ) over ρ. ρ = 0 means that there is no linear correlation between variables, while nonlinear or non-monotone dependence may exist. Although R 2 n is just a sample counterpart of R, we believe that R 2 n keeps more information than ρ while characterizing an image region, which is clarified in the subsequent sections.

C. Efficient algorithm for computing Brownian descriptor

Let I be an image and L = {L 1 , L 2 , . . . , L n } be a set of feature layers defined by mapping φ. In other words, after applying mapping φ on I, each pixel z of the image can be expressed as the following feature vector:

φ(I, z) = [L 1 (z), L 2 (z), . . . , L n (z)]. (14)
The task is to provide a discriminative representation of a given image region R containing Z pixels. We propose to treat each layer L i as a point in a Zdimensional space and to express the Brownian descriptor as R 2 n (L, L). We design the Brownian descriptor defining an algorithm (See Algorithm 1), in which we employ the computing formula for the sample distance covariance V 2 n (see Sec. III-B2). The final descriptor is expressed by the standardized version R 2 n (L, L). Note that R 2 n (L, L) is actually a scalar value -Eq. [START_REF] Bak | Brownian descriptor: a rich metafeature for appearance matching[END_REF]. Rather than representing an image region by a scalar value, we keep distance coefficients in the form of matrix (R 2 kl ). We believe that this provides finer and more distinctive representation.

Similarly to the classical covariance matrix, the Brownian descriptor is represented by a positive definite symmetric matrix and it provides a natural way of fusing multiple features. This descriptor does not contain any information regarding the order and the number of pixels. This implies a certain scale and rotation invariance over the image regions in different images as long as layers L i are invariant (similarly to the classical covariance descriptor [START_REF] Tuzel | Region covariance: A fast descriptor for detection and classification[END_REF]).

Intuitively, the difference between the classical covariance descriptor and our Brownian is that covariance computes correlation with respect to µ of each feature layer (see Eq. ( 2)), while Brownian statistics are based on distances between all feature layers (a kl ).

1) Extraction complexity

The computation time and memory complexity for both Brownian descriptor and the classical covariance matrix [START_REF] Tuzel | Region covariance: A fast descriptor for detection and classification[END_REF] is the same; the computation complexity for both descriptors is O(n 2 Z), where n is the number of feature layers and Z is the Algorithm 1: Brownian descriptor algorithm number of pixels. For fast descriptor computation, similarly to [START_REF] Tuzel | Region covariance: A fast descriptor for detection and classification[END_REF], we can construct integral images that need to be extracted for each |L k -L l | and for each in Algorithm 1. After computing integral images, the descriptor can be computed in constant time O(1).

Data: Layers L = {L 1 , L 2 , . . . , L n }, L i ∈ R Z Result: Brownian descriptor R 2 kl begin Compute the Euclidean distance matrix (a kl ) (a kl ) = (|L k -L l |) Let A kl = a kl -āk• -ā•l + ā•• , where āk• = 1 n n l=1 a kl (mean of the k-th row) ā•l = 1 n n k=1 a kl (mean of the l-th column) ā•• = 1 n 2 n k,l=1 a kl (mean of distances) Let V 2 n (L) = 1 n 2 n k,l=1 A 2 kl R 2 kl = A 2 kl V 2 n (L)
(-G)=0 = ∑ 𝑫 𝒊𝒋 a) b) c) 2 L = {L 1 , L 2 , . . . , L n } L i 2 R p R 2 (a kl ) (a kl ) = (|L k L l | p A kl = a kl āk• ā•l + ā•• • = 1 n P n l=1 a kl = 1 n P n k=1 a kl = 1 n 2 P n k,l=1 a kl V 2 n (L) = 1 n 2 P n k,l=1 A 2 kl l = A 2 kl V 2 n (L)
(-G)=0 = ∑ 𝑫 𝒊𝒋 a) b) c) ⇢ R 2 n ⇢ = 0 R 2 n > 0 O(n 2 p) |L k L l | R 2 n ⇢ ⇢ R 2 n R 2 n 𝜇 Dj L = {L 1 , L 2 , . . . , L n } L i 2 R p R 2 (a kl ) (a kl ) = (|L k L l | p A kl = a kl āk• ā•l + ā•• āk• = 1 n P n l=1 a kl ā•l = 1 n P n k=1 a kl ā•• = 1 n 2 P n k,l=1 a kl V 2 n (L) = 1 n 2 P n k,l=1 A 2 kl R 2 kl = A 2 kl V 2 n (L)
(-G)=0 = ∑ 𝑫 𝒊𝒋 a) b) c) ⇢ R 2 n ⇢ = 0 R 2 n > 0 O(n 2 p) |L k L l | P R 2 n ⇢ ⇢ R 2 n R 2 n > 0
2) Matching complexity Instead of using geodesic distance or tangent plane projections at the identity matrix, we can directly employ an Euclidean metric for expressing distance between two Brownian descriptors (see Sec. IV-A3 for elaboration). This makes our descriptor computationally efficient in opposite to the classical covariance descriptors. The descriptor performance with respect to several metrics is evaluated in Sec. IV-A. Its efficiency is discussed in Sec. IV-C.

