Lecture Present

8/2/2019 Lecture Present

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Bayesian Modeling of Uncertaintyin Low-Level Vision

Paper by Richard Szeliski

International Journal of Computer Vision

(1990)

Presentation by Michael Ross


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Bayesian methods for intrinsicimages

Dense fields and inverse problems (depthmaps, orientation, stereo matching).

Traditionally solved by energy minimization

or regularization methods.

Replace these methods with Bayesianmethods to get explicit modeling ofuncertainty, better understanding of models.


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Intrinsic images

Finding denseinformation fieldsfrom sparse or

uncertain data.

Interpolation andaccounting for

uncertainty inmeasurement.


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Energy methods

E u 1 E d u , d E p u

Ed u , d 12 i

c i u x i, y i d i

2

E p u

1

2u x

2u y

2dx dy


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Discrete energy methods

E u 1

2

uTA u u

Tb c

Ed u , d

1

2 u

d

T

A d u

d

E p u

1

2

uTA p u

E u 1

2u u '

TA u u ' k


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Energy implies probability

Squared error terms imply Gaussian noisemodels.

Smoothness (and discontinuity) can be

modeled as a Markov random field.

Using probability directly allows us access toinformation hidden by energy models(uncertainty modeling, model accuracy).


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Markov random fields

p u i

u p u i

u j j

N i

A Markov random chain intwo dimensions. Mostcommonly used to calculatea MAP estimate, the

assignment whichmaximizes p(u|d).


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Prior models

Describes p(u), our model of the world givenno data. In principle, we can calculate it fromthe MRF terms.

Gibbs distributions and sampling make thesecalculations simple and tractable. (Geman &Geman)

p u 1

Z pexp E p u

T p E p u E c u


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Graph cliques


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Sampling from prior models

p u i

u 1

Z pexp E p u i

u

T p

Thin plate on data. Sample thin plate.


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Coarse to fine sampling.

Thin plate on data. Coarse-fine thin plate sample.

Why is this so much better?


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Coarse to fine sampling.

Fourier analysis reveals that the surfacesdefined by thin-plate models are inherentlyfractal (self-similar at different resolutions).

Gibbs sampling at a single resolution is not apractical way to propagate the low-frequencycomponents of the random field.

Coarse to fine sampling is faster and producesmore representative samples.


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Sensor models

Directly incorporated and easy to swap.

p d

u 1

Zdexp Ed u , d E d u , d

E di

u i, d i

Typical Gaussian sensor model (assuminguncorrelated errors).

p d i

u 1

2

i

exp

u i

d i 2

2

i


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Contaminated gaussians

p d i

u 1

2

1

exp

u i

d i2

2

1

2

2

2

exp

u i

d i2

2

2

2


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Virtual sensors

Use a lower-level vision algorithm (opticalflow, for example) as a dense data source.

Anandan and Weiss found that optical flow

errors depend on the type of local dataavailable (none, line, or corner).

This can be used to construct a higher-level

model that produces better flow estimates byexplicitly modeling these dependencies in thesensor distribution.


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Posterior models

The output (intrinsic image) given the data -used to compute the MAP estimate of p(u | d).

Described by a Gibbs distribution (MRF) with

energy:E u E p u

E d u , d

Priormodel

Sensormodel


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Loss functions

L

L u , u ' p u

d du

L

u , u ' p u

d du

Generic:

Map:

Loss functions allow for top-down influence.


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Uncertainty estimatation

We can compute the covariance of theposterior models.

In the Gaussian sensor model case:

E u

1

2u u '

TA u u '

k

cov u

A

1


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Stochastic variance estimation

Run the Gibbs sampler and estimate thevariance with a Monte Carlo algorithm.

Time averages are ergodic only over long

time frames, unless we use multiresolutionsampling.

1000 iterations 100 iterations


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Dynamic models

Kalman filtering is a natural part of the MRFframework.

u

N u 0,P 0

d k H ku k

rk

u k Fku k 1

q k

Prior model

Sensor model

System model


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Dynamic models

This work applies Kalman filtering to denseinformation, by using the sparse informationmatrices rather than the dense covariancematrices.

Ak

'

'

Ak

' '

Hk

TR

k

1H

k

T

b k'

'

b k' '

H kT

R k

1d k

uk

Ak

1b

k


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Dynamic models

Why are the information matrices sparse?

In the prior model, we have local smoothness,so any column of the information matrix only

has non-zero values for the locallyneighboring points.

In the sensor model, we usually assume that

the sensor errors are uncorrelated, so thesensor information matrix is diagonal.


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Applications

Incremental depth from motion, using theKalman filtering approach.

Motion estimation without correspondence

(discover the estimate that maximizes thelikelihood that two sets of points are drawnfrom the same smooth surface).

Maximum likelihood estimation for theregularization parameter.


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Incremental depth from motion

Compute a displacement estimate usingcorrelation between frames.

Transform to disparity map using camera

motion, integrate with predicted map (Kalmanfiltering).

Regularize map.

Predict next map using known cameramotion.


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Motion without correspondence

Surface through set #1 Surface through set #2 Surface through both

Minimize the distance between point set #2 and the surface through set#1 and the surface through both sets.


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ML parameter estimation

p u

d exp

E d u , d E p u

p

2

p d 2

H P0H

T R

1

2exp

1

2d

TH P

0H

T R

1d

P 0 1

p

2

A p

Take the log and maximize...


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Conclusion

Probability models encompass energy models.

We can draw samples to check their validity.

We can model sensor behavior easily.

Flexibility in choosing loss functions.

Direct calculation of error.

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