A synthetic camera model to test calibration procedures A four step procedure (last slide) based on an initial position (LookAt) and 13 parameters: (  x,  y  z ) – local Euler rotations (dt x dt y dt z ) – local displacement (f x f y s h  1  2 c x c y ) – camera internal parameters

ococ at c up c Camera initial extrinsic parameters: Initial extrinsic parameters: user input at c ococ up c xwxw ywyw zwzw The camera must start in a given position (not too far from the final solution!). The module must have an initialization procedure.

ococ Initial extrinsic parameters: first transformation xwxw ywyw zwzw Notation: origin system vector

Incremental rotation with Euler angles:  x  y  z xx x y z yy x y z zz x y z where: c y = cos  y, s x = sin  x, etc.. for small  ’s:

Incremental translation: dt x dt y dt z ococ t t0t0 xwxw ywyw zwzw dt

Transformation from global to local system xcxc ycyc zczc q y cd x cd next step: camera projection

Projection into pixels sxsx sysy pixel dimensions inclined projections where: xdxd ydyd projected point written in the coordinate system (x d,y d ) [in pixels] center of distortion axis in the middle of the image

Taking radial deformation into account xdxd ydyd Note that  1 and  2 are not the same as Tsai’s k 1 and k 2. The map shown here is in the inverse direction.

Changing the origin of the image x im y im xdxd ydyd xdxd ydyd

Putting all together: (x w, y w, z w ) T  (x c, y c, z c ) T  (x cd,y cd ) T  (x d, y d ) T  (x im,y im ) T Given and initial position given by L at and: (  x,  y  z dt x dt y dt z f x f y s h  1  2 c x c y ) we can compute x im and y im by: (step 1) (step 2) (step 3) (step 4)