Modelling data static data modelling Hidden variable cascades

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 TCA Mixture model Transformed latent image Transformed mixture model PCA/FA MTCA

PGMs for image motion analysis where with or equivalently Latent image Explicit density fn: with prob. so Mixture model (Frey and Jojic, 99/00)

PGMs for image motion analysis with prob. and Transformed latent image and Overall: PCA/FA A AA and

PGMs for image motion analysis Latent image Mixture model Transformed latent image PCA/FA TCA A Transformed mixture model MTCA

PGMs for image motion analysis Latent image TCA Mixture model Transformed mixture model Transformed latent image MTCA (Frey and Jojic, 99/00) PCA/FA Transformed HMM

Results: image motion analysis by THMM video summary image stabilisation image segmentation T sensor noise removal data

PCA as we know it Data mean Data covariance matrix eigenvalues/vectors Model: with or even

Probabilistic PCA (Tipping & Bishop 99) and AA PCA params are Overall: Since Need: But so: AA

Probabilistic PCA AA MLE estimation should give: and AA (data covariance matrix) ? ? eigenvalues -- in fact set eigenvals of AA to be and

EM algorithm for FA Still true that but anisotropic – kills eigenvalue trick for MLE with Instead do EM on : hidden Log-likelihood linear in the “sufficient statistics”:

. . . EM algorithm for FA Given sufficient statistics E-step: compute expectation using: -- just “fusion” of Gaussian dists: M-step Compute substituting in

EM algorithm for TCA Put back the transformation layer so now we have and define A Aso: hidden A A and need -- to be used as before in E-step. Lastly, compute transformation “responsibilities”: where (using “prediction” for Gaussians): AA M-step as before.

TCA Results PCA Components TCA Components PCA Simulation TCA Simulation

Observation model for video frame-pairs (Jepson Fleet & El Maraghi 2001) Observation: --- eg wavelet output -- hidden State: Lost Wandering Stable Prior: Likelihoods: mixture

Observation model for video frame-pairs WSL model

. . . could also have mentioned § Bayesian PCA § Gaussian processes § Mean field and variational EM § ICA (Simoncelli, Weiss) § Manifold models

where are we now? ü static data modelling. üHidden variable cascades: build in invariance (eg affine) üEM: general framework for inference with hidden vars. • On to modelling of sequences -temporal and spatial -discrete and continuous