Given the fact that the displacement vector is usually a small value (normally several pixels), Eq. (2) can be approximated using first-order Taylor expansion with the higher-order terms ignored as follows: Display Formula
(3)With two unknowns, and , in one equation, the linear least squares (LS) estimator minimizes the square errors: Display Formula
(4)As a linear LS problem, the minimum of can be found by setting its derivatives with respect to are zero: Display Formula
(5)Equation (5) can be written in matrix form Display Formula
(6)in which is the difference matrix and denotes the gradient matrix Display Formula
(7)where refers to the number of pixels in selected templates.