Jia-Bin Huang
email: jbhuang@iis.sinica.edu.tw

Chu-Song Chen

Learning Moving Cast Shadows for Foreground Detection

We present a new algorithm for detecting foreground and moving shadows in surveillance videos. For each pixel, we use the Gaussian Mixture Model (GMM) to learn the behavior of cast shadows on background surfaces. The pixelbased model has the advantages over regional or global model for their adaptability to local lighting conditions, particularly for scenes under complex illumination conditions. However, it would take a long time for convergence if motion is rare on that pixel. We hence build a global shadow model that uses global-level information to overcome this drawback. The local shadow models are updated through confidence-rated GMM learning, in which the learning rate depends on the confidence predicted by the global shadow model. For foreground modeling, we use a nonparametric density estimation method to model the complex characteristics of the spatial and color information. Finally, the background, shadow, and foreground models are built into a Markov random field energy function that can be efficiently minimized by the graph cut algorithm. Experimental results on various scene types demonstrate the effectiveness of the proposed method.

Introduction

Moving object detection is at the core of many computer vision applications, including video conference, visual surveillance, and intelligent transportation system. However, moving cast shadow points are often falsely labeled as foreground objects. This may severely degrade the accuracy of object localization and detection. Therefore, an effective shadow detection method is necessary for accurate foreground segmentation.

Moving shadows in the scene are caused by the occlusion of light sources. Shadows reduce the total energy incident at the background surfaces where the light sources are partially or totally blocked by the foreground objects. Hence, shadow points have lower luminance values but similar chromaticity values. Also, the texture characteristic around the shadow points remains unchanged since shadows do not alter the background surfaces.

Much research has been devoted to moving cast shadow detection. Some methods are based on the observation that the luminance values of shadowed pixels decrease respect to the corresponding background while maintaining chromaticity values. For examples, Horprasert et al. [START_REF] Horprasert | A statistical approach for real-time robust background subtraction and shadow detection[END_REF] used a computational color model which separates the brightness from chromaticity component and define brightness and chromaticity distortion. Cucchiara et al. [START_REF] Cucchiara | Detecting moving objects, ghosts, and shadows in video streams[END_REF] and Schreer et al. [START_REF] Schreer | Fast and robust shadow detection in videoconference applications[END_REF] addressed the shadow detection problem in HSV and YUV color space respectively and detected shadows by exploiting the color differences between shadow and background. Nadimi et al. [START_REF] Nadimi | Physical models for moving shadow and object detection in video[END_REF] proposed a spatial-temporal albedo test and a dichromatic reflection model to separate cast shadows for moving objects.

Texture features extracted from spatial domain had also been used to detect shadows. Zhang et al. [START_REF] Wei | Moving cast shadows detection using ratio edge[END_REF] proved that ratio edge is illumination invariant and the distribution of normalized ratio edge difference is a chi-square distribution. A significance test was then used to detect shadows. Fung et al. [START_REF] Fung | Effective moving cast shadow detection for monocular color traffic image sequences[END_REF] computed a confidence score for shadow detection based on the characteristics of shadows in luminance, chrominance, gradient density, and geometry domains. However, the above-mentioned methods require to set parameters for different scenes and can not handle complex and time-varying lighting conditions. A comprehensive survey of moving shadow detection approaches was presented in [START_REF] Prati | Detecting moving shadows: algorithms and evaluation[END_REF].

Recently, the statistical prevalence of cast shadows had been exploited to learn shadows in the scenes. In [START_REF] Martel-Brisson | Learning and removing cast shadows through a multidistribution approach[END_REF], Martel-Brisson et al. used the Gaussian mixture model (GMM) to describe moving cast shadows. Proikli et al. [START_REF] Porikli | Shadow flow: a recursive method to learn moving cast shadows[END_REF] proposed a recursive method to learn cast shadows. Liu et al. [START_REF] Liu | Cast shadow removal combining local and global features[END_REF] presented to remove shadow using multi-level information in HSV color space. The major drawback in [START_REF] Martel-Brisson | Learning and removing cast shadows through a multidistribution approach[END_REF] and [START_REF] Porikli | Shadow flow: a recursive method to learn moving cast shadows[END_REF] is that their shadow models need a long time to converge while the lighting conditions should remain stable. Moreover, in [START_REF] Martel-Brisson | Learning and removing cast shadows through a multidistribution approach[END_REF], the shadow model is merged into the background model. The Gaussian states for shadows will be discarded in the background model when there are no shadows for a long period. Therefore, these two approaches are less effective in a real-world environment. Liu et al. [START_REF] Liu | Cast shadow removal combining local and global features[END_REF] attempted to improve the convergence speed of pixel-based shadow model by using multi-level information. They used region-level information to get more samples and globallevel information to update a pre-classifier. However, the method still suffers from the slow learning of the conventional GMM [START_REF] Stauffer | Adaptive background mixture models for real-time tracking[END_REF], and the pre-classifier will become less discriminative in scenes having different types of shadows, e.g. light or heavy.

Another foreground and shadow segmentation approaches are through Bayesian methods, which construct background, shadow, and foreground model to evaluate the data likelihood of the observed values [START_REF] Wang | A dynamic conditional random field model for foreground and shadow segmentation[END_REF], [START_REF] Benedek | Bayesian foreground and shadow detection in uncertain frame rate surveillance videos[END_REF]. These two approaches used global parameters to characterize shadows in an image, which are constant in [START_REF] Wang | A dynamic conditional random field model for foreground and shadow segmentation[END_REF] and probabilistic in [START_REF] Benedek | Bayesian foreground and shadow detection in uncertain frame rate surveillance videos[END_REF]. Although the global models are free from the convergence speed problem, they lose the adaptability to local characteristics and cannot handle scenes with complex lightings.

In this paper, we propose a Bayesian approach to moving shadow and object detection. We learn the behavior of cast shadows on surfaces by GMM for each pixel. A global shadow model (GSM) is maintained to improve the convergence speed of the local shadow model (LSM). This is achieved by updating the LSM through confidencerated GMM learning, in which the learning rates are directly proportional to the confidence values predicted by the GSM. The convergence speed of LSM is thus significantly improved. We also present a novel description for foreground modeling, which uses local kernel density estimation to model the spatial color information instead of temporal statistics. The background, shadow, and foreground are then segmented by the graph cut algorithm, which minimizes a Markov Random Field (MRF) energy. A flow diagram of the proposed algorithm is illustrated in Fig. 1.

The remainder of this paper is organized as follows. The formulation of the MRF energy function is presented in Section 2. In Section 3, we describe the background and shadow model, primarily focusing on how to learn cast shadows. Section 4 presents a nonparametric method for estimating foreground probability. Experimental results are presented in Section 5. Section 6 concludes the paper.

Energy Minimization Formulation

We pose the foreground/background/shadow segmentation as an energy minimization problem. Given an input video sequence, a frame at time t is represented as an array z t = (z t (1), z t (2), ...z t (p), ...z t (N )) in RGB color space, where N is the number of pixels in each frame. The segmentation is to assign a label l p to each pixel p ∈ P , where P is the set of pixels in the image and the label The global labeling field L = {l p |p ∈ P } is modeled as MRF [START_REF] Geman | Stochastic relaxation, gibbs distributions, and the bayesian restoration of images[END_REF]. The first order energy function of MRF can be decomposed as a sum of two terms:

E(L) = E data (L) + E smooth (L), (1)
= p∈P [D p (l p ) + q∈Np V p,q (l p , l q )]
in which the first term E data (L) evaluates the likelihood of each pixel belonging to one of the three classes, the second term E smooth (L) imposes a spatial consistency of labels through a pairwise interaction MRF prior, D p (l p ) and V p,q (l p , l q ) are the data and smoothness energy terms respectively, and the set N p denotes the 4-connected neighboring pixels of point p. The energy function in (1) can be efficiently minimized by the graph cut algorithm [START_REF] Boykov | Fast approximate energy minimization via graph cuts[END_REF], so that an approximating maximum a posteriori (MAP) estimation of the labeling field can be obtained.

The Potts model [START_REF] Potts | Some generalized order-disorder transformations[END_REF] is used as a MRF prior to stress spatial context and the data cost E data (L) is defined as

E data (L) = p∈P D p (l p ) = p∈P -log p(z p |l p ), (2)
where p(z p |l p ) is the likelihood of a pixel p belonging to background, shadow, or foreground. In the following sections, we will show how to learn and compute these likelihoods.

Background and Shadow Model

Modeling the Background

For each pixel, we model the background color information by the well-known GMM [START_REF] Stauffer | Adaptive background mixture models for real-time tracking[END_REF] with K BG states in RGB color space. The first B states with higher weights and smaller variances in the mixture of K BG distributions are considered as background model. The index B is determined by

B = argmin b b k=1 ω BG,k > T BG , (3)
where T BG is the pre-defined weight threshold, and ω BG,k is the weight of the k th Gaussian state in the mixture model. The likelihood of a given pixel p belonging to the background can be written as (the subscript time t is ignored):

p(z(p)|l p = BG) = 1 W BG B k=1 ω BG,k G(z(p), µ BG,k , Σ BG,k ), (4
) where W BG = B j=1 ω BG,j is a normalization constant, and G(z(p), µ BG,k , Σ BG,k ) is the probability density function of the k th Gaussian with parameters θ BG,k = {µ BG,k , Σ BG,k }, in which µ BG,k is the mean vector and Σ BG,k is the covariance matrix.

