05 cv mil_normal_distribution

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05 cv mil_normal_distribution

  1. 1. Computer vision:models, learning and inference Chapter 5 The normal distribution Please send errata to s.prince@cs.ucl.ac.uk
  2. 2. Univariate Normal DistributionFor short we write: Univariate normal distribution describes single continuous variable. Takes 2 parameters and 2>0 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 2
  3. 3. Multivariate Normal DistributionFor short we write:Multivariate normal distribution describes multiple continuousvariables. Takes 2 parameters • a vector containing mean position, • a symmetric “positive definite” covariance matrix Positive definite: is positive for any real Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 3
  4. 4. Types of covariance SymmetricComputer vision: models, learning and inference. ©2011 Simon J.D. Prince 4
  5. 5. Diagonal Covariance = Independence Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 5
  6. 6. Decomposition of CovarianceConsider green frame of reference:Relationship between pink and green framesof reference:Substituting in: Full covariance can be decomposed Conclusion: into rotation matrix and diagonal Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 6
  7. 7. Transformation of VariablesIf and we transform the variable asThe result is also a normal distribution:Can be used to generate data from arbitrary Gaussians from standard one Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 7
  8. 8. Marginal Distributions Marginal distributions of a multivariate normal are also normalIf then Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 8
  9. 9. Conditional DistributionsIfthen Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 9
  10. 10. Conditional DistributionsFor 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 10
  11. 11. Product of two normals (self-conjugacy w.r.t mean)The product of any two normal distributions in the samevariable is proportional to a third normal distributionAmazingly, the constant also has the form of a normal! Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 11
  12. 12. Change of VariablesIf the mean of a normal in x is proportional to y then this can be re-expressed asa normal in y that is proportional to x where Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 12
  13. 13. 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 13

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