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ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

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Introduction to MAP Inference in Discrete Models Andrew Blake, Microsoft Research Cambridge Modern probabilistic modelling has revolutionized the design and implementation of machine vision systems. There are now numerous instances of systems that can see stereoscopically in depth, or separate foreground from background, or accurately pinpoint objects of a particular class, all in real time. The underlying advances owe a lot to probabilistic frameworks for inference in images. In particular, the Markov Random Field (MRF), borrowed originally from statistical physics, first appeared in image processing in the 70s. It has staged a resounding comeback in the last decade, for very interesting reasons. Lecturers: Pushmeet Kohli, Microsoft Research Cambridge M. Pawan Kumar, Stanford University Carsten Rother, Microsoft Research Cambridge
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Page 1: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Introduction to MAP Inference in Discrete Models

Andrew Blake, Microsoft Research Cambridge

Modern probabilistic modelling has revolutionized the design and

implementation of machine vision systems. There are now numerous instances

of systems that can see stereoscopically in depth, or separate foreground from

background, or accurately pinpoint objects of a particular class, all in real time.

The underlying advances owe a lot to probabilistic frameworks for inference in

images. In particular, the Markov Random Field (MRF), borrowed originally

from statistical physics, first appeared in image processing in the 70s. It has

staged a resounding comeback in the last decade, for very interesting reasons.

Lecturers:

Pushmeet Kohli, Microsoft Research Cambridge

M. Pawan Kumar, Stanford University

Carsten Rother, Microsoft Research Cambridge

Page 2: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Course programme

9.30-10.00 Introduction (Andrew Blake)

10.00-11.00 Discrete Models in Computer Vision (Carsten Rother)

15min Coffee break

11.15-12.30 Message Passing: DP, TRW, LP relaxation (Pawan Kumar)

12.30-13.00 Quadratic pseudo-boolean optimization (Pushmeet Kohli)

1 hour Lunch break

14:00-15.00 Transformation and move-making methods (Pushmeet Kohli)

15:00-15.30 Speed and Efficiency (Pushmeet Kohli)

15min Coffee break

15:45-16.15 Comparison of Methods (Carsten Rother)

16:30-17.30 Recent Advances: Dual-decomposition, higher-order, etc.

(Carsten Rother + Pawan Kumar)

All online material will be online (after conference):http://research.microsoft.com/en-us/um/cambridge/projects/tutorial/

Page 3: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

An account of how vision might work

Having the ability to test hypotheses

Dealing with the ambiguity of the visual world

Having the ability to “fuse” information

Having the ability to learn

Reasoning with probabilities

Page 4: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Bottom-Up Segmentation (Berkeley)

Page 5: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Better defined problem: Foreground Segmentation

Page 6: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Separability of colour pallettes

Red

Gre

en

Red

Gre

en

Page 7: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Pixelwise independent decisions?

Maximum

likelihood

estimate:

log P(zi|xi=0) log P(zi|xi=1)

likelihoods

Page 8: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

... Segmentation in camouflage

Page 9: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Need for spatial priors

Data in Colour Space Mixture of Gaussian Models

-- f/b distributions intertwined

Page 10: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Connecting up pixels

How can you express coherence in a way that is practical?

N

N

Page 11: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Markov models

1st order Markov chain

Textbook example:

Mon Tues Wed Thur

?

Page 12: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

"As I've commented before, really relating to

someone involves standing next to impossible."

"Oh, sorry. Nevermind. I am afraid of it becoming

another island in a nice suit."

Markov models

2nd order Markov chain

Predictive text

Dasher

(Ward, Blackwell & Mackay 2000)

Page 13: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

1st order with stochastic observations – Hidden Markov Model

HMMs – ubiquitous in speech recognition(Rabiner, 89; Jelinek 98) HTK (Young,Woodland et al. 97)

tractable –

Dynamic Programming

etc.

time

Page 14: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Factorial Hidden Markov Model(Ghahramani and Jordan 1995)

Page 15: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Temporal HMMs in vision

Page 16: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Probabilistic Graphical Models (Pearl 1988)

•Probabilistic graphical models

•Inference by message passing

Generalisation (Lauritzen and Spiegelhalter, 1988)

Factor Graphs (Kschischang, Frey, Loeliger, 2001)

-- see also (Bishop, 2006)

Page 17: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

2D Markov model?

