An edge detection method starts by finding the brightness gradient (Ex, Ey) at each picture cell using some local estimator of the partial derivatives of image brightness E (x, y). The magnitude of the brightness gradient is then computed. Next, “non-maximum suppression’’ keeps for further consideration only pixels where the magnitude of the gradient is a local maximum in the direction of the local brightness gradient. The maximum of