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Modelling data

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).

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Modelling data

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  1. Modelling data • static data modelling. • Hidden variable cascades: build in invariance (eg affine) • EM: general framework for inference with hidden vars.

  2. Accounting for data variability Active shape models (Cootes&Taylor, 93) Active appearance models (Cootes, Edwards &Taylor, 98)

  3. Mixture model Latent image Transformed latent image PCA/FA Transformed mixture model TCA MTCA Hidden variable modelling

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

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

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

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

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

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

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

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

  12. 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”:

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

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

  15. TCA Results PCA Components TCA Components TCA Simulation PCA Simulation

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

  17. Observation model for video frame-pairs WSL model

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

  19. 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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