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2. Outline: A brief introduction to Kalman filter Classical Kalman Filter Kalman & Stereo vision localization Different extensions of Kalman Filter Image and Video De-noising (Kalman approach) Kalman Tracking for Robust pose estimation 3. Introduction: Kalman is a recurrent algorithm relying on series of uncertain measurements to optimally estimate an unknown variable. The Kalman filter is 50 years old but is still a popular method for sensory information analysis . The Kalman applications span area of navigation systems, aerospace, robotics, information processing , computer vision, etc Main properties: Requires small computational cost State space representation and ease of implementation Recursive algorithm which only uses last state and current observation 4. Formulation:Where F is transition matrix, G is a control input matrix . H is output matrix. Random variables w and v are process and measurement noise respectively. 5. Kalman stages Predict or time updateCorrect or measurement update 6. Multivariate Gaussians: posterior priorMeasurement evidenceMarkov model:p( x | z ) p( z | x) p( x) 7. Multivariate Gaussians: posterior Measurement evidenceprior 1 p ( z | x) exp ( x z )T 1 ( x z ) 2 1 p ( x) exp ( x )T 1 ( x ) 2 new = ( 1 + 1 ) 1 ( 1 + 1 z )( new = 1 + 1)1 8. Multivariate Gaussians:CS373 - Programming a Robotic Car - Unit 2 Kalman Filter Prediciton - CS373 Unit 2 - Udacity By Sebastian Thrun 9. Kalman & Stereo Vision Localization: Stereo localization requires both cameras information at each time step. 10. Kalman & Stereo Vision Localization: There is a small time delay between the two cameras image capturing. This time lag causes an error in estimation of object position in Cartesian coordinates. 11. Non-linearity & Kalman Filter: [Refer to Video1] Python and Blenderused for data acquisition which brings Physics engine to generate exact trajectory. Blender communicate with matlab over UDP protocol. 12. Non-linearity & Kalman Filter: Extended Kalman FilterFigures from CS-223 (Stanford) 13. Non-linearity & Kalman Filter: Extended Kalman FilterFigures from CS-223 (Stanford) 14. Non-linearity & Kalman Filter: Extended Kalman FilterFigures from CS-223 (Stanford) 15. Adaptive Kalman Filter: Adaptive Fading Kalman Filter Innovation based Adaptive Kalman Filter 16. Kalman Filter CategoriesKalmanEKFNon-linearityAEKFAKFAdaption 17. SV ResultsNoise level n = 0.05Noise level n = 0.5 18. Image and Video de-noising: noise is assumed to be signal-independent at each pixel 19. Image and Video de-noising: Wiener Filter Bilateral Filter Median Filter Non-Local Means Filtering 20. Image and Video de-noising: Wiener Filter Assume Stationary signal Signal and noise are independent with constant PSD 21. Image and Video de-noising: Wiener Filter 22. Image and Video de-noising: Wiener Filter n = 15 23. Image and Video de-noising: Wiener FilterFiltered 24. Image and Video de-noising: Non-Local Means Filter Each pixel is replaced by weighted average of its neighborhoods. Larger patch size leads to expensive computation. while Smaller patch size results in poor de-noising effect. All weights should be normalized . 25. Image and Video de-noising: Non-Local Means Filter n = 15 26. Image and Video de-noising: Non-Local Means Filter 27. Image and Video de-noising: Comparison originalWiener filterNoisy n = 15NLM filter 28. Image and Video de-noising: ComparisonPSNR (db) 14 12 10 8 6 4 2 0PSNR (db) 29. Video de-noising: Each pixel is replaced by 3D spatio-temporal weighted average of its neighborhoods. 30. Motion estimation & Kalman Filter Optical Flow Algorithms Block Matching Algorithms Full Search Algorithm (FSA) Three Step Algorithm (TSA) Four Step Algorithm (FSA) Diamond Search Algorithm (DSA) Adaptive Rood Pattern Search 31. Motion estimation & Kalman Filter Why Kalman Filter? Motion estimation inaccuracy when illumination is changed Inaccuracy based on noise The size of moving object may differ from searching region size Fails in presence of occlusion 32. Motion estimation & Kalman Filter 2D Causal AR model:B(m,n-1)B(m-1,n)B(m,n)vi , x (m, n) = a01vi , x (m, n 1) + a10vi , x (m 1, n) + wi , x (m, n) vi , y (m, n) = b01vi , y (m, n 1) + b10vi , y (m 1, n) + wi , y (m, n) 33. Motion estimation & Kalman Filter State space representation : vi , x (m, n) a01 a11 0 0 vi , x (m 1, n) q v (m 