Uncalibrated Epipolar - Calibration Jana Kosecka CS223b

Uncalibrated Camera calibrated coordinates Linear transformation pixel CS223b

Overview Calibration with a rig Uncalibrated epipolar geometry CS223b

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

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

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

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

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

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 Solve for translation CS223b

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

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

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

Barrel and Pincushion Distortion wideangle tele CS223b

Models of Radial Distortion distance from center CS223b

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

Barrel distortion CS223b

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

Calibration Software: Matlab CS223b

Calibration Software: OpenCV CS223b

Calibration by nonlinear Least Squares Least Mean Square Gradient descent: CS223b

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 N 1 3 4 6 K CS223b

Hint: may not be co-linear 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

The Calibration Problem Quiz 1 3 4 6 K No Yes need (N-3)K ≥ 2 Hint: may not be co-linear CS223b

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