# lecture5 - VisualSimulation CAP6938 Dr.HassanForoosh...

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Visual Simulation CAP 6938 Dr. Hassan Foroosh  Dept. of Computer Science UCF © Copyright Hassan Foroosh 2002

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Today Last Lecture Feature Tracking Structure from Motion Tomasi and Kanade Extensions Today Camera Calibration Bundle adjustment
Camera calibration Determine camera parameters from  known  3D  points or calibration object(s) 1. internal  or  intrinsic  parameters such as focal  length, optical center, aspect ratio: what kind of camera? 2. external  or  extrinsic  (pose) parameters: where is the camera? 3. How can we do this?

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Camera calibration –  approaches Possible approaches: 1. linear regression (least squares) 2. non-linear optimization 3. vanishing points 4. multiple planar patterns 5. panoramas (rotational motion)
Image Formation  Equations u ( X c ,Y c ,Z c ) u c f

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Calibration matrix Is this form of K good enough? non-square pixels (digital video) skew radial distortion
Camera matrix Fold  intrinsic  calibration matrix  K  and  extrinsic   pose parameters ( R , t ) together into a camera matrix M  =  K  [ R  |  t  ] (put 1 in lower r.h. corner for 11 d.o.f.)

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Camera matrix calibration Directly estimate 11 unknowns in the  matrix  using known 3D points ( X i , Y i , Z i ) and measured  feature positions ( u i , v i )
Camera matrix calibration Linear regression: Bring denominator over, solve set of (over-determined)  linear equations.  How? Least squares (pseudo-inverse) Is this good enough?

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Optimal estimation Feature measurement equations Likelihood of  M  given {( u i , v i )}
Optimal estimation Log likelihood of  M  given {( u i , v i )} How do we minimize  C ? Non-linear regression (least squares), because  û i   and  v i  are non-linear functions of  M

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Levenberg-Marquardt Iterative non-linear least squares [Press’92] Linearize measurement equations Substitute into log-likelihood equation:  quadratic cost  function in  m
Iterative non-linear least squares [Press’92] Solve for minimum Hessian: error: Does this look familiar…? Levenberg-Marquardt

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What if it doesn’t converge? Multiply diagonal by (1 +  λ ), increase  λ until it does Halve the step size  m  (my favorite) Use line search Other ideas? Uncertainty analysis:  covariance  Σ  = A -1 Is  maximum  likelihood the best idea? How to start in vicinity of global minimum?
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## This note was uploaded on 06/13/2011 for the course CAP 6938 taught by Professor Staff during the Spring '08 term at University of Central Florida.

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lecture5 - VisualSimulation CAP6938 Dr.HassanForoosh...

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