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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 the Perspective projection equation Reading: Chapter 2 of FP, Chapter 2 of S.

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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 the Perspective projection equation • Reading: Chapter 2 of FP, Chapter 2 of S

  3. Quantitative Measurements and Calibration Euclidean Geometry

  4. Euclidean Coordinate Systems

  5. Planes homogenous coordinate

  6. Coordinate Changes: Pure Translations OBP = OBOA + OAP ,BP = BOA+ AP

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

  8. Coordinate Changes: Rotations about the z Axis

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

  10. A rotation matrix is characterized by the following properties: • 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. Coordinate Changes: Pure Rotations

  13. Coordinate Changes: Rigid Transformations

  14. Block Matrix Multiplication What is AB ? Homogeneous Representation of Rigid Transformations

  15. Rigid Transformations as Mappings

  16. Rigid Transformations as Mappings: 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 • Ratios of distance along a line • |p2-p1|/|p3-p2| is preserved

  19. Shear matrix Horizontal shear Vertical shear

  20. 2D planar transformations

  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. The Intrinsic Parameters of a Camera Units: k,l :pixel/m f :m a,b : pixel Physical Image Coordinates (f ≠1) Normalized Image Coordinates

  27. The Intrinsic Parameters of a Camera Calibration Matrix 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 • Focal length may vary for zoom lenses • Optical axis may not be perpendicular to image plane • Change focus affects the magnification factor • From now on, assume camera is focused at infinity

  29. Extrinsic Parameters

  30. Explicit Form of the 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 x 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 the Projection Matrix Note: : i-th row of R 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

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