1, n) 1 0 0 0 vi , x (m, n 1) 0 i,x = * + vi , x (m, n) 0 0 b01 b11 vi , x (m 1, n) q vi , x (m 1, n) 0 0 1 0 vi , x (m, n 1) 0 vi , x (m, n) r vi , x (m, n) 1 0 0 0 vi , x (m 1, n) 0 + = z= * vi , y (m, n) 0 0 1 0 vi , x (m, n) r vi , x (m 1, n) 0 q 2 0 Q= 2 q 0 0 q2 0 0 0 q2r 2 0 R= 2 r 0 0 r2 0 0 0 r200000 0 0 0 0 0 0 0 34. Motion estimation & Kalman Filter 35. Motion estimation & Kalman Filter 36. Motion estimation & Kalman Filter 37. Motion estimation & Kalman Filter 38. Motion estimation & Kalman Filter 39. Motion estimation & Kalman Filter Green arrows show motion vectors Red arrows show corrected motion vectors Kalam uses corrected version of neighboring vectors if available 40. Motion estimation Results [Refer to Video2] Results are generated using the following approaches: 2D 3x3 spatial domain NLM 3D 3x3x3 spatio-temporal domain NLM 3D 5x3x3 spatio-temporal domain NLM 41. Motion estimation Results [Refer to Video2] 42. Kalman Tracking for Robust pose estimation Object Recognition Object recognition requires set of 2D correspondences between target image and video frame. Robust RANSAC (RANdom SAmple Consensus) is used to robustly estimate the pose against wrong matches. Robust pose estimation brings an exhaustive computational cost. 43. Kalman Tracking for Robust pose estimation Tracking Feature points tracking Tracking the region of search for matches. Keep tracking of estimated pose after pose estimation process. Keep tracking of inliers (best matches) found from RANSAC loop. (My Contribution) 44. Kalman Tracking for Robust pose estimation Pose estimation: Video Frame pointsTarget Points Point MatchingRANSAC LoopM-estimator (LSM)Kalman tracker 45. Kalman Tracking for Robust pose estimation Pose estimation: [Refer to Video3] 46. Implementation Programs are written with C++ using OpenCV library C++ implementation of generic Kalman Filter algorithm C++ implementation of Non-Local Means algorithm C++ implementation of FSA Block Matching algorithm C++ implementation of Wiener filter for image denoising Matlab full implementation of Conventional, Extended , Adaptive Fading and Innovation-based Adaptive Kalman filterWill be available for download soon on https://github.com/hamidb 47. Conclusion: We studied Kalman Filter as an adaptive filter which can be applied for Smoothing, prediction , filtering and tracking Three main extensions of Kalman filter including conventional, extended and adaptive Kalman filter are reviewed. Applying Kalman Filter in Video de-noising and compared the result with other filters like Wiener filter Applying Kalman filter in computer vision applications such as multiple view geometry and pose estimation 48. References: 1.Dan Simon. Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches. Wiley-Interscience, 1 edition, August 2006. 2. S. Sangsuk-Iam and T. E. Bullock. Analysis of discrete time kalman filtering under incorrect noise covariances. Automatic Control, IEEE Transactions 3. Qijun Xia, Ming Rao, Yiqun Ying, and Xuemin Shen. Adaptive fading kalman filter with an application. Automatica 4. Buades, A., Coll, B., Morel, J.M.: A non-local algorithm for image denoising. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR). (2005) 5. Buades, A., Coll, B., Morel, J.M.: Denoising image sequences does not require motion estimation. In: IEEE International Conference on Advanced Video and Signal Based Surveillance. (2005) 6. Han, Yu Bing and Chen, Rushan: Efficient video denoising based on dynamic nonlocal means . Journal of Image Vision Comput. (2012) 7. Chung-Ming Kuo; Chaur-Heh Hsieh; Yue-Dar Jou; Hsieh-Cheng Lin; Po-Chiang Lu, "Motion estimation for video compression using Kalman filtering," Broadcasting, IEEE Transactions on , Jun 1996 8. Bazargani, H.; Omidi, E.; Talebi, H.A., "Adaptive Extended Kalman Filter for asynchronous shuttering error of stereo vision localization," Robotics and Biomimetics (ROBIO), 2012 IEEE International Conference on , vol., no., pp.2254,2259, 11-14 Dec. 2012 9. Ruiz, V.; Fotopoulos, V.; Skodras, A.N.; Constantinides, A.G., "An 88-block based motion estimation using Kalman filter," Image Processing, 1997. 10. Antoni Buades and Bartomeu Coll and Jean Michel Morel, On image denoising methods,Technical Note, CMLA (Centre de Mathematiques et de Leurs Applications,(2004) 49. Thank you for your Attention.http://www.slideshare.net/hamidbazargani/venndiagrams-29181371