Learning Moving Cast Shadow

This subsection presents how to construct the shadow models from the background model. Since different foreground objects often block the light sources in a similar way, cast shadows on background surfaces are thus similar and independent of foreground objects. The ratio of shadowed and illuminated value of a given surface point is considered to be nearly constant. We exploit this regularity of shadows to describe the color ratio of a pixel under shadow and normal illumination. We first use the background model to detect moving pixels, which might consist of real foregrounds and cast shadows. The weak shadow detector in Section 3.2.1 is then employed as a pre-filter to decide possible shadow points. These samples are used to train the LSM in Section 3.2.2 and a GSM in Section 3.2.3 over time. The slow convergence speed of LSM is refined by the supervision of the GSM through confidence-rated learning in Section 3.3.

Weak Shadow Detector

The weak shadow detector evaluates every moving pixels detected by the background model to filter out some impossible shadow points. The design principle of the weak shadow classifier is not to detect moving shadows accurately, but to determine whether a pixel value fits with the property of cast shadows. For simplicity, we pose our problem in RGB color space. Since cast shadows on a surface reduce luminance values and change the saturation, we define the potential shadow values fall into the conic volume around the corresponding background color [START_REF] Porikli | Shadow flow: a recursive method to learn moving cast shadows[END_REF], as illustrated in Fig. 2. For a moving pixel p, we evaluate the relationship of the observation pixel values z t (p) and the corresponding background model b t (p). The relationship can be characterized by two parameters: luminance ratio r l (p) and angle variation θ(p), and can be written as:

r l (p) = b t (p) z t (p) cos(θ(p)) , (5)
θ(p) = arccos z t (p), b t (p) z t (p) b t (p) , (6)
where •, • is the inner product operator, and • is the norm of a vector. A pixel p is considered as a potential cast shadow point if it satisfies the following criteria:

r min < r l (p) < r max , θ(p) < θ max , (7)
where r max < 1 and r min > 0 define the maximum brightness and darkness respectively, and θ max is the allowed maximum angle variation.

Local Shadow Model (LSM)

The color ratio between shadowed and illuminated intensity values of a given pixel p is represented as a vector of random variables:

r t (p) = ( z r t (p) b r t (p) , z g t (p) b g t (p) , z b t (p) b b t (p) ), (8)
where b i t (p), i = r, g, b are the means of the most probable Gaussian distributions (with the highest ω/||Σ|| value) of the background model in three channels, and z i t (p), i = r, g, b are the observed pixel values. The weak shadow detector is applied to moving regions detected by background model to obtain possible shadow points. We call these possible shadow points as "shadow candidates", which form the set Q. At the pixel level, we model r t (p), p ∈ Q using the GMM as the LSM to learn and describe the color feature of shadows.

Once the LSM is constructed at pixel p, it can serve as the shadow model to evaluate the likelihood of a local observation z t (p). Since we describe shadow by its color ratio of the pixel intensity values between shadow and background, the likelihood of an observation z(p) can be derived by the transformation from "color ratio" to "color". Let s(p) be a random variable describing the intensity values of a pixel p when the pixel is shadowed. Then, s(p) can be characterized by the multiplication of two random variables: s(p) = r(p)b(p), where b(p) is the most probable Gaussian distribution with parameter {µ BG,1 , Σ BG,1 }, and r(p) is the color ratio modeled as GMM with parameter settings {ω CR,k , µ CR,k , Σ CR,k }, in which CR denotes "color ratio". When the pixel p is shadowed, the likelihood can be written as:

p(z(p)|l p = SD) = 1 W SD S k=1 ω SD,k G(z(p), µ SD,k , Σ SD,k ), (9)
where normalization constant W SD and S are determined in similar ways as in (3) with threshold T SD ; the weight ω SD,k , mean vector µ SD,k and the diagonal covariance matrix Σ SD,k of the k th are:

ω SD,k = ω CR,k (10)
µ SD,k = µ BG,1 * µ CR,k (11)
Σ SD,k = µ 2 BG,1 Σ CR,k + µ 2 CR,k Σ BG,1 + (12)
Σ CR,k Σ BG,1

Benedek et al. [START_REF] Benedek | Bayesian foreground and shadow detection in uncertain frame rate surveillance videos[END_REF] also defined color features in CIELuv color space in a similar form and performed a backgroundshadow color value transformation. However, they did not take the uncertainty of the background model into consideration. Thus, the parameter estimation of the shadow model might become inaccurate for scenes with high noise level.

As gathering more samples, the precision of the LSM will become more accurate. With the online learning ability, the LSM can not only adapt to local characteristics of the background, but also to global time-varying illumination conditions. However, the LSM requires several training samples for the estimated parameters to converge. In other words, a single pixel should be shadowed many times while under the same illuminating condition. This assumption is not always fulfilled, particularly on the regions where motion is rare. Thus, the LSM on its own is not effective enough to capture the characteristics of shadows.

Global Shadow Model (GSM)

To increase the convergence rate, we propose to update the LSM using samples along with their confidence values. This is achieved by maintaining a global shadow model, which adds all shadow candidates in an image (r t (p), p ∈ Q) into the model. In contrast to the LSM, the GSM obtains much more samples (every pixel p ∈ Q), and thus does not suffer from slow learning. We characterize shadows of GSM in an image also by GMM.

To evaluate the confidence value of a pixel p ∈ Q, we first compute the color feature r t (p) by ( 8) and check against every states in the GSM. If r t (p) is associated with the m th state in the mixture of GSM, described as G m , the confidence value can be approximated by the probability of G m being considered as shadows:

C(r t (p)) = p(l p = SD|r t (p)) = p(l p = SD|G m ). (13)
In GMM, Gaussian states with higher prior probabilities and smaller variances would be considered as shadows. Therefore, we approximate p(SD|G m ) using logistic regression similar to that in [START_REF] Lee | Effective gaussian mixture learning for video background subtraction[END_REF] for background subtraction. On the other hand, if there are no states in GSM associated with the input color feature value r t (p), the confidence value is set to zero.

For each pixel p ∈ Q, we evaluate the confidence value C(r t (p)) predicted by the GSM and then update the LSM through confidence-rated learning (presented in Section 3.3). With the proposed learning approach, the model needs not to obtain numerous samples to converge, but a few samples having high confidence value are sufficient. The convergence rate is thus significantly improved.

Confidence-rated Gaussian Mixture Learning

We present an effective Gaussian mixture learning algorithm to overcome some drawbacks in conventional GMM learning approach [START_REF] Stauffer | Adaptive background mixture models for real-time tracking[END_REF]. Let α ω and α g be the learning rates for the weight and the Gaussian parameters (means and variances) in the LSM, respectively. The updating scheme follows the the formulation of the combination of incremental EM learning and recursive filter:

α ω = C(r t ) * ( 1 -α def ault Σ K j=1 c j,t ) + α def ault , (14)
α g = C(r t ) * ( 1 -α def ault c k,t ) + α def ault , (15)
where c k,t is the number of match of the k th Gaussian state, and α def ault is a small constant, which is 0.005 in this paper. In the initial learning stage, the total number of match of the pixel Σ K j=1 c j,t is small and thus the learning rate for weight is relatively large. As time goes on, the pixel becomes stable and the learning rate approaches to the type of recursive filter. Similarly, the learning rate for Gaussian parameters α g is relatively large for newly-generated states. Instead of blind update scheme, which treats each sample in the same way, we propose to update a sample with its confidence value. The two learning rates are controlled by a confidence value C(r t ), which indicates how confident the sample belongs to the class. Observations with higher confidence values will converge faster than those with low ones. The confidence-rated learning procedure in one dimension is described in Algorithm 1.

Here, we describe the drawbacks of the conventional GMM learning [START_REF] Stauffer | Adaptive background mixture models for real-time tracking[END_REF] for background modeling and how does the proposed learning approach overcome these disadvantages. Firstly, the method suffers from slow learning in the initial stage. If the first value of a pixel is a foreground object, the background model will have only one Gaussian state with weight equaling to unity. It will take a long time for the true pixel values to be considered as part of background. In the initial learning stage, our method will follow the incremental EM learning and approaches to recursive filter over time. Therefore, we do not suffer from the slow learning in the initial stage. Secondly, the method faces the trade-off problem between model convergence speed and stability. In order to maintain the system stability, a very small learning rate will be chosen to preserve a long learning history. However, a small learning rate results in slow convergence speed. While larger learning rate improves the convergence speed, the model becomes unstable. In the proposed method, the adaptability of the learning rate for Gaussian parameters allows fast convergence for the newly-generated states without degrading the stability. Lastly, there are tradeoffs regarding where to update in the image. Typically, there are two ways to update the model: selective update and blind update Selective update only adds samples being considered as background and thus enhances the detection of foreground. Unfortunately, this approach would fall into a deadlock situation whenever the incorrect update decision was made. On the other hand, the blind update adds all samples into the model, thus it might result in more false negatives. By updating all samples with corresponding confidence, the trade-off regarding how to update in the image can be avoid.

