Camera Calibration from Planar Patterns courtesy JeanYves Bouguet

Camera Calibration from Planar Patterns (courtesy: Jean-Yves Bouguet, Intel) Mitul Saha Homework 2 Help Session CS 223 b Stanford University

Camera Calibration Object Space Image Space M yc m xc m = [Camera Projection Matrix] M fx alpha* f x ox 0 fy oy 0 0 1 camera intrinsics A [R t] extrinsics

Camera Calibration Object Space Image Space M yc m xc m = [Camera Projection Matrix] M fx alpha* f x ox 0 fy oy 0 0 1 camera intrinsics A [R t] extrinsics • Camera calibration is about finding the camera intrinsics • But, why do we need them?

Camera Calibration • Common approach Non-planar pattern Planar pattern

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: observed estimate: A [R t] M

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

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

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 V= B = A –T A -1 b is the eigenvector of VTV with smallest eigenvalue

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 yields the intrinsic parameter matrix A. Rotation matrix [r 1 r 2 r 3] and translation t is computed from:

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 yields the intrinsic parameter matrix A. Rotation matrix [r 1 r 2 r 3] 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

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