Robust Online Trajectory Clustering

Robust Online Trajectory Clustering without Computing Trajectory Distances Michael Ulm,Norbert Br¨a ndleAustrian Institute of Technology,Mobility Department {Michael.Ulm|Norbert.Braendle}@ait.ac.atAbstractWe propose a novel trajectory clustering algorithm which is suitable for online processing of pedestrian or vehicle trajectories computed with a vision-based tracker.Our approach does not require defining dis-tances between trajectories,and can thus process bro-ken trajectories which are inevitable in most cases when object trackers are applied to real world video footage.Clusters are defined as smooth vectorfields on a bounded connected set,and cluster distance is based on pairwise distances between vector sets.The results are illustrated on a trajectory set from the Edinburgh In-formatics Forum Pedestrian Dataset,on a trajectory set from a public transport junction,and trajectories from an experimental setup in a corridor.1.IntroductionRecent years saw a dramatic increase in use of video technology for movement analysis in applications such as security and traffic analysis.Monitoring and analyz-ing such video data has become an exceedingly chal-lenging task.End users often desire automatic methods to distinguish between normal and abnormal behavior of individuals.Since people movement in an examined location may shift over time,such abnormal behavior detectors should cope with dynamic scenes.Input data often comes from video devices as a continuous stream, requiring online methods that are incremental in nature.Many approaches are based on clustering trajectory data extracted with a detector/tracker combination and concentrate on distance measures between trajectories [1,2,6,7].Such trajectory clustering approaches work well in simple scenes,but have problems in many real-world scenes,for which currently available object track-ers produce noisy and broken trajectories.Figure1 shows a snapshot of the results of a commercially avail-able HOG-inspired people tracker[9].More globalap-Figure1.Noisy and broken people trajec-tories computed on corridor surveillancefootageproaches targeting at motion segmentation can usually deal with more noisy data[10,11,12],but are difficult to generalize to dynamic scenes.A classic approach to online clustering is described in[8].Their idea of a rep-resentative trajectory for each cluster,however,is not applicable to most real-world data from object trackers. An online algorithm capable of dealing with noisy and broken trajectories is provided in[4].Their two-step approach,however,throws away global information on the trajectories and so can easily lead to false positives.In the following we present a novel approach that is able to deal with broken and noisy trajectories.Sec-tion2describes the concept of the algorithm,and Sec-tion3provides experimental results on three datasets.2.Proposed AlgorithmIn order to overcome the problem of computing dis-tances between trajectories,we introduce a new inter-pretation of a trajectory cluster.We define a trajectory cluster as a bounded,connected setΩtogether with a21st International Conference on Pattern Recognition (ICPR 2012) November 11-15, 2012. Tsukuba, Japansmooth vectorfieldφ:Ω→R.We represent a cluster as a set of unit length vectors,each given by a starting point and a direction.In addition,each vector records the number of trajectories that were used in its compu-tation as a weight factor.The clusters are constructed in such a way that the starting points are distributed evenly throughout the support of the cluster.A trajectory can then be trans-formed into a cluster via spatial subsampling.Given two vector sets V1and V2,we define the di-rected distance¯δ(V1,V2)between them as¯δ(V1,V2)=11v1∈V1minv2∈V2d(v1,v2),(1)where d is a weighted Euclidean distance between the vectors.The distanceδ(V1,V2)between two vector sets V1and V2is then defined asδ(V1,V2)=¯δ(V1,V2)+¯δ(V2,V1).(2) Note that the functionδdoes not define a metric on the set of non-empty vector sets,as it does not satisfy the triangle inequality.Algorithm1describes the update for online trajec-tory clustering.The algorithm takes as input a list L of clusters and a trajectory T.The parameters are the difference threshold c d,the merge threshold c m and the merge frequency f m.The output of the algorithm is the updated list of clusters L.Merging of clusters is a rare but necessary step,and is only performed after a given number of trajectories have been processed.Clusters merging is described in Algorithm2.It has as input two clusters C1and C2and as output the merged cluster C M.Every processed trajectory is assigned to a cluster–it is therefore essential to distinguish between clusters that show stable trends(in the following denoted as ma-ture clusters),and clusters that were created from rare occurrences.We use a simple threshold on the number of trajectories in the cluster to distinguish between the two cases.3.Experimental ResultsThe algorithm has been implemented in C++,and performance was measured on an Intel Core2Duo CPU with2.8GHz.There are no standard clustering data sets available for comparison,making it hard to com-pare the algorithm performance in an objective manner. We demonstrate the results of our proposed clustering on three different datasets with varying properties.The first dataset is a subset from the Edinburgh Informat-ics Forum Pedestrian Database[5],and is denoted as Algorithm1Update cluster list LCreate cluster C T out of Tfor all C i∈L dod i:=δ(C T,C i)Increase age of C iend forFind k and d k such that d k=min i d iif d k<c d thenC k←Merge(C k,C T)Decrease age