Foreground Model and Segmentation

The foreground model has been described as a uniform distribution [START_REF] Wang | A dynamic conditional random field model for foreground and shadow segmentation[END_REF], providing a weak description of the fore-Algorithm 1: Confidence-Rated Learning Procedure in One Dimension User-defined Variables : K, ω init , σ init , α def ault Initialization : ∀ j=1...K ω j = 0 , µ j = Inf , σ j = σ initial , c j = 0 while new observed data r t with confidence C(r t ) do

α ω = C(r t ) * ( 1-α def ault Σ K j=1 cj,t ) + α def ault if r t is associated with the k th Gaussian then for j ← 1to K do if j=k then M j,t =1 else M j,t = 0 ω j,t = (1 -α ω ) • ω j,t-1 + α ω • M j,t end c k,t = c k,t-1 + 1 α g = C(r t ) * ( 1-α def ault c k,t ) + α def ault µ k,t = (1 -α g ) • µ k,t-1 + α g • r t σ 2 k,t = (1 -α g ) • σ 2 k,t-1 + α g • (r t -µ k,t-1 ) 2 else for j ← 1to K do ω j,t = (1 -α ω ) • ω j,t-1 µ j,t = µ j,t-1 , σ 2 j,t = σ 2 j,t-1 end k = argmin j { ωj,t σj,t } c k,t = 1 ω ,t = C(z t ) * ( 1-ωinit Σ K j=1 cj,t ) + ω init µ k,t = r t , σ 2 k,t = σ 2
initial end end ground. In [START_REF] Sheikh | Bayesian modeling of dynamic scenes for object detection[END_REF], they exploited the temporal persistence as a property to model foreground using joint domain-range nonparametric density function. However, the foreground model in the previous frame cannot provide an accurate description for foreground in the current frame when large motion occurs or new moving objects enter into the scene. Therefore, instead of exploring temporal statistics, we turn to spatial color information in the current frame. We use the background and shadow model to generate a partial foreground mask, i.e. a set of pixels that can be confidently labeled as foreground. Nonparametric method is used to estimate the foreground probability because the spatial augmented GMM might face the problem of choosing the correct number of modes. Details are introduced below.

In the partial foreground generation stage, we aim not to find all foreground pixels, but some samples from foreground. This is achieved by finding pixels whose intensity values are impossible to be generated from the existing background and shadow model. In other words, the partial foreground mask F can be generated by simple thresholding:

F = {p|E BG (z t (p)) > κ, E SD (z t (p)) > κ}, (16)
where E(.) is the minus log of the likelihood of a pixel belonging to background or shadow, κ is a threshold.

We can now estimate the foreground probability of a query pixel p using samples from foreground. We use the neighborhood pixels around a query point p to gather M relevant samples, which forms the set

G p = {g 1 , g 2 , • • • , g M }.
When a pixel p is in the foreground, the probability can be estimated using kernel density estimation method [START_REF] Wand | Kernel Smoothing[END_REF]:

p(z p |l p = F G) = 1 M M i=1 K H (z p -g i ), (17)
where K H (x) = |H| -1/2 K(H 1/2 x). The kernel, K, is taken to be a 3-variate density function with K(w)dw = 1, wK(w)dw = 0, and ww T K(w)dw = I 3 , and H is a symmetric positive definite 3x3 bandwidth matrix. We choose a common Gaussian density as our kernel K,

K H (x) = |H| -1/2 (2π) -d/2 exp(- 1 2 x T H -1 x). (18)
Although variable bandwidth kernel density estimators can lead to improvement over kernel density estimators using global bandwidth [START_REF] Sain | Multivariate locally adaptive density estimation[END_REF], the computational cost of choosing an optimal bandwidth is expensive for a real-time surveillance system. Hence, the bandwidth matrix H is assumed to be diagonal, in which only two parameters are defined: variances in spatial σ s , and in color domain σ c .

After computing the data likelihood of the local observation z t (p) of three classes, we can perform maximum a posteriori (MAP) estimator for the label field L. The energy function of Eq. 1 can be efficiently minimized by the graph cut algorithms.

A concrete description of the learning process and detection process are shown in Algorithms 2 and 3, respectively.

Experimental Results

We have implemented and test the proposed approach on various scene types. In the LSM and GSM, three and five Gaussian states are assigned, respectively. The initial variance of both LSM and GSM are set as 0.0025. We use three Gaussian models for the background, and the covariance matrix is assumed diagonal.

Here, we describe the test sequences we used in this paper.

• "Highway" video is a sequence from the benchmark set [START_REF] Prati | Detecting moving shadows: algorithms and evaluation[END_REF] . This sequence shows a traffic environment. The shadows appearing in the video are dark and cover large area in the scene.

• "Laboratory" video is a typical indoor sequence with light shadow. But, the lighting conditions are more complex than the outdoor scenes.

• "Intelligent Room" [START_REF] Prati | Detecting moving shadows: algorithms and evaluation[END_REF] is an indoor sequence with low shadow strength.

• "Entrance" [START_REF] Benedek | Bayesian foreground and shadow detection in uncertain frame rate surveillance videos[END_REF] are typical surveillance videos captured in different time of a day, thus with different lighting conditions. The effect of GSM is illustrated by Figure 5. The detection result without using the GSM is presented in Figure 5(b), in which some portions of real foreground are miss labeled as shadows. This is due to the slow learning of the LSM. In Figures 5(c 5 and 5 show the quantitative comparison with previous approaches. The shadow detection and discriminative rate η and ξ follow the metrics described in [START_REF] Prati | Detecting moving shadows: algorithms and evaluation[END_REF]. The readers should refer to [START_REF] Prati | Detecting moving shadows: algorithms and evaluation[END_REF] for exact equations. Table 1. Quantitative comparison on "Highway" sequence.

Method η% ξ% Proposed 76.76% 95.12% SNP [START_REF] Prati | Detecting moving shadows: algorithms and evaluation[END_REF] 81.59% 63.76% SP [START_REF] Prati | Detecting moving shadows: algorithms and evaluation[END_REF] 59.59% 84.70% DNM1 [START_REF] Prati | Detecting moving shadows: algorithms and evaluation[END_REF] 69.72% 76.93% DNM2 [START_REF] Prati | Detecting moving shadows: algorithms and evaluation[END_REF] 75.49% 62.38%

Figure 1 .

 1 Figure 1. Flow chart of the proposed algorithm

Figure 2 .

 2 Figure 2. The weak shadow detector. The observation value z t (p) will be considered as shadow candidate if it falls into the shaded area

Figure. 3

 3 Figure. 3 and 4 demonstrate the effectiveness of the proposed method in both outdoor and indoor scenes. (a) shows the original image in the sequence. (b) shows the moving pixels detected by the background model and the gray regions of (b) is labeled as shadow points by the weak shadow detector. Note that the weak shadow detector has

  ), the foreground objects are accurately detected. Figure. 6 demonstrates the adaptability of the proposed algorithm to different lighting conditions. Figure. 6(a),(c), and (e) show the same scene captured in different periods of the day: morning, noon, and afternoon, respectively. As shown in the Figure. 6(b),(d), and (f), our method can detect the foreground and shadows correctly. Table

Figure 3 . 6 .Figure 4 .Figure 5 .Figure 6 .

 36456 Figure 3. Outdoor sequence: "Highway". (a) Frame in the sequence. (b) Detection result by the weak shadow detector. (c) The confidence map by GSM. (d) Foreground and Segmentation result

Acknowledgment

This research was supported by the National Science Council of Taiwan under Grant No. NSC 96-3113-H-001-011 and NSC 95-2221-E-001-028-MY3.


Gérard Lassibille

Organisation and Efficiency of Education Systems: some empirical findings

Using a sampk of countries chosen for their similar level of development, this article shows the extent to which the organisation of primary and secondary education differs from one country to another, notably with respect to the way in which systems differentiate and select pupils for specialised curricula. It also explores the question of whether an education system that sorts pupils at a very early age is more cost-effective than a system that does not sort pupils during compulsory education. The stylised description of national education systems is based on various sources of information. The data used in the cost analysis come from the Organisation for Economic Cooperation and Development (OECD) and the United Nations Educational, Scientific, and Cultural Organisation (UNESCO). Due to data limitations, the analysis of system performance is based on student achievement only in mathematics and science.

Introduction

Numerous studies have examined the quantitative aspects of national education systems and synthesised their relative positions with respect to conventional statistical indicators (see, for example, [START_REF] Wolhuter | Classification of national education systems: a multivariate approach[END_REF]. In this article, we follow a different approach because we are more interested in the qualitative dimensions of education systems and the effect that the organisation of systems has on the cost and quality of schooling. Indeed, the structure of education systems differs widely across countries. The duration of basic, secondary, and post-secondary education varies from one country to another. At each level of education, schooling systems are more or less diversified. Furthermore, there are differences in the way and the age at which they sort and select students into separate schools or separate curricula.

The key question of this article is the extent to which one kind of organisation is more efficient than another. In order to try to answer this question, we describe the education systems of 29 countries with a similar level of development [1]. Our analysis is restricted to primary and secondary education, because it is still difficult to describe in detail the structure of higher education in a great number of countries. After examining the education systems of these countries, which can be divided into two broad categories, we compare their costs and their performance measured through student achievement in mathematics and science.

The Organisation of Education Systems

Table 1 shows some of the characteristics of primary and secondary education systems in the sample of countries considered. The data corne from various sources (Organisation for Economie Co-operation and Development (OECD), 1996;[START_REF] Postelthwaite | International Encyclopedia of National Systems of Education 2nd ed[END_REF]the European Commission, 1996). The information refers to the most recent year for which the organisation of the education system in each country is known. The structure of schooling systems is described using the criteria of the International Standard Classification of Education (ISCED). ISCED is designed to be universally valid and invariant to the particular circumstances of a national education system. However, using this system of classification to describe the organisation of education systems at an international level may introduce some biases, as the countries may classify differently a similar type of diploma. Yet, in our case the bias is probably small because we focus only on primary and secondary education. At each level of education, the number of streams shows the number of choices available for pupils. This number does not take into account options that pupils can follow within a stream; in fact in many countries, the number of streams corresponds to the number of institutions that pupils may attend during their schooling. Of course, this number underestimates the degree of diversity in education; the bias is probably even greater for second-cycle secondary education than for first-cycle secondary education. The age of entry into a level of education does not necessarily mean that pupils can choose the different streams that are available for that level at that age; in fact, in some countries the choice at level 2, for example, is made after a student enters the level. Finally, in Table 1, as in what follows, special needs education is not taken into account. The data show that during the last three decades, compulsory schooling lengthened, sometimes markedly. Infact, while on average the duration of compulsory education grew by about 1 year, in some countries it increased by 3-5 years. No countries lag behind, and indeed countries are in a much more similar situation today than they were 30 years ago. However, considerable differences persist in the organisation of their education systems.

