Camera Calibration Coordinate change Translation P’ = P – O Rotation P’ = R P General transformation P’ = R[P - O] What is a rotation matrix

Perspective Projection Projection to normal coordinates A line from (0,0,0) to P Z=1 plane: x = X/Z , y = Y/Z p  P Camera as a box. A line from (0,0,0) to P Z=1 plane: x = X/Z , y = Y/Z Lenses

From normal coordinates to pixels If focal length is f : x = fX/Z y = fY/Z

Internal calibration K External calibration O & R Calibration options: Full calibration M Internal calibration K External calibration O & R

Calibration Use linear methods to recover M Extract from them K, R , O What to do about Radial distortion

Radial Distortion

Radial distortion Define d as the distance from the center of the image on the normalized image plane

Radial distortion Solve the whole thing in a non-linear fashion. Bad idea Assume u0 and v0 are known (center of image). Try to get rid of . Solve what you can Return to solve the rest in a non-linear fashion.

Calibration from checkers pattern

Calibration from checkers pattern

Planes Planes have a one-to-one relationship from the world to the image Planes have a one-to-one relationship from two images of the same plane

Vanishing points

Vanishing points

Special cases of  If s = 0 then w(1,2)-w(2,1)=0. If fx=fy then w(1,1)=w(2,2). Need less equations. Only two homographies.

Points and Lines

Points and Lines

3 Points on a line

The fundamental matrix