Distance Functions for Polygons/Trajectories

1. Distance Functions for Polygons/Trajectories Chun-Sheng & Vadeerat 10/09/2009

2. Type of Distance Function for Polygons/Trajectories • Similarity Measurement • Focus on the similarity of shapes • Remedy for • Scaling • Rotation • Alignment • Classification problem (speech recognition, image matching, hand writing recognition … ) • Geographical Distance • Scaling, rotation, and alignment are all considered in the calculation of the distance • This is what we concerned for spatial data mining DMML Group Meeting

3. Distance function for trajectories • Discrete Trajectories: • Most of the researches are based on discrete trajectories • Connect observations at different time by a line segment Polylines 6 2 5 1 3 4 A discrete trajectory of size equal to 6 ((x1,y1),(x2,y2),(x3,y3),(x4,y4),(x5,y5),(x6,y6)) DMML Group Meeting

4. Distance Function for trajectories Trajectories • Euclidean distance[2] (trajectories): • average distance between corresponding points • Trajectories need to be the same size (length) • PCA+Euclidean distance[3] (trajectories) : • “a trajectory is first represented as a 1-D signal by concatenating the x- and the yprojections. Then the signal is converted into the first few PCA (Principle Components Analysis) coefficients. The trajectory similarity is computed as the Euclidean distance between the PCA coefficients” • HMM(Hidden Markov Model)-based distance[5] (trajectories) • ERP(Edit distance with Real Penalty)[6] (trajectories) • LCSS distance (Longest Common Subsequence)[7] (trajectories) • DTW distance (Dynamic Time Warping)[7] (trajectories) • Area Distance (trajectories) : Area enclosed by trajectories DMML Group Meeting

5. Dynamic Time Warping i i i+2 i i time time Any distance (Euclidean, Manhattan, …) which aligns the i-th point on one time series with the i-th point on the other will produce apoor similarity score. A non-linear (elastic) alignment produces amore intuitive similarity measure, allowing similar shapes to match even if they are out of phase in the time axis. DMML Group Meeting

6. Area between Trajectories (Geo) D(A, B) = Sum(Area(1)+Area(2)+Area(3)) / (Length(A)+Length(B)) • A and B can be different length • Lower computation cost: O(m+n) A 3 1 2 B DMML Group Meeting

7. Distance Function for Polygons Polygons • Hausdorff distance[4] (trajectories/polygons) • Fréchet Distance [8][9][10] (trajectories/polygons) DMML Group Meeting

8. Hausdorff distance • Hausdorff distance is the “maximum distance of a set to the nearest point in the other set ” • Hausdorff distance from A to B Brute force algorithm :O(n m) 1.  h = 02.  for every point ai of A,      2.1  shortest = Inf ;      2.2  for every pointbj of Bdij = d (ai , bj)                    if dij< shortest then                              shortest = dij      2.3  if shortest > h then                    h = shortest DMML Group Meeting

9. Hausdorff distance • Hausdorff distance is oriented: • Hausdorff distance between polygons(convex)/trajectories • H(A, B) applies to all defining points of these lines or polygons, and not only to their vertices • Sensitive to position: DMML Group Meeting

10. FréchetDistance (Vadeerat) DMML Group Meeting

11. Fréchet Distance α(t) and β(t) range over continuous and increasing functions with α(0) = 0, α(1) = N, β(0) = 0 and β(1) = M only DMML Group Meeting

12. References [1] Zhang Zhang, Kaiqi Huang, and Tieniu Tan. Comparison of similarity measures for trajectory clustering in outdoor surveillance scenes. In Proc. 18th International Conference on Pattern Recognition ICPR 2006, volume 3, pages 1135–1138, 2006. [2] Z.Fu, W.Hu, T.Tan, “Similarity Based Vehicle Trajectory Clustering and Anomaly Detection”, in Proc. Intl. Conf. on Image Processing (ICIP’05), vol 2, pp 602-605, 2005 [3] F.I.Bashir, A.A.Khokhar, D.Schonfeld, “Segmented trajectory based indexing and retrieval of video data”, in Proc. Intl. Conf. on Image Processing (ICIP’03), vol 2, pp.623-626, 2003 [4] G. Rote (1991). Computing the minimum Hausdorff distance between two point sets on a line under translation. Information Processing Letters, v. 38, pp. 123-127 [5] F. Porikli, “Trajectory distance metric using hidden markov model based representation,” in IEEE European Conference on Computer Vision, PETS Workshop, 2004 [6] L. Chen and R. Ng. On the marriage of edit distance and Lp norms. In Proc. VLDB, 2004 [7] Chen, L., Özsu, M. T., and Oria, V. 2005. Robust and fast similarity search for moving object trajectories. In Proceedings of the 2005 ACM SIGMOD international Conference on Management of Data (Baltimore, Maryland, June 14 - 16, 2005). SIGMOD '05. ACM, New York, NY, 491-502 [8] Buchin, K., Buchin, M., and Wenk, C. 2006. Computing the Fréchet distance between simple polygons in polynomial time. In Proceedings of the Twenty-Second Annual Symposium on Computational Geometry (Sedona, Arizona, USA, June 05 - 07, 2006). SCG '06. ACM, New York, NY, 80-87 [9] Alt, H., Knauer, C., and Wenk, C. 2001. Matching Polygonal Curves with Respect to the Fréchet Distance. In Proceedings of the 18th Annual Symposium on theoretical Aspects of Computer Science (February 15 - 17, 2001). A. Ferreira and H. Reichel, Eds. Lecture Notes In Computer Science, vol. 2010. Springer-Verlag, London, 63-74 [10] Computing the Fréchet distance between two polygonal curves http://www.cim.mcgill.ca/~stephane/cs507/Project.html DMML Group Meeting