Learning and Vision: Generative Methods

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Learning and Vision Generative Methods

• Andrew Blake, Microsoft Research
• Bill Freeman, MIT
• ICCV 2003
• October 12, 2003

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Learning and vision Generative Methods

• Machine learning is an important direction in
  computer vision.
• Our goal for this class
• Give overviews of useful techniques.
• Show how these methods can be used in vision.
• Provide references and pointers.
• Note afternoon companion class
• Learning and vision Discriminative
  Methods,
• taught by Chris Bishop and Paul Viola.

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Learning and VisionGenerative Models

• Intro roadmap for learning and inference in
  vision (WTF)
• Probabilistic inference introduction
  integration of sensory data
• applications color constancy, Bayes Matte
• 900 Basic estimation methods (WTF)
• MLE, Expectation Maximization
• applications two-line fitting
• Learning and inference in temporal and spatial
  Markov processes
• Techniques
• 9.25 (1) PCA, FA, TCA (AB)
• inference linear (Wiener) filter
• learning by Expectation Maximization (EM)
• applications face simulation, denoising,
  Weisss intrinsic images
• and furthermore Active Appearance Models,
  Simoncelli, ICA non-Gaussianity, filter banks
• 10.00 (2) Markov chain HMM (AB)
• inference - MAP by Dynamic Programming, Forward
  and Forward-Backward (FB) algorithms
• learning by EM Baum-Welch
• representations pixels, patches
• applications stereo vision

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What is the goal of vision?
If you are asking, Are there any faces in this
image?, then you would probably want to use
discriminative methods.
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What is the goal of vision?
If you are asking, Are there any faces in this
image?, then you would probably want to use
discriminative methods.
If you are asking, Find a 3-d model that
describes the runner, then you would use
generative methods.
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Modeling

• So we want to look at high-dimensional visual
  data, and fit models to it forming summaries of
  it that let us understand what we see.

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The simplest data to modela set of 1d samples
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Fit this distribution with a Gaussian
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How find the parameters of the best-fitting
Gaussian?
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How find the parameters of the best-fitting
Gaussian?
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Derivation of MLE for Gaussians
Observation density
Log likelihood
Maximisation
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Basic Maximum Likelihood Estimate (MLE) of a
Gaussian distribution
Mean
Variance
Covariance Matrix
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Basic Maximum Likelihood Estimate (MLE) of a
Gaussian distribution
Mean
Variance
For vector-valued data, we have the Covariance
Matrix
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Model fitting example 2 Fit a line to observed
data
y
x
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Maximum likelihood estimation for the slope of a
single line
Maximum likelihood estimate
where
gives regression formula
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Model fitting example 3Fitting two lines to
observed data
y
x
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MLE for fitting a line pair
(a form of mixture dist. for )
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Fitting two lines on the one hand
If we knew which points went with which lines,
wed be back at the single line-fitting problem,
twice.
y
x
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Fitting two lines, on the other hand
We could figure out the probability that any
point came from either line if we just knew the
two equations for the two lines.
y
x
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Expectation Maximization (EM) a solution to
chicken-and-egg problems
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MLE with hidden/latent variablesExpectation
Maximisation
General problem
data
parameters
hidden variables
For MLE, want to maximise the log likelihood
The sum over z inside the log gives a complicated
expression for the ML solution.
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The EM algorithm
We dont know the values of the labels, zi , but
lets use its expected value under its posterior
with the current parameter values, ?old. That
gives us the expectation step
E-step
Now lets maximize this Q function, an expected
log-likelihood, over the parameter values, giving
the maximization step
M-step
Each iteration increases the total log-likelihood
log p(y?)
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Expectation Maximisation applied to fitting the
two lines
associate data point
Hidden variables
with line
and probabilities of association are



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EM fitting to two lines
with
/2
E-step
and
repeat
Regression becomes
M-step
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Experiments EM fitting to two lines
(from a tutorial by Yair Weiss,
http//www.cs.huji.ac.il/yweiss/tutorials.html)
Line weights
line 1
line 2
Iteration
1
2
3
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Applications of EM in computer vision

• Structure-from-motion with multiple moving
  objects
• Motion estimation combined with perceptual
  grouping
• Multiple layers/or sprites in an image.