Visual Tracking via Geometric Particle Filtering on the Affine Group with Optimal Importance Functions

Transcription

1 Monday, June 22 Visual Tracking via Geometric Particle Filtering on the Affine Group with Optimal Importance Functions Junghyun Kwon 1, Kyoung Mu Lee 1, and Frank C. Park 2 1 Department of EECS, 2 School of MAE Seoul National University, Korea

2 Affine Motion Tracking Estimation of via particle filtering (PF) Initial frame y y k-th frame x x Affine motion tracking Non-linear filtering Particle filtering Effective for non-linear filtering

3 Issues in Affine Motion Tracking via PF Adequate state representation State space A set of 2-D affine matrices Use of optimal importance function (OIF) PF Sampling from the importance function OIF is essential for robust tracking?

4 Two Different State Representations State Conventional Geometric (Kwon et al, 2008) 6-D vector by local coordinates 2-D Affine matrix itself as a Lie group (Aff(2)) State Equation Ignores geometry of the underlying space We take this Geometric approach!

5 Drawback of Conventional Approach Toy example State farther from I Need Larger perturbation Need more particles Inefficiency

6 Advantage of Geometric Approach For the same example More efficient than the conventional approach Affine motion tracking via Geometric PF on Aff(2)!

7 Remaining Issue: Use of OIF Popular importance function Direct sampling from the state equation Simple but inefficient because of missing y k Optimal importance function Maintaining the largest number of effective particles Increased effective particles Increased performance

8 How to Use OIF in Practice Difficulties in using OIF for Affine Motion Tracking D1. Only approximation to OIF is possible D2. Our state 2-D affine group Aff(2) Our contribution Approximation of OIF for geometric PF on Aff(2) Consideration of the geometry of Aff(2)

9 OIF Approximation for Vector State Gaussian approximation by (Doucet et al, SC2000) k-1 k Predicting p(x k, y k x k-1 ) by Jacobian of y k w.r.t. x k p(x k x k-1, y k ) by correcting using y k

10 Questions in OIF Approx. on Aff(2) Q1. What is Gaussian on Aff(2)?? Q2. How to obtain Jacobian of y k w.r.t. X k? Our answer Taylor expansion on Aff(2) is required Use of exponential coordinates!

11 Geometric View to Affine Matrix Affine matrix 2-D Affine group Aff(2) Lie group Group + Differentiable manifold Lie algebra Tangent space at the identity (aff(2)) Exp Log Origin Identity Aff(2) aff(2) Exp: aff(2) Aff(2) Log: Aff(2) aff(2)

12 Local Diffeomorphism of Exp Map Local diffeomorphism of Exp: aff(2) Aff(2) One-to-one and onto sufficiently near the identity Origin One-to-one & onto aff(2) Identity Aff(2)

13 Exponential Coordinates By local diffeomorphism of Exp map Neighborhood of Aff(2) Exp Origin aff(2) X Aff(2) Identity

14 A1. Gaussian on Aff(2) Motivation Gaussian is well defined on aff(2)! N Aff(2) (X,S) Exponential of Gaussian on aff(2) S Exp aff(2) X Aff(2) Constraint Sufficiently small S

15 A2. Measurement Jacobian Taylor expansion on Aff(2) by Exp coordinates J with respect to the exponential coordinates J of PCA-based measurement by the chain rule

16 OIF Approximation on Aff(2) k-1 Prediction by J Correction Exp Exp k

17 Experiment 1 Geometric approach vs Conventional approach Our tracker vs Tracker of (Ross et al, IJCV 2008) Same measurement, Same importance function 3x speed playback 2x speed playback

18 Experiment 2 Approx. OIF vs State prediction density Our tracker with Approx. OIF vs with p(x k X k-1 ) Same # of particles, Same covariance for dw k

19 Experiment 2 Approx. OIF vs State prediction density Number of effective particles [ (w k (i) ) 2 ] -1 Importance function Cube Vase Toy Approx. OIF p(x k X k-1 )

20 Conclusions Geometric framework to approx. OIF for PF on Aff(2) Use of Exponential coordinates Approx. Gaussian and Taylor expansion on Aff(2) Experimental validation Efficiency of geometric PF on Aff(2) Efficiency of OIF for geometric PF on Aff(2)

21 Thanks for your attention!

22 Additional Slides

23 Conventional Approach 6-D Vector representation using local coordinates Via Euclidean Embedding Via Singular Value Decomposition

24 Physical Illustration

25 PCA-based Measurement Explicit representation to apply the chain rule DFFS k-th frame y DIFS x w -1 y I x Initial coordinates

26 Application of the Chain Rule Straight-forward application and calculation Derivative of DFFS and DIFS Derivative of warping function Image gradient Now OIF approximation for PF on Aff(2) possible!

27 Derivative of PCA Term Derivative of DFFS term Derivative of DIFS term

28 Approx. OIF for PF on Aff(2) Prediction and Correction Step 1 Prediction of p(x k, y k X k-1 ) by J Step 2 p(x k X k-1, y k ) by correction