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Transformation-invariant clustering using the EM algorithm

This paper discusses an unsupervised learning approach for clustering images regardless of transformation. It proposes a probabilistic description of the data and utilizes the EM algorithm for clustering. The method incorporates both pre- and post-transformation noise and can be extended for time series data and more transformations.

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Transformation-invariant clustering using the EM algorithm

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  1. Transformation-invariant clustering using the EM algorithm Brendan Frey and Nebojsa Jojic IEEE Trans on PAMI, 25(1) 2003 yan karklin. cns presentation 08/10/2004

  2. Goal • unsupervised learning of image structure regardless of transformation • probabilistic description of the data • clustering as density modeling – grouping “similar” images together Invariance • manifold in data space • all points on manifold “equivalent” • complex even for basic transformations • how to approximate? yan karklin. cns presentation 08/10/2004

  3. Approximating the Invariance Manifold • discrete set of points • sparse matrices Ti map cannonical feature z into transformed feature x (observed) • as a Gaussian probability model, • all possible transformations T enumerated yan karklin. cns presentation 08/10/2004

  4. This is what it would look like for... • a 2x3 image with pixel-shift translations (wrap-around) z = {T1...T6} = x = yan karklin. cns presentation 08/10/2004

  5. The full statistical model • for one feature (one cluster): • data, given latent repr: • joint of all variables: • Gaussian post-transformation with noise Ψ • Gaussian pre-transformation with noise Φ • for multiple features (clusters), mixture model: yan karklin. cns presentation 08/10/2004

  6. The full statistical model • the generative equation: • for each “feature”, have a cannonical mean and cannonical variance • image contains one of the cannonical features (mixture model) that has undergone one transformation yan karklin. cns presentation 08/10/2004

  7. Inference and is Gassian • marginals for inferring parameters T, c, z: yan karklin. cns presentation 08/10/2004

  8. Adapting the rest of parameters • pre-transformation noise • post-tranformation noise • all learned with EM • E-step: assume known params, infer P(z, T, c) • M-step: update parameters yan karklin. cns presentation 08/10/2004

  9. Experiments recovering 4 clusters 4 clusters w/o transform. yan karklin. cns presentation 08/10/2004

  10. Pre/post transformation noise yan karklin. cns presentation 08/10/2004

  11. Pre/post transformation noise mean variance single Gaussian model of image μ Φ transformation-invariant model, no post-t noise μ Φ Ψ transformation-invariant model, with post-t noise yan karklin. cns presentation 08/10/2004

  12. Conclusions • fast (uses sparse matrices, FFT) • incorporates pre- and post-transformation noise • works on artificial data, clustering simple image sets, cleaning up somewhat contrived examples • can be extended to make use of time series data, account for more transformations • poor transformation model • fixed, pre-specified transformations • must be sparse • poor feature model • Gaussian representation of structure yan karklin. cns presentation 08/10/2004

  13. yan karklin. cns presentation 08/10/2004

  14. yan karklin. cns presentation 08/10/2004

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