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