Probability Review

(many slides from Octavia Camps)

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 trialsWhere N(a) is the number that event a happens in n trials

More Formal:

• is the Sample Space:– Contains all possible outcomes of an experiment

• 2 is a single outcome• A 2 is a set of outcomes of interest

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

Conditional Probability

• One of the most useful concepts!

AABB

Bayes Theorem

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

Using Partitions:

• If events Ai are mutually exclusive and partition

BB

Random Variables

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

X(X())

Random Variables Distributions

• Cumulative Probability Distribution (CDF):

• Probability Density Function (PDF):Probability Density Function (PDF):

Random Distributions:

• From the two previous equations:

Uniform Distribution

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

XX11 XX22

Gaussian (Normal) Distribution

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

Statistical Characterizations

• Expectation (Mean Value, First Moment):

•Second Moment:Second Moment:

Statistical Characterizations• Variance of X:

• Standard Deviation of X:

Mean Estimation from Samples

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

Variance Estimation from Samples

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

Image Noise Model

• Additive noise: – Most commonly used

),(),(),(ˆ jiNjiIjiI

Additive Noise Models

• Gaussian – Usually, zero-mean, uncorrelated

•Uniform

Measuring Noise

• Noise Amount: SNR = s/ n

• Noise Estimation: – Given a sequence of images I0,I1, … IN-1

1

0

1

0

21

0

1

0

),(1

)),(),((1

1),(

),(1),(

R

i

C

jn

k

N

k

N

kk

jiRC

jiIjiIN

ji

jiIN

jiI

Good estimatorsData values z are random variablesA parameter describes the distributionWe have an estimator z) of the unknown parameter

If E(z) or E(z) ) = E(the estimatorz) is unbiased

Balance between bias and variance

Mean squared error as performance criterion

Least Squares (LS)

If errors only in b

Then LS is unbiased

But if errors also in A (explanatory variables)

Errors in Variable Model

Least Squares (LS)

biasLarger variance in A,,ill-conditioned A,u oriented close to the eigenvector of the smallest eigenvalue increase the biasGenerally underestimation

(a) (b)

Estimation of optical flow

(a) Local information determines the component of flow perpendicular to edges(b) The optical flow as best intersection of the flow constraints is biased.

Optical flow

• One patch gives a system:

0

0000

1

2

1

22

11

ts

t

t

t

yx

yx

yx

IuI

I

II

vu

II

IIII

nnn

tyx IvIuI

Noise model• additive, identically, independently distributed,

symmetric noise:

iii

iii

iii

ttt

yyy

xxx

NII

NII

NII

22 )()()( tttsyyxx iiiiiiNNENNENNE