Multiple Speaker Tracking with the Factorial von Mises- Fisher Filter
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1 Multiple Speaker Tracking with the Factorial von Mises- Fisher Filter IEEE International Workshop on Machine Learning for Signal Processing Sept 21-24, 2014 Reims, France Johannes Traa, Paris Smaragdis University of Illinois at Urbana-Champaign
2 Outline Motivation Sequential Bayesian Inference Linear Dynamical System o Kalman Filter Directional Statistics o von Mises-Fisher distribution o 3D Rotations Spherical Dynamical System o Particle Filter o von Mises-Fisher Filter (vmff) o Factorial vmff (FvMFF) Experiments
3 Motivation Tracking on the sphere o Human-computer interfaces with compact arrays
4 Motivation Sequential Bayesian Inference o Powerful framework for tracking Traditional techniques o Kalman Filter Ignores topology of sphere o Particle Filter Computationally intensive Proposed method o von Mises-Fisher Filter Accurate Deterministic
5 Sequential Bayesian Inference Dynamic Bayesian Network (DBN) o State transition o Measurement x t p (x t x t 1 ) y t p (y t x t ) x t 1 x t x t+1 y t 1 y t y t+1
6 Sequential Bayesian Inference Bayesian Filtering Equations (BFE) o Predict (convolution) p (x t y 1:t 1 ) = o Correct (Bayes rule) Z p (x t x t 1 ) p (x t 1 y 1:t 1 ) d x t 1 p (x t y 1:t ) / p (y t x t ) p (x t y 1:t 1 ) Predict x t 1 x t Correct y t
7 Linear Dynamical System (LDS) Probabilistic model Kalman Filter o Optimal on-line inference for LDS x t N (Ax t 1, x ) y t N (Bx t, y ) x t 1 x t x t+1 y t 1 y t y t+1
8 Kalman Filter KF ignores unique topology of DOA manifold o 3D state tracking o Posterior lies off of manifold
9 Directional Statistics von-mises Fisher distribution o Unit Sphere S 2 = x : x 2 R 3, kxk 2 =1 o Probability density function p (x µ,apple)= apple 4 sinh (apple) eapple x > µ
10 Directional Statistics Rotations on the sphere o Axis (unit vector) a a o Angle o Rotation matrix R (a, )= a 3 a 2 a 3 0 a 1 a 2 a sin ( )+ I aa > cos ( )+aa > o Rotation = linear transformation x 0 = R (a, ) x = R ( a) x
11 Spherical Dynamical System (SDS) Probabilistic Model o DOA state transition o Rotation state transition o Measurement x t x t 1, r t 1 vmf (R (r t 1 ) x t 1,apple x ) r t r t 1 N (Ar t 1, r ) y t x t vmf (x t,apple y ) o Rotation vector: r = a o Full state vector: s t = apple xt r t
12 Particle Filter Approximate inference for the SDS o Stochastic o Sequential variant of Monte Carlo o Approximate BFEs with particles (weighted point estimates) S t = n s (l) t o,w (l) t o Maintain particle representation of filtered state distribution p (s t y 1:t ) LX l=1 w (l) t s (l) t
13 von Mises- Fisher Filter (vmff) Factored representation p (s t 1 y 1:t 1 ) = vmf x t 1 bµ t 1, bapple t 1 N r t 1 b t 1, b t 1 Predict step Position Rotation o DOA state à approximate via convolution of 2D wrapped Normals Convolution p (x t y 1:t 1 ) = Z S 2 p (x t s t 1 ) p (x t 1 y 1:t 1 ) d x t 1 Mean Concentration bµ t = R b t 1 bµ t 1 A bapple t A (bapple t 1 ) A (bapple x ) A 1/ 2 r =0 A (apple) = 1 tanh (apple) 1 apple Position noise Rotation noise o Rotation state à regular KF prediction
14 von Mises- Fisher Filter (vmff) Correct step o DOA state à closed-form solution for vmf o Rotation state à approximate via auxiliary observation Bayes rule p (r t y 1:t ) / p (y t r t ) p (r t y 1:t 1 ) Emission density (not explicitly defined in SDS) Auxiliary observation: rotation vector required for bµ t = y t y r t = cos 1 bµ t 1 y t bµ t 1 y t kbµ t 1 y t k 2 Degree of rotation Axis of rotation
15 Factorial vmff (FvMFF) Probabilistic Data Association (PDA) o Probabilistic assignment of observations to speakers o EM-like de-coupling of correct steps for the speakers o Include component to handle clutter Outlier distribution Source 2 Source 1
16 Experiments Accuracy o Comparison between: 3D Kalman filter (KF) Particle filter on SDS (vmfpf) o 50 particles Proposed method (vmff) o Average angular error: E = 1 T TX cos 1 t=1 True DOA x > t bµ t Mean of filtered distribution Error (radians) Error (radians) Error (radians) k x = 100 KF vmfpf vmff k y k x = 200 KF vmfpf vmff k y k x = 500 KF vmfpf vmff k y
17 Experiments Number of particles and run time o Particle filter matches vmff performance with 150 particles o However, it runs 60x more slowly Error (radians) vmfpf computation time (milliseconds per iteration) KF vmfpf vmff # particles apple x = 200,apple y = 30
18 Experiments Measurement extraction Microphone 1 Microphone M Time domain x (1) t [1...N] x (M) t [1...N] Discrete Fourier Transform Frequency domain X (1) 1:N,t X (M) 1:N,t Inter- channel features Inter- Channel Time Differences 1,t N 2,t Map ITDs to DOAs DOA measurements y 1,t... yn 2,t
19 Experiments Multiple speaker tracking o Square array with M =4 microphones (side length = 2 centimeters) o 2- to 3-second sentences from TSP corpus o Speakers moved around array o ~ 0 db mixture o T 60 reverb time: 100 milliseconds True vmff 1 vmff 2 o Successful tracking requires gating procedure to avoid model mismatch
20 Thank you
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