Probability Review

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Probability Review
(many slides from Octavia Camps)
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Intuitive Development

• Intuitively, the probability of an event a could
  be defined as

Where N(a) is the number that event a happens in
n trials
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More Formal

• W is the Sample Space
• Contains all possible outcomes of an experiment
• w2 W is a single outcome
• A 2 W is a set of outcomes of interest

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Independence

• The probability of independent events A, B and C
  is given by
• P(ABC) P(A)P(B)P(C)

A and B are independent, if knowing that A has
happened does not say anything about B happening
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Conditional Probability

• One of the most useful concepts!

W
A
B
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Bayes Theorem

• Provides a way to convert a-priori probabilities
  to a-posteriori probabilities

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Using Partitions

• If events Ai are mutually exclusive and partition
  W

W
B
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Random Variables

• A (scalar) random variable X is a function that
  maps the outcome of a random event into real
  scalar values

W
X(w)
w
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Random Variables Distributions

• Cumulative Probability Distribution (CDF)

• Probability Density Function (PDF)

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Random Distributions

• From the two previous equations

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Uniform Distribution

• A R.V. X that is uniformly distributed between x1
  and x2 has density function

X1
X2
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Gaussian (Normal) Distribution

• A R.V. X that is normally distributed has density
  function

m
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Statistical Characterizations

• Expectation (Mean Value, First Moment)

• Second Moment

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Statistical Characterizations

• Variance of X

• Standard Deviation of X

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Mean Estimation from Samples

• Given a set of N samples from a distribution, we
  can estimate the mean of the distribution by

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Variance Estimation from Samples

• Given a set of N samples from a distribution, we
  can estimate the variance of the distribution by

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Image Noise Model

• Additive noise
• Most commonly used

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Additive Noise Models

• Gaussian
• Usually, zero-mean, uncorrelated

• Uniform

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Measuring Noise

• Noise Amount SNR ?s/ ?n
• Noise Estimation
• Given a sequence of images I0,I1, IN-1

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Good estimators
Data values z are random variables A parameter q
describes the distribution We have an estimator j
(z) of the unknown parameter q. If E(j
(z) - q ) 0 or E(j (z) ) E(q)
the estimator j (z) is unbiased
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Balance between bias and variance
Mean squared error as performance criterion
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Least Squares (LS)
If errors only in b
Then LS is unbiased
But if errors also in A (explanatory variables)
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Errors in Variable Model
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Least Squares (LS)
bias
Larger variance in dA,,ill-conditioned A, u
oriented close to the eigenvector of the
smallest eigenvalue increase the bias Generally
underestimation
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Estimation of optical flow
(a)
(b)

1. Local information determines the component of
   flow perpendicular to edges
2. The optical flow as best intersection of the flow
   constraints is biased.

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Optical flow

• One patch gives a system

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Noise model

• additive, identically, independently distributed,
  symmetric noise