Numerical computation II. Reprojection error Bundle adjustment Family of Newtonʼs methods Statistical background Maximum likelihood estimation

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1 Numerical computation II Reprojection error Bundle adjustment Family of Newtonʼs methods Statistical background Maximum likelihood estimation

2 Reprojection error Reprojection error = Distance between the observation and its estimation (reproduction) measured on the image We search for x that minimizes E(x); Called bundle adjustment Minimization is performed by Newtonʼs method etc. Good initial values necessary

3 Bundle adjustment Minimizing the sum of reprojection errors with all the unknown parameters, i.e., camera parameters and point coordinates x (i) j = Input h x (i) j i =1,..., m, 2mn i > y (i) j 1 j =1,...,n Geometric model x (i) Output j / P i X j X j = X j Y j Z j 1 > 3n P i = K i Ri t i 11m

4 Example: Calibration of multiple surveillance cameras Observation: point correspondences among multi-view images Parameters to estimate: Poses and focal lengths of the cameras plus the 3D coordinates of scene points

5 Example: Fitting an ellipse to a set of points Observation: points To estimate: the shape of ellipse and the true positions of the points

6 Minimization of Any numerical method for minimization can generally be used Family of Newtonʼs method Other approaches Reprojection error has the form of a sum of squares; nonlinear least squares methods are a natural choice Standard methods: Gauss-Newton method Levenberg-Marquardt method

7 Using Newtonʼs method for minimization We approximate the local shape of E(x) by a quadratic function; then we find the minimum of the quadratic function Starting an initial value, we repeat this procedure until convergence

8 Using Newtonʼs method for minimization We approximate the local shape of E(x) by a quadratic function; then we find the minimum of the quadratic function Starting an initial value, we repeat this procedure until convergence Gradient: Hessian: Compute the gradient and Hessian Update solution

9 Gauss-Newton method Approximates the Hessian utilizing E being a sum of squares Or equivalently where Parameter update:

10 Levenberg-Marquardt method Update the parameter with [Levenberg1944, Marquardt1963] is called the damping factor à coincides with the Gauss-Newton update à coincides with the steepest descent method is adaptively chosen depending on if E(x) decreases Increase when Decrease when E(x + x) >E(x) E(x + x) <E(x)

11 Updating parameters Parameter update à Solve a linear equation Matrix inverse should not be used; inefficient, inaccurate (NG!) A standard approach = Cholesky decomposition + forward/backward substitution 1) 2) 3) = = Remark: Preconditioned Conjugate Gradient (PCG) method is preferred when A is very large

12 Statistical explanation of bundle adjustment Observation inevitably has errors observation true val. error Statistical model of the errors They are random variables following a Gaussian distribution

13 What is good estimation? There are an infinite number of estimating methods Observation Method Method A A Estimates of unknown params. A shared view of Whatʼs good estimate? Mean of the estimates coincides with the true value Variance of the estimates is as small as possible There indeed exists a theoretical lower bound for the variance, named the Cramer-Rao lower bound Distribution of estimates True value

14 Maximum likelihood estimation Maximum likelihood = Selects the parameter value that maximizes the likelihood as an estimate: The likelihood is defined as parameter all observations The estimate is called the maximum likelihood estimate It can be shown that it attains the Cramer-Rao bound asymptotically as the number of observations goes to infinity

15 Bundle adjustment as maximum likelihood Model of an observation: z i = z i + " i where the error " i N(0, 2 I) Then, the PDF of an observation is given by (zi z i ) 2 p(z i )=Cexp 2 2 C = 1 (2 ) d/2 d The PDF of all observations z 1,...,z N NY (zi z i ) 2 p(z 1,...,z N )= C exp i=1 2 2 Maximize the likelihood L( z 1,..., z N ; z 1,...,z N ) is equivalent to minimizing negative log-likelihood: NX (z i z i ) 2 i=1