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