1. Camera Calibration from Planar Patterns Homework 2 Help SessionCS223bStanford University Mitul Saha (courtesy: Jean-Yves Bouguet, Intel)

2. Camera Calibration Object Space Image Space x c y c M m m = [ Camera Projection Matrix ] M A [R t] camera intrinsics extrinsics f x f y f x alpha* o x o y 0 001

3. Camera Calibration Object Space Image Space x c y c M m m = [ Camera Projection Matrix ] M A [R t] camera intrinsics extrinsics f x f y f x alpha* o x o y 0 001 Camera calibration is about finding the camera intrinsics But, why do we need them?

4. Camera Calibration Common approach Planar pattern Non-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] observed estimate: A [R t] M Minimize:

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] observed estimate: A [R t] M Minimize: Two steps: Find an initial solution for A [R t] Minimize the objective function using the initial solution

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

8. 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 V T V with smallest eigenvalue V = B = A –T A -1

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: R T R=RR T =I. One can it enforce by: min||R new - R||, [U D V] = SVD(R), R new = UV T V b = 0

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