Adaptive Particle Filter based on the Kurtosis of Distribution

Master’s Thesis Adaptive Particle Filter based on the Kurtosis of Distribution Songlin Piao Hanyang Universty, Graduate School February 2011 Master’s Thesis Adaptive Particle Filter based on the Kurtosis of Distribution Songlin Piao Hanyang Universty, Graduate School February 2011 Adaptive Particle Filter based on the Kurtosis of Distribution by Songlin Piao A Thesis Presented to the FACULTY OF THE GRADUATE SCHOOL HANYANG UNIVERSITY In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE in the Department of Electrical and Computer Engineering February 2011 Copyright 2010 Songlin Piao Adaptive Particle Filter based on the Kurtosis of Distribution by Songlin Piao Approved as to style and content by: Sang-Won Nam (Co-Chair of Committee) Jong-Il Park (Member) Whoi-Yul Kim (Member) Hanyang Universty, Graduate School February 2011 TABLE OF CONTENTS TABLE OF CONTENTS I ABSTRACT V I Introduction 1.1 Background 1.2 Related work II Particle filter 2.1 Auxiliary particle filter 2.2 Gaussian particle filter 2.3 Unscented particle filter 2.4 Rao-Blackwellized particle filter III Proposed method 11 3.1 Basic concept 12 3.2 Concept of Kurtosis 15 3.3 Proposed method in 1D 17 3.4 Proposed method in 2D 19 3.5 Proposed method in 3D 20 3.6 Proposed method in general case 22 IV Experiment 24 4.1 Simulation in 1D 24 4.2 Simulation in 2D 27 I 4.3 Simulation in 3D 28 4.4 Real particle tracking 31 4.5 Face tracking 36 V Conclusion 41 VI Appendix 43 BIBLIOGRAPHY 44 Acknowledgment 52 II LIST OF FIGURES 3.1 Transition example 13 3.2 Example in 2D case 14 3.3 Kurtosis of Gaussians 16 3.4 Proposed distribution in 1D 18 3.5 Sampling from the specific probability density function 18 3.6 Proposed pdf looks similar with water wave 20 3.7 Motion vector in spherical system 21 3.8 Proposed pdf looks similar with shockwave 22 4.1 Simulation of fluctuation case 25 4.2 Simulation of non fluctuation case 27 4.3 Simulation in 2D 29 4.4 Simulation in 3D 30 4.5 Particles detection result 31 4.6 Motions in each frame 33 4.7 Motion angle 33 4.8 Tracking result 34 4.9 RMS error comparison 35 4.10 Face is tracked and detected 37 4.11 Face is tracked but not detected 38 4.12 Motion Angle 38 4.13 Speed 39 4.14 Face tracking result 39 4.15 Analysis data 40 III LIST OF TABLES 3.1 Kurtosis of Gaussians 16 6.1 Random number generation test using 5000000 samples 43 IV ABSTRACT Adaptive Particle Filter based on the Kurtosis of Distribution Songlin Piao Department of Electrical and Computer Engineering, Hanyang University Directed by Professor Whoi-Yul Kim Kurtosis based adaptive particle filter is presented in this paper The concept of belief is proposed to each particle sampling and the distribution of particles can be adaptively changed according to the belief and motion information so that particles could track object in higher accuracy The belief and motion information could be defined as a distance function of observation vector In order to achieve this goal, we change the way of normal re-sampling technique We introduce a framework that particles are re-sampled based on the distance function We demonstrate the advantages of proposed method in two steps First, we did strict simulation tests in 1D, 2D and 3D spaces to show that our method can give better result Furthermore, we did the experiments in the real cases One is real particle tracking in the hydraulic engineering area and the other is normal face tracking based on the color feature We compared the result in each step to the result obtained from standard particle filter V I Introduction 1.1 Background The analysis and making inference about a dynamic system arise in a wide variety of applications in many disciplines The Bayesian framework is the most commonly used method for the study of dynamic systems There are two components needed in order to describe Bayesian framework First, a process model describing the evolution of a hidden state of the system and second, a measurement