This paper is concerned with the inference of marginal densities based on MRF models. The optimization algorithms for continuous variables are only applicable to a limited number of problems, whereas those for discrete variables are versatile. Thus, it is quite common to convert the continuous variables into discrete ones for the problems that ideally should be solved in the continuous domain, such as stereo matching and optical flow estimation. In this paper, we show a novel formulation for this continuous-discrete conversion. The key idea is to estimate the marginal densities in the continuous domain by approximating them with mixtures of rectangular densities. Based on this formulation, we derive a mean field (MF) algorithm and a belief propagation (BP) algorithm. These algorithms can correctly handle the case where the variable space is discretized in a non-uniform manner. By intentionally using such a non-uniform discretization, a higher balance between computational efficiency and accuracy of marginal density estimates could be achieved. We present a method for actually doing this, which dynamically discretizes the variable space in a coarse-to-fine manner in the course of the computation. Experimental results show the effectiveness of our approach.

1. Discrete MRF Inference of Marginal Densities for
Non-uniformly Discretized Variable Space
Masaki Saito, Takayuki Okatani, Koichiro Deguchi
Tohoku University

2. Outline
1. Introduction
2. Algorithms for a non-uniformly discretized variable space
3. Dynamic discretization method
4. Experimental results

3. Outline
1. Introduction
2. Algorithms for a non-uniformly discretized variable space
3. Dynamic discretization method
4. Experimental results

4. Markov Random Fields
MAP inference
• Directly computes the
maximum of
• Equivalent to
minimizing
MPM inference
• Estimates the marginal density
(then computes its maximum )
• Necessary in some problems
– e.g. Learning CRF models

5. Continuous and Discrete MRF
• Mean Field(MF) and Belief Propagation(BP) are only choices
for the estimation of marginal densities
• Two formulations: Discrete and Continuous
– The variable x at each site takes a continuous or a discrete
value
Applicability
Computational
cost (per iter.)
Discrete Wide O((# of labels)^2)
Continuous Limited Efficient

6. Continuous and Discrete MRF
Discretization
• A common practice is to convert a continuous problem to a
discrete one by discretizing the variable space uniformly
– A continuous density is approximated by a discrete one

7. Continuous and Discrete MRF
• What if the variable space can be non-uniformly discretized so
that the accuracy is maximized with a minimum number of
discretization bins?
– Will achieve a better trade-off between accuracy vs.
computational cost
– Will be useful for non-Euclidean variable spaces
uniform non-uniform Spherical surface

8. Outline
1. Introduction
2. Algorithms for a non-uniformly discretized variable space
3. Dynamic discretization method
4. Experimental results

9. • ‘s may have arbitrary positions and sizes
• ‘s may have different number of mixtures
Remark:
Our approach
1. Represent the marginal density as a mixture of rectangular
densities
2. Follow the variational method to derive MF/BP algorithms
weight rectangular density

10. Our approach
1. Represent the marginal density as a mixture of rectangular
densities
2. Follow the variational method to derive MF/BP algorithms
Introduce P such that
marginal densities are
easily computed
Find P minimizing
the KL distance
between P and Q
Compute the
marginal
densities
MF
[Textbook of Graphical Models by Koller-Friedman]

11. Conventional continuous-discrete conversion
1. Discretize the variable space into discrete labels (uniformly or
non-uniformly)
2. Define an energy E(x) in the discrete domain
 Original continuous energy (if any) is discarded
 Step 2 does not care about how the space was discretized
conventional
define a new discrete
energy function

12. Approximate joint
density
Models for the marginal
densities
Constraints
MF
BP
Derivation of new MF & BP algorithms

13. New MF and BP updating rules
• A few terms are newly added; they compensate for the non-
uniformity of descretization
– They vanish when the discretization is uniform
New updating rules Conventional updating rules
MF
BP
&
: newly added terms

14. Outline
1. Introduction
2. Algorithms for a non-uniformly discretized variable space
3. Dynamic discretization method
4. Experimental results

15. Dynamic discretization of the variable space
• If we have a prior knowledge about where is important in the
variable space, the application of our method is
straightforward
• Our “dynamic discretization method” is targeted at the case
where no such knowledge is available
1D
2D

16. Coarse-to-fine block subdivision
1. Discretize the variable space
coarsely (maybe uniformly)
2. Perform MF or BP iterations to
obtain the estimates of the
marginal densities
3. Identify the block whose mixture
weight is the largest, and
divide the block into subblocks
– Weights and messages are adjusted
4. Go to 2 until a desired result is
obtained

17. Outline
1. Introduction
2. Algorithms for a non-uniformly discretized variable space
3. Dynamic discretization method
4. Experimental results

18. Effect of non-uniform discretization
• A simple Gaussian grid MRF
• Non-uniform discretization; asymmetric with respect to the
origin
5
5
-1-2 210
64 16
64+16 = 80

19. Effect of non-uniform discretization
Mean Field
Belief Propagation

20. Stereo matching
• The proposed method is used to adaptively and recursively
discretize the variable space
• Conventional: divide the variable space into 16 fixed blocks
• Proposed: initially divide into 8 blocks, and finally divide into
16 blocks with 8 subdivision procedures
Conventional Proposed Input images
Ground truthx8

21. Stereo matching(BP)
…
Conventional
Blocks=16
Proposed
Blocks=8 Blocks=10 Blocks=16
Input images Ground truth
Blocks=12
Estimated marginal densities
Our method gives a better result than conventional one
with only 10 blocks
Disparity maps (maximum of marginal)

22. Conclusions
• We have derived new MF and BP algorithms for estimating
marginal densities for pairwise MRFs that can properly deal
with non-uniform discretization
• We have also presented a method that can dynamically
discretize the variable space
• These tools are expected to make it possible to achieve a
better trade-off between accuracy and computational cost
• The idea is to represent the marginal densities with a mixture
of rectangular densities and then apply the variational method
• Our approach also gives a rigorous method for converting
continuous MRFs to discrete ones