D. R 2

kl vs. ρ kl In the Brownian descriptor ρ kl is replaced by coefficients of R 2 kl for measuring dependence between image features. We claim that R 2 kl coefficients keep more information on dependence between features included in the mapping φ.

Fig. 2 illustrates a comparison between R 2 kl and ρ kl , while handling non-monotone dependency between two feature layers (red and green channels). We can notice that ρ kl ignores non-monotone correlation due to mean-dependent computation (see Eq. ( 2)). It results in ρ kl = 0. This is the fundamental problem of covariance, in which ρ kl may go very close to zero even if the two variables are highly correlated. In contrary, R 2 kl keeps information on the dependence between features even when they exhibit non-monotone or nonlinear correlation. Fig. 1 illustrates a real case where we observe non-monotone correlation between features.

IV. EXPERIMENTAL RESULTS

This section focuses on evaluating the Brownian descriptor on two vision tasks: pedestrian detection in Sec. IV-A and person re-identification in Sec. IV-B. We concentrate on a comparison of the Brownian descriptor with the classical covariance descriptor. Additionally, we carry out an analysis of feature-based descriptors, e.g. HOG, LBP, illustrating superiority of relation-based descriptors. Sec. IV-C discusses the efficiency of the proposed descriptor.

A. Pedestrian detection

Pedestrian detection is an important and complex task in Computer Vision [START_REF] Dollar | Pedestrian detection: An evaluation of the state of the art[END_REF], representing one of the most basic operations in many significant applications such as car assistance [START_REF] Xu | Detection of sudden pedestrian crossings for driving assistance systems[END_REF], video-surveillance, robotics and content-based image/video retrieval. The articulated structure and variable appearance of the human body, combined with illumination and pose variations, different point of views and low image resolution contribute the complexity of the problem in realworld applications. Furthermore, in case of a moving camera in a dynamic environment, changing backgrounds and partial occlusions may cause additional problems.

In this section, we explore the Brownian descriptor and employ it for detecting pedestrians. We carry out our experiments on two challenging data, Daimler Multi-Cue Occluded Pedestrian Classification Benchmark Dataset [START_REF] Enzweiler | Multi-cue pedestrian classification with partial occlusion handling[END_REF] and INRIA Pedestrian Dataset [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] that provide different low-level features (e.g. depth, motion). Figure 3 shows some examples from these datasets.

We evaluate five cases: (1) BROWNIAN; (2) BROWN-IAN P roj. ; (3) COVARIANCE; (4) COVARIANCE P roj. and (5) HOG. By label P roj., we indicate that a descriptor is assumed to be an element of a Riemannian manifold. In this case, we project the descriptor on the tangent plane at the identity matrix [START_REF] Tosato | Characterizing humans on riemannian manifolds[END_REF]. In all cases, we employ a linear SVM [START_REF] Fan | Liblinear: A library for large linear classification[END_REF] for classification.

1) Daimler Multi-Cue Pedestrian Dataset [START_REF] Enzweiler | Multi-cue pedestrian classification with partial occlusion handling[END_REF] This dataset contains 77720 unoccluded positive and 48700 negative samples. These are split into 52122 positive and 32465 negative samples for training and 25608 positive and 16235 negative samples for testing. Each image, of size 96 × 48, is composed of three image modalities: standard visible gray scale image V (x, y), depth D(x, y), and motion flow M (x, y). For each image, we have the following dense feature map F(x, y):

F(x, y) = F V (x, y), F D (x, y), F M (x, y), x, y , (15)
where each 96 × 48 map F V (x, y), F D (x, y) and F M (x, y) represents low-level features extracted from the visible, depth, and motion flow modality, respectively; x and y are the horizontal and vertical pixel coordinates. These two last features are particularly interesting, since they allow to instantiate relations that hold between particular cues and their spatial position. In particular, on each modality we extract the following low-level features:

F i (x,y) = I i , I i x , I i y , I i xx , I i yy , I i 2 x + I i 2 y , LBP (I i ) , (16)
where I i , I i

x , I i y , I i xx and I i yy , are the intensity, first-and second-order derivatives of the three image modalities, and the last term represents the LBP [START_REF] Zheng | Effective pedestrian detection using center-symmetric local binary/trinary patterns[END_REF] 1 . For the depth and motion flow modalities, the depth value and the module of the motion flow are considered as image intensities. Therefore, the resulting number of feature layers is n = 23.

In the first pedestrian detection experiment, we decided to use a simple object model and a simple classifier to demonstrate descriptors' performance (BROWNIAN, BROWNIAN P roj. , COVARIANCE, COVARIANCE P roj. , HOG).

For each image, BROWNIAN and COVARIANCE descriptors are extracted on a set of patches of size 12×12, fusing together the different modalities, resulting in 13 × 5 matrices. The global feature vector fed to a linear SVM classifier is given by (n + n 2 )/2 (276) elements of the vectorized descriptor multiplied by the total number of patches (65). HOG descriptor is extracted in the same way, following the procedure of [START_REF] Enzweiler | A multilevel mixture-of-experts framework for pedestrian classification[END_REF].

