Tracking H akan Ard o February 22, 2012 H akan Ard o Tracking - PowerPoint PPT Presentation

Introduction Tracking Tracking H˚ akan Ard¨ o February 22, 2012 H˚ akan Ard¨ o Tracking February 22, 2012 1 / 51

Introduction Tracking Sliding Window Detection Outline Introduction 1 Sliding Window Detection Tracking 2 Greedy Kalman filter Particle filter H˚ akan Ard¨ o Tracking February 22, 2012 2 / 51

Introduction Tracking Sliding Window Detection Sliding window detectors H˚ akan Ard¨ o Tracking February 22, 2012 3 / 51

Introduction Tracking Sliding Window Detection Detection probability H˚ akan Ard¨ o Tracking February 22, 2012 4 / 51

Introduction Tracking Greedy Kalman filter Particle filter Greedy tracker H˚ akan Ard¨ o Tracking February 22, 2012 5 / 51

Introduction Tracking Greedy Kalman filter Particle filter STC Lecture Series An Introduction to the Kalman Filter Greg Welch and Gary Bishop University of North Carolina at Chapel Hill Department of Computer Science http://www.cs.unc.edu/~welch/kalmanLinks.html UNC Chapel Hill Computer Science Slide 1 H˚ akan Ard¨ o Tracking February 22, 2012 6 / 51

Introduction Tracking Greedy Kalman filter Particle filter HUMAN AND SYSTEMS ENGINEERING: Gentle Introduction to Particle Filtering Sanjay Patil 1 and Ryan Irwin 2 Graduate research assistant 1 , REU undergrad 2 Human and Systems Engineering URL: www.isip.msstate.edu/publications/seminars/msstate/2005/particle/ H˚ akan Ard¨ o Tracking February 22, 2012 7 / 51

Introduction Tracking Greedy Kalman filter Particle filter Some Intuition UNC Chapel Hill Computer Science Slide 5 H˚ akan Ard¨ o Tracking February 22, 2012 8 / 51

Introduction Tracking Greedy Kalman filter Particle filter First Estimate Conditional Density Function z 1 σ 2 , z 1 1 = z 1 x ˆ N( z 1 , σ z 1 2 ) σ 1 = σ 2 2 ˆ z 1 -2 0 2 4 6 8 10 12 14 UNC Chapel Hill Computer Science Slide 6 H˚ akan Ard¨ o Tracking February 22, 2012 9 / 51

Introduction Tracking Greedy Kalman filter Particle filter Second Estimate Conditional Density Function z 2 σ 2 , z 2 2 ) N( z 2 , σ z 2 2 = ...? x ˆ σ 2 = ...? 2 ˆ -2 0 2 4 6 8 10 12 14 UNC Chapel Hill Computer Science Slide 7 H˚ akan Ard¨ o Tracking February 22, 2012 10 / 51

Introduction Tracking Greedy Kalman filter Particle filter Combine Estimates [ ] z 1 + σ z 1 [ ] z 2 ( 2 + σ z 2 ) ( 2 + σ z 2 ) = σ z 2 σ z 1 σ z 1 x ˆ 2 2 2 2 2 [ ] = ˆ 1 + K 2 z 2 − ˆ x x 1 where ( 2 + σ z 2 ) K 2 = σ z 1 σ z 1 2 2 UNC Chapel Hill Computer Science Slide 8 H˚ akan Ard¨ o Tracking February 22, 2012 11 / 51

Introduction Tracking Greedy Kalman filter Particle filter Combine Variances 1 σ 2 = 1 σ z 1 ( ) + 1 σ z 2 ( ) 2 2 2 UNC Chapel Hill Computer Science Slide 9 H˚ akan Ard¨ o Tracking February 22, 2012 12 / 51

Introduction Tracking Greedy Kalman filter Particle filter Combined Estimate Density Conditional Density Function ˆ σ 2 ) x, ˆ N( x = ˆ x ˆ 2 2 = σ 2 σ ˆ 2 -2 0 2 4 6 8 10 12 14 UNC Chapel Hill Computer Science Slide 10 H˚ akan Ard¨ o Tracking February 22, 2012 13 / 51

Introduction Tracking Greedy Kalman filter Particle filter Add Dynamics dx / dt = v + w where v is the nominal velocity w is a noise term (uncertainty) UNC Chapel Hill Computer Science Slide 11 H˚ akan Ard¨ o Tracking February 22, 2012 14 / 51

Introduction Tracking Greedy Kalman filter Particle filter Particle filtering algorithm step-by-step (1) Particle Filtering – Gentle Introduction and Implementation Demo Page 7 of 20 H˚ akan Ard¨ o Tracking February 22, 2012 15 / 51

Introduction Tracking Greedy Kalman filter Particle filter Particle filtering step-by-step (2) Particle Filtering – Gentle Introduction and Implementation Demo Page 8 of 20 H˚ akan Ard¨ o Tracking February 22, 2012 16 / 51

Introduction Tracking Greedy Kalman filter Particle filter Particle filtering step-by-step (3) Particle Filtering – Gentle Introduction and Implementation Demo Page 9 of 20 H˚ akan Ard¨ o Tracking February 22, 2012 17 / 51

