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Advanced Techniques in Hidden Variable Modeling and Image Motion Analysis

This document discusses various methodologies in hidden variable modeling, including mixture models and active shape and appearance models. Techniques for handling data variability through probabilistic PCA, expectation-maximization (EM) algorithms, and transformed latent images are explored. The text also covers image motion analysis using graphical models and highlights advancements like TCA and MTCA. Additionally, approaches to video frame pair observations, Bayesian PCA, and variational EM are included to emphasize the importance of various models in capturing complex data relationships.

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Advanced Techniques in Hidden Variable Modeling and Image Motion Analysis

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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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