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3d tracking : chapter2-1 mathematical tools

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