We show the classification performance in figure 4(a) using the Detection Trade-Off (DET) curve that expresses the Miss Rate #F alseBackground #T otalP edestrians against the False Positives Rate (FPRate) #F alseP edestrians #T otalBackgrounds on a log scale. One can notice two important results: first, the performances of BROWNIAN and BROWNIAN P roj. are almost similar. This may be due to the fact that a Brownian manifold is sufficiently flat and no projection is needed, saving computational cost time (see section IV-A3 for elaboration). Second, the performance of BROWNIAN and COVARIANCE P roj. are even comparable, demonstrating the quality of our descriptor. In fact, considering a false positives rate of 10 -2 , the miss rate is equal to 0.054 for COVARIANCE P roj. and 0.055 for BROWNIAN, with no statistical differences between the two descriptors. Furthermore, both descriptors are better than HOG, which has a miss rate greater than BROWNIAN and COVARIANCE, equal to 0.093. The HOG descriptor was computed following the same protocol of [START_REF] Enzweiler | A multilevel mixture-of-experts framework for pedestrian classification[END_REF]. As expected, the performance of COVARIANCE without projection is lower than the others descriptors with a miss rate value over 0.1. Similar result was reported in [START_REF] Tuzel | Region covariance: A fast descriptor for detection and classification[END_REF]. This confirms that manifold projection is crucial for achieving good performance by the covariance descriptor.

2) INRIA Pedestrian Dataset [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] This dataset contains 1, 774 pedestrian annotations (3, 548 with reflections) and 1, 671 person-free images. The pedestrian annotations are scaled into fixed size window of 64 × 128 pixels (with a margin of 16 pixels around the pedestrians). We divide the data into two: 2, 416 pedestrian annotations and 1, 218 person-free images for training, and 1, 126 pedestrian annotations and 453 person-free images for testing. Detection on the INRIA pedestrian dataset is challenging since it includes subjects with variations in pose, clothing, illumination, background, and partial occlusions. The framework used to evaluate our descriptor is the same of [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] and [START_REF] Tuzel | Pedestrian detection via classification on riemannian manifolds[END_REF]. We detect pedestrians on each test image (positives and negatives) in all positions and scale, computing the descriptors on a patch size of 16 × 16 pixels with a step size of 8 × 8 and a scale factor of 1.2. Multi-scale and nearby detections are merged using non maximal suppression and a list of detected bounding boxes are given out. Evaluation on the list of detected bounding boxes is done using the PASCAL criterion which counts a detection to be correct if the overlap of the detected bounding box and ground truth bounding box is greater than 0.5. For each sliding window, we have the following dense feature map F(x, y):

[x, y, Φ R , Φ G , Φ B , M R , M G , M B , R, G, B, LBP (I)] (17)
where x and y represent horizontal and vertical pixel coordinates, Φ represents the first-order gradient vector, M its magnitude, R, G B are the channels values and LBP is computed on intensity. As in the previous experiment, the performances are evaluated by adopting the Detection Error Trade-Off (DET) curve. Putting a threshold on the SVM scores, between -5 and 5, we obtain the DET curves (figure 4(b)). In this dataset, we compared our BROWNIAN descriptor with COVARIANCE and HOG. Covariance descriptor was already compared to HOG in [START_REF] Tuzel | Pedestrian detection via classification on riemannian manifolds[END_REF], showing that it outperforms HOG [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF]. However, due to time complexity and sophisticated manifold learning required for the covariance, HOG has been applied more often to pedestrian detection task.

The results clearly show that BROWNIAN descriptor outperforms COVARIANCE and HOG (experiments for HOG and COVARIANCE are done following the same framework of [START_REF] Tuzel | Pedestrian detection via classification on riemannian manifolds[END_REF] and [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] and results are reported in Fig 4(b)) and Table I. Considering a false positives rate per image equal to 10 0 , the equivalent miss rate values for BROWNIAN, COVARIANCE P roj. and HOG are 0.34, 0.38 and 0.45, respectively. This result illustrates that with the same number of false positive per image, BROWNIAN is able to detect 4% more pedestrians with respect to COVARIANCE, and 11% more with respect to HOG. The good performance of Brownian demonstrates good encoding of nonlinear relations that covariance fails to capture.

3) Manifold curvature analysis The previous result surprisingly develops that Brownian descriptor, an element of Sym + d , performs relatively good in a Euclidean space. To further investigate this phenomenon, we employ a quantitative measure of nonflatness of the manifold, that is the sectional curvature κ p [START_REF] Tosato | Characterizing humans on riemannian manifolds[END_REF].

Given a Riemannian manifold (M, , ), its sectional curvature κ p (X p , Y p ) at p ∈ M, if X p and Y p are linearly independent tangent vectors at p, is given by: where R is denoting the Riemann curvature operator.

κ p (X p , Y p ) = R(X p , Y p )X p , Y p p X p , X p p Y p , Y p p -X p ,
If we use the identity matrix as a projection point p = I d , we can re-write the formula 18 as:

κ I d (X, Y ) = R(X, Y )X, Y X 2 Y 2 -X, Y 2 = 2 T r((XY ) 2 -X 2 Y 2 ) T r(X 2 )T r(Y 2 ) -(T r(XY )) 2 , (19)
where T r is the trace operator. The sectional curvature for Sym + d is nonpositive at any point. The lower κ I d , the stronger a Riemannian differs from a flat one (i.e. Euclidean).

