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約3年前 (2013/07/29)にアップロードinテクノロジー

This paper is concerned with the inference of marginal densities based on MRF models. The optimiz...

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.

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- Discrete MRF Inference of Marginal Densities for

Non-uniformly Discretized Variable Space

Masaki Saito, Takayuki Okatani, Koichiro Deguchi

Tohoku University - Outline

1. Introduction

2. Algorithms for a non-uniformly discretized variable space

3. Dynamic discretization method

4. Experimental results - Outline

1. Introduction

2. Algorithms for a non-uniformly discretized variable space

3. Dynamic discretization method

4. Experimental results - Markov Random Fields

MAP inference

MPM inference

• Directly computes the

• Estimates the marginal density

maximum of

(then computes its maximum )

• Equivalent to

• Necessary in some problems

minimizing

– e.g. Learning CRF models - 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 - Continuous and Discrete MRF

• 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

Discretization - 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

– Wil be useful for non-Euclidean variable spaces

uniform

non-uniform

Spherical surface - Outline

1. Introduction

2. Algorithms for a non-uniformly discretized variable space

3. Dynamic discretization method

4. Experimental results - Our approach

1. Represent the marginal density as a mixture of rectangular

densities

2. Fol ow the variational method to derive MF/BP algorithms

weight

rectangular density

Remark:

• ‘s may have arbitrary positions and sizes

• ‘s may have different number of mixtures - Our approach

1. Represent the marginal density as a mixture of rectangular

densities

2. Fol ow the variational method to derive MF/BP algorithms

[Textbook of Graphical Models by Koller-Friedman]

Introduce P such that

Find P minimizing

Compute the

marginal densities are

the KL distance

marginal

easily computed

between P and Q

densities

MF - 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

define a new discrete

conventional

energy function - Derivation of new MF & BP algorithms

Approximate joint

Models for the marginal

density

densities

Constraints

MF

BP - 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

1. Introduction

2. Algorithms for a non-uniformly discretized variable space

3. Dynamic discretization method

4. Experimental results- 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 - 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

1. Introduction

2. Algorithms for a non-uniformly discretized variable space

3. Dynamic discretization method

4. Experimental results- Effect of non-uniform discretization

• A simple Gaussian grid MRF

5

5

• Non-uniform discretization; asymmetric with respect to the

origin

64+16 = 80

64

16

-2

-1

0

1

2 - Effect of non-uniform discretization

Mean Field

Belief Propagation - 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

x8

Ground truth - Stereo matching(BP)

Disparity maps (maximum of marginal)

Input images

Ground truth

Conventional Proposed

Blocks=16

Blocks=8

Blocks=10

Blocks=12

Blocks=16

…

Estimated marginal densities

Our method gives a better result than conventional one

with only 10 blocks - 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 dynamical y

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