The 9 years that pupils spend on average in compulsory education are structured in very different ways. Primary education, which most students enter at the age of 6 years, lasts about 6 years in most countries, but only 4 years in some countries and in others 8 years. The differences observed at the primary level explain some of the differences that exist in the structure of lower secondary education between countries. Generally, the longer the primary education, the shorter the first level of secondary education, but the correlation between the duration of both stages is far from being complete [2]. On average, lower secondary education lasts 3 years, but it lasts for only 2 years in some countries and 6 years in others. At this stage of education, the number of educational programmes that pupils may attend varies widely between countries. Whereas in about half the countries pupils all follow the same curriculum, in the other half pupils are sorted and differentiated very early in different streams. In these countries, pupils may choose between two and four general streams; in a large number of countries pupils may also enter into technical or vocational education at the beginning of lower secondary education. Entry into upper secondary education takes place generally at 15 years of age. This level is more diverse than the previous one; pupils can choose between one and seven general streams or between one and five technical streams. Countries that provide the most technical streams at this stage generally also have the most technical streams at level 2. On average, the second stage of general and vocational secondary education lasts 3 years. However, large differences exist between countries, with the duration of schooling varying from 2 to 5 years for general and from 1 to 7 years for vocational upper secondary education. These first elements of comparison show that education systems are structured in very different ways and, in particular, that countries select their pupils for specialised courses at a more or less early age. To describe better these differences, we construct three indicators for each country. The first, I 1 , is computed by dividing the duration of the common-core syllabus [3] by the total duration of compulsory education; the lower the value of I 1 , the lower the duration of the common-core curriculum in proportion to the duration of all compulsory education. I 2 and I 3 relate the minimum theoretical age at which pupils can be sorted into general or vocational streams (of level 2 or 3) compared with the theoretical age at which they can exit from compulsory education. When I 1 and I 2 are equal to 1, the systems do not differentiate pupils during compulsory education, but they do differentiate them during post-compulsory education, in which case, I 1 = I 2 ≤ I 3 From these results, it appears that in almost half the systems, compulsory education is organised on the basis of a common-core curriculum from the time of entry into primary education to the end of compulsory education [4]. In these so-called non-differentiated systems, pupils spend on average 6 years in primary education and 3-5 years in secondary education (Table II). Specialisation in general or technical streams only occurs at the end of compulsory education. At that moment, pupils may choose between one and three general streams and between one and four vocational streams; these studies at level 3 last between 1 and 5 years. The fact that pupils cannot enter vocational education until the end of their compulsory education may explain why demand for vocational education is comparatively weaker in non-differentiated systems than in differentiated ones [5]. The absence of selection into the different streams of the upper level of secondary education (see Organisation and Performance of the Systems, below) implies that pupils have less incentive to enrol in vocational and technical education, which have lower rates of return in the labour market. In differentiated systems [6], the period of compulsory education is longer on average (9.6 years compared with 8.8 years). Pupils follow a common-core curriculum lasting between 50 and 60% of the total duration of their compulsory education. At the end of these 5 years of common-core curriculum, pupils enter secondary education. Here they are sorted into three general streams. With a few exceptions, pupils are enrolled in technical education before the end of their compulsory education. Most often, pupils enter lower secondary vocational education at around 12 years of age, after taking 1-2 years of general education at this level, but in certain countries, pupils can enter vocational education at the same age as general education level 2. In more than half these countries, pupils leave compulsory education with training at level 3; general and vocational education are more diversified at this level in differentiated than in non-differentiated systems (Table Il).

Organisation and the Cost of Systems

Many studies have attempted to explain the differences in educational costs between countries (see, for example, [START_REF] Schultz | Expansion of public school expenditures and enrollments: intercountry evidence on the effects of income, prices, and population growth[END_REF][START_REF] Ram | Public expenditures in the United States: an analytical comparison with other industrialized countries[END_REF]. In these studies, the organisation of the education system is not considered a major element of educational expenditure. However, given that education systems are structured in very different ways, the question is whether, all things being equal, one type of organisation is, on average, more expensive than another.

To answer this question, we compare the expenditure per pupil in secondary education in our sample of countries. In order to take into account the differences in prices of the factors that enter into the production of education, unitary expenditure is expressed as a percentage of gross domestic product (GDP) per capita. The cost data are taken from OECD (1996); they refer to 1993 and cover both public and private educational institutions. From these data, it appears that unit costs are, on average, very similar in both types of systems; they represent nearly 26% of GDP per capita (Table III). However, differences between countries are large, mainly among those with a differentiated education system. In order to shed more light on the differences, Figure 1 shows the relationship between unit costs and the degree of differentiation of systems as measured by the I 1 index. As we can see, some of the non-differentiated systems have very high expenditure per pupil (Denmark, Switzerland, Finland), while others have relatively low unit costs (Greece, Portugal, Japan). In non-differentiated systems, unit costs are twice as high in some countries as in others. In differentiated systems, they can be three rimes as high, and it is clear that the more differentiation, the higher the unit costs. These cost differences are due not only to the way schooling systems are organised but also to the technical characteristics of the process of education and probably also to the way schooling systems are managed. To illustrate the analysis on these points, we regress the unit cost of education, expressed as a percentage of GDP per capita, on the degree of differentiation of the systems (I 1 index), their pupil/teacher ratio, and their degree of decentralisation, approximated by the share of public expenditures on education financed by the central administration (Table IV). The data are taken from OECD (1996). According to these data, the share of funding by central administration is 51% on average. However, differentiated systems seem, on average, to be more centralised than the others: in the former, 56% of funding is by the central administration compared with 47% in the latter. The pupil/teacher ratio in secondary education is on average 14, but it is slightly larger in differentiated systems (14.8) than in non-differentiated systems (13.0).

As far as the management of education systems is concemed, Table IV shows that centralised systems have lower expenditure per pupil than decentralised systems. It seems that, other things being equal, management on a regional or local level gives rise to high administrative costs, which may reduce the share of funds allocated to teaching expenditures. This is true of both differentiated and non-differentiated systems. In addition, organising the curricula according to a common-core curriculum is significantly less expensive than differentiating pupils into streams. Indeed, all else remaining the same, differentiated systems, because of their greater diversity, are capable of fewer economics of scale. However, because the results do not take into account the performance of the systems, it is difficult to say whether non-differentiated systems are underfinanced and are less efficient than the others or whether, in fact, they operate more efficiently.

Organisation and Performance of the Systems

For the last 10 or so years, numerous studies have tried to analyse the academic achievements of pupils (see, for example, [START_REF] Hanushek | The economics of schooling production and efficiency in public schools[END_REF]. Although there is no theoretical framework that defines precisely all the significant variables in the process of education, these empirical studies generally recognise the influence of at least three factors: the personal characteristics of pupils, including their level of performance in the previous grades, their social characteristics, and the environment in which learning takes place (characteristics of the school, qualifications of teachers, pupil/teacher ratio, teaching equipment, teaching methods, and so forth). Most of these studies analyse the influence of these factors for one given type of organisation of education system, because they deal with only one country. Although these factors play a role at the international level, it is clear that the differences observed in the pupils' performance can also be explained by the organisation of the education system. For example, differences in performance at the international level can be explained by the duration of primary and secondary education, the timing of selection and specialisation of pupils into streams, the size of the private sector, or the level of decentralisation of the education system. To estimate the effect that differentiation has on school performance, we analysed the results obtained in standardised mathematics and science tests by pupils in grade 7 and grade 8; that is, by pupils who are theoretically 13 and 14 years old. These results were obtained in 1994-1995 by the International Association for the Evaluation of Educational Achievement (IEA) in its Third International Mathematics and Science Study (TIMSS). Twenty-four countries are included in our analysis; among these, 12 have differentiated education systems (Austria, the Czech Republic, France, Germany, Hungary, Ireland, Korea, The Netherlands, New Zealand, Sweden, England and Wales, and USA), and 12 have non-differentiated education systems (Australia, Canada, Denmark, Greece, Iceland, Japan, Norway, Portugal, the Russian Federation, Scotland, Spain, Switzerland). Theoretically, when they took the tests, the pupils in differentiated systems had all completed the common-core syllabus and been sorted into several streams, while in non-differentiated systems, they ail had completed the same curriculum. On average, pupils in differentiated systems perform better than pupils in non-differentiated systems; for example, in grade 7, pupils in the first system achieved a score of 502, while the second achieved only 477 (Table V). However, these average test scores mask a large diversity (Figure 2). Some non-differentiated systems achieve relatively high results Gapan, Switzerland, the Russian Federation), while others may achieve very weak results (Portugal, Greece, Spain). Among differentiated schooling systems, the differences between countries are smaller; nevertheless, there is a 20% gap in the scores between countries with the best and the worst performances. Assuming that the distribution of abilities is homogeneous over all countries, the observed differences may be explained by the internal characteristics of each kind of system. Due to lack of appropriate data, it is not possible to measure these characteristics precisely and fully. However, existing information makes it possible to determine the effect that certain aspects of the differentiation of systems have on school achievement, notably the selection practices, the pupil/teacher ratio, the instructional time, the size of the private sector, and the degree of centralisation of the system.

