EECS 274 Computer Vision Geometric Camera Calibration Geometric

EECS 274 Computer Vision Geometric Camera Calibration

Geometric camera calibration • • Camera calibration problem Least-squares techniques Linear calibration from points Analytical photogrammetry • Reading: Chapter 1 and 22 of FP, Chapters 2, 6 of S

Calibration • Determine the intrinsic and extrinsic parameters • Assume that the camera observes a set of features (points, or lines) with known positions • Calibration: modeled as an optimization to minimize the discrepancy between the observed image features and their theoretical projections (using the perspective projection equations)

Calibration problem Given n points, P 1, …, Pn with known positions and their images points, p 1, …, pn, find ξ

Linear systems Square system: A x = b • unique solution • Gaussian elimination Rectangular system ? ? A x = b • underconstrained: infinity of solutions • overconstrained: no solution Minimize ||Ax-b||2

Overconstrained problems

In matrix form Can be derived from the perspective of projection matrix

Homogenous linear systems Square system: A x = 0 • unique solution: 0 • unless Det(A)=0 Rectangular system ? ? • 0 is always a solution A x = 0 2 Minimize |Ax| 2 under the constraint |x| =1

Overconstrained homogenous linear systems The solution is e 1.

Example: linear fitting Problem: minimize with respect to (a, b, d). • Minimize E with respect to d: • Minimize E with respect to a, b: where • Solution is the unit eigenvector with minimum eigenvalue

Note • Matrix of second moments of inertia • Axis of least inertia in mechanics

Linear camera calibration

When M is known Once M is known, need to recover the intrinsic and extrinsic parameters This is a decomposition problem, not an estimation problem r ρ: scale factor • Intrinsic parameters • Extrinsic parameters

Decomposition of M As the recovered Orthonormal basis vector θ is close to π/2 and has positive sine

Degenerate point configuration Are there other solutions besides M ? • One solution: (l, m, n )=(m 1, m 2, m 3) • Consider the points Pi all lie in some plane, s. t. , P∙Pi=0 for some P • Coplanar points: choose (l, m, n )=(P, 0, 0) or (0, P, 0) or (0, 0, P ), or any linear combination of these vectors yields a solution Does not (usually) happen for 6 or more random points!

Radial distortion • Depends on the distance between the image center and an image point, d Barrel distortion Corners are detected by fitting lines in each square Using estimated distortion parameters

Correct radial distortion 11+q parameters • Tsai’s algorithm (1987) exploits radial alignment constraints for estimating extrinsic parameters

Analytic photogrammetry Given n points, P 1, …, Pn with known positions and their images situations, p 1, …, pn, find ξ Non-Linear Least-Squares Methods • Newton • Gauss-Newton • Levenberg-Marquardt Iterative, quadratically convergent in favorable situations

Application Mobile Robot Localization (Devy et al. , 1997)

Calibration • Numerous ways that exploits properties of projective geometry – e. g. calibration using lines, calibration circular controlled points

Camera calibration toolbox • Excellent MATLAB toolbox by Jean-Yves Bouguet http: //www. vision. caltech. edu/bouguetj/calib_doc/ • Steps: – Generate calibration board – Collect images under different views – Select extreme points – Find corner points – Solve optimization problem

Calibration images

Extreme points

Guessed grid corners

Corner extraction

Repeat for all other images

Solving optimization problem

Reprojected corners

Camera centered view

World centered view

Applications • • Augmented reality Image registration Image stitching Panoramic image

Panoramic image

Notes • • Camera pose estimation Multi-camera calibration Auto/self calibration Multi-camera self calibration Projective geometry Multi-view geometry RANSAC (RANdom Sample Consensus)