Single-view geometry

1. Single-view geometry Odilon Redon, Cyclops, 1914

2. X? X? X? Our goal: Recovery of 3D structure • Recovery of structure from one image is inherently ambiguous x

3. Our goal: Recovery of 3D structure • Recovery of structure from one image is inherently ambiguous

4. Our goal: Recovery of 3D structure • Recovery of structure from one image is inherently ambiguous

5. Ames Room http://en.wikipedia.org/wiki/Ames_room

6. Our goal: Recovery of 3D structure • We will need multi-view geometry

7. Recall: Pinhole camera model • Principal axis: line from the camera center perpendicular to the image plane • Normalized (camera) coordinate system: camera center is at the origin and the principal axis is the z-axis

8. Recall: Pinhole camera model

9. Principal point • Principal point (p): point where principal axis intersects the image plane • Normalized coordinate system: origin of the image is at the principal point • Image coordinate system: origin is in the corner • How to go from normalized coordinate system to image coordinate system? py px

10. Principal point offset principal point: py px

11. Principal point offset principal point: calibration matrix

12. Pixel coordinates • mx pixels per meter in horizontal direction, my pixels per meter in vertical direction Pixel size: m pixels pixels/m

13. Camera rotation and translation • In general, the camera coordinate frame will be related to the world coordinate frame by a rotation and a translation • Conversion from world to camera coordinate system (in non-homogeneous coordinates): coords. of point in camera frame coords. of camera center in world frame coords. of a pointin world frame

14. Camera rotation and translation Note: C is the null space of the camera projection matrix (PC=0)

15. Camera parameters • Intrinsic parameters • Principal point coordinates • Focal length • Pixel magnification factors • Skew (non-rectangular pixels) • Radial distortion

16. Camera parameters • Intrinsic parameters • Principal point coordinates • Focal length • Pixel magnification factors • Skew (non-rectangular pixels) • Radial distortion • Extrinsic parameters • Rotation and translation relative to world coordinate system

17. Camera calibration

18. Xi xi Camera calibration • Given n points with known 3D coordinates Xi and known image projections xi, estimate the camera parameters

19. Camera calibration: Linear method Two linearly independent equations

20. Camera calibration: Linear method • P has 11 degrees of freedom • One 2D/3D correspondence gives us two linearly independent equations • Homogeneous least squares: find p minimizing ||Ap||2 • Solution given by eigenvector of ATA with smallest eigenvalue • 6 correspondences needed for a minimal solution

21. Camera calibration: Linear method • Note: for coplanar points that satisfy ΠTX=0,we will get degenerate solutions (Π,0,0), (0,Π,0), or (0,0,Π)

22. Camera calibration: Linear method • Advantages: easy to formulate and solve • Disadvantages • Doesn’t directly tell you camera parameters • Doesn’t model radial distortion • Can’t impose constraints, such as known focal length and orthogonality • Non-linear methods are preferred • Define error as squared distance between projected points and measured points • Minimize error using Newton’s method or other non-linear optimization