model on noisy observations related to the hidden state If the noise and the prior distribution of the state variable is Gaussian, the predicted and posterior densities can be described by Gaussian densities Kalman filter is one of the cases, which yields the optimized solution in MMSE But there are two big problems when people apply Bayesian framework to the real world One is that a realistic process and measurement model for a dynamic system in the real world is often nonlinear, and the other one is that the process noise and measurement noise sources could be non-Gaussian Simultaneous localization and mapping (SLAM) [1] problem in robotic research area is a typical example Kalman filter performs poor when the linear and Gaussian conditions are not satisfied This has motivated intensive research for nonlinear filters for over 40 years Nonlinear filters have involved finding suboptimal solutions and may be classified into two major approaches: a local approach, approximating the posterior density function by some particular form, and a global approach, computing the posterior density function without making any explicit assumptions about its form [2] Figure 4.13 Speed (a) First test trajectory (b) First test RMS (c) Second test trajectory (d) Second test RMS Figure 4.14 Face tracking result 39 Figure 4.15 Analysis data Fig 4.15 shows the analysis data sheet in order to explain why the RMS error in Fig 4.14(b) and in Fig 4.14(d) changed abruptly at about frame 92 You can see the speed at frame 92 is 11.40175425, which changed abruptly from the previous speed This is because the distance function in the proposed method considers only first derivative of the observation vector, so the proposed model seems a little weak to the abrupt change of the speed 40 V Conclusion Kurtosis based adaptive particle filter is proposed in this article This new method uses a new sampling method, which changes particles’ probability density function adaptively according the motion information, which is the special case of the distance function defined previously We show the accuracy of the new method by doing simulations in 1D, 2D and 3D, respectively We also applied the new method in real cases in order to show the potential applicable possibility in real world problems The result has shown that the new method has a good performance as we expected This method is very useful when the object state transition function is unknown But there is one thing that I want to emphasize here is the performance of the proposed method depends on the measurement noise and Gaussian distribution noise in the original particle filter Because proposed method and original particle filter are both depends on the measurement noise, we could ignore it here So the proposed method only depends on the noise of the original particle filter relatively There is some threshold value in this noise value If the noise of the original particle filter is less then this threshold, then the original particle filter performs better than the proposed method But if the noise of the original particle filter is greater than the threshold value, then the proposed method performs better It is meaningful when we would apply the proposed method into the real world cases Because, with the too small noise value, we cannot tracking object robustly The future work includes expanding this framework to the higher dimensional space We are doing 41 research on the convergence problem when dimension becomes higher Of course, the proposed method has also some limitations, for example, it is a little weak to deal with the abrupt changes as it was shown in face tracking case This is because the distance function considers only the first derivative of the observation vector So we need to add some other factors to handle the abrupt changes We could consider second derivative, third order derivative or curvature in the future The parameters we used in the experiments was obtained by test, so how to get the optimized parameters is