of C kelseAdd C T to Lend ifRemove clusters that are too oldif size(L)=0mod(f m)thenrepeatfor all C i∈L,C j∈L with i=j dod i,j:=δ(C i,C j).Find k,l such that d k,l=min i,j d i,j.if d k,l<c m thenC k←Merge(C k,C l)Delete C l from Lend ifend foruntil No clusters were mergedend ifEIFPD.The image sequences of EIFPD were captured from an overhead camera with640×480pixels.The trajectories computed from the images sequences are noisy but continuous,with little or no false trajectories (see Fig.2a).This dataset comprises the smoothed tra-jectories from2009-09-18to2009-09-29.The second dataset is composed of people trajecto-ries processed from image sequences at a public trans-port hub in Graz,Austria and is denoted as Puntigam. The1024×768pixel image sequences have been cap-tured from an elevated view,and the trajectories have been computed with a commercially available real-time Algorithm2Merge clustersMC1and C2Set the age of C M as the minimum of the ages of C1 and C2for all P on the support of C M doSet the direction of P to be the weighted mean of the directions of C1and C2at P.Set the weight at P as the sum of the weights of P at C1and C2.end for(a)EIFPD(b)PuntigamFigure 2.Typical trajectories from the EIFPD and Puntigam datasetsimplementation of a HOG inspired people tracker [9].This dataset is composed of smooth and usually contin-uous trajectories,with only occasional breaks and skips,but several false trajectories (see Fig.2b).The third dataset (LASE )comprises people trajec-tories in the corridor shown in Fig.1.The 640×480image sequences have been captured from the low ceil-ing,and the trajectories have been computed with same people tracker [9]as in the Puntigam dataset.The tra-jectories in this dataset are very noisy and highly frag-mented,but without false trajectories.We have used similar parameter values for all three scenes –the only adaptation was to accomodate the dif-ferent temporal and spatial scales of the scenes.The results of the test runs are summarized in Table 1.Figures 3a and 3b show the development of the num-ber of clusters and number of mature clusters,respec-tively,in relation to the number of processed trajecto-ries.It can be seen,that after some initial rise,the num-ber of clusters levels off and remains approximately at a constant level.The sudden drops in the EIFPDdataset(a)AllClusters(b)Mature ClustersFigure 3.Development of Clustersare caused by the lack of trajectories during the nights.The mature clusters of each dataset are visualized in Figure 4.The algorithm is clearly able to recognize the main motion trends.A shortcoming of the method can be seen in the EIFPD and LASE data sets,where gen-eral movements are visible,but the algorithm does not discern between different exits that are close to each other.The method of [3]could be employed to over-come this problem.Table 1.Summary of Clustering Results Dataset EIFPD Puntigam LASE Nr.trajectories 106151149512Nr.points 8890634497413476Nr.clusters 1427081Nr.mature clusters 2658Running time (s)2172.613.7 5.7(a)EIFPD(b)Puntigam(c)LASEFigure 4.Visualization of mature clusters of the three datasets4.ConclusionWe have presented a new method of online trajec-tory clustering.The key idea was to represent clusters as smooth vector fields on a bounded connected set.The resulting clustering implementation is fast,robust and easily configurable.This makes it suitable for real world applications that need real time performance.References[1]Z.Fu,W.Hu,and T.Tan.Similarity based vehicle tra-jectory clustering and anomaly detection.In Proceede-ing IEEE International Conference on Image Process-ing,2005.ICIP 2005,volume 2,2005.[2]S.Gaffney and P.Smyth.Trajectory Clustering with Mixtures of Regression Models.In Proc.5th ACM SIGKDD International Conference on Knowledge Dis-covery and Data Mining .ACM,1999.[3]J.Lee,J.Han,and K.Wang.Trajectory Clustering:A Partition-and-Group Framework.In Proc 2007ACM SIGMOD International Conference on Management of Data ,pages 593–604,2007.[4]Z.Li,J.Lee,X.Li,and J.Han.Incremental Cluster-ing for Trajectories.In Proc.2010Database Systems for Advanced Applications (DASFAA’10),volume 5982,pages 32–46,Tsukuba,Japan,2010.[5]B.Majecka.Statistical models of pedestrian behaviour in the Forum.Master’s thesis,School of Informatics,University of Edinburgh,2009.[6]B.Morris and M.Trivedi.Learning Trajectory Patterns by Clustering:Experimental Studies and Comparative Evaluation.In Proc.IEEE International Conference on Computer Vision and Pattern Recognition (CVPR2009),2009.[7]N.Pelekis,I.Kopanakis,E.E.Kotsifakos,E.Frent-zos,and Y .Theodoridis.Clustering Uncertain Trajec-tories.Knowledge and Information Systems (KAIS),28(1):117–147,2011.[8]C.Piciarelli and G.L.Foresti.On-line trajectory clus-tering for anomalous events detection.Pattern Recog-nition Letters ,27(15):1835–1842,2006.[9]O.Sidla.Object tracking by combining detection,mo-tion estimation,and verification.In Proc.of the SPIE.Intelligent Robots and Computer Vision XXVII:Algo-rithms and Techniques.,volume 7539,2010.[10]P.Widhalm and N.Br¨a ndle.Learning Major Pedestrian Flows in Crowded Scenes.In Proc.20th International Conference on Pattern Recognition (ICPR2010),pages 4064–4067,Aug.2010.[11]S.Wu,B.E.Moore,and M.Shah.Chaotic Invariants of Lagrangian Particle Trajectories for Anomaly Detec-tion in Crowded Scenes.In Proc.International Con-ference on Computer Vision and Pattern Recognition (CVPR2010),2010.[12]M.Zeppelzauer,M.Zaharieva,D.Mitrovic,and C.Bre-iteneder.A Novel Trajectory Clustering Approach for Motion Segmentation.In Proc.16th International Con-ference on Advances in Multimedia Modeling ,pages 433–443.Springer,2010.。