As far as selection procedures are concerned, systems that sort and differentiate pupils into streams appear to be more selective than others (Table VI). Indeed, in 75% of differentiated systems, pupils enter lower secondary school according to their academic performance, while in only 13% of nondifferentiated systems do school results play a role. In the latter, however, parents are more often involved in the selection procedures, taking part in the process of selection in 13% of cases compared with only 5% in differentiated systems. In the second level of secondary education, the same criteria are generally applied in each type of system, but compared with the previous level of education, differentiated systems accord a larger place to parental choice. Due to these selection practices, repeating a grade and dropping out are quite rare in non-differentiated systems. However, in differentiated systems, repeating a grade and dropping out are more frequent, especially in the first level of secondary education, where the choices that affect most pupils' future are made (Table VI). If the mode of selection in differentiated systems results in higher test scores, these systems have a lower level of equity than non-differentiated ones. Indeed, as indicated in Table VI, the social characteristics of the pupils play a role in 62% of differentiated systems and 13% of nondifferentiated systems insofar as selection processes are concemed in lower secondary education. In both types of system, the weight of social factors is more important in upper secondary education, but social factors once again play a greater role in differentiated systems.

Beyond these aspects, the two systems differ quite clearly in the instructional time that pupils receive annually in the different levels of education. Taking together the duration of compulsory education and the annual number of school hours in each level of studies, students receive on average 7515 hours of instruction in differentiated systems and only 7200 hours in nondifferentiated systems [7]. The differences between the two systems are mainly at the first level of secondary education. At age 13 years, pupils from differentiated systems have received on average 20 more hours of education. However, between certain countries the difference can be as much as 240 hours a year. It is generally assumed that the most efficient schools are those where school heads have autonomy in the hiring and assignment of teachers, teachers have autonomy in the choice of pedagogical practices, and all the relevant actors-heads, teachers, and students-face explicit incentives for good performance. Private schools usually have more autonomy in areas that matter for student achievement. If true, this may explain the better performance of differentiated systems. In fact, in differentiated systems 21% of primary pupils and 19% of secondary students are enrolled in private schools, whereas only 8 and 11%, respectively, are enrolled in private schools in non-differentiated systems [8]. Many empirical studies have shown that the pupil/teacher ratio has a significant impact on student performance. In this respect, there are considerable differences between education systems. Although on average there are 17 pupils per teacher in all countries, this ratio is much higher in differentiated systems than in non-differentiated systems (18 compared with 14) [9]. In order to show the extent to which the characteristics of the two systems affect pupils' achievement, in Table VII we regress the mathematics test scores in grades 7 and 8 on the following variables: the differentiation index of systems, the pupil/teacher ratio, the instructional time from the first year of primary education up to the age when pupils took the tests, the percentage of pupils enrolled in the private sector, and the degree of decentralisation of systems, estimated on the basis of the share of central government funding. The test scores in the 7th and 8th grades are adjusted within the framework of a fixed-effect model. To take into account the correlation between the results obtained in each grade, the variance-covariance matrix is corrected for heteroscedasticity by clustering observations by country (see, for example, [START_REF] Greene | Econometric Analysis[END_REF]. Given the availability of data, the estimation covers 14 countries, yielding 28 observations. Among these, seven countries have differentiated and seven have non-differentiated education systems; the countries belonging to each type of system have performance profiles similar to those described in Table V. Differentiated systems obtain better results than non-differentiated systems, all things being equal. The effect of the structure of the system is far from negligible, because the timing of differentiation of pupils into streams explains about 15% of the difference in scores between pupils. The generally accepted idea that non-differentiated systems are the best way to maximise the production of knowledge is probably false. Apparently, their only advantage is to reduce social inequalities at an early age. However, this advantage may only be apparent, because even if they do reduce inequalities of access to education, they do not compensate for social inequalities that occur within the process of education itself. The results in Table VII also show that the pupil/teacher ratio has a negative impact on pupils' scores. This result is similar to those obtained by other studies (see, for example, [START_REF] Hanushek | The economics of schooling production and efficiency in public schools[END_REF]. The total number of school hours, which differs widely across countries, has no significant effect on achievement in mathematics. In fact, according to regression results, the instructional time in primary education explains the good performance of certain countries more than the instructional time in secondary education. These results probably confirm the importance of basic education and the need for pupils to acquire the basics of reading, writing, and arithmetic. Regarding the organisation of education systems, these results should encourage countries with a relatively short period of basic education to lengthen it slightly. Combined with an early differentiation of pupils in the first level of secondary education, such a measure could maximise student performance. Furthermore, the regression results indicate that countries where private education is more widespread perform significantly better than countries where it is more limited. The result showing the private sector to be more efficient is similar to those found in other contexts with individual data (see, for example, [START_REF] Psacharopoulos | Public versus private schools in developing countries: evidence from Colombia and Tanzania[END_REF][START_REF] Jimenez | The relative efficiency of private and public schools in developing countries[END_REF]. This finding should convince countries to reconsider policies that reduce the role of the private sector in the field of education. However, the empirical evidence based on our regressions is only preliminary, because we do not take into account the fact that pupils in the private sector have different characteristics from pupils in the public sector. In other words, we do not exclude the possibility of significant selectivity effects in the adjustment of student tests scores (see, for example, [START_REF] Heckman | Sample selection bias as a specification error[END_REF].

Finally, centrally managed education systems perform as well as decentralised ones. The idea that decentralised systems are more efficient from the point of view of student achievement seems erroneous. Decentralised organisation, which is becoming more common, combines two disadvantages: it is relatively expensive, and it is not as efficient as other organisations in relationship to student achievement.

Conclusions

In this article we used aggregate data to measure the degree of differentiation of schooling systems for a sample of countries chosen for their similar level of development. Obviously, comparative analysis in this field poses some problems. In particular, it is possible that the degree of differentiation on which the analysis is based somewhat underestimates the reality. Furthermore, owing to data limitations our results cover the scholastic achievement of pupils in one particular subject, when it is known that not ail systems accord the same importance to the subject in question. Despite these problems, the analysis has clearly shown that primary and secondary education systems are structured very differently across countries. The age of entry into and duration of basic education vary from one country to another. However, much greater differences exist in secondary education. In some countries, pupils follow the same curriculum to the end of their compulsory education; in others, they are differentiated and sorted into different streams very early on. The particularities that appear in this respect have repercussions for the cost of systems. The analysis shows that, ail things being equal, differentiated systems have significantly higher unit costs than non-differentiated ones. However, they also seem to be considerably more efficient, at least as far as learning mathematics is concemed. These results suggest that basic education should be slightly reinforced when it covers only a comparatively short period of rime, because a good grasp of basic arithmetic, and undoubtedly also of reading, guarantees high achievement in secondary education and probably further on, given that educational attainment is clearly cumulative. Combining this with early selection of pupils into streams could markedly improve the internal efficiency of systems, at least in as far as student achievement in mathematics and science is concemed.

Acknowledgements

This research is part of the Targeted Socio-economic Research (TSER) programme on Schooling, Training and Transition. We are grateful to Louis Lévy-Garboua, a TSER referee and anonymous reviewers for helpful comments.

The data are drawn from the Third

NOTES

[l] The countries are Australia, Austria, Francophone Belgium, Flemish Belgium, Canada, the Czech Republic, Denmark, Finland, France, Germany, Greece, Hungary, heland, Italy, Japan, Korea, Mexico, The Netherlands, New Zealand, Norway, Poland, Portugal, the Russian Federation, Spain, Sweden, Switzerland, Turkey, the UK and the USA.

[2] The rank correlation coefficient between the duration of primary education and the duration of the first cycle of secondary education is 0.70.

[3] This number is calculated from the time that pupils enter primary education up to the age when a first choice is suggested to them.

[4] Australia, Canada, Denmark, Finland, Greece, Italy, Japan, Norway, Poland, Portugal, the Russian Federation, Spain, Sweden, and Switzerland.

[5] In non-differentiated systems 52% of secondary pupils are in vocational education compared with 56% in differentiated systems (1992 data).

[6] Austria, Francophone Belgium, Flemish Belgium, the Czech Republic, France, Germany, Hungary, Ireland, Korea, Mexico, The Netherlands, New Zealand, Turkey, the UK and the USA.

[7] Calculated on the basis of International Bureau of Education (1997), OECD (1996) and UNESCO (1996). Of course, these estimates only have a relative value, because they are based on the hours of study observed in specific years of each cycle of schooling, and obviously there is no evidence that teaching time is identical in the different years of study that make up the same cycle of education.

[8] Percentages calculated on the basis of UNESCO (1996).

[9] Calculated on the basis of OECD (1996).


Alessio Dore
email: dore@dibe.unige.it

Majid Asadi
email: asadi@dibe.unige.it

Carlo S Regazzoni

Online Discriminative Feature Selection in a Bayesian Framework using Shape and Appearance

This paper presents a probabilistic Bayesian framework for object tracking using the combination of a cornerbased model and local appearance to form a locally enriched global object shape representation. A shape model is formed by corner information and it is rendered more robust and reliable by adding local descriptors to each corner. Local descriptors contribute to estimation by filtering out some irrelevant observations, making it more reliable. The second contribution of this paper consists in introducing an online feature adaptation mechanism that enables to automatically select the best set of features in presence of time varying and complex background, occlusions, etc. Experimental results on real-world videos demonstrate the effectiveness of the proposed algorithm.

