Applied Bayesian Nonparametrics 5. Spatial Models via Gaussian Processes, not MRFs Tutorial at CVPR 2012 Erik Sudderth Brown University NIPS 2008: E. Sudderth & M. Jordan, Shared Segmentation of Natural Scenes using Dependent Pitman-Yor Processes. CVPR 2012: S. Ghosh & E. Sudderth, Nonparametric Learning for Layered Segmentation of Natural Images.
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Applied Bayesian Nonparametrics
5. Spatial Models via Gaussian Processes, not MRFs
Tutorial at CVPR 2012Erik Sudderth Brown University
NIPS 2008: E. Sudderth & M. Jordan, Shared Segmentation of Natural Scenes using Dependent Pitman-Yor Processes. CVPR 2012: S. Ghosh & E. Sudderth, Nonparametric Learning for Layered Segmentation of Natural Images.
Human Image Segmentation
BNP Image Segmentation
•! How many regions does this image contain? •! What are the sizes of these regions?
Segmentation as Partitioning
•! Huge variability in segmentations across images •! Want multiple interpretations, ranked by probability
Why Bayesian Nonparametrics?
BNP Image Segmentation
Inference !!Stochastic search &
expectation propagation
Model !!Dependent Pitman-Yor processes
!!Spatial coupling via Gaussian processes
Results !!Multiple segmentations of
natural images
cesses
Learning !!Conditional covariance
calibration
Feature Extraction
•! Partition image into ~1,000 superpixels •! Compute texture and color features:
Texton Histograms (VQ 13-channel filter bank) Hue-Saturation-Value (HSV) Color Histograms
•! Around 100 bins for each histogram
Pitman-Yor Mixture Model
Observed features (color & texture)
Assign features to segments
PY segment size prior
Visual segment appearance model
Color: Texture:
π
z1 z2
z3z4
x1x2
x3x4xc
i ∼ Mult(θczi)
xsi ∼ Mult(θszi)
zi ∼ Mult(π)
πk = vk
k−1∏
�=1
(1− v�)
vk ∼ Beta(1− a, b+ ka)
Dependent DP&PY Mixtures
Observed features (color & texture)
Visual segment appearance model
Color: Texture:
z1 z2
z3z4
x1x2
x3x4xc
i ∼ Mult(θczi)
xsi ∼ Mult(θszi)
π1 π2
π3π4
Assign features to segments
zi ∼ Mult(πi)
Some dependent prior with DP/PY
“like” marginals
Kernel/logistic/probit stick-breaking process,
order-based DDP, !
Example: Logistic of Gaussians
•! Pass set of Gaussian processes through softmax to get probabilities of independent segment assignments
•! Nonparametric analogs have similar properties Figueiredo et. al., 2005, 2007