PC_Perspective(pcpatch/pcpoint, float[16]) : pcpatch/pcpoint and specializations

[mbredif]

This is the most general perspective transform with 16 parameters.

Interesting specialization could be implemented in C or SQL :

• PC_ProjectPinhole

fx, s, cx, 0
0, fy, cy, 0
0, 0, 0, 1
0, 0, 1, 0

fx and fy are the focal lengths expressed in horizontal and vertical pixel sizes, cx and cy are the pixel coordinates of the optical axis, s is the amount of shearing). Usually, fx=fy, (cx,cy) is close to the image center and s=0. It takes a metric 3D point where the x axis is along the image rows, the y axis along the image columns and the z axis along the optical axis, and outputs a point (c,r,d) where (c,r) are the column and row coordinates in pixels and d is 1/depth along the optical axis (d=0 for a point at infinity and d>0 for points in front of the camera

• PC_UnprojectPinhole (the inverse of the above)

fy, -s, 0, cy.s-cx.fy
0, fx, 0, -cy.fx
0, 0, 0, fx.fy
0, 0, fx.fy, 0

• PC_Scale

sx, 0, 0, 0
0, sy, 0, 0
0, 0, sz, 0
0, 0, 0, 1

• PC_TransScale

sx, 0, 0, tx
0, sy, 0, ty
0, 0, sz, tz
0, 0, 0, 1

• PC_RotateX
• PC_RotateY
• PC_RotateZ
• PC_RotateQuaternionUnnormalized (works even if quaternion (x,y,z,w) is not normalized)

 x*x-y*y-z*z+w*w, 2*x*y-2*z*w, 2*x*z+2*y*w, 0
2*x*y+2*z*w, -x*x+y*y-z*z+w*w, 2*y*z-2*x*w, 0
2*x*z-2*y*w, 2*y*z+2*x*w,-x*x-y*y+z*z+w*w, 0
0,0,0,x*x+y*y+z*z+w*w