The numerical evaluation of the curvature κ I d in correspondence to training samples of a particular descriptor allows us to understand concavity of the related manifolds. Having extracted BROWNIAN and COVARIANCE descriptors for INRIA dataset, we compute the mean value and the standard deviation of κ I d for both descriptors (see Table II). The mean value obtained for Brownian manifold is twice larger than for the Covariance manifold with also smaller standard deviation. That confirms our hypothesis that the manifold of the Brownian is flatter than the one of the Covariance. This might suggest that the Brownian manifold is sufficiently flat and no projection is needed for achieving good performance.

B. Person re-identification

Person re-identification is a visual search of the same person across a network of non-overlapping cameras. This task requires models dealing with significant appearance changes caused by variations in lighting conditions, pose changes and sensor scarce resolution. It is crucial that these models are based on visual features, which show a good trade-off between their discriminative power and invariance to camera changes. This trade-off can be learned [START_REF] Koestinger | Large scale metric learning from equivalence constraints[END_REF] but it requires significant amount of labeled data which might be unattainable in a large camera network. Alternatively, it has been shown that relation-based descriptors perform relatively well in the re-identification scenario [START_REF] Hirzer | Person re-identification by descriptive and discriminative classification[END_REF], [START_REF] Bak | Multiple-shot human reidentification by mean riemannian covariance grid[END_REF], [START_REF] Bak | Boosted human reidentification using riemannian manifolds[END_REF], [START_REF] Bak | Learning to match appearances by correlations in a covariance metric space[END_REF], [START_REF] Cai | Matching groups of people by covariance descriptor[END_REF] but they usually involve a high computational cost. In this section, we show that our descriptor captures distinctive information while showing practical invariance to appearance changes and keeping computational efficiency. We carry out experiments on four various re-identification datasets evaluating descriptor performance on different challenges: significant variations in illumination -PRID2011 [START_REF] Hirzer | Person re-identification by descriptive and discriminative classification[END_REF]; low resolution images -CAVIAR4REID [START_REF] Cheng | Custom pictorial structures for re-identification[END_REF]; cluttered environments with occlusions -i-LIDS [START_REF] Zheng | Associating groups of people[END_REF]; and serious perspective and pose changes -SAIVT-SOFTBIO database [START_REF] Bialkowski | A database for person re-identification in multi-camera surveillance networks[END_REF].

1) Experimental setup

In the past few years the re-identification problem has been the focus of intense research bringing proper metrics and datasets for evaluation. Re-identification performance is analyzed in terms of recognition rate, using the averaged cumulative matching characteristic (CMC) curve [START_REF] Gray | Evaluating Appearance Models for Recognition, Reacquisition, and Tracking[END_REF]. The CMC represents the expectation of finding the correct match in the top matches. The nAUC is a scalar obtained by normalizing the area under the CMC curve.

Every human annotation is scaled into a fixed size window of 64 × 192 pixels. The set of rectangular sub-regions is produced by shifting 32 × 32 regions with a step size of 16 pixels in either direction. This results in 33 overlapping rectangular sub-regions. From each sub-region, we extract 5 descriptors; three histogram-based descriptors: [START_REF] Bay | Speeded-up robust features (surf)[END_REF] A3) in these experiments we assume BROWNIAN to be an element of a Euclidean space for avoiding expensive projections on the tangent plane.

For each subject we compute signatures using randomly selected K consecutive images. We evaluated both singleshot (K = 1) and multiple-shot (K > 1) scenario. In multiple-shot case, descriptor values are simply averaged to encode a set of K images depicting the same subject. Every signature is used as a query to the gallery set of signatures from different cameras. The procedure is repeated 10 times to produce average CMC curves and nAUC values.

2) PRID2011 dataset [START_REF] Hirzer | Person re-identification by descriptive and discriminative classification[END_REF] The PRID2011 dataset consists of person images recorded from two different static surveillance cameras. Images are extracted from trajectories providing roughly 50 to 100 images per subject and camera view. Characteristic challenges of this dataset are significant differences in illumination, viewpoint and pose changes (see Fig. 5). Although, one camera view contains up to 749 subjects, only 200 person appear in both cameras. In our evaluation we used only these 200 subjects. We selected K = 1 and K = 20. Fig. 6 illustrates a comparison between different descriptors. We can notice that COLOR histograms are less discriminative than other descriptors. In particular, significant difference is found for multi-image signatures (K = 20). This effect can be explained by strong illumination changes (compare rows in Fig. 5) and reasonable image quality. Although all images are re-scaled to a uniform size (64 × 192), the original person images were typically 100 to 200 pixels high. This yields better quality images containing edge and texture information. The best performance among all descriptors is obtained by the BROWNIAN descriptor. The recognition accuracy per rank is given in Table III.

