3D geometry and camera calibration

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1 3D geometry and camera calibration

2 camera calibration determine mathematical relationship between points in 3D and their projection onto a 2D imaging plane 3D scene imaging device 2D image [Tsukuba]

3 stereo vision

4 principal application: 3D scene reconstruction [Zhu and Humphreys]

5 camera calibration take pictures of scene with known 3D structure and solve for free parameters in projection: y x z

6 camera calibration big questions (today s lecture): how do we model the way a camera projects 3D points onto a 2D image? how many parameters are involved in this projection and what is their relationship (linear/non-linear)? which measurements/how many allow us to robustly estimate these parameters?

7 3D coordinate systems right-handed vs. left-handed:

8 2D coordinate systems y axis up vs. y axis down origin at center vs. corner! often use (u,v) for image coordinates u v v v u u

9 3D geometry basics 3D points = column vectors Transformations = pre-multiplied matrices:

10 rotations rotation about the z-axis: rotation about x- and y- axes are similar (permutations of above matrix)

11 arbitrary rotation any rotation is a composition of rotations about x, y, z (Euler angles) composition of transformations = matrix multiplication (watch the order!) result: orthonormal matrix each row, column has unit length and mutually orthogonal inverse of matrix = transpose

12 scale scaling along each axis:

13 shear shear parallel to xy plane:

14 translation can translation be represented by multiplying by a 3x3 matrix? No. Proof?

15 homogeneous coordinates add a fourth dimension to each point: to get real 3D coordinates, divide by w:

16 translation in homogeneous coordinates after divide by w, this is just a translation by

17 perspective projection what does 4th row of matrix do? after divide:

18 perspective projection projection onto z=1 plane (x,y,z) (x/z,y/z,1) (0,0,0) z=1 add scaling, etc. => pinhole camera model

19 putting it all together: a camera model translate to image center perspective projection camera location homogeneous divide to get (u,v) coords scale to pixel size camera orientation 3D point (homogeneous)

20 putting it all together: a camera model intrinsics extrinsics camera pose

21 putting it all together: a camera model image coordinates pixel coordinates eye coordinates camera coordinates normalized device coordinates world coordinates

22 more general camera model multiply matrices together to form single transformation that maps 3D position => 2D image location don t care about z after transformation how many free parameters?

23 radial distortion

24 radial distortion these nonlinear effects cannot be represented by a matrix additional intrinsic parameter

25 camera calibration take pictures of scene with known 3D structure and solve for free parameters in projection: y x z

26 calibration algorithms particular method used depends on: what data is available! intrinsics only vs. extrinsics only vs. both camera model being used (pinhole, weak-perspective, affine, etc.)

27 camera calibration: example #1 given: 3D <=> 2D correspondences general perspective camera model (11-parameter, no radial distortion) write equations:....

28 camera calibration: example #1 linear equation in unknowns overconstrained (equations > unknowns) underconstrained (rank deficient matrix)

29 camera calibration: example #1 standard linear least squares methods for Ax=0 will give the solution x=0 instead, look for solution with x =1 (recall our solution is valid up to global scale) that is, minimize Ax ^2 subject to x ^2=1

30 camera calibration: example #1

31 camera calibration: example #1 to minimize subject to set and all others to punchline: least squares solution is eigenvector of corresponding to minimum eigenvalue (what algorithm?)

32 camera calibration: example #2 incorporating additional constraints into camera model no shear, no scale (rigid-body motion) square pixels these lead to nonlinear constraints on camera parameters

33 camera calibration: example #2 option 1: solve for general perspective model as before, then find closest solution that satisfies constraints option 2: nonlinear least squares usually gradient descent techniques common implementations available (e.g., MATLAB s optimization toolbox)

34 camera calibration: example #3 incorporating radial distortion option 1: find distortion first (use straight lines in calibration target) warp image to eliminate distortion run (simpler) perspective calibration option 2: nonlinear least squares

35 camera calibration: example #4 what if 3D points are not known? Structure from Motion! unknowns: 3N + 6C (extrinsics only) knowns: 2NC overconstrained when: 2NC > 3N + 6C

36 multi-camera geometry epipolar geometry - relationship between observed positions of points in multiple cameras assumptions: 2 cameras known intrinsics and extrinsics (i.e., fully calibrated rig)

37 epipolar geometry

38 epipolar geometry

39 epipolar geometry

40 epipolar geometry Epipolar line Epipoles

41 epipolar geometry epipoles can lie at infinity all epipolar lines intersect epipole corresponding points must lie on conjugated epipolar lines

42 epipolar geometry goal: derive equation for key observation: determine a plane

43 epipolar geometry extrinsics: key observation:

44 epipolar geometry extrinsics: key observation: Essential Matrix

45 Essential Matrix E depends only on camera geometry given E, can derive equation for line

46 Fundamental Matrix can define fundamental matrix analogously, operating on pixel coordinates instead of camera coordinates: advantage: can sometimes estimate F without knowing camera calibration allows you to reconstruct epipolar geometry w/o any information of intrinsic or extrinsic parameters

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