Introduction

In literature, corner information has been widely used for tracking. For example, in [START_REF] Asadi | Tracking using continuous shape model learning in the presence of occlusion[END_REF] a probabilistic Bayesian framework has been introduced that estimates the new position of the object and its new shape. The work takes advantage of just corners information and the object motion within the last two image frames. Gabriel et al. [START_REF] Gabriel | Object tracking using color interest points[END_REF] used Kalman filter to predict the position of the object. Then, corners around the predicted position are extracted and they are matched with model corners using the Mahalanobis distance. Du and Piater have presented a paper where a mixture particle filter is exploited to analyze the feature points clusters [START_REF] Du | Tracking by cluster analysis of feature points using a mixture particle filter[END_REF].

To improve corner or, more generally, feature points matching the usage of local descriptors is a very popular and successful approach in computer vision and video processing. Several typologies of descriptors can be found in the literature that can be categorized into two groups: a) intensity-based (e.g. color histograms on texture image patches [START_REF] Pérez | Color-based probabilistic tracking[END_REF]) and b) orientation-based (e.g. SIFT (Scale Invariant Feature Transform) [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF] or Histogram of Gradients -HoG [START_REF] Triggs | Proc. of CVPR. Histograms of oriented gradients for human detection[END_REF]). SIFT [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF] is one of the most well known matching technique that exploits orientation descriptors. In particular SIFT descriptors are vectors of gradients processed in order to guarantee invariance to shape and orientation and, partially, to viewpoint and illumination changes.

In recent years several works can be found in the literature dealing with feature adaptation. Noguer et al. fuse multiple cues to segment an object from its background [START_REF] Moreno-Noguer | Dependent multiple cue integration for robust tracking[END_REF]. An adaptive multifeature statistical target model was employed by Maggio et al. [START_REF] Maggio | Adaptive multifeature tracking in a particle filtering framework[END_REF] to combine features in a single particle filter. A reliability measure, derived from the particle distribution in the state space, estimates the reliability of the information by measuring the spatial uncertainty of features. Bagdanov et al. [START_REF] Bagdanov | Improving the robustness of particle filter-based visual trackers using online parameter adaptation[END_REF] estimate uncertainty in the particle filter using a continuously adaptive parameter estimation approach. Collins et al. presented an online discriminative feature selection mechanism for evaluating multiple features while tracking an object [START_REF] Collins | On-line selection of discriminative tracking features[END_REF]. To this end, the color histograms of different linear combinations of the RGB channels are computed both for the object and the background. Log-likelihood and variance ratio are used to find the best color combination for mean shift tracking. An improved approach based on this technique is also proposed in [START_REF] Leung | Optimizing distribution-based matching by random subsampling[END_REF]. Woodley et al. propose another online feature selection algorithm [START_REF] Woodley | Tracking using online feature selection and a local generative model[END_REF]. It uses a local generative appearance model to select features in the classifier. The model is computed by local non-negative matrix factorization. Wang and Yagi extend the standard mean-shift algorithm to an adaptive tracker [START_REF] Wang | Integrating color and shape-texture features for adaptive real-time object tracking[END_REF]. Multicue shape and color features represented by color and gradient histograms are selected using the variance ratio. Wang et al. [START_REF] Wang | Online selecting discriminative tracking features using particle filter[END_REF] select a subset of Haar wavelet features to construct a classifier ensemble for appearance model. The feature selection procedure is embedded into the particle filtering process and Fisher discriminant is employed to rank the classification capacity of each feature.

The rest of the paper is organized as follows. Section 2 describes a Bayesian model-based tracking approach enriched with local appearance descriptors .In section 3 an online feature adaptation strategy is integrated in the tracking mechanism. Experimental results are shown in section 4 and finally, section 5 provides conclusions.

2 Non Linear Shift Estimator (NLSE) Algorithm with Local Descriptors

Shape Model Representation

The state of the object at time t, X t = {X p,t , X s,t } is defined as composed by its global position X p,t = (x p,t , y p,t ) and by its shape model X s,t . The shape model, in turn, is composed by M model elements and their descriptors. The model elements are corners, extracted inside a bounding box B t surrounding the object. Any extracted corner m, with absolute coordinates (x m t , y m t ), is represented in the model using its relative coordinates dX m t = (dx m t , dy m t ) with respect to the object global position, where dx m t = x m t -x p,t and dy m t = y m t -y p,t . In addition the corner m is associated to a vector, where the first element is called persistence, P m t , initialized to a minimum value P I that is used for updating the shape model (see Sect. 2.5). The second element of this vector is a representation of an l × l color patch centered at the corner. The representation is derived using a 1D gradient angle histogram, H m t = {H m t,k } k=1,...,K of K bins. To obtain the histogram, the gradient angle of each pixel inside the color patch is calculated using Roberts mask. Then, H m t is normalized with respect to the gradient angle of the central pixel of the patch (i.e., corner m). The normalization (see [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF]) is done using subtracting the gradient angle of that pixel from the gradient angle of the central point.. Finally, the normalized angle is mapped into one bin,k , of the histogram. Therefore, the object shape model is defined as a set of (3+K)-D vectors:

X s,t = X m s,t = [dx m t , dy m t , P m t , H m t ] m=1,...,M .

Observation Representation

At a given time t, starting from the object global position at the previous time t -1, all corners, say N , inside a search area S t (larger than the bounding box B t that surrounded the target in the previous time) centered at that position are extracted. Then, the observation set is formed using the corners absolute coordinates in the image plane Z n c,t = (x n t , y n t ) along with the gradient angle histogram of the corners, H n t , as a set of N (2+K)-D vectors Z t = {Z n t = [x n t , y n t , H n t ]} n=1,...,N .

Bayesian Framework

In the probabilistic framework, having the object status at time t -1 and the observation set at time t, the goal of the tracker is to estimate the posterior p(X t |Z t , X t-1 ). Using Bayesian filtering approach one can write:

p(X t |Z t , X t-1 ) = p(X p,t , X s,t |Z t , X p,t-1 , X s,t-1 ) = p(X p,t |Z t , X p,t-1 , X s,t-1 )• p(X s,t |Z t , X p,t , X p,t-1 , X s,t-1 ) (1)
Maximizing each of the two terms at the right side of (1) separately, provides a suboptimal solution to the problem of maximizing the posterior of X t . The two terms in (1) are related to the posteriors of the current global position model (tracking) and the current global shape-based model (model updating). First, the object global position (tracking), and then the shape model (model updating) are estimated. Expanding each term in [START_REF] Asadi | Tracking using continuous shape model learning in the presence of occlusion[END_REF] with the hypothesis of statistical independence between the shape X s,t and global motion X p,t of a given tracked object, a Bayesian network of dependencies between involved variables can be written such that (1) becomes (see [START_REF] Asadi | Tracking using continuous shape model learning in the presence of occlusion[END_REF]):

p(X t |Z t , X t-1 ) = k • p(X p,t |X p,t-1 )p(Z t |X p,t , X p,t-1 , X s,t-1 ) p(X s,t |X s,t-1 )p(Z t |X p,t , X s,t , X p,t-1 , X s,t-1 ) (2)
where the variable k is a constant and the first two term in (2) are obtained from an expansion of the first term in [START_REF] Asadi | Tracking using continuous shape model learning in the presence of occlusion[END_REF], and hence they are related to the object global position estimation. In the same way, the last two terms in (2) are related to the object global shape estimation. More in detail, the first term in (2) considers some conditions on the object movement, such as the object speed having some a priori knowledge, by giving probability values to different positions in an area around the object last position [START_REF] Asadi | Tracking using continuous shape model learning in the presence of occlusion[END_REF]. The second term evaluates the probability of different positions having the contribution of observations. Using this term, it is possible to define different functions that relates the contribution of the observations and the previous shape model to any given position in the area defined by the first term. This contribution is called voting procedure, where the observation set votes for different positions based on the previous model. One of the contributions of the paper involves in this term by filtering out some observation to prevent the irrelevant observations to vote for different positions; and hence, it improves the probability function. This part is investigated in the next sub-section. Again, in a manner similar to the first term, the third term, having fixed the object new position, applies some a priori conditions on the observation set to filter out different possible configurations of the observations and assign them probabilities. The last term, similar to the second term, considers a function to relate the observation set around the new position to the previous object shape model to choose one configuration among all possible ones, to be considered as the new object shape model and to update the model [START_REF] Asadi | Tracking using continuous shape model learning in the presence of occlusion[END_REF]. The filtering procedure that is applied on the second term, is also helpful here. Simplifying the situation by considering equal probabilities for all positions in the first term, and equal probabilities for all possible configurations in the third term, one can investigate the voting procedure (second term) and the updating module (fourth term).