3) CAVIAR4REID dataset [START_REF] Cheng | Custom pictorial structures for re-identification[END_REF] This dataset comes from the CAVIAR project 2 . Video clips are recorded from two different points of view in a shopping center in Lisbon. The resolution is half-resolution PAL standard (384 × 288 pixels, 25 frames per second). Small appearances and a very low resolution in one of the two cameras make the appearance matching very challenging (see Fig. 7). Of the 72 different individuals identified (with images varying from 17 ⇥ 39 to 72 ⇥ 144), 50 are captured by both views and 22 from only one camera. In our evaluation we only used these 50 people leaving out the remaining 22. The dataset provides up to 10 images per subject and camera, thus we assumed K = 1 and K = 5 for evaluation. The average CMC curves are displayed in Fig. 8.

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n)
Interestingly, neither BROWNIAN nor COVARIANCE obtained the best performance. The best accuracy is achieved by simple COLOR histogram. We believe that this is due to the significant changes in camera resolution and illumination conditions. In Fig. 7, we can notice that low resolution images barely contain edge and texture information. Note that BROWNIAN and COVARIANCE maps (Eq. ( 20)) mainly focus on extracting correlations between gradient-based layers in opposite to quantitative descriptors like color histograms. This experiment illustrates that both, BROWNIAN and COVARIANCE are quite sensitive to low resolution images (some persons in this dataset are less than 40 pixels height). The low resolution explains as well low performance of remaining descriptors, which mostly exploit texture information that in fact is not present in these images. However, although BROWNIAN has slightly worse performance than COLOR histogram, it outperforms all remaining descriptors for both K = 1 and K = 5 modalities. Table IV provides detailed results with respect to the considered rank and the number of K images.

4) SAIVT-SOFTBIO database [START_REF] Bialkowski | A database for person re-identification in multi-camera surveillance networks[END_REF] This dataset consists of 152 people moving through a network of 8 cameras. Subjects travel in uncontrolled manner thus most of subjects appear only in a subset of the camera network. This provides a highly unconstrained environment reflecting a real-world scenario. In average, each subject is registered by 400 frames spanning up to 8 camera views in challenging surveillance conditions (significant illumination, pose and view point changes, see Fig. 9(a)). Each camera captures data at 25 frames per second at resolution of 704 ⇥ 576 pixels. Although some cameras have overlap, we do not use this information while testing re-identification algorithms. Authors [START_REF] Bialkowski | A database for person re-identification in multi-camera surveillance networks[END_REF] provide XML files with annotations given by coarse bounding boxes indicating the location of the subjects. For each subject we randomly select the first frame in such way that we can create the signature from the next K = 75 frames. Every signature is used as a query to the gallery set of signatures from the other cameras. This procedure has been repeated 10 times to obtain averaged CMC results.

As SAIVT-SOFTBIO consists of several cameras, we display the CMC results using 3D bar-charts (see Fig. 10). The horizontal axis corresponds to recognition accuracy, while on the vertical axis the first 25 ranks are presented for each camera pair (i.e. having 8 cameras we actually can produce 56 CMC bar series that present recognition accuracy for each camera pair). We also color the CMC bars with respect to recognition accuracy and display it as a top-view image of 3D bar. In the result we can see that re-identification accuracy might be strongly associated with a particular pair of cameras (similar/non-similar camera view, resolution, the number of registered subjects). Fig. 10(a-e) illustrates the retrieval results for each descriptor. From the results it is apparent that Brownian descriptor outperforms the rest of descriptors. Table V shows the averaged (among all 56 camera pairs) recognition accuracy with respect to the rank and Fig. 11(a) illustrates averaged CMC curves. We can see that the Brownian descriptor consistently achieves the best performance for all ranks.

5) i-LIDS dataset [45]

Since the achieved performance showed the advantage of the Brownian descriptor, we have decided to employ MRCG appearances and a very low resolution in one of the two cameras make the appearance matching very challenging (see Fig. 7). Of the 72 different individuals identified (with images varying from 17 × 39 to 72 × 144), 50 are captured by both views and 22 from only one camera. In our evaluation we only used these 50 people leaving out the remaining 22. The dataset provides up to 10 images per subject and camera, thus we assumed K = 1 and K = 5 for evaluation. The average CMC curves are displayed in Fig. 8.

Interestingly, neither BROWNIAN nor COVARIANCE obtained the best performance. The best accuracy is achieved by simple COLOR histogram. We believe that this is due to the significant changes in camera resolution and illumination conditions. In Fig. 7, we can notice that low resolution images barely contain edge and texture information. Note that BROWNIAN and COVARIANCE maps (Eq. ( 20)) mainly focus on extracting correlations between gradient-based layers in opposite to quantitative descriptors like color histograms. This experiment illustrates that both, BROWNIAN and COVARIANCE are quite sensitive to low resolution images (some persons in this dataset are less than 40 pixels height). The low resolution explains as well low performance of remaining descriptors, which mostly exploit texture information that in fact is not present in these images. However, although BROWNIAN has slightly worse performance than COLOR histogram, it outperforms all remaining descriptors for both K = 1 and K = 5 modalities. Table IV provides detailed results with respect to the considered rank and the number of K images.

