Nov 19, 2012 - inputs X can be any kind of objects. â output y is a real number. â Prediction of complex outputs. â. Structured output y is complex (images, ...
Structured output models for image segmentation Aurelien Lucchi Machine Learning Workshop (MLWS) IDIAP - EPFL Monday November 19th, 2012 Collaborators: Yunpeng Li, Kevin Smith, Raphael Sznitman, Bohumil Maco, Graham Knott, Pascal Fua.
Outline 1.Review Conditional Random Fields (CRF) 2.Maximum likelihood training for CRFs 3.Maximum Margin Training for CRFs 1.Cutting plane (Structured SVM) 2.Online subgradient descent
1. Review CRF
Structured prediction ●
Non structured output ● ●
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inputs X can be any kind of objects output y is a real number
Prediction of complex outputs
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Structured output y is complex (images, text, audio...) Ad hoc definition of structured data: data that consists of several parts, and not only the parts themselves contain information, but also the way in which the parts belong together Slide courtesy: Christoph Lampert
Structured prediction for image segmentation
Histograms, Filter responses, ...
CRF for image segmentation
Maximum-a-posteriori (MAP) solution :
Data (D)
Unary likelihood
Pair-wise Terms
MAP Solution
Boykov and Jolly [ICCV 2001], Blake et al. [ECCV 2004] Slide courtesy : Pushmeet Kohli
CRF for image segmentation
Maximum-a-posteriori (MAP) solution :
Data (D)
Unary likelihood
Pair-wise Terms
MAP Solution
Boykov and Jolly [ICCV 2001], Blake et al. [ECCV 2004] Slide courtesy : Pushmeet Kohli
CRF for image segmentation
Pair-wise Terms Favors the same label for neighboring nodes.
CRF for image segmentation
Maximum-a-posteriori (MAP) solution :
Data (D)
Unary likelihood
Pair-wise Terms
MAP Solution
Boykov and Jolly [ICCV 2001], Blake et al. [ECCV 2004] Slide courtesy : Pushmeet Kohli
Energy minimization ●
MAP inference for discrete graphical models:
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Dynamic programming –
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Graph-cuts (Boykov, 2001) –
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Exact on non loopy graphs Optimal solution if energy function is submodular
Belief propagation (Pearl, 1982) –
No theoretical guarantees on loopy graphs but seems to work well in practice.
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Mean field (root in statistical physics)
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...
Training a structured model ? ●
First rewrite the energy function as:
Log-linear model
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Efficient Learning/Training – need to efficiently learn parameters w from training data ?
Training a structured model ? ●
Energy function is parametrized by vector w
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-1
1
-1
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1
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Training a structured model ? ●
Energy function is parametrized by vector w
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-1
1
-1
0
1
1
1
0
Low energy
High energy
2. Maximum likelihood training
Maximum likelihood
Note: We assumed that p is a Gibbs distribution
Maximum likelihood
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L(w) is differentiable and convex (it has a positive definite Hessian) so gradient descent can find the global optimum.
Maximum likelihood
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For general CRFs, there is still a problem with the computation of the derivative because the number of possible configurations for y is typically (exponentially) large.
Training a structured model ? ●
Other solutions exist: ●
Pseudo-likelihood
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Variational approximation
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Contrastive divergence
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Maximum-margin framework (e.g. Structured SVM)
3.1. Maximum Margin Training of Structured Models: cutting plane (structured SVM)
Structured SVM
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Given a set of N training examples with ground truth labels , we can write ≡ Energy for the correct labeling at least as low as energy of any incorrect labeling..
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Broadcasting and television are now entering the era of High Definition (HD) â a transition as ... Several European countries have also used the introduction.
