3d tracking : chapter2-1 mathematical tools


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3d tracking : chapter2-1 mathematical tools

  1. 1. Monocular Model-Based 3D Tracking of Rigid Objects: A Survey<br />2008. 12. 04.백운혁<br />Chapter 2. Mathematical Tools<br />
  2. 2.
  3. 3. Agenda<br />Monocular Model-Based 3D Tracking of Rigid Objects : A Survey<br />Chapter 2. Mathematical Tools<br />2.1 Camera Representation<br />2.2 Camera Pose Parameterization<br />2.3 Estimating the External Parameters Matrix<br />2.4 Least-Squares Minimization Techniques<br />2.5 Robust Estimation<br />2.6 Bayesian Tracking<br />
  4. 4. the standard pinhole camera model<br />2.1 Camera Representation<br />
  5. 5. 2.1.1 The Perspective Projection Model<br />Image coordinate system<br />World Coordinates<br />Image Coordinates<br />(in the image)<br />Projection Matrix<br />
  6. 6. 2.1.2 The Camera Calibration Matrix<br />internal parameters<br />focal length<br />principal point<br />skew parameter<br />the number of pixels per unit distance in the u<br />the number of pixels per unit distance in the v<br />
  7. 7. 2.1.2 The Camera Calibration Matrix<br />projection<br />focal length<br />Image Plane<br />
  8. 8. 2.1.2 The Camera Calibration Matrix<br />projection to image<br />principal point (center of image plane)<br />the number of pixels per unit distance in the u<br />the number of pixels per unit distance in the v<br />
  9. 9. 2.1.2 The Camera Calibration Matrix<br />skew<br />field of view<br />referred as the skew, usually<br />image plane size and field of view are assumed to be fixed,<br />but not fixed focal length<br />
  10. 10. 2.1.3 The External Parameters Matrix<br />world coordinate to camera coordinate<br />The 3x4 external parameters<br />rotation matrix<br />translation vector<br />in the world coordinate system<br />in the camera coordinate system<br />
  11. 11. 2.1.3 The External Parameters Matrix<br />
  12. 12. 2.1.4 Estimating the Camera Calibration Matrix<br />internal parameters are assumed to be fixed<br />make use of a calibration pattern of known sizeinside the field of view<br />correspondence between the 3D points and the 2D image points<br />
  13. 13. 2.1.5 Handling Lens Distortion<br />(usually ignored)<br />radial distortion<br />tangential distortion<br />
  14. 14. can be avoided by locally re-pametrizing the rotation<br />2.2 Camera Pose Parameterization<br />
  15. 15. 2.2.1 Euler Angles<br /><ul><li>α,β,γ to be rotation angles around the Z, Y, and X axis respectively yields
  16. 16. one rotation has no effect
  17. 17. gimbal lock problem</li></li></ul><li>2.2.2 Quaternions<br />A rotation about the unit vector by an angle <br /><ul><li>a scalar plus a 3-vector</li></li></ul><li>2.2.3 Exponential Map<br />A rotation about the unit vector by an angle <br /><ul><li>Let be a 3D vector</li></li></ul><li>2.2.3 Exponential Map<br />Rodrigues’ formula<br /><ul><li>the exponential map represents a rotation as a 3-vector that gives its axis and magnitude.</li></ul>is the skew-symmetric matrix<br />
  18. 18. 2.2.4 Linearization of Small Rotations<br />
  19. 19. estimated camera positions<br />(when the internal parameters are known)<br />2.3 Estimating the External parameters Matrix<br />
  20. 20. 2.3.1 How many Correspondences are necessary?<br />n=3 known correspondences produce 4 possible solution (P3P Problem)<br />n>=4 known correspondences produce 2 possible solution<br />n>=4 known correspondences (points are coplanar) produce unique solution<br />n>=6 known correspondences produce unique solution<br />
  21. 21. 2.3.2 The Direct Linear Transformation (DLT)<br />to estimate the whole matrix P by solving a linear system even when the internal parameters are not known<br />Each correspondence gives rise to two linearly independent equations<br />
  22. 22. 2.3.2 The Direct Linear Transformation (DLT)<br />Stacking all the equation into B yields the linear system : <br />
  23. 23. 2.3.2 The Direct Linear Transformation (DLT)<br />is the eigen vector of B corresponding to the smallest eigenvalue of B<br />6 correspondences must be known<br />for 3D tracking , using a calibrated camera and estimating only its orientation and position<br />
  24. 24. 2.3.3 The Perspective-n-Point (PnP) Problem<br />
  25. 25. 2.3.4 Pose estimation from a 3D Plane<br />The relation between a 3D plane and its image projection can be represented by a homogeneous 3x3 matrix (homography matrix)<br />Let us consider the plane<br />
  26. 26. 2.3.4 Pose estimation from a 3D Plane<br />The matrix H can be estimated from four correspondences using a DLT algorithm<br /> the translation vector <br />last column is given by the cross-product<br />since the columns of R must be orthonormal<br />
  27. 27. 2.3.5 non-Linear Reprojection Error<br />
  28. 28. finding the pose that minimizes a sum of residual errors <br />2.4 Least-Squares Minimization Techniques<br />
  29. 29. 2.4.1 Linear Least-Squares<br />the function is linear<br />the camera pose parameters <br />the unknowns of a set of linear equations in matrix form as<br /> can be estimated as <br />pseudo-inverse of A<br />
  30. 30. 2.4.2 Newton-Based Minimization Algorithms<br />the function is not linear<br />algorithms start from an initial estimate of the minimum and update it iteratively<br /> is chosen to minimize the residual at iteration and estimated by approximating to the first order<br />
  31. 31. 2.4.2 Newton-Based Minimization Algorithms<br />Jacobian matrix the partial derivatives of all these functions<br />stabilizes the begavior<br />
  32. 32. inliers<br /> data whose distribution can be explained by some set of model parameters<br />outliers<br /> which are data that do not fit the model<br /> the data can be subject to noise<br />M-estimators good at finding accurate solutions require an initial estimate to converge correctly<br />RANSAC does not require such an initial estimate does not take into account all the available data lacks precision<br />2.5 Robust Estimation<br />
  33. 33. 2.5.1 M-Estimators<br />least-squares estimation<br />the assumption that the observations are independent and have a Gaussian distribution<br />Instead of minimizing<br />are residual errors<br />is an M-estimator that reduce the influence of outliers<br />
  34. 34. 2.5.1 M-Estimators<br />Huber estimator<br />Tukey estimator<br />
  35. 35. <ul><li>Huber estimator : linear to reduce the influence of large residual errors
  36. 36. Tukey estimator : flat so that large residual errors have no influence at all</li></ul>2.5.1 M-Estimators<br />
  37. 37. 2.5.2 RANSAC<br /> samples of data pointsare randomly selected<br />estimate model parameters<br />find the subset of points (consistent with the estimate)<br />the largest is retained and refined by least-squares minimization<br />the model parameters require a minimum of<br />a set of measurements<br />
  38. 38. 2.5.2 RANSAC<br />linear least-square estimation<br />
  39. 39. 2.5.2 RANSAC<br />random sampling<br />
  40. 40. 2.5.2 RANSAC<br />random sampling<br />
  41. 41. 2.5.2 RANSAC<br />random sampling<br />
  42. 42. 2.5.2 RANSAC<br />random sampling<br />
  43. 43. estimating the density of successive states in the space of possible camera poses.<br />2.6 Bayesian Tracking<br />
  44. 44.
  45. 45. Thank you for your attention<br />