Voting Procedure and Position Estimation

The probability density function of the second term in ( 2) is defined as follows:

p(Z t |X p,t , X p,t-1 , X s,t-1 ) = K Z (X p,t , Z t , X p,t-1 , X s,t-1 ) Z K Z (X p,t , Z t , X p,t-1 , X s,t-1 ) (3)
where:

K Z (X p,t , Z t , X p,t-1 , X s,t-1 ) = exp(V t (X p,t , Z t , X p,t-1 , X s,t-1 ) -1) exp(V t (X p,t , Z t , X p,t-1 , X s,t-1 ) (4)
The function K Z (•) is a kernel on the shape subspace that filters the observations based on different possible object positions. The function V t (•) is the number of votes for a potential object position It is clear that the higher the number of votes is, the higher will be the probability in [START_REF] Collins | On-line selection of discriminative tracking features[END_REF]. Note that the current object position at the current time is a variable for which different observations configurations are obtained. Equation (3) says that if there is no vote for a given position (i.e. V t (•) = 0) the probability value will equal zero. Instead, if the number of votes for a given position tends to infinity, the probability value will be equal to one. The function V t (•) is defined here as follows:

V t (X p,t , Z t , X p,t-1 , X s,t-1 ) = N n=1 S n (X p,t , Z n t , X p,t-1 , X s,t-1 ) (5)
where N is the number of observation set elements. Equation [START_REF] Triggs | Proc. of CVPR. Histograms of oriented gradients for human detection[END_REF] indicates that the total number of votes acquired by V t (•) is equal to the summation of the contribution of all observations. The contribution of each observation depends on the function S n (•), where different observations may have different contributions since some of them may be considered as irrelevant observations for a given model element. The function S n (•) is defined as follows:

S n (X p,t , Z n t , X p,t-1 , X s,t-1 ) = M m=1 (u ρ(H n t , H m t-1 ) -thr K R d m,n (X m s,t-1 , Z n t ) (6)
where ρ(•) indicates the Bhattacharyya coefficient that evaluates the similarity between two histograms as:

ρ(H n t , H m t-1 ) = K k=1 H n t,k • H m t-1,k (7)
When the Bhattacharyya coefficient is higher than a defined threshold thr, 0 < thr < 1, the observation n doesn't match the model element m and hence, it doesn't vote for the position identified by the m-th model element. Therefore the local appearance is used to filter out misleading votes by the step function u(•):

u(x -thr) = 1 x ≥ thr 0 otherwise (8)
Instead, if the Bhattacharyya coefficient of the histograms of the observation n and the model element m is above the threshold, then Z n t contributes to the position X p,t based on the metric d m,n (X m s,t-1 , Z n t ) and the kernel function K R (•). The metric is defined as:

d m,n (X m s,t-1 , Z n t ) = dX m t-1 -(Z n c,t -X p,t ) (9)
Equation ( 9) can be interpreted as follows. If X p,t is the new position where observation n votes for, the relative coordinates of observation n with respect to the new position will be (Z n c,t -X p,t ). Therefore [START_REF] Isard | Condensation -conditional density propagation for visual tracking[END_REF] shows the distance between model element m and observation n if the old and the new object positions are considered as origin of the Cartesian coordinate system. Consequently, the distance shows the distortion of corner in two successive frames. This distance has a role in voting. Its influence is taken into account using the kernel function K R (•). The kernel used here is a uniform kernel:

K U (d m,n ) = 1 if d m,n ≤ R R 0 otherwise (10)
According to [START_REF] Leung | Optimizing distribution-based matching by random subsampling[END_REF] if the distance is less than the kernel radius R R , it can be deduced that model corner m is the same as observation n with a distortion less than R R . In this case function S n (•) in ( 6) increases by one. Worth of note is that it would be possible to use different types of kernels. For example, choosing a Kronecker delta kernel indicates that contribution happens just if the distance is zero. In other words, it allows no distortion in the model corners. Choosing a Gaussian kernel, instead, changes the amount of contribution of an observation based on its distance from a model corners. After voting, the position that maximizes the probability in (3) is chosen as the new object global position, since it shows the highest contribution among the observations set and the shape model.

Therefore, the object new global position is considered as X p,t = X p,t . Finally, the dimension of the bounding box B t is adapted according to the distance metric in (9) related to all observations and model elements that contributed to X p,t . All contributing model elements and observations to X p,t are also stored in a list of pairs for using them in the model-updating phase.

Model Updating

After estimating the object new global position, the next step is to update the model. The goal of the model updating is to find unique matches for model corners. To this end, the list of pairs stored in the previous step is used. If observation n and model m corner contribute to X p,t (i.e., the pair (m, n) exists in list of pairs), and m does not contribute with any other observation to vote for position X p,t (i.e., m appears in the list just once), m will be associated with n Therefore, m is updated using information about n as follows:

dX m t = Z n c,t -X p,t ; H m t = H n t ; P m t = P m t + 1 (11)
In the case that there is a dense situation in observation and J observations, n j , j = 1, . . . , J, contribute to the model element m voting for X p,t , just the most similar observation to model element m, based on Bhattacharyya criterion, is considered as the unique association for model element m:

if d m,nj ≤ R R ∧ u ρ(H nj t , H m t-1 ) -thr = 1 ⇒ j = arg max j ρ(H nj t , H m t-1 ); j = 1, . . . , J (12)
After finding all unique associations for model corners, the persistence of un-associated model corners decreases by one:

P m t = P m t-1 -1 (13)
In this case, if the persistence goes below a threshold P th , the corner will be removed from the model. All remaining observations are added to the model using their relative coordinates with respect to X p,t and a minimum persistence P I :

dX M +1 t = Z n c,t -X p,t ; H M +1 t = H n t ; P m t = P I ( 14
)
3 Adaptive Feature Selection for Cornerbased Trackers

The performances of the large majority of tracking algorithms strictly depend on the quality of target observations, intended as the capability of describing the target in an unambiguous way with respect to background and other objects in the scene. In fact both model-based algorithms (e.g. Mean Shift [START_REF] Comaniciu | Kernel-based object tracking[END_REF]) and prediction/correction-based approaches (e.g. CONDENSATION [START_REF] Isard | Condensation -conditional density propagation for visual tracking[END_REF]) must rely on an appropriate description of the measured object characteristics considering their temporal variability. Lighting changes, background variation, occlusions, object deformation, etc. are typical causes of performance fall that are tied to the lack of robustness of the object representation with respect to these situations. To cope with these problems a thorough and time consuming process of parameters setting is to be accomplished on the algorithms that provide target features to the tracking algorithms. However, for applications supposed to work 24/7 with, if possible, no human presence, this procedure is usually not sufficient because of the the scene changes. In this scenarios automatic and adaptive feature selection algorithm are of fundamental importance to design efficient and flexible tracking algorithms.

Corner Extraction from Weighted Images

In this work the attention is focused on corner-based trackers as the one introduced in Sect. 2 in order to render it more flexible with respect to the low-level module computing the set of corners and descriptors Z t .

The voting procedure in Sect. 2.4 allows to compute the target position by evaluating the contribution of observations to the previously estimated model. Therefore, since this task is performed by matching observed corner descriptors extracted in the analyzed frame and the ones stored in the target model, two basic considerations are taken into account for feature adaptation:

• observations Z obj t related to the target should be as discriminative as possible with respect to the ones associated to the background Z bg t • observations Z obj t related to the target should enable robust matching with the target model X s,t-1 , filtering out possible distractors.

In order to follow these purposes, the set of corners and descriptors are extracted and computed in a collection of single channel (gray-level) images obtained through different color channel linear combinations (see [START_REF] Collins | On-line selection of discriminative tracking features[END_REF]). Therefore, single channel images I c1,c2,c3 i,t , i = 1, . . . , 49 are obtained as:

I c1,c2,c3 i,t = Φ i (I t ) Φ i (I t ) = {c 1 R I + c 2 G I + c 3 B I } c j ∈ [-2, -1, 0, 1, 2] (15)
where R I , B I and G I are respectively the red, blue and green color channel of the image frame I t at time t; the number of possible combinations is 49 since the linear dependent ones are not considered. Applying a corner extractor (e.g. Harris [START_REF] Harris | Combined corner and edge detector[END_REF], Kanade-Lucas-Tomasi -KLT detector [START_REF] Tomasi | Detection and tracking of point features[END_REF], etc.) on each of the i-th images I c1,c2,c3 i,t , a set of observations Z i,t , i = 1, . . . , 49 is obtained characterized by different properties deriving from the linear transformation Φ i (I t ).

Selection of the best Sets of Features

In order to select the best representative sets of features to track the target, at each frame the observations Z i,t extracted in one of the i-th gray images are separated into two groups the one considered as belonging to target Z obj i,t and the one associated to the background Z bg i,t . To this end assuming that the target position X p,t and the related bounding box B t are reliably estimated, the observations belonging to the object Z obj i,t are considered as the ones inside B t . Instead the Z bg i,t are the ones internal to the search area S t but outside B t , i.e. Z obj i,t = Z i,t ∈ B t and Z bg i,t = (Z i,t ∈ S t ) ∧ (Z i,t / ∈ B t ) (denoting ∈ and / ∈ respectively as operators that check if the corner Z n c,i,t is inside or outside a bounding box). It is reasonably assumed that the corner of the background that are inside B t are not relevant in number and that they don't affect the selection procedure.