also the problem we need to solve in the future We did experiments using only Gaussian distributions currently, but it would be extended to any random distribution in the future 42 VI Appendix Whatever you are doing simulation or implementing the proposed method in the real cases, random number generation is very important There are many libraries we can use For example, when I did my simulation test, I used Matlab to generate the desired distribution directly But in the case of the real world applications, people have to implement the proposed method using C language in order to maximize the speed I did a small test which compares the random number generation performance among my own implemented functions, random number generation functions from boost library, random number generation functions from GNU scientific library and random number generation functions from open computer vision library As uniform distribution and gaussian distribution are the most important distributions in particle filter, I compared the performance focused on these two distributions using the mentioned four libraries Table 6.1 shows the performance results using each library The performance of open computer vision library is the best among these methods Table 6.1 Random number generation test using 5000000 samples Own function Boost function GSL function opencv opencvForloop 150ms 220ms 90ms 60ms 892ms 290ms 230ms 682ms 70ms 934ms 43 BIBLIOGRAPHY [1] G Grisetti, G D Tipaldi, C Stachniss, W Burgard, and D Nardi, “Fast and accurate slam with rao-blackwellized particle filters,” Robotics and Autonomous Systems, vol 55, no 1, pp 30 – 38, 2007, simultaneous Localisation and Map Building [Online] Available: http://www.sciencedirect.com/science/article/ B6V16-4KXWJX2-1/2/f9af0753c6f167226e4bf5865d38e369 [2] I Arasaratnam, S Haykin, and R Elliott, “Discrete-time nonlinear filtering algorithms using gauss-hermite quadrature,” Proceedings of the IEEE, vol 95, no 5, pp 953 –977, May 2007 [3] D Alspach and H Sorenson, “Nonlinear bayesian estimation using gaussian sum approximations,” IEEE Trans on Automatic Control, no 17, pp 438–448, 1972 [4] C Shan, T Tan, and a mean shift ing using tern Recognition, vol [Online] Available: 40, Y Wei, “Real-time hand track- embedded particle filter,” Pat- 1970, 2007 no 7, pp 1958 – http://www.sciencedirect.com/science/article/ B6V14-4MRN9NH-2/2/c73d054b0c9edc452bafb7bde59a0b6b [5] I.-C based vol Chang on In a and S.-Y progressive Press, Accepted [Online] Available: Lin, particle “3d human filter,” Manuscript, motion Pattern pp tracking Recognition, 3621–3635, 2010 http://www.sciencedirect.com/science/article/ B6V14-501FPFP-3/2/f68a83ab803b3322f0554934239a55f5 44 ´ [6] A Gil, Oscar Reinoso, M Ballesta, and M Juli´a, “Multi-robot visual slam using a rao-blackwellized particle filter,” Robotics and Autonomous Systems, vol 58, no 1, pp 68 – 80, 2010 [Online] Available: http://www.sciencedirect.com/science/article/ B6V16-4WXHBVY-2/2/11c1074061fbc4f66c12297b5bdf9671 [7] L Jing and P Vadakkepat, “Interacting mcmc particle filter for tracking maneuvering target,” Digital Signal Processing, vol 20, no 2, pp 561 – 574, 2010 [Online] Available: http://www.sciencedirect.com/science/ article/B6WDJ-4WY6JXG-1/2/6899f03ab9df97dd36f3f8a125cc41cd [8] R Lakaemper and M Sobel, “Using the particle filter approach to building partial correspondences between shapes,” Int J Comput Vision, vol 88, no 1, pp 1–23, 2010 [9] A Mukherjee and A Sengupta, “Likelihood function modeling of particle filter in presence of non-stationary non-gaussian measurement noise,” Signal Processing, vol 90, no 6, pp 1873 – 1885, 2010 [Online] Available: http://www.sciencedirect.com/science/article/ B6V18-4Y0D637-1/2/cc4c6c6dd78542cda93bb3030cc91287 [10] H Pistori, V V V A Odakura, J B O Monteiro, W N Gon¸calves, A R Roel, J de Andrade Silva, and B B Machado, “Mice and larvae tracking using a particle filter with an auto-adjustable observation model,” Pattern Recognition Letters, vol 31, no 4, pp 