Introduction Tracking Greedy Kalman filter Particle filter Particle filtering step-by-step (4) Particle Filtering – Gentle Introduction and Implementation Demo Page 10 of 20 H˚ akan Ard¨ o Tracking February 22, 2012 18 / 51

Introduction Tracking Greedy Kalman filter Particle filter Particle filtering step-by-step (5) Particle Filtering – Gentle Introduction and Implementation Demo Page 11 of 20 H˚ akan Ard¨ o Tracking February 22, 2012 19 / 51

Introduction Tracking Greedy Kalman filter Particle filter Particle filtering step-by-step (6) Particle Filtering – Gentle Introduction and Implementation Demo Page 12 of 20 H˚ akan Ard¨ o Tracking February 22, 2012 20 / 51

Introduction Tracking Greedy Kalman filter Particle filter Some Details = + x x A A x x w w = z z H H x x . . . . . . . . UNC Chapel Hill Computer Science Slide 13 H˚ akan Ard¨ o Tracking February 22, 2012 21 / 51

Introduction Tracking Greedy Kalman filter Particle filter Discrete Kalman Filter Maintains first two statistical moments z process state (mean) y error covariance x UNC Chapel Hill Computer Science Slide 14 H˚ akan Ard¨ o Tracking February 22, 2012 22 / 51

Introduction Tracking Greedy Kalman filter Particle filter Necessary Models dynamic model previous state next state image plane ( u , v ) measurement model state measurement UNC Chapel Hill Computer Science Slide 16 H˚ akan Ard¨ o Tracking February 22, 2012 23 / 51

Introduction Tracking Greedy Kalman filter Particle filter The Process Model Process Dynamics x k+1 = Ax k + w k Measurement z k = Hx k + v k UNC Chapel Hill Computer Science Slide 17 H˚ akan Ard¨ o Tracking February 22, 2012 24 / 51

Introduction Tracking Greedy Kalman filter Particle filter Process Dynamics x k+1 = Ax k + w k state vector x k ∈ R n contains the states of the process UNC Chapel Hill Computer Science Slide 18 H˚ akan Ard¨ o Tracking February 22, 2012 25 / 51

Introduction Tracking Greedy Kalman filter Particle filter Process Dynamics x k+1 = Ax k + w k state transition matrix n x n matrix A relates state at time step k to time step k+1 UNC Chapel Hill Computer Science Slide 19 H˚ akan Ard¨ o Tracking February 22, 2012 26 / 51

Introduction Tracking Greedy Kalman filter Particle filter Process Dynamics x k+1 = Ax k + w k process noise w k ∈ R n models the uncertainty of the process w k ~ N(0, Q ) UNC Chapel Hill Computer Science Slide 20 H˚ akan Ard¨ o Tracking February 22, 2012 27 / 51

Introduction Tracking Greedy Kalman filter Particle filter Measurement z k = Hx k + v k measurement vector z k ∈ R m is the process measurement UNC Chapel Hill Computer Science Slide 21 H˚ akan Ard¨ o Tracking February 22, 2012 28 / 51

Introduction Tracking Greedy Kalman filter Particle filter Measurement z k = Hx k + v k state vector UNC Chapel Hill Computer Science Slide 22 H˚ akan Ard¨ o Tracking February 22, 2012 29 / 51

Introduction Tracking Greedy Kalman filter Particle filter Measurement z k = Hx k + v k measurement matrix m x n matrix H relates state to measurement UNC Chapel Hill Computer Science Slide 23 H˚ akan Ard¨ o Tracking February 22, 2012 30 / 51

Introduction Tracking Greedy Kalman filter Particle filter Measurement z k = Hx k + v k measurement noise z k ∈ R m models the noise in the measurement v k ~ N(0, R ) UNC Chapel Hill Computer Science Slide 24 H˚ akan Ard¨ o Tracking February 22, 2012 31 / 51

Introduction Tracking Greedy Kalman filter Particle filter State Estimates a priori state estimate x – ˆ k a posteriori state estimate x k ˆ UNC Chapel Hill Computer Science Slide 25 H˚ akan Ard¨ o Tracking February 22, 2012 32 / 51

Introduction Tracking Greedy Kalman filter Particle filter Estimate Covariances a priori estimate error covariance – = E[( x k - x k ˆ – )( x k - x k ˆ – ) T ] P k a posteriori estimate error covariance P k = E[( x k - x k )( x k - x k ) T ] ˆ ˆ UNC Chapel Hill Computer Science Slide 26 H˚ akan Ard¨ o Tracking February 22, 2012 33 / 51

Introduction Tracking Greedy Kalman filter Particle filter Filter Operation Time update ( a priori estimates) Project state and covariance forward to next time step, i.e. predict Measurement update ( a posteriori estimates) Update with a (noisy) measurement of the process, i.e. correct UNC Chapel Hill Computer Science Slide 27 H˚ akan Ard¨ o Tracking February 22, 2012 34 / 51

Introduction Tracking Greedy Kalman filter Particle filter Time Update (Predict) state error covariance UNC Chapel Hill Computer Science Slide 28 H˚ akan Ard¨ o Tracking February 22, 2012 35 / 51

Introduction Tracking Greedy Kalman filter Particle filter Measurement Update (Correct) UNC Chapel Hill Computer Science Slide 29 H˚ akan Ard¨ o Tracking February 22, 2012 36 / 51