Camera Calibration from Planar Patterns

1. Camera Calibration fromPlanar Patterns Homework 2 Help Session (courtesy: Jean-Yves Bouguet, Intel) Mitul Saha CS223b Stanford University

2. c c y x alpha* f f o o f y x x x y 0 0 0 1 Camera Calibration Object Space Image Space M m m= [Camera Projection Matrix] M A [R t] extrinsics camera intrinsics

3. c c y x alpha* o f f f o x y x y x 0 0 0 1 Camera Calibration Object Space Image Space M m m= [Camera Projection Matrix] M • Camera calibration is about • finding the camera intrinsics • But, why do we need them? A [R t] extrinsics camera intrinsics

4. Camera Calibration • Common approach Non-planar pattern Planar pattern

5. Camera Calibration from Planar Patterns • ICCV Zhang’99: “Flexible Calibration by Viewing a Plane From Unknown Orientations” m= [Camera Projection Matrix] M A [R t] Minimize: estimate: A [R t] M observed

6. Camera Calibration from Planar Patterns • ICCV Zhang’99: “Flexible Calibration by Viewing a Plane From Unknown Orientations” m= [Camera Projection Matrix] M A [R t] • Two steps: • Find an initial solution • for A [R t] • Minimize the objective function • using the initial solution Minimize: estimate: A [R t] M observed

7. [u, v, 1]T Camera Calibration from Planar Patterns • Finding an initial solution • First step • Estimate the image homography matrix H for each image Minimize: Initial solution for minimization: L x is the eigenvector of LTL with smallest eigenvalue

8. V = B = A –T A -1 Camera Calibration from Planar Patterns • Finding an initial solution • First step • Estimate the image homography matrix H for each image • Second step • Solve for b in the linear system: V b = 0 b is the eigenvector of VTV with smallest eigenvalue

9. Camera Calibration from Planar Patterns • Finding an initial solution • First step • Estimate the image homography matrix H for each image • Second step • Solve for b in the linear system: • b yields the intrinsic parameter matrix A. Rotation matrix [r1 r2 r3] and translation t is computed from: V b = 0

10. Camera Calibration from Planar Patterns • Finding an initial solution • First step • Estimate the image homography matrix H for each image • Second step • Solve for b in the linear system: • b yields the intrinsic parameter matrix A. Rotation matrix [r1 r2 r3] and translation t: • But the computed rotation matrix does not satisfy the properties of rotation matrix: RTR=RRT=I. One can it enforce by: min||Rnew - R||, [U D V] = SVD(R), Rnew = UVT V b = 0

11. Camera Calibration from Planar Patterns m= [Camera Projection Matrix] M A [R t] • Two steps: • Find an initial solution • for A [R t] • Minimize the objective function • using the initial solution Minimize: use “lsqnonlin” in Matlab estimate: A [R t] M observed