337 – 346, 2010, 20th SIBGRAPI: Advances in Image Processing and Computer Vision [Online] Available: http://www.sciencedirect.com/science/ article/B6V15-4WKKTPH-1/2/5d4ae128c34ade76e94fedd314249e04 [11] R Sajeeb, C Manohar, and D Roy, “A semi-analytical particle filter for identification of nonlinear oscillators,” Probabilistic 45 Engineering Mechanics, [Online] Available: vol 25, no 1, pp 35 – 48, 2010 http://www.sciencedirect.com/science/article/ B6V4M-4WGK4MR-1/2/1c853d33959a93f5398e9a24dc83a717 [12] X Shao, B Huang, and J M Lee, “Constrained bayesian state estimation - a comparative study and a new particle filter based approach,” Journal of Process Control, vol 20, no 2, pp 143 – 157, 2010 [Online] Available: http://www.sciencedirect.com/science/ article/B6V4N-4XV44SP-1/2/befc1f07c8313ef37a5d64bda550a955 [13] B Xu, J Zhu, and H Xu, “An ant stochastic decision based particle filter and its convergence,” Signal Processing, vol 90, no 9, pp 2731 – 2748, 2010 [Online] Available: http://www.sciencedirect.com/science/ article/B6V18-4YPT1K4-2/2/2515af8c78afaa3e1cfec9825e6b4e4f [14] C F Chung and T Furukawa, “Coordinated pursuer control using particle filters for autonomous search-and-capture,” Robotics and Autonomous Systems, vol 57, no 6-7, pp 700 – 711, 2009 [Online] Available: http://www.sciencedirect.com/science/article/ B6V16-4TYYT75-1/2/bd9141c9a448ba653b7dd6600db3e36b [15] R van Handel, carlo particle plications, “Uniform time average consistency of monte filters,” vol 119, [Online] Available: Stochastic no 11, Processes pp 3835 and – their 3861, Ap2009 http://www.sciencedirect.com/science/article/ B6V1B-4X6MT1G-3/2/9a323f2ef6877a2cba759cef22dba375 [16] P Hlinomaz and track-before-detect puter Modelling, L Hong, particle vol 49, “A filter,” no 46 1-2, multi-rate multiple Mathematical pp 146 – model and Com- 162, 2009 [Online] Available: http://www.sciencedirect.com/science/article/ B6V0V-4S9P5NW-2/2/6745a8a30f777d606982b2db830ab36d [17] K Hotta, “Adaptive weighting of local classifiers by particle filters for robust tracking,” Pattern Recognition, vol 42, no 5, pp 619 – 628, 2009 [Online] Available: http://www.sciencedirect.com/science/ article/B6V14-4TP49JJ-3/2/54bf07f3d87c2a47aaa5856a6f59561c [18] W.-L Lu, K Okuma, and J J Little, “Tracking and recognizing actions of multiple hockey players using the boosted particle filter,” Image and Vision Computing, 189 – 205, vol 27, no 1-2, pp 2009, canadian Robotic Vision 2005 and 2006 [Online] Available: http://www.sciencedirect.com/science/article/ B6V09-4S0JMXW-1/2/f390ab32a967897ec47e01b01d561d82 [19] E Maggio and A Cavallaro, “Learning scene context for multiple object tracking,” Image Processing, IEEE Transactions on, vol 18, no 8, pp 1873 –1884, aug 2009 [20] F Moreno, J Blanco, and J Gonzalez, cific models for particle filter-based slam,” tonomous Systems, [Online] Available: vol 57, no 9, pp “Stereo vision speRobotics and Au955 – 970, 2009 http://www.sciencedirect.com/science/article/ B6V16-4VXMPR5-1/2/45238efee3416d5522006c33d161407f [21] Z Wang, X Yang, Y Xu, and S Yu, “Camshift guided particle filter for visual tracking,” Pattern Recognition Letters, vol 30, no 4, pp 407 – 413, 2009 [Online] Available: http://www.sciencedirect.com/science/ article/B6V15-4TY9MJ3-1/2/516e085802a15aa44d596cdb1a589d53 [22] W Zheng and S M Bhandarkar, “Face detection and tracking using a boosted adaptive particle filter,” Journal of Visual Communication 47 and Image Representation, vol 20, no 1, pp – 27, 2009 [Online] Available: http://www.sciencedirect.com/science/article/ B6WMK-4TG35JF-1/2/5f97c103e2f0db09cec87db6a3b82ea3 [23] S Choi and D Kim, “Robust head tracking using 3d ellipsoidal head model in particle filter,” Pattern Recognition, vol 41, no 9, pp 2901 – 2915, 2008 [Online] Available: http://www.sciencedirect.com/science/ article/B6V14-4RW434C-3/2/a96f82674094e76e7daad9fa8adaf6e0 [24] Y Li, H Ai, T Yamashita, S Lao, and M Kawade, “Tracking in low frame rate video: A cascade particle filter with discriminative observers of different life spans,” Pattern Analysis and Machine Intelligence, IEEE Transactions on, vol 30, no 10, pp 1728 –1740, oct 2008 [25] Y.