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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 trials 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 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 Conditional Probability • One of the most useful concepts! W B A Bayes Theorem • Provides a way to convert a-priori probabilities to a-posteriori probabilities: Using Partitions: • If events Ai are mutually exclusive and partition W W B 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 Random Variables Distributions • Cumulative Probability Distribution (CDF): • 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: X1 X2 Gaussian (Normal) Distribution • A R.V. X that is normally distributed has density function: m Statistical Characterizations • Expectation (Mean Value, First 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: Iˆ(i, j ) I (i, j ) N (i, j ) – Most commonly used 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 N 1 1 I (i, j ) N I (i, j ) 1 N 1 2 ( I ( i , j ) I ( i , j ) ) k N 1 k 0 n 1 RC k 0 R 1 C 1 k (i, j ) (i, j ) i 0 j 0 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 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) bias Larger variance in dA,,ill-conditioned A, u oriented close to the eigenvector of the smallest eigenvalue increase the bias Generally underestimation Estimation of optical flow (a) (b) (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 I xu I y v I t • One patch gives a system: I x1 I y1 I t1 0 I 0 I I u y2 t2 x2 v 0 I I xn I tn1 0 yn I su I t 0 Noise model • additive, identically, independently distributed, symmetric noise: I xi I xi N xi I yi I yi N yi I ti I ti N ti E ( N xi N xi ) E ( N yi N yi ) 2 s E ( N ti N ti ) 2 t