Camera Model Calibration Robot Vision Systems Camera Model Calibration

“perspectograph” Alberti’s Grid

Pinhole Camera scene image plane iris, optical center

Pinhole Camera “image coordinates”

Pinhole Camera “Alberti’s Grid”

Pinhole Camera Classical pin-hole x f r’ z

Classical Pinhole Camera Similar Triangles x f c’ z

Classical Pinhole Camera Similar Triangles x f c’ z Image coordinates Point expressed in the camera frame Camera matrix (projection) Projective coordinates

Classical Pinhole Camera Similar Triangles x f Convert pixels to mm: c’ z world

Camera Calibration Used to determine the elements in the camera matrix Use pairs of known world points and their corresponding image points e.g. use calibration grid

Camera Model – Perspective Projection Need to determine the parameters!

Camera Calibration camera matrix

Camera Calibration Projective Equivalence Two equations in 12 unknowns

Camera Calibration Have 6 point pairs (c,r) and (x,y,z) Correspondence known World coordinates known accurately

Camera Calibration Method 1: assume a34 = 0 and solve for aij Solve using pseudo-inverse

Camera Calibration Method 2: make no assumption about a34 and solve for aij The vector a = [a11, …, a34]T is in the nullspace of the design matrix

Camera Calibration Ba = 0, Use SVD, then a is the column of V corresponding to the null singular value of B One property of the SVD is that the columns of V corresponding to the zero singular value span the null space of B The vector a = [a11, …, a34]T is in the nullspace of the design matrix

Camera Calibration Given the camera matrix solved w.r.t. the robot base frame

Using the Camera Matrix Projection ( is 3 x 4)– cannot invert!

Measurements from Images Must have relationship between the image “pixels” and the world 2D imaging the image plane and the “world” plane are in 1-1 correspondence

Using the Camera Matrix in 2D If all world points are on a plane Then z is a linear function of x & y

Using the Camera Matrix in 2D Now the projection equations Can be written

Using the Camera Matrix in 2D Now the projection equations Can be written

Using the Camera Matrix in 2D Now the projection equations Can be written 2 equations, 9 unknowns

Using the Camera Matrix in 2D Four known points, i=1,..4 Linear equations -- can be solved 8 equations, 9 unknowns up to a scale (8 unknowns)

Using the Camera Matrix in 2D

Using the Camera Matrix in 2D Now the projection equations are simpler … and we can “invert” (map image points back to the world plane)

Measurements from Images

Robot to Plane Homogeneous transformation from base (or end-effector) frame to the work-plane of the imaging system Use “known” corresponding points to solve for the elements of T (6 unknowns)