-L Lin, W.-D Chang, and J.-G Hsieh, “A particle swarm optimization approach to nonlinear rational filter modeling,” Expert Systems with Applications, vol 34, no 2, pp 1194 – 1199, 2008 [Online] Available: http://www.sciencedirect.com/science/article/ B6V03-4MSR767-5/2/14feda8f2a3644d6abc9d784ed979915 [26] J Olsson and T Ryd´en, “Asymptotic properties of particle filter-based maximum likelihood estimators for state space models,” Stochastic Processes and their Applications, vol 118, no 4, pp 649 – 680, 2008 [Online] Available: http://www.sciencedirect.com/science/ article/B6V1B-4NTJH4K-1/2/26464dd28f25136d31ba438dca52fb2b [27] J Pantrigo, A S´ anchez, A Montemayor, and A Duarte, “Multidimensional visual tracking using scatter search particle filter,” Pattern Recognition Letters, vol 29, no 8, pp 1160 – 1174, 2008, pattern Recognition in Interdisciplinary Perception and Intelligence - PRint- 48 Perclntel [Online] Available: http://www.sciencedirect.com/science/ article/B6V15-4RKMHTN-1/2/94ab6e0c1753b5f58d1f8f0a2e886950 [28] A Sankaranarayanan, A Srivastava, and R Chellappa, “Algorithmic and architectural optimizations for computationally efficient particle filtering,” Image Processing, IEEE Transactions on, vol 17, no 5, pp 737 –748, may 2008 [29] M Simandl and O Straka, “Functional sampling density design for particle filters,” Signal Processing, vol 88, no 11, pp 2784 – 2789, 2008 [Online] Available: http://www.sciencedirect.com/science/ article/B6V18-4SPYKCC-1/2/3483214367f3eb9360886b5044747957 [30] H Wu, F Sun, and H Liu, “Fuzzy particle filtering for uncertain systems,” IEEE Trans Fuzzy Syst., vol 16, no 5, pp 1114–1129, 2008 [31] D Yee, J P Reilly, T Kirubarajan, and K Punithakumar, “Approximate conditional mean particle filtering for linear/nonlinear dynamic state space models,” IEEE Trans Signal Process., vol 56, no 12, pp 5790–5803, 2008 [32] L Hong olutional nal and D particle Processing, Wicker, filter vol [Online] Available: 87, with no “A spatial-domain thresholded 6, pp multires- wavelets,” 1384–1401, Sig2007 http://www.sciencedirect.com/science/article/ B6V18-4MSR760-1/2/6f91b2c80bbf63fe739f3429c6dc2b0c [33] J Li and C.-S Chua, “Transductive local exploration particle filter for object tracking,” Image and Vision Computing, vol 25, no 5, pp 544 – 552, 2007 [Online] Available: http://www.sciencedirect.com/science/ article/B6V09-4KBX4GR-1/2/06794f30096ba897fad4242a84b77f66 49 [34] S McKenna and H Nait-Charif, “Tracking human motion using auxiliary particle filters and iterated likelihood weighting,” Image and Vision Computing, vol 25, no 6, pp 852 – 862, 2007 [Online] Available: http://www.sciencedirect.com/science/article/ B6V09-4KGX84W-2/2/220f0915973165f496a6351ddb18a08f [35] Y Rathi, N Vaswani, A Tannenbaum, and A Yezzi, “Tracking deforming objects using particle filtering for geometric active contours,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol 29, pp 1470–1475, 2007 [36] D E Clark and J Bell, “Convergence results for the particle phd filter,” IEEE Trans Signal Process., vol 54, no 7, pp 2652–2661, 2006 [37] M Emoto, A Hayashi, and N Suematsu, “Efficient posture estimation using the particle filter,” International Congress Series, vol 1291, pp 205 – 208, 2006, brain-Inspired IT II: Decision and Behavioral Choice Organized by Natural and Artificial Brains Invited and selected papers of the 2nd International Conference on Brain-inspired Information Technology held in Hibikino, Kitakyushu, Japan between and October 2005 [Online] Available: http://www.sciencedirect.com/science/ article/B7581-4K51GH7-1R/2/a9d7007d3eba2e8912edd16005cb0dc6 [38] P Fearnhead, O Papaspiliopoulos, and G O Roberts, “Particle filters for partially observed diffusions,” Tech Rep., 2006 [39] H Tamimi, H Andreasson, A Treptow, T Duckett, and A Zell, “Localization of mobile robots with omnidirectional