4) SAIVT-SOFTBIO database [START_REF] Bialkowski | A database for person re-identification in multi-camera surveillance networks[END_REF] This dataset consists of 152 people moving through a network of 8 cameras. Subjects travel in uncontrolled manner thus most of subjects appear only in a subset of the camera network. This provides a highly unconstrained environment reflecting a real-world scenario. In average, each subject is registered by 400 frames spanning up to 8 camera views in challenging surveillance conditions (significant illumination, pose and view point changes, see Fig. 9(a)). Each camera captures data at 25 frames per second at resolution of 704 × 576 pixels. Although some cameras have overlap, we do not use this information while testing re-identification algorithms. Authors [START_REF] Bialkowski | A database for person re-identification in multi-camera surveillance networks[END_REF] provide XML files with annotations given by coarse bounding boxes indicating the location of the subjects. For each subject we randomly select the first frame in such way that we can create the signature from the next K = 75 frames. Every signature is used as a query to the gallery set of signatures from the other cameras. This procedure has been repeated 10 times to obtain averaged CMC results.

As SAIVT-SOFTBIO consists of several cameras, we display the CMC results using 3D bar-charts (see Fig. 10). The horizontal axis corresponds to recognition accuracy, while on the vertical axis the first 25 ranks are presented for each camera pair (i.e. having 8 cameras we actually can produce 56 CMC bar series that present recognition accuracy for each camera pair). We also color the CMC bars with respect to recognition accuracy and display it as a top-view image of 3D bar. In the result we can see that re-identification accuracy might be strongly associated with a particular pair of cameras (similar/non-similar camera view, resolution, the number of registered subjects). Fig. 10(a-e) illustrates the retrieval results for each descriptor. From the results it is apparent that Brownian descriptor outperforms the rest of descriptors. Table V shows the averaged (among all 56 camera pairs) recognition accuracy with respect to the rank and Fig. 11(a) illustrates averaged CMC curves. We can see that the Brownian descriptor consistently achieves the best performance for all ranks.

5) i-LIDS dataset [45]

Since the achieved performance showed the advantage of the Brownian descriptor, we have decided to employ MRCG model [START_REF] Bak | Multiple-shot human reidentification by mean riemannian covariance grid[END_REF] that consists of a dense grid of covariances, and replace the classical covariance with BROWNIAN (without pro-

C. Descriptor effectiveness and efficiency

The significant improvement can be noticed on person reidentification. This confirms that the Brownian descriptor is less dependent on camera parameters than the covariance. We believe that it is due to the descriptor design based on statistics computed on distances between all feature layers. It bears out that this descriptor reaches sufficient trade off between discriminative power and camera invariance. Thus, we recommend Brownian as a valuable descriptor for vision tasks that require camera independence.

Moreover, other main benefits of using Brownian come from the significant speed-up in matching/classification without losing descriptive properties. Instead of projecting the descriptor on the tangent plane or using the geodesic distance, we can directly use a Euclidean metric (see Sec. IV-A3). In the results, we speedup 4 times the whole pedestrian detection framework in comparison with the classical covariance (feature extraction & classification).

For the same set of features, the proposed Brownian descriptor achieves similar or better accuracy than the classical covariance based descriptor. Importantly, the matching accuracy by Brownian is achieved at n 2 times faster speed than classical covariance (see Fig. 12(a)). Note that the larger number of feature layers n in a descriptor, the bigger speedup is achieved (theoretically speedup is lower bounded by o(n 2 ) due to SVD computation in geodesic distance for covariance). This has a tremendous impact on the re-identification task, where speedup in matching has a direct effect on the whole retrieval framework. Fig. 12 illustrates comparison of distance computation while using a Euclidean distance, distance with a projection on a tangent plane at the identity matrix and a geodesic distance. Time complexity in Fig. 12(b) was computed on Intel quadcore 2.4GHz without applying any hardware-dependent optimization routines (e.g. no block operations optimized for architecture). We can notice that matching is significantly faster applying a Euclidean metric.

V. CONCLUSIONS In this paper, we introduced a novel descriptor based on mathematical statistics related to Brownian covariance. This new descriptor can be seen as a natural extension of the classical covariance descriptor. While the classical covariance measure is limited to model only linear dependencies between features, the Brownian descriptor is capable to measure all kinds of possible relationships between low-level features of visual entities.

The advantages of the proposed descriptor were presented by the theoretical analysis and the experimental evaluation on different vision tasks. We extensively evaluated the proposed approach, outperforming covariance on INRIA pedestrian detection dataset and bringing novel state-of-the-art performance in person re-identification. The significant improvement on person re-identification task w.r.t. the classical covariance suggests that Brownian descriptor indeed helps in correlating non linearly related features and hence can be applied to many vision tasks requiring camera invariant descriptors.

We shown not only the effectiveness of the Brownian descriptor, but also elaborate its efficiency. For computing the distance between two Brownian descriptors, we can use a Euclidean metric that is o(n 2 ) times faster than the classical geodesic distance, where n is the number of feature layers.

We believe that this descriptor is valuable beyond the scope of the presented applications and can be used in many diverse scenarios. In future, we plan to integrate Brownian descriptors with alternative classifiers (e.g. boosting, decision trees) and apply it to other vision tasks. Furthermore, detailed analysis of layer-significance w.r.t. the vision task will be investigated. ACKNOWLEDGMENT This work has also been supported by PANORAMA European project.

