# Stereo Rectification when Extrinsics are known

Suppose I have two cameras $$C_1$$ and $$C_2$$ arranged in a stereo configuration. Let us assume that the camera is free of distortion with ideal intrinsics and that the camera extrinsics $$R$$ and $$t$$ are known. Let $$C_1$$ be the reference for the extrinsics of the system (i.e $$R$$ and $$t$$ are defined w.r.t to $$C_1$$). The two cameras have different focal lengths $$f_{c_1}$$ and $$f_{c_2}$$

Now, in order to calculate disparity given the images $$I_1$$ and $$I_2$$ of the cameras $$C_1$$ and $$C_2$$, it is necessary that the images need to be rectified.

I have understood rectification as making the the images captured lie in the same plane in space. The baseline of the stereo $$b$$ is equal to the norm of the translation vector $$t$$. Let ($$x_2$$, $$y_2$$) be a pixel in the Image $$I_2$$ and ($$x_1$$, $$y_1$$) a pixel in $$I_1$$. I would like to rectify Image $$I_1$$ such that the it lies in the same plane as $$I_2$$.

Let $$P_1 = \begin{pmatrix}{x_1} \\{y_1} \\f_{c_1}\end{pmatrix}$$ and $$P_2 = \begin{pmatrix}{x_2} \\{y_2} \\f_{c_2}\end{pmatrix}$$ Would it be correct to say a pixel $$P_2$$ in $$I_2$$ is given by:

$$P_2 = \frac{f_{c_2}} {f_{c_1}}.(R.P_1 + t)$$

where $$P_1$$ is a pixel in $$I_1$$? We could then extend this and write $$P_1 = R^T.(\frac{f_{c_1}} {f_{c_2}}P_2 - t)$$ and that $$P_{1_{rectified}} = R^T.(\frac{f_{c_1}} {f_{c_2}}P_2 - t) + t$$ since $$t$$ defines the baseline in the stereo system

Would the difference between $$P_{1_{rectified}}$$ and $$P_2$$ for all points $$(x2, y2)$$ in $$P_2$$ give me disparity? If not what would be the correct way to calculate disparity given the camera extrinsics and two cameras of different focal lengths. I have calculated the above expressions from my understanding of the geometry of the system. Please correct me if there are errors.