vision using particle filter and iterative sift,” Robotics and Autonomous Systems, vol 54, no 9, pp 758 – 765, 2006, selected papers from the 2nd European Conference on Mobile Robots (ECMR ’05) 50 [Online] Available: http://www.sciencedirect.com/science/article/ B6V16-4KJ0SSH-1/2/a0cb5ac27fe99eec909f6767d486baa2 [40] M Bolic, P M Djuric, and S Hong, “Resampling algorithms and architectures for distributed particle filters,” IEEE Trans Signal Process., vol 53, no 7, pp 2442–2450, 2005 [41] Z Khan, T Balch, and F Dellaert, “Mcmc-based particle filtering for tracking a variable number of interacting targets,” IEEE Trans Pattern Anal Mach Intell., vol 27, no 11, pp 1805–1819, 2005 [42] S S¨ akk¨ a, A Vehtari, and J Lampinen, “Rao-blackwellized particle filter for multiple target tracking,” Information Fusion, vol 8, no 1, pp – 15, 2007, special Issue on the Seventh International Conference on Information Fusion-Part II, Seventh International Conference on Information Fusion [Online] Available: http://www.sciencedirect.com/science/ article/B6W76-4HJRRX5-1/2/c4d5356937753f21dc812b7943ffd6cd [43] T Schon, F Gustafsson, and P.-J Nordlund, “Marginalized particle filters for mixed linear/nonlinear state-space models,” IEEE Trans Signal Process., vol 53, no 7, pp 2279–2289, 2005 [44] M Pitt and N Shephard, “Filtering via simulation: Auxiliary particle filters,” Journal of the American Statistical Association, vol 94, no 446, pp 590–599, Jun 1999 [45] J Kotecha and P Djuric, “Gaussian particle filtering,” Signal Processing, IEEE Transactions on, vol 51, no 10, pp 2592 – 2601, oct 2003 [46] S Bonnabel, P Martin, and E Salaun, “Invariant extended kalman filter: theory and application to a velocity-aided attitude estimation problem,” in Decision and Control, 2009 held jointly with the 2009 28th 51 Chinese Control Conference CDC/CCC 2009 Proceedings of the 48th IEEE Conference on, 2009, pp 1297 –1304 [47] J Mart´ınez-del Rinc´ on, C Orrite-Uru˜ nuela, and G Rogez, “Raoblackwellized particle filter for human appearance and position tracking,” in Pattern Recognition and Image Analysis, ser Lecture Notes in Computer Science, J Mart´ı, J Bened´ı, A Mendon¸ca, and J Serrat, Eds Springer Berlin / Heidelberg, 2007, vol 4477, pp 201–208 [Online] Available: http://dx.doi.org/10.1007/978-3-540-72847-4 27 [48] G Casella and C P Robert, “Rao-blackwellisation of sampling schemes,” Biometrika, vol 83, no 1, pp 81–94, 1996 [Online] Available: http://biomet.oxfordjournals.org/content/83/1/81.abstract [49] Q Cheng and P Bondon, “A new unscented particle filter,” in Proc IEEE Int Conf Acoustics, Speech and Signal Processing ICASSP 2008, 2008, pp 3417–3420 [50] S C Chapra, Applied Numerical Methods with MATLAB for Engineering and Scientists, 2nd ed MC Graw Hill [51] “Department of ocean engineering - university of rhode island,” http: //www.oce.uri.edu/areasofstudy.shtml [52] “Mvtec software gmbh building vision for business,” http://www.mvtec com/ [53] A Bruhn, J Weickert, and C Schn¨orr, “Lucas/kanade meets horn/schunck: Combining local and global optic flow methods,” International Journal of Computer Vision, vol 61, pp 211–231, 2005 52 Acknowledgment I would like to express my sincere gratitude to my advisor Prof Whoi Yul Kim for the continuous support during my master study and research I am also thankful to my mother Xing Ai Li, my young brother Xiang Yu Piao and my girl friend Mei Xian Fang who supported me all through with their love I would like to thank to all my fellow friends who came to Hanyang University together from Shanghai Jiao Tong University for their support during my study I would like to thank to my friend Sheng Zhu Kim who gave me support from the side of the mathematics for my master thesis All my colleagues in Image Engineering Lab have supported me in my research work during my master study period more or less I want to thank them for their help, support and valuable hints I would like to thank to BK21 for their financial support during my master study period 53