Fig. 1 .

 1 Fig. 1. Nonlinear dependency between two features extracted from a patch. Intensity values and gradient magnitudes are plotted together to illustrate the relation.

ρ

  RG =R*G+(--R*0)+R*(--G)=0 RG = (-R*0)+R*

  (-R*0)+R*

  (-R*0)+R*

Fig. 2 .

 2 Fig. 2. Comparison of R 2 kl vs. ρ kl . For the sake of simplicity we consider only 3 pixel values of two feature layers -Red and Green channel. ρ kl = 0 and R 2 kl > 0, while actually layers Red and Green are in non-monotone correlation.

Fig. 3 .

 3 Fig. 3. Positive and negative examples extracted from the two pedestrian datasets.

Fig. 5 .

 5 Fig. 5. The sample images from PRID2011 dataset. Top and bottom lines correspond to images from different cameras. We illustrate the first 10 subjects from the dataset.

Fig. 6 .

 6 Fig. 6. Comparison of different descriptors using CMC curves on PRID2011 dataset: signatures have been computed using (a) K = 1 and (b) K = 20 images.

Fig. 7 .

 7 Fig. 7. The sample images from CAVIAR dataset. Pairs of images highlight appearance changes due to different camera resolution and illumination conditions.

Fig. 8 .

 8 Fig. 8.Comparison of different descriptors using CMC curves on CAVIAR4REID dataset: signatures have been computed using (a) K = 1 and (b) K = 5 images.

Fig. 7 .

 7 Fig. 7. The sample images from CAVIAR dataset. Pairs of images highlight appearance changes due to different camera resolution and illumination conditions.

Fig. 8 .

 8 Fig. 8.Comparison of different descriptors using CMC curves on CAVIAR4REID dataset: signatures have been computed using (a) K = 1 and (b) K = 5 images.

Fig. 11 .

 11 Fig. 11. Performance comparison using CMC curves: (a) averaged values among all 56 pairs of cameras; (b) our vs. MRCG[START_REF] Bak | Multiple-shot human reidentification by mean riemannian covariance grid[END_REF], CPS[START_REF] Cheng | Custom pictorial structures for re-identification[END_REF], SDALF[START_REF] Bazzani | Symmetry-driven accumulation of local features for human characterization and re-identification[END_REF], Group Context [49].

Fig. 12 .

 12 Fig.[START_REF] Lowe | Object recognition from local scale-invariant features[END_REF]. Comparison of time complexity with respect to a chosen metric: a Euclidean metric, a tangent projection at the identity matrix and a geodesic distance.

  Pedestrian detection, DET curves for comparison between BROWNIAN, BROWNIAN P roj. , COVARIANCE, COVARIANCE P roj. and HOG [3]: (a) Daimler Multi-Cue dataset; (b) INRIA dataset.

	Miss Rate	0.1 0.15 0.2 0.25 0.3 0.35		Brownian Brownian Covariance Proj Covariance Proj HOG	Miss Rate	1 0.4 0.6 0.8	Brownian Brownian Proj Covariance Covariance Proj HOG
		0.05
		10 -3	10 -2 False positives per window (FPPW)	10 -1		10 -1 0.2	10 0 False positives per image (FPPI)	1 10
			(a) Daimler Multi-Cue dataset				(b) INRIA dataset
	Fig. 4. DESCRIPTOR BROWNIAN BROWNIAN P roj. COVARIANCE COVARIANCE P roj. HOG	Miss Rate 0.343 0.355 0.673 0.379 0.448
	TABLE I PEDESTRIAN DETECTION. MISS RATE VALUES ON INRIA DATASET FOR A FALSE POSITIVE RATE EQUAL TO 10 0 :

  Y p 2

	MANIFOLD	mean	standard deviation
	BROWNIAN	-0.1000	±0.0811
	COVARIANCE	-0.2066	±0.1398
	TABLE II CURVATURE ANALYSIS OF Sym + d .
	(18)
	p

  COLOR RGB histogram, (2) LBP histogram and (3) HOG histogram, and two correlation-based descriptors: (4) COVARIANCE P roj and (5) BROWNIAN. We employ 11-dimensional feature map from [28]:

	x, y, R xy , G xy , B xy , ∇ R xy , θ R xy , ∇ G xy , θ G xy , ∇ B xy , θ B xy	(20)

where x and y are pixel location, R xy , G xy , B xy are RGB channel values and ∇ and θ corresponds to gradient magnitude and orientation in each channel, respectively. Motivated by the curvature analysis of Sym + d (section IV-

  CAVIAR webpage: http://homepages.inf.ed.ac.uk/rbf/CAVIAR

	K=1	DESCRIPTOR BROWNIAN COLOR LBP HOG COVARIANCE	r = 1 11.43% 8.16% 9.98% 9.62% 11.41%	r = 5 28.14% 17.92% 23.11% 20.83% 25.55%	r = 10 40.17% 27.12% 29.22% 28.14% 35.39%	r = 25 58.93% 40.09% 48.41% 42.79% 52.13%
	K=20	BROWNIAN COLOR LBP HOG COVARIANCE	r = 1 24.31% 11.32% 23.51% 24.02% 23.73%	r = 5 49.13% 26.33% 39.64% 44.81% 42.55%	r = 10 62.22% 34.55% 51.51% 54.92% 52.19%	r = 25 78.11% 86.32% 64.91% 68.61% 69.43%

2

TABLE III DESCRIPTOR

 III

PERFORMANCE COMPARISON ON PRID2011 DATASET AT DIFFERENT RANKS r.

