The real voyage of discovery consists not in seeking new landscapes, but in having new eyes.

Transcription

1 The real voyage of discovery consists not in seeking new landscapes, but in having new eyes. - Marcel Proust

2 University of Texas at Arlington Camera Calibration (or Resectioning) CSE Vision-based Robot Sensing, Localization and Control Dr. Gian Luca Mariottini, Ph.D. Department of Computer Science and Engineering University of Texas at Arlington WEB :

3 Pinhole Camera Calibration: Definition We derived the generalized pinhole camera model Intrinsic parameters Extrinsic parameters Our goal => Pinhole Camera Calibration Definition (Pinhole Camera Calibration): A process of determining the intrinsic and the extrinsic parameters of a static camera (or a group of cameras). Intrinsic parameters: Geometric and optical characteristics of the camera; Extrinsic parameters: 3-D pose (position and orientation) of the camera with respect to the world frame.

4 Why is Camera Calibration important? 3-D world coordinates Image coordinates Image coordinates 3-D world coordinates Multi-camera Relative Pose (N.Navab) Multi-cameras wrt the 3-D field Single camera wrt 3-D object

5 The Camera Calibration Setup The pinhole-camera model is: By taking the cross-product by we obtain: known unknown This motivates the typical setup for a camera calibration scenario (aka resection): {C} {W}

6 Resectioning: Linear-form Expression (1) We obtained an expression linear in We want to express it as a linear function of the unknown : where is the i-th row of...

7 Resectioning: Linear-form Expression (2) We note that each point has 2 coordinates (in pixels), so we can only use the first two equations!...so we can finally write: 2x12 12x1 2x1 If we have n 3-D to 2-D correspondences, we can write it as: 2nx12 2nx1 (i) Our goal then reduced to finding the camera parameters in homogeneous linear expression in (i) that verify the

8 Homogeneous Linear Equations: Existence of Solution (1) Let us consider two lines in 2-D: 2x3 3x1 Does (ii) always admit a solution? Homogeneous Linear Equation - (ii) - we are not interested in the trivial solution so we need to add a constraint: - In the above case (#rows=#cols-1) a unique solution is found! - If #rows of is equal #columns of, and these rows are linearly dependent, then a unique solution is still found! It corresponds to adding the line: linearly dep. (try it!) that still intersects all the other lines at

9 Homogeneous Linear Equations: Existence of Solution (2) - if #rows of is equal #columns of, and these rows are linearly independent, then no solution to is found! What if we want just to make as small as possible? These 3 rows are all linearly independent! The 3 rd row corresponds to a line that does not pass through the same point (1,1). Let us recap: Given a homogeneous linear system, we want to find the solution s.t. - If #rows=#cols-1 then a unique solution exists. - If #rows=#cols, and these rows are linearly dependent, then a unique solution exists. - if #rows=#cols, and these rows are linearly independent, then no solution exists.

10 Least Squares Solution for Homogeneous Linear Equations For ease of notation let us rewrite the Least Squares cost function to be minimized as: Let us note that is always possible to decompose a matrix via Singular Value Decomposition (SVD) (ii) where: + is an mxn matrix; + is mxn,while is nxn; + and are orthonormal, i.e., ; + is a diagonal matrix with nonnegative real numbers (singular values) on the diagonal, in decreasing order Try in MATLAB (see example Ex_SVD.m) Substituting (ii) into (i) yields: (i) and, if we call, then the original minimization problem becomes: Last column of

11 Resectioning: Solution (1) We want to solve: in which: According to what we have found for the solution of homogeneous linear equations: 1) Compute the SVD of matrix 2) The SVD will return three matrices,, and 3) The solution can be computed as: (see example Ex_LeastSquares.m ) The vector will contain the rows of the projection matrix (see example Ex_CameraCalibration.m )

12 Once an estimate of be computed. Notice that: Resectioning: Solution (2) is obtained, the intrinsic and extrinsic calibration need to so that an estimate of the translation vector (from {W} to {C}) is given by: - An estimate of the rotation (from {W} to {C}) and of the intrinsic calibration are given by: Orthonormal Upper Triangular