In order to select between the features Z i,t the most discriminative ones we compute the matrix D i for the i-th single channel image:

D i =          D 11,i . . . D 1K,i . . . . . . D jk,i . . . . . . D J1,i . . . D JK,i          ( 16
)
where each element is:

D jk,i = ρ(H obj j,i , H bg k,i ) d(Z j,obj c,i,t , Z k,bg c,i,t ) (17)
The expression ρ(H obj j,i , H bg k,i ) is the Bhattacharyya coefficients (see [START_REF] Gabriel | Object tracking using color interest points[END_REF]) evaluating the similarity between the de-scriptor H obj h,i related to the h-th observation target corner and the descriptor H bg k,i related to the k-th observation background corner, both of them computed in the i-th transformed single channel image. The expression d(Z j,obj c,i,t , Z k,bg c,i,t ) indicates the Euclidean distance between the j-th observation target corner and the k-th observation background corner. With J and K respectively the cardinality of the object observation set and the background observation set are indicated, i.e. J = |Z obj j,t | and K = |Z bg k,t |. Hence, the numerator of ( 17) accounts for the similarity between the target and the background descriptors associated to the extracted corners whereas the denominator considers the vicinity between the two corners. Therefore, if the result of ( 17) is close to 1 it means that a corner of the background is similar and close to one of the object and, then, it can be a distractor in the voting procedure [START_REF] Triggs | Proc. of CVPR. Histograms of oriented gradients for human detection[END_REF]. In fact it is probable that applying (5), the background corner votes for one of the local maxima due to small object deformations or rotation leading; this can lead to a situation where this local maxima become a global maximum and are then considered as the new object position. On the other hand background corner descriptors that are similar but far from the object corners are not considered problematic since the voted center is not a maxima (local or global) since, likely, it is not supported by other votes deriving from the matching procedure between the model and the observations. Once the 49 matrices D i , i = 1, . . . , 49 are computed for each single channel transformed image it is possible to select the best set of discriminative features Z * p,t as the ones extracted from the p-th image such that the mean value of the elements D jk,p , j = 1, . . . , J p ; k = 1, . . . , K p of the matrix D p is the maximum between all of the D i , i.e.:

p = arg max i 1 J i K i Ji j=1 Ki k=1 D hk,i → Z * p,t (18)
Worth of note is that this methodology can be applied to other transformations that provide different set of observations Z i,t , i = 1, . . . , C. For example, instead of using the generation of multiple weighted images the Z i,t can be obtained applying different values of the corner extractor parameters in order to select online the best choice. Moreover in the case that other descriptors of local corner appearance are to be used instead of the SIFT-like ones (e.g. color patches, wavelets, etc.) it is sufficient to change the Bhattacharrya coefficient in [START_REF] Wang | Integrating color and shape-texture features for adaptive real-time object tracking[END_REF] with a related appropriate similarity function.

Online Feature Adaptation Strategy

It can be noticed that the feature selection method is quite computationally onerous, in fact its complexity is O(JKC), where C is the number of observation sets obtained by the possible image transformations, in this case C = 49. Therefore for a tracking application, it has been decided to perform it only when necessary (e.g. after a light changing, an occlusion, etc.). In order to accomplish this task online a procedure is developed to detect automatically when there is the need to update the set of features. To do so when a target is detected T trackers as the one described in Sect. 2 are initialized with the T best sets of observations Z i,t , i = 1, . . . , T selected using [START_REF] Woodley | Tracking using online feature selection and a local generative model[END_REF], i.e. finding the T matrices S i with the lower elements mean value. From now on we will assume T an odd number (e.g. T = 3). Therefore, each tracker provides at each step an estimate X t,i = { X p,t,i , X s,t,i }, i = 1, . . . , T obtained by applying the tracking procedure with the observations Z i,t , i = 1, . . . , T . According to these estimates, the position of the target is computed as a weighted mean of the X p,t,i where the weight is the number of votes related to that position normalized with respect to the number of corner detected inside the searching area S t (i.e. J i + K i ):

X * t = T i=1 w i X p,t,i T i=1 w i (19)
where the weight factor is w i = V t,i /(J i + K i ) with V t,i (see [START_REF] Triggs | Proc. of CVPR. Histograms of oriented gradients for human detection[END_REF]) is the number of votes associated to X p,t,i .

The feature selection procedure is instantiated every time the one of the estimated position X p,t,e starts diverging consistently with respect to the overall estimated position X * t , i.e. when:

X * t -X p,t,e > th ( 20
)
where th is a threshold under which we consider X p,t,e reliable. When (20) is verified for E = {e 1 , . . . , e L }, 1 ≤ L ≤ T /2 -1 trackers then the selection strategy is applied, excluding from the search the transformations Φ i related to the observations used by the unreliable L trackers. After that the L trackers are reinitialized with the new observations Z i,t , i = 1, . . . , L obtained by applying the corner extractor and computing the descriptors on the single channel images related to the best matrices D i , i = 1, . . . , L. This approach to online feature selection then allows to have a more accurate estimated position X * t together with multiple shape models X t,i due to the combination of multiple trackers and at the same time to enable a reasonable strategy to automatically identify the need of the feature adaptation procedure.

Experimental results

The contribution introduced in this paper are twofold: 1) introduction of a local appearance descriptor to the tracker [START_REF] Asadi | Tracking using continuous shape model learning in the presence of occlusion[END_REF] in order to improve the accuracy and to reduce the complexity of the model storing only the more reliable corners; 2) an online strategy to select a good set of features in order to improve the tracking performance and to automatically adapt to environmental changes. Presented experiments are proposed in order to demonstrate this features.

Results of NLSE tracker with Local Descriptors

In this section experimental results obtained with the described tracking method will be proposed. Several testing sequences have been tested that proposed different difficulties as partial occlusion of non-rigid targets (PETS2006), moving camera (Scooter), and oscillating image with targets moving non uniformly (i LIDS AVSS2007). Kanade-Lucas-Tomasi (KLT) detector has been used to extract corner from the sequences resized to 352x288 pixels. No change detection or motion detection module is used to reduce outliers. The KLT parameters affects the tracking results in terms of sparseness or denseness of the created model and then it has to be chosen accurately. A too low number of feature will lead to a poor informative model and, on the other hand, if too many corners are added in the model the shape matching capacities are reduced. The average computational complexity of the method for a nonoptimized code is of 5 frames/sec with a 3.0 Ghz CPU with 2.0 GB of RAM. In Figure 1 AVSS sequence is presented. The sequence is oscillating due to camera movements and this leads to corners instability, i.e. from frame to frame some corners disappear and others appear. The vehicle also slows down when it arrives close to the parked car and after it accelerates again. Despite of these issues the tracker doesn't fail; the scale adaptation is accomplished successfully also in presence of a relevant size reduction of the target. In Figure 1(b) the estimated model X s,t is represented showing with different color the stability with respect to previous frames, i.e. the persistence of the corners P t . Without the usage of local information we have found that the number of element composing the model is averagely three times greater than the one here presented, 30 corners in our model, 100 corners without the patches. Moreover also the persistence tends to be higher in our model demonstrating more robust matching between model and observation elements. In Figure 2(a) the proposed method handles a partial occlusion without being distracted by the outlier corners arisen while the target is hidden by other people. The estimated model X s,t in Figure 2(b) shows that though the object deformation leads to model elements with lower persistence P t with respect to rigid object as in Figure 1(b), X s,t represents quite well the human shape and the occlusion don't affect significantly the model.

Feature Adaptation Results

Different experiments have been performed on several sequence comparing the tracking method without the local descriptors (NLSE), the one using corners local appearance described in Sect. 2 (D-NLSE) and, finally the D-NLSE with the feature adaptation approach of Sect 3 (called A-NLSE). In the following experiments T = 3 trackers are contemporary used in the ANLSE. In Figure 3 the comparison of the three methods is proposed. It can be noticed that the ANLSE (Fig. 3 (see [START_REF] Tomasi | Detection and tracking of point features[END_REF]) before approaching to the crossing stripes were (1,1,1), (1,-1,-1) and (2,-2,1). When the car move closer the stripes, the background changes lead the tracker using the observations extracted from I 1,1,1 i,t to fail ((20) is verified with th = 8pixels). The feature selection process is instantiated and I 1,1,1 i,t is substituted with the color combination (0,1,2), i.e. I 0,1,2 i,t . Though the bounding box is not properly adapted due to the persistence of two corners in the upper right are the accuracy in the stripes region is significantly better than the one of DNLSE and NLSE. In Figure 4 the trajectories of the target centroid are drawn in order to compare them to the ground truth. The accuracy improvement due to the contemporary usage of multiple trackers in the ANLSE is outlined in the first part of the trajectory where the background is simple and no light change occurs. The average tracking error with respect to the ground truth over the sequence of 85 frames are shown in Table 1. Other sequences, some of them with mobile camera, have been tested showing similar improvements with respect to DNLSE and NLSE but can

Conclusion

In this paper a Bayesian framework for object tracking using a corner-based shape model enriched with local appearance descriptors has been proposed. The usage of additional information aims at improving the position estimation and the model construction. An online feature adaptation strategy is integrated in this framework to automatically select the best set of features to pass to the trackers coping with different source of noise affecting them (e.g. light changes, complex background, etc.). Good performances in experimental results on difficult sequences have demonstrated the effectiveness of the algorithm.

Figure 1 .

 1 Figure 1. (a) Tracking result on i LIDS AVSS sequence for frames 40, 80, 164; (b) estimated models X s,t for i LIDS AVSS sequence for frames 40, 80, 164. White points: persistence = [1,2); Green points: persistence: [2,5) ; Red points: persistence: [5,10); Cyan points: persistence > 10

Figure 2 .

 2 Figure 2. (a) Tracking result on i LIDS AVSS sequence for frames 40, 80, 164; (b) estimated models X s,t for PETS 2006 sequence for frames 40, 80, 164. White points: persistence = [1,2); Green points: persistence: [2,5) ; Red points: persistence: [5,10); Cyan points: persistence > 10

  (a)) performs better than DNLSE (Fig. 3(b)) and NLSE (Fig. 3(c)) being able to cope with the background change due to the crossing stripes and the local luminosity change caused by the lights of the other cars. During the sequence the three (R,G,B) color combinations used to compute the weight image I c1,c2,c3 i,t

Figure 3 .

 3 Figure 3. (a) ANLSE tracking, (b) DNLSE tracking, (c) NLSE tracking.The wider bounding box is S t , i.e. the searching area.

Figure 4 .

 4 Figure 4. Trajectories obtained from different tracking algorithm: blue) ANLSE; green) DNLSE, pink) NLSE, cyan) ground truth

Table 1 . Root Mean Square Error -RMSE in pixels for the sequence in Fig. 3 (352x288) for the ANLSE, DNLSE and NLSE method

 1

	Tracking method RMSE
	ANLSE	9.3
	DNLSE	13.1
	NLSE	21.8