Computer vision: models, learning and inference

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Presentation transcript:

Computer vision: models, learning and inference Chapter 5 The normal distribution Please send errata to s.prince@cs.ucl.ac.uk

Univariate Normal Distribution For short we write: Univariate normal distribution describes single continuous variable. Takes 2 parameters m and s2>0 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

Multivariate Normal Distribution For short we write: Multivariate normal distribution describes multiple continuous variables. Takes 2 parameters a vector containing mean position, m a symmetric “positive definite” covariance matrix S Positive definite: is positive for any real Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

Types of covariance Symmetric Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

Diagonal Covariance = Independence Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

Decomposition of Covariance Consider green frame of reference: Relationship between pink and green frames of reference: Substituting in: Full covariance can be decomposed into rotation matrix and diagonal Conclusion: Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

Transformation of Variables If and we transform the variable as The result is also a normal distribution: Can be used to generate data from arbitrary Gaussians from standard deviation of one Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

Marginal Distributions Marginal distributions of a multivariate normal are also normal If then Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

Conditional Distributions If then Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

Conditional Distributions Renormalize each scan line! For spherical / diagonal case, x1 and x2 are independent so all of the conditional distributions are the same. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

Product of two normals (self-conjugacy w.r.t mean) The product of any two normal distributions in the same variable is proportional to a third normal distribution Amazingly, the constant also has the form of a normal! Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

Change of Variables If the mean of a normal in x is proportional to y then this can be re-expressed as a normal in y that is proportional to x where Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

Conclusion Normal distribution is used ubiquitously in computer vision Important properties: Marginal dist. of normal is normal Conditional dist. of normal is normal Product of normals prop. to normal Normal under linear change of variables Computer vision: models, learning and inference. ©2011 Simon J.D. Prince