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EECS 274 Computer Vision

EECS 274 Computer Vision. Geometric Camera Models. Geometric Camera Models. Elements of Euclidean geometry Intrinsic camera parameters Extrinsic camera parameters General form of perspective projection Reading: Chapter 1 of FP, Chapter 2 of S. Geometric camera calibration.

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EECS 274 Computer Vision

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  1. EECS 274 Computer Vision Geometric Camera Models

  2. Geometric Camera Models • Elements of Euclidean geometry • Intrinsic camera parameters • Extrinsic camera parameters • General form of perspective projection • Reading: Chapter 1 of FP, Chapter 2 of S

  3. Geometric camera calibration Euclidean Geometry

  4. Euclidean coordinate system

  5. Planes homogenous coordinate

  6. Pure translation OBP = OBOA + OAP ,BP = BOA+ AP AP: point P in frame A

  7. Pure rotation 1st column: iA in the basis of (iB, jB, kB) 3rd row: kB in the basis of (iA, jA, kA)

  8. Rotation about z axis

  9. Rotation matrix Elementary rotation R=R x R y R z , described by three angles

  10. Properties of rotation matrix • Its inverse is equal to its transpose, R-1=RT , and • Its determinant is equal to 1. Or equivalently: • Its rows (or columns) form a right-handed • orthonormal coordinate system.

  11. Rotation group and SO(3) • Rotation group: the set of rotation matrices, with matrix product • Closure, associativity, identity, invertibility • SO(3): the rotation group in Euclidean space R3 whose determinant is 1 • Preserve length of vectors • Preserve angles between two vectors • Preserve orientation of space

  12. Pure rotations

  13. Rigid transformation

  14. Block matrix manipulation What is AB ? Homogeneous Representation of Rigid Transformations

  15. Rigid transformations as mappings

  16. Rotation about the k Axis

  17. Affine transformation • Images are subject to geometric distortion introduced by perspective projection • Alter the apparent dimensions of the scene geometry

  18. Affine transformation • In Euclidean space, preserve • Collinearity relation between points • 3 points lie on a line continue to be collinear • Ratio of distance along a line • |p2-p1|/|p3-p2| is preserved

  19. Shear matrix Horizontal shear Vertical shear

  20. 2D planar transformations See Szeliski Chapter 2

  21. 2D planar transformations

  22. 2D planar transformations

  23. 3D transformation

  24. Idealized coordinate system

  25. Camera parameters • Intrinsic: relate camera’s coordinate system to the idealized coordinated system • Extrinsic: relate the camera’s coordinate system to a fix world coordinate system • Ignore the lens and nonlinear aberrations for the moment

  26. Intrinsic camera parameters Units: k,l :pixel/m f :m (See EXIF tags) a,b: pixel Physical Image Coordinates (f ≠1) Normalized Image Coordinates Scale parameters: k, l (image sensor may not be square) Offset: u0, v0 Manufacturing error: θ

  27. Intrinsic camera parameters Calibration matrix κ The perspective projection Equation

  28. In reality • Physical size of pixel and skew are always fixed for a given camera, and in principal known during manufacturing • Some parameters often available in EXIF tag • Focal length may vary for zoom lenses when optical axis is not perpendicular to image plane • Change focus affects the magnification factor • From now on, assume camera is focused at infinity

  29. Extrinsic camera parameters

  30. Explicit form of projection Matrix denotes the i-th row of R, tx, ty, tz, are the coordinates of t can be written in terms of the corresponding angles R can be written as a product of three elementary rotations, and described by three angles M is 3 × 4 matrix with 11 parameters 5 intrinsic parameters: α, β, u0, v0, θ 6 extrinsic parameters: 3 angles defining R and 3 for t

  31. Explicit form of projection Matrix : i-th row of R Note: M is only defined up to scale in this setting!!

  32. Theorem (Faugeras, 1993)

  33. Projection equation • The projection matrix models the cumulative effect of all parameters • Useful to decompose into a series of operations identity matrix intrinsics projection rotation translation Camera parameters • A camera is described by several parameters • Translation T of the optical center from the origin of world coords • Rotation R of the image plane • focal length f, principle point (x’c, y’c), pixel size (sx, sy) • blue parameters are called “extrinsics,” red are “intrinsics” • Definitions are not completely standardized • especially intrinsics—varies from one book to another

  34. Camera calibration toolbox • Matlab toolbox by Jean-Yves Bouguet http://www.vision.caltech.edu/bouguetj/calib_doc/ • Extract corner points from checkerboard

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