  SOFTBIO DATASET. VALUES CORRESPOND TO THE RECOGNITION ACCURACY AVERAGED AMONG ALL 56 PAIRS OF CAMERAS AT DIFFERENT RANKS r.

	TABLE IV DESCRIPTOR PERFORMANCE COMPARISON ON CAVIAR4REID DATASET
		AT DIFFERENT RANKS r.
	DESCRIPTOR BROWNIAN COLOR LBP HOG COVARIANCE	r = 1 16.06% 8.12% 11.30% 12.02% 15.83%	r = 5 37.53% 22.09% 25.62% 26.73% 33.65%	r = 10 49.37% 32.79% 38.92% 40.64% 45.09%	r = 25 72.09% 54.60% 59.91% 62.89% 66.13%
	TABLE V DESCRIPTOR PERFORMANCE COMPARISON ON SAIVT-

TABLE V DESCRIPTOR

 V PERFORMANCE COMPARISON ON SAIVT-SOFTBIO DATASET. VALUES CORRESPOND TO THE RECOGNITION ACCURACY AVERAGED AMONG ALL 56 PAIRS OF CAMERAS AT DIFFERENT RANKS r.

  Sample images from: (a) SAIVT-SOFTBIO dataset. Rows correspond to 2 first subjects from the database, while columns show the appearance from different camera views (notice significant difference in appearance); (b) i-LIDS dataset. Top and bottom lines correspond to images from different cameras. Columns illustrate the same person. Fig.10. Descriptor performances as CMC bars for 56 camera pairs (a-e) of SAIVT-SOFTBIO dataset. nAUC is a weighted (by gallery size) average of nAUC obtained by each pair of cameras. For each descriptor the top view and 3D chart is presented. Red color indicates high recognition accuracy. For each descriptor we can notice the red region on the top view (see rows 7 -14). This is the retrieval result for the second camera in which only few subjects were registered (29 out of 152). The rest of cameras is more balanced (about 100 subjects per camera). (f) illustrates the difference between BROWNIAN and COVARIANCE P roj . We can notice that BROWNIAN performed better for most of camera pairs (bluish color correspond to opposite case).

	C1	C3	C4	C5	C6	C7	C8
	ID = 1
	ID = 2
		(a) SAIVT-SOFTBIO dataset			(b) i-LIDS dataset
	Fig. 9. (a) COLOR, nAUC=0.65			(b) LBP, nAUC=0.70	(c) HOG, nAUC=0.69
	(d) COV P roj , nAUC=0.73			(e) BROW, nAUC=0.79	(f) (e)-(d), ∆nAUC=0.06
	jection). This allows us to compare the descriptor with state-of-the-art approaches on i-LIDS data. This dataset contains 476 images with 119 individuals registered by two different cameras. It is very challenging dataset since there are many (see Fig. 9(b)). We reproduce the same experimental settings as occlusions and often only the top part of the person is visible	METHOD OUR MRCG[28] CPS[44] SDALF[48] GROUP[49]	r = 1 48.75% 45.79% 44.02% 38.93% 25.98%	r = 5 75.09% 66.80% 69.30% 64.61% 45.03%	r = 10 83.75% 75.21% 76.31% 74.19% 55.07%	r = 25 96.70% 85.71% 86.09% 85.13% 70.01%

[START_REF] Bak | Multiple-shot human reidentification by mean riemannian covariance grid[END_REF]

. Signatures are generated using N = 2 images. Fig.

11(b

) illustrates a comparison with re-identification state-of-the-art approaches. We see that combination of MRCG technique with the Brownian descriptor outperforms state-of-the-art performance. Table VI provides the recognition accuracy with respect to the considered rank.

TABLE VI PERFORMANCE

 VI COMPARISON ON I-LIDS DATASET WITH STATE-OF-THE-ART APPROACHES AT DIFFERENT RANKS r.

Note that here LBP is employed as a low-level feature which provides just a single value, from 0 to

255, for each image pixel.

[49] Schwartz, W.R., Davis, L.S.: Learning discriminative appearance-based models using partial least squares. In: Conference on Graphics, Patterns and Images (SIBGRAPI). ( 2009


With some of my colleagues, I'd like to put musical scores online. But I don't know what I can do and how to proceed! Can you help me? Well Tork, use the SSK! I think the scenario "Collaborative Digital Edition of a Musical Corpus" can be very useful for you. It will guide you through all the necessary steps to put these scores online, and present the best practices associated to each of these steps.

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  With some of my colleagues, I'd like to put musical scores online. But I don't know what I can do and how to proceed! Can you help me? Well Tork, use the SSK! I think the scenario "Collaborative Digital Edition of a Musical Corpus" can be very useful for you. It will guide you through all the necessary steps to put these scores online, and present the best practices associated to each of these steps.

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