Shelf coordinates:
Extrinsic Transform:
R=1000cos(θ)−sin(θ)0sin(θ)cos(θ)
R =
\begin{bmatrix}
1 & 0 & 0\\
0 & \cos{(\theta)} & \sin{(\theta)}\\
0 & -\sin{(\theta)} & \cos{(\theta)}
\end{bmatrix}
t=0−Hccos(θ)Hcsin(θ)
t = \begin{bmatrix}
0\\
- H_c \cos{(\theta)}\\
H_c \sin{(\theta)}
\end{bmatrix}
For p=[x,y,z] we have on the camera coordinates:
Rp+t=[x,cos(θ)(y−Hc)+sin(θ)z,sin(θ)(Hc−y)+cos(θ)z]T
Rp + t = \left[x, \cos{(\theta)} (y-H_c) + \sin{(\theta)} z, \sin{(\theta)} (H_c-y) + \cos{(\theta)} z \right]^T
Camera intrinsic parameters:
K=fx000fy0cxcy1
K =
\begin{bmatrix}
f_x & 0 & c_x\\
0 & f_y & c_y\\
0 & 0 & 1
\end{bmatrix}
- fx and fy are focal length for each direction.
- (cx,cy) are the center of the image plane.
If pc=Rp+t=:[xc,yc,zc]T then the point in the image plane is:
u=[fxzcxc+cx,fyzcyc+cy]
u = \left[f_x \frac{x_c}{z_c} + c_x, f_y \frac{y_c}{z_c} + c_y\right]
Suppose ptf and pbf are the top and bottom points for the first row object.
- They have the same x and z coordinates;
- ybf=0,ytf=Ho= "height of the object";
- Call z=d= "distance between camera and the first object.
The distance on the y coordinate in the image plane is given by:
Δuy=fy(dcos(θ)+sin(θ)(Hc−Ho)dsin(θ)−cos(θ)(Hc−Ho)−dcos(θ)+sin(θ)Hcdsin(θ)−cos(θ)Hc)
\Delta u_y = f_y \left( \frac{ d \sin{(\theta)} - \cos{(\theta)}(H_c - H_o) }{ d\cos{(\theta)} + \sin{(\theta)}(H_c - H_o)}
- \frac{ d \sin{(\theta)} - \cos{(\theta)}H_c }{ d\cos{(\theta)} + \sin{(\theta)}H_c}
\right)
=sin(θ)2Hc(Hc−Ho)+sin(θ)cos(θ)d(2Hc−Ho)+cos(θ)2d2fyHod
= \frac{f_y H_o d}{\sin{(\theta)}^2 H_c (H_c- H_o) + \sin{(\theta)}\cos{(\theta)}d(2H_c - H_o) + \cos{(\theta)}^2 d^2}
Case 1: first row
Suppose pf and pb are the front and back points for the last row object.
- They have the same x and y coordinates;
- yb=yf=Ho= "height of the object";
- Call zf=D "distance between camera and the last object, and zb=D+Wo (from width).
The distance on the y coordinate in the image plane is given by:
Δuy=fy((D+Wo)cos(θ)+sin(θ)(Hc−Ho)(D+Wo)sin(θ)−cos(θ)(Hc−Ho)−Dcos(θ)+sin(θ)(Hc−Ho)Dsin(θ)−cos(θ)(Hc−Ho))
\Delta u_y = f_y \left( \frac{ (D+W_o) \sin{(\theta)} - \cos{(\theta)}(H_c - H_o) }{ (D+W_o)\cos{(\theta)} + \sin{(\theta)}(H_c - H_o)}
- \frac{ D \sin{(\theta)} - \cos{(\theta)}(H_c - H_o) }{ D\cos{(\theta)} + \sin{(\theta)}(H_c - H_o)}
\right)
=sin(θ)2(Hc−Ho)2+sin(θ)cos(θ)(2D+Wo)(Hc−Ho)+cos(θ)2D(D+Wo)fy(Hc−Ho)Wo
= \frac{f_y (H_c - H_o) W_o}{\sin{(\theta)}^2 (H_c- H_o)^2 + \sin{(\theta)}\cos{(\theta)}(2D+W_o)(H_c - H_o) + \cos{(\theta)}^2 D(D+W_o)}
Case 2: last row (no inclination)
Shelf coordinates: