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Hierarchical Image-Motion Segmentation using Swendsen-Wang Cuts

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Hierarchical Image-Motion Segmentation using Swendsen-Wang Cuts

Adrian Barbu

Siemens Corporate Research

Princeton, NJ

Acknowledgements: S.C. Zhu , Y.N. Wu, A.L. Yuille et al.

- The Swendsen-Wang Cuts algorithm
- The original Swendsen-Wang algorithm
- Generalization to arbitrary probabilities

- Multi-Grid and Multi-Level Swendsen-Wang Cuts
- Application: Hierarchical Image-Motion Segmentation
- Conclusions and future work

Swedsen-Wang (1987) is an extremely smart idea that flips a patch at a time.

Each edge in the lattice e=<s,t> is associated a probability q=e-b.

1. If s and t have different labels at the current state, e is turned off.

If s and t have the same label, e is turned off with probability q.

Thus each object is broken into a number of connected components (subgraphs).

2. One or many components are chosen at random.

3. The collective label is changed randomly to any of the labels.

Pros

- Computationally efficient in sampling the Ising/Potts models
Cons:

- Limited to Ising / Potts models and factorized distributions
- Not informed by data, slows down in the presence of an external field (data term)

Swendsen Wang Cuts

- Generalizes Swendsen-Wang to arbitrary posterior probabilities
- Improves the clustering step by using the image data

Theorem (Barbu,Zhu ‘03). The acceptance probability for the Swendsen-Wang Cuts algorithm is

Theorem (Metropolis-Hastings) For any proposal probability q(AB) and probability p(A), if the Markov chain moves by taking samples from q(A B) which are accepted with probability

then the Markov chain is reversible with respect to p and has stationary distribution p.

1. Initialize a graph partition 2. Repeat, for current state A= π

3. Repeat for each subgraph Gl=<Vl, El>, l=1,2,...,n in A 4. For e El turn e=“on” with probability qe. 5. Partition Gl into nl connected components: gli=<Vli, Eli>, i=1,...,nl 6. Collect all the connected components in CP={Vli: l=1,...,n, i=1,...,nl}.

8. Propose to reassign V0 to a subgraph Gl’, l' follows a probability q(l'|V0,A)

State A

State B

CP

Swendsen-Wang Cuts: SWCInput: Go=<V, Eo>, discriminative probabilities qe, e Eo, and generative posterior probability p(W|I).Output: Samples W~p(W|I).

The initial graphGo

7. Select a connected component V0CP at random

9. Accept the move with probability α(AB).

- Our algorithm bridges the gap between the specialized and generic algorithms:
- Generally applicable – allows usage of complex models beyond the scope of the specialized algorithms
- Computationally efficient – performance comparable with the specialized algorithms
- Reversible and ergodic – theoretically guaranteed to eventually find the global optimum

Three-level representation:

- Level 2: Intensity regions are grouped into
moving objects Oiwith motion parametersqi

- Level 1: Atomic regions are grouped into
intensity regions Rijof coherent motion

with intensity models Hij

- Level 0: Pixels are grouped into atomic regions
rijkof relatively constant motion and intensity

- motion parameters (uijk,vijk)
- intensity histogram hijk

Select an attention window ½ G.

Cluster the vertices within and select a connected component R

Swap the label of R

Accept the swap with probability , using as boundary condition.

- Select a level s, usually in an increasing order.
- Cluster the vertices in G(s) and select a connected component R
- Swap the label of R
- Accept the swap with probability, using the lower levels, denoted by X(<s), as boundary conditions.

Modeling occlusion

- Accreted (disoccluded) pixels
- Motion pixels

Bayesian formulation

Accreted pixels

Motion pixels explained by motion

Intensity segmentation factor with generative and histogram models.

- Main motion for each object

- Boundary length

- Number of labels

The prior has factors for

- Smoothness of motion

Level 1:

Histogram Hi

Histogram Hj

Level 2:

Motion histogram Mi

Motion histogram Mj

- Level 0:
- Pixel similarity
- Common motion

Input sequence

Image Segmentation

Motion Segmentation

Input sequence

Image Segmentation

Motion Segmentation

Input sequence

Image Segmentation

Motion Segmentation

Input sequence

Image Segmentation

Motion Segmentation

Two extensions:

- Swendsen-Wang Cuts
- Samples arbitrary probabilities on Graph Partitions
- Efficient by using data-driven techniques
- Hundreds of times faster than Gibbs sampler

- Marginal Space Learning
- Constrain search by learning in Marginal Spaces
- Six orders of magnitude speedup with great accuracy
- Robust, complex statistical model by supervised learning

- Algorithm Boosting
- Any algorithm has a success rate and an error rate
- Can combine algorithms into a more robust algorithm by supervised learning
- Proof of concept for Image Registration

- Hierarchical Computing
- Efficient representation of Top-Down and Bottom-Up communication using specialized dictionaries
- Robust integration of multiple MSL paths by Algorithm Boosting

- Applications to medical imaging
- 3D curve localization and tracking
- Brain segmentation
- Lymph node detection

- A. Barbu, S.C. Zhu. Generalizing Swendsen-Wang to sampling arbitrary posterior probabilities, IEEE Trans. PAMI, August 2005.http://www.stat.ucla.edu/~abarbu/Research/partition-pami.pdf
- A. Barbu, S.C. Zhu. Generalizing Swendsen-Wang for Image Analysis. To appear in J. Comp. Graph. Stat. http://www.stat.ucla.edu/~abarbu/Research/jcgs.pdf
Thank You!