Geometric Motion Constraints

Recall brightness constancy equation: $$I_x \cdot u + I_y \cdot v + I_t = 0 \\ $$ Substitute affine transformation: $$I_x (a_1 + a_2x +a_3y) + I_y (a_4 + a_5x +a_6y) + I_t = 0$$ Use least square minimization: $$Err(\overrightarrow a) = \sum [I_x (a_1 + a_2x +a_3y) + I_y (a_4 + a_5x +a_6y) + I_t]^2$$ Minimize squared error: