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Uncalibrated Epipolar - Calibration

Uncalibrated Epipolar - Calibration. Jana Kosecka CS223b. calibrated coordinates. Linear transformation. pixel coordinates. Uncalibrated Camera. Overview. Calibration with a rig Uncalibrated epipolar geometry. Calibrated camera. Image plane coordinates

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Uncalibrated Epipolar - Calibration

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  1. Uncalibrated Epipolar - Calibration Jana Kosecka CS223b

  2. calibrated coordinates Linear transformation pixel coordinates Uncalibrated Camera CS223b

  3. Overview • Calibration with a rig • Uncalibrated epipolar geometry CS223b

  4. Calibrated camera • Image plane coordinates • Camera extrinsic parameters • Perspective projection Uncalibrated camera • Pixel coordinates • Projection matrix Uncalibrated Camera CS223b

  5. Taxonomy on Uncalibrated Reconstruction • is known, back to calibrated case • is unknown • Calibration with complete scene knowledge (a rig) – estimate • Uncalibrated reconstruction despite the lack of knowledge of • Autocalibration (recover from uncalibrated images) • Use partial knowledge • Parallel lines, vanishing points, planar motion, constant intrinsic • Ambiguities, stratification (multiple views) CS223b

  6. Calibration with a Rig Use the fact that both 3-D and 2-D coordinates of feature points on a pre-fabricated object (e.g., a cube) are known. CS223b

  7. Given 3-D coordinates on known object • Eliminate unknown scales • Solve for translation Calibration with a Rig • Recover projection matrix • Factor the into and using QR decomposition CS223b

  8. More details • Direct calibration by recovering and decomposing the projection matrix 2 constraints per point CS223b

  9. Solve for translation More details • Recover projection matrix • Collect the constraints from all N points into matrix M (2N x 12) • Solution eigenvector associated with the smallest eigenvalue • Unstack the solution and decompose into rotation and translation • Factor the into and using QR decomposition CS223b

  10. Calibration with a planar pattern To eliminate unknown depth, multiply both sides by CS223b

  11. Calibration with a planar pattern Because are orthogonal and unit norm vectors of rotation matrix We get the following two constraints • We want to recover S • Unknowns in K (S) Skew is often close 0 -> 4 unknowns • S is symmetric matrix (6 unknowns) in general we need at least 3 views • To recover S (2 constraints per view) - S can be recovered linearly • Get K by Cholesky decomposition of directly from entries of S CS223b

  12. Alternative camera models/projections Orthographic projection Scaled orthographic projection Affine camera model CS223b

  13. Barrel and Pincushion Distortion wideangle tele CS223b

  14. Models of Radial Distortion distance from center CS223b

  15. Tangential Distortion cheap CMOS chip cheap lens image cheap glue cheap camera CS223b

  16. Barrel distortion CS223b

  17. Distorted Camera Calibration • Set k1=k2=0, solve for undistorted case • Find optimal k1,k2 via nonlinear least squares • Iterate Tends to generate good calibrations CS223b

  18. Calibration Software: Matlab CS223b

  19. Calibration Software: OpenCV CS223b

  20. Calibration by nonlinear Least Squares • Least Mean Square • Gradient descent: CS223b

  21. The Calibration Problem Quiz • Given • Calibration pattern with N corners • K views of this calibration pattern • How large would N and K have to be? • Can we recover all intrinsic parameters? NO CS223b

  22. Constraints • N points • K images  2NK constraints • 4 intrinsics (distortion: +2) • 6K extrinsics  need 2NK≥ 6K+4  (N-3)K ≥ 2 Hint: may not be co-linear CS223b

  23. The Calibration Problem Quiz need (N-3)K ≥ 2 Hint: may not be co-linear CS223b

  24. Problem with Least Squares • Many parameters (=slow) • Many local minima! (=slower) CS223b

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