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
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    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)
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    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
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    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 )
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    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 )}
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    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
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    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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