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```									Hierarchical Image-Motion Segmentation
using Swendsen-Wang Cuts

Siemens Corporate Research
Princeton, NJ
Acknowledgements: S.C. Zhu , Y.N. Wu, A.L. Yuille et al.

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Talk Outline
   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

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Swendsen-Wang for Ising / Potts Models
Swedsen-Wang (1987) is an extremely smart idea that flips a patch at a time.

V2

V0

V1

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.
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The Swendsen-Wang Algorithm
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

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SW Cuts: the Acceptance Probability
Theorem (Metropolis-Hastings) For any proposal probability q(AB) 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.

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

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The Swendsen-Wang Cuts Algorithm
Swendsen-Wang Cuts: SWC
Input: Go=<V, Eo>, discriminative probabilities qe, e Eo,
and generative posterior probability p(W|I).
Output: Samples W~p(W|I).
1. Initialize a graph partition
2. Repeat, for current state A= π
The initial graph Go                                       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
V2
V2
6. Collect all the connected components in
x x

x
x
x
x
CP={Vli: l=1,...,n, i=1,...,nl}.
7. Select a connected component V0CP at random
x
x                   V0      V0
V0
x                                           x

V1
V1       x
x
x
x
x
8. Propose to reassign V0 to a subgraph Gl’,
x       x
l' follows a probability q(l'|V0,A)
CP
State AB
State                                                              9. Accept the move with probability α(AB).

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Advantages of the SW Cuts Algorithm

   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

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Hierarchical Image-Motion Segmentation
Three-level representation:
X2 – Level 2: Intensity regions are grouped into
moving objects Oi with motion parameters qi

X1 – Level 1: Atomic regions are grouped into
intensity regions Rij of coherent motion
with intensity models Hij
X0 – Level 0: Pixels are grouped into atomic regions
rijk of relatively constant motion and intensity
– motion parameters (uijk,vijk)
– intensity histogram hijk

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Multi-Grid SWC
State XA                        State XB

V1            x    x       x             V1
x
   x                                                          x

x                                                           x
R                              R           x
x                                                   x

x
x

V2                                 V3   V2                                     V3

1.   Select an attention window  ½ G.
2.   Cluster the vertices within  and select a connected component R
3.   Swap the label of R
4.   Accept the swap with probability , using     as boundary condition.

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Multi-Level SWC

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

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Hierarchical Image-Motion Segmentation
Modeling occlusion
 Accreted (disoccluded) pixels
 Motion pixels

Bayesian formulation                              Accreted pixels

   Motion pixels explained by motion

   Intensity segmentation factor       with generative and
histogram models.

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Hierarchical Image-Motion Segmentation
The prior has factors for
   Smoothness of motion

   Main motion for each object

   Boundary length

   Number of labels

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Designing the Edge Weights
    Level 0:
   Pixel similarity
   Common motion

   Level 1:
Histogram Hi

Histogram Hj

   Level 2:
Motion histogram Mi

Motion histogram Mj
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Experiments

Input sequence    Image Segmentation    Motion Segmentation

Input sequence   Image Segmentation    Motion Segmentation

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Experiments

Input sequence    Image Segmentation   Motion Segmentation

Input sequence    Image Segmentation   Motion Segmentation

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Conclusion
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

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Future Work
   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

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References
   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!

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