Tree of connected pixels(Veksler 2005)

Markov Random Field (MRF) –

1st order(Geman & Geman 84; Besag, 1974, 1986)

Independence:

where

neighbours of i

Page 18: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

2D MRF – 1st order Example

Ising Model

where

and

Binary variables:

Joint probability distribution:

Page 19: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

K=0.4

2D MRF simulation (Swensden Wang MCMC)

Ising Model

K=0.5

K=0.55

Page 20: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

2D Hidden MRF

Page 21: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

2D Hidden MRF

priorobservation

likelihood

Inference – MAP:

Page 22: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Simple segmentation --- Ising prior

MRF – expressed as additive energy terms

where “energy”

(-ve) log-prior V(x)

and

energy/cost minimization

colour

observations

with and

MAP

?? How to compute ie

Page 23: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Segmentation artefacts --- Ising prior

?? How to overcome artefacts

(Boykov and Kolmogorov ICCV 2003)

Page 24: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Boykov-Jolly contrast-sensitive segmentation

• Conditional Random Field -- CRF

where now

with

(Lafferty et al. 2001; Kumar and Hebert 2003)

log-”prior” V(x,z)

data-dependence

(Boykov and Jolly 2001; Rother et al. 2004; Li, Shum et al. 2004 )

Page 25: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Approximate variational extremum [Mumford and Shah 1985,9]

MAP estimation for Markov Random Fields

– Energy Minimization

Iterated conditional Modes [Besag 1986]

Simulated annealing [Metropolis, Rosenbluth, Rosenbluth, Teller and Teller, 1953]

Graduated nonconvexity [Blake and Zisserman 1987]

Graph cut [Greig, Porteous and Seheult, 1989]

Gibbs sampling [Geman and Geman 1984]

Generally NP-hard, so approximate:

Loopy Belief Propagation [Freeman and Pasztor, 1999]

“Modern” graph cut [Boykov, Veksler and Zabih, 2001]

Page 26: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Marginals

Inference – MAP:

whole distribution?:

pixelwise marginals?: i

Page 27: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

MRFs – a generic framework for low-level vision

Medical Imaging

Image Restoration

Stereo Vision

Recognition

Analysis of Structure

Page 28: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Medical Imaging

Medical image segmentation GeoS (Criminisi et al 08)

Page 29: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Image restoration

Page 30: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Stereo matching – solved problem – variational algorithmsDynamic programming (Ohta & Kanade, 85; Belhumeur and Mumford 92)

Graph cut (Roy and Cox 98; Boykov et al. 00)

Page 31: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Live Stereo Segmentation

Page 32: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Background substitution

Kolmogorov, Criminisi, Blake, Cross and Rother (CVPR 2005, PAMI 2006)

Page 33: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Recognising and segmenting objects(Winn and Jojic, 2005)

Page 34: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Unwrap mosaic(Rav-Acha, Kohli, Rother, Fitzgibbon 2008)

Page 35: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Forthcoming book!

“Advances in Markov Ramdom Fields for Computer Vision”

MIT Press, summer 2010

Topics of this course and much, much more

Contributors: usual suspects – lecturers on this course + Boykov,

Kolmogorov, Weiss, Freeman, ....

one for the office and one for home

www.research.microsoft.com/vision/MRFbook

Page 36: ICCV2009: MAP Inference in Discrete Models: Part 1: Introduction

Course programme

9.30-10.00 Introduction (Andrew Blake)

10.00-11.00 Discrete Models in Computer Vision (Carsten Rother)

15min Coffee break

11.15-12.30 Message Passing: DP, TRW, LP relaxation (Pawan Kumar)

12.30-13.00 Quadratic pseudo-boolean optimization (Pushmeet Kohli)

1 hour Lunch break

14:00-15.00 Transformation and move-making methods (Pushmeet Kohli)

15:00-15.30 Speed and Efficiency (Pushmeet Kohli)

15min Coffee break

15:45-16.15 Comparison of Methods (Carsten Rother)

16:30-17.30 Recent Advances: Dual-decomposition, higher-order, etc.

(Carsten Rother + Pawan Kumar)

All online material will be online (after conference):http://research.microsoft.com/en-us/um/cambridge/projects/tutorial/


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