given $\theta_1$ and $\theta_2$ as affine transformations for $I_1$ and $I_2$
$$
\theta_1 = \left(\begin{array}{ccc}r_{0,1}&r_{1,1}&t_{0,1}\r_{2,1}&r_{3,1}&t_{1,1}\end{array} \right) = \left( R_1 | T_1 \right)
$$
$$
\theta_2 = \left(\begin{array}{ccc}r_{0,2}&r_{1,2}&t_{0,2}\r_{2,2}&r_{3,2}&t_{1,2}\end{array} \right) = \left( R_2 | T_2 \right)
$$
$$
\theta_2^{-1}= \left( R_2^{-1} | -R_2^{-1}T_2 \right)
$$
Optical Flow is given by the function $F$ such as
$$
F(x,y) \in \mathbb R^2 ,~ I_1 \left(\begin{array}{c}
x\y
\end{array}\right) = I_2\left(\left(\begin{array}{c}
x\y
\end{array}\right) + F\right)
$$
we want to go from $(x,y)$ in $I'_1$ to $(x_3,y_3)$ in $I'_2$. Main idea is to go from warped $I'_1$ to unwarped $I_1$ to unwarped $I_2$ (through $F$) to warped $I'_2$ .
NB : $(x,y)$ is homogeneous, obtained from $(u-u_0, v-v_0)$
To get corresponding $(x_1,y_1)$ in unwarped image, we apply
$$
\left(\begin{array}{c}
x_1\y_1
\end{array}\right) = R_1\left(\begin{array}{c}x\y\end{array}\right) + T_1
$$
$(x_2,y_2)$ in unwarped $I_2$, is obtained with $F$, evaluated on $(x_1, y_1)$ (named $F_1$)
$$
\left(\begin{array}{c}
x_2\y_2
\end{array}\right) = \left(\begin{array}{c}
x_1\y_1
\end{array}\right) + F \left(\begin{array}{c}
x_1\y_1
\end{array}\right) = R_1\left(\begin{array}{c}x\y\end{array}\right) + T_1 + F\left(R_1\left(\begin{array}{c}x\y\end{array}\right) + T_1 \right) = R_1\left(\begin{array}{c}x\y\end{array}\right) + T_1 + F_1
$$
$(x_3,y_3)$ is then obtained from $(x_2,y_2)$ with
$$
\left(\begin{array}{c}
x_2\y_2
\end{array}\right) = R_2\left(\begin{array}{c}x_3\y_3\end{array}\right) + T_2
$$
$$
\left(\begin{array}{c}
x_3\y_3
\end{array}\right) = R_2^{-1}\left(\begin{array}{c}x_2\y_2\end{array}\right) - R_2^{-1}T_2
$$
$$
\left(\begin{array}{c}
x_3\y_3
\end{array}\right) = R_2^{-1}\left(R_1\left(\begin{array}{c}x\y\end{array}\right) + T_1 + F_1\right) -R_2^{-1}T_2
$$
$$
\left(\begin{array}{c}
x_3\y_3
\end{array}\right) = R_2^{-1}R_1\left(\begin{array}{c}x\y\end{array}\right) + R_2^{-1}(T_1-T_2) + R_2^{-1}F_1
$$
$$
F_{final} = \left(\begin{array}{c}
x_3\y_3
\end{array}\right) - \left(\begin{array}{c}
x\y
\end{array}\right) = (R_2^{-1}R_1 - I_2)\left(\begin{array}{c}x\y\end{array}\right) + R_2^{-1}(T_1-T_2) + R_2^{-1}F_1
$$
