Auto-regressive dynamical models

1
Auto-regressive dynamical models

• Continuous form of Markov process
• Linear Gaussian model
• Hidden states and stochastic observations
  (emissions)
• Statistical filters Kalman, Particle
• EM learning
• Mixed states

2
Auto-regressive dynamical model

• Configuration
• AR model
• Parametric shape/texture model,
• eg curve model

ARP order
driven by independent noise
possibly nonlinear
3
Deformable curve model
curve model
Planar affine learned warps
Active shape models (CootesTaylor, 93) Residual
PCA (Active Contours, Blake Isard, 98) Active
appearance models (Cootes, Edwards Taylor, 98)
4
Linear AR model
(Active Contours, Blake and Isard, Springer
1998)

• Configuration
• Linear Gaussian AR model
• Prior shape
• Steady state prior

(1st order)
5
Gaussian processes for shape motion
intra-class
single object
(Reynard, Wildenberg, Blake Marchant, ECCV
96)
6
Kalman filter
(Gelb 74)

• Stochastic observer
• Kalman filter (Forward filter)
• Kalman smoothing filter (Forward-Backward)

independent noise
also
etc.
7
Classical Kalman filter
8
Visual clutter
9
Visual clutter ? observational nonlinearity
10
Particle Filter Non-Gaussian Kalman filter
www.research.microsoft.com/ablake/talks/MonteCarl
o.ppt
11
Particle Filter (PF)
continue
12
JetStream cut-and-paste by particle filtering

• particles sprayed along the contour

13
Propagating Particles

• particles sprayed along the contour
• contour smoothness prior

14
Branching
15
MLE Learning of a linear AR Model

• Direct observations Classic Yule-Walker
• Learn parameters
• by maximizing
• which for linear AR process ? minimizing
• Finally solve
• where sufficient statistics are

16
Handwriting
Scribble
-- simulation of learned ARP model
-- disassembly
17
Simulation of learned Gait
-- simulation of learned ARP model
18
Walking Simulation (ARP)
19
Walking Simulation (ARP HMM)
(Toyama Blake 2001)
20
Dynamic texture
(S. Soatto, G. Doretto, Y. N. Wu, ICCV 01 A.
Fitzgibbon, ICCV01)
21
Speech-tuned filter
(Blake, Isard Reynard, 1985)
22
EM learning

• Stochastic observations z

unknown -- hidden
unavailable classic EM

• M-step

i.e.

• E-step

FB smoothing
23
PF forward only
24
PF forward-backward
continue
25
Juggling (North et al., 2000)
26
Learned Dynamics of Juggling
State lifetimes and transition rates also learned
27
Juggling
28
Perception and Classification
Ballistic (left)
Catch, carry, throw (left)
29
Underlying classifications
30
Learning Algorithms
EM-P
31

• 1D ? Markov models
• 1D ?? Markov models

• 2D Markov models

32
EM-PF Learning

• Forward-backward particle smoother (Kitagawa 96,
  Isard and Blake, 98) for non-Gaussian problems

• Generates particles with weights

• Autocorrelations

• Transition Frequencies