Modelling data
static data modelling.
Hidden variable cascades: build in invariance (eg affine)
EM: general framework for inference with hidden vars.
Accounting for data variability
Active shape models (Cootes&Taylor, 93)Active appearance models (Cootes, Edwards &Taylor, 98)
Hidden variable modelling
Latent image
Mixturemodel
TCA
Transformedlatent image
PCA/FA
Transformedmixture model
MTCA
PGMs for image motion analysis (Frey and Jojic, 99/00)
Latent image
Mixturemodel
where with
or equivalently
Explicit density fn:
with prob.
so
PGMs for image motion analysis
Transformedlatent image
PCA/FA
with prob.
and
andand
Overall:
A
AA
PGMs for image motion analysis
Latent image
Mixturemodel
TCA
Transformedlatent image
PCA/FA
Transformedmixture model
MTCA
A
PGMs for image motion analysis (Frey and Jojic, 99/00)
Latent image
Mixturemodel
TCA
Transformedlatent image
PCA/FA
Transformedmixture model
MTCA Transformed HMM
Results: image motion analysis by THMM
video summary
image segmentation
sensor noise removal
image stabilisation
data
T
PCA as we know it
Data mean
Model:
Data covariance matrix
eigenvalues/vectors
Data
with
or even
Probabilistic PCA
Since PCA params are
Need:
so: AA
(Tipping & Bishop 99)
andand
Overall: AA
A
But
Probabilistic PCA
MLE estimation should give:
and??
-- in fact set eigenvals of to be
and
(data covariance matrix)
AA
AA
AA
eigenvalues
EM algorithm for FA
Log-likelihood linear in the “sufficient statistics”:
Still true that
but anisotropic – kills eigenvalue trickfor MLE with
Instead do EM on :
hidden
...EM algorithm for FA
Given sufficient statistics
E-step:
M-step
compute expectation using:
-- just “fusion” of Gaussian dists:
Compute substituting in
EM algorithm for TCAPut back the transformation layer
and define so:
and need -- to be used as before in E-step.
M-step as before.
Lastly, compute transformation “responsibilities”:
A A
A A
A A
where (using “prediction” for Gaussians):
so now we have
hidden
TCA Results
PCA Components
TCA Components
PCA Simulation TCA Simulation
Observation model for video frame-pairs
State:
(Jepson Fleet & El Maraghi 2001)
Observation: --- eg wavelet output
Wandering
Stable
Lost
Prior:
Likelihoods:
-- hidden
mixture
Observation model for video frame-pairsWSL model
... could also have mentioned
Bayesian PCA
Gaussian processes
Mean field and variational EM
ICA
Manifold models
(Simoncelli, Weiss)
where are we now?
static data modelling.
Hidden variable cascades: build in invariance (eg affine)
EM: general framework for inference with hidden vars.
• On to modelling of sequences
-temporal and spatial
-discrete and continuous
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