Integration and Graphical Models

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Integration and Graphical Models

• Derek Hoiem
• CS 598, Spring 2009
• April 14, 2009

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Why?

• The goal of vision is to make useful inferences
  about the scene.
• In most cases, this requires integrative
  reasoning about many types of information.

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Example 3D modeling
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Object context
From Divvala et al. CVPR 2009
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How?

• Feature passing
• Graphical models

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Class Today

• Feature passing
• Graphical models
• Bayesian networks
• Markov networks
• Various inference and learning methods
• Example

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Properties of a good mechanism for integration

• Modular different processes/estimates can be
  improved independently
• Symbiotic each estimate improves
• Robust mistakes in one process are not fatal for
  others that partially rely on it
• Feasible training and inference is fast and easy

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Feature Passing

• Compute features from one estimated scene
  property to help estimate another

Image
X Estimate
X Features
Y Estimate
Y Features
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Feature passing example
Use features computed from geometric context
confidence images to improve object detection
Features average confidence within each window
Above
Object Window
Below
Hoiem et al. ICCV 2005
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Feature Passing

• Pros and cons
• Simple training and inference
• Very flexible in modeling interactions
• Not modular
• if we get a new method for first estimates, we
  may need to retrain
• Requires iteration to be symbiotic
• complicates things
• Robust in expectation but not instance

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Probabilistic graphical models

• Explicitly model uncertainty and dependency
  structure

Directed
Undirected
Factor graph
a
a
a
b
b
b
c
d
c
d
c
d
Key concept Markov blanket
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Directed acyclical graph (Bayes net)
Arrow directions matter
a
a
c independent of a given b d independent of a
given b
a,c,d dependent when conditioned on b
b
b
c
d
c
d
P(a,b,c,d) P(cb)P(db)P(ba)P(a)
P(a,b,c,d) P(ba,c,d)P(a)P(c)P(d)
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Directed acyclical graph (Bayes net)

• Can model causality
• Parameter learning
• Decomposes learn each term separately (ML)
• Inference
• Simple exact inference if tree-shaped (belief
  propagation)

a
b
c
d
P(a,b,c,d) P(cb)P(db)P(ba)P(a)
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Directed acyclical graph (Bayes net)

• Can model causality
• Parameter learning
• Decomposes learn each term separately (ML)
• Inference
• Simple exact inference if tree-shaped (belief
  propagation)
• Loops require approximation
• Loopy BP
• Tree-reweighted BP
• Sampling

a
b
c
d
P(a,b,c,d) P(cb)P(da,b)P(ba)P(a)
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Directed graph

• Example Places and scenes

Place office, kitchen, street, etc.
Objects Present
Fire Hydrant
Car
Person
Toaster
Microwave
P(place, car, person, toaster, micro, hydrant)
P(place) P(car place) P(person place)
P(hydrant place)
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Directed graph

• Example Putting Objects in Perspective

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Undirected graph (Markov Networks)

• Does not model causality
• Often pairwise
• Parameter learning difficult
• Inference usually approximate

x1
x2
x3
x4
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Markov Networks

• Example label smoothing grid

Binary nodes
Pairwise Potential
0 1 0 0 K 1 K 0
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Factor graphs

• A general representation

Factor Graph
a
Bayes Net
a
b
b
c
d
c
d
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Factor graphs

• A general representation

Factor Graph
a
b
c
d
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Factor graphs
Write as a factor graph
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Inference Belief Propagation

• Very general
• Approximate, except for tree-shaped graphs
• Generalizing variants BP can have better
  convergence for graphs with many loops or high
  potentials
• Standard packages available (BNT toolbox, my
  website)
• To learn more
• Yedidia, J.S. Freeman, W.T. Weiss, Y.,
  "Understanding Belief Propagation and Its
  Generalizations, Technical Report, 2001
  http//www.merl.com/publications/TR2001-022/

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Inference Graph Cuts

• Associative edge potentials penalize different
  labels
• Associative binary networks can be solved
  optimally (and quickly) using graph cuts
• Multilabel associative networks can be handled by
  alpha-expansion or alpha-beta swaps
• To learn more
• http//www.cs.cornell.edu/rdz/graphcuts.html
• Classic paper What Energy Functions can be
  Minimized via Graph Cuts? (Kolmogorov and Zabih,
  ECCV '02/PAMI '04)

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Inference Sampling (MCMC)

• Metropolis-Hastings algorithm
• Define transitions and transition probabilities
• Make sure you can get from any state to any other
  (ergodicity)
• Make proposal and accept if rand(1) lt P(new
  state)/P(old state) P(backward transition) /
  P(transition)
• Note if P(state) decomposes, this is easy to
  compute
• Example Image parsing by Tu and Zhu to find
  good segmentation

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Learning parameters maximize likelihood

• Simply count for Bayes network with discrete
  variables
• Run BP and do gradient descent for Markov network
• Often do not care about full likelihood

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Learning parameters maximize objective

• SPSA (simultaneous perturbation stochastic
  approximation) algorithm
• Take two trial steps in a random direction, one
  forward and one backwards
• Compute loss (or objective) for each and get a
  pseudo-gradient
• Take a step according to results
• Refs
• Li and Huttenlocher, Learning for Optical Flow
  Using Stochastic Optimization, ECCV 2008
• Various papers by Spall on SPSA

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Learning parameters structured learning
See also Tsochantaridis et al.
http//jmlr.csail.mit.edu/papers/volume6/tsochant
aridis05a/tsochantaridis05a.pdf
Szummer et al. 2008
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How to get the structure?

• Set by hand (most common)
• Learn (mostly for Bayes nets)
• Maximize score (greedy search)
• Based on independence tests
• Logistic regression with L1 regularization for
  finding Markov blanket

For more www.autonlab.org/tutorials/bayesstruct05
.pdf
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Graphical Models

• Pros and cons
• Very powerful if dependency structure is sparse
  and known
• Modular (especially Bayesian networks)
• Flexible representation (but not as flexible as
  feature passing)
• Many inference methods
• Recent development in learning Markov network
  parameters, but still tricky

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Which techniques have I used?

• Almost all of them
• Feature passing (ICCV 2005, CVPR 2008)
• Bayesian networks (CVPR 2006)
• In factor graph form (ICCV 2007)
• Semi-naïve Bayes (CVPR 2004)
• Markov networks (ECCV 2008, CVPR 2007, CVPR 2005
  HMM)
• Belief propagation (CVPR 2006, ICCV 2007)
• Structured learning (ECCV 2008)
• Graph cuts (CVPR 2008, ECCV 2008)
• MCMC (IJCV 2007 didnt work well)
• Learning Bayesian structure (2002-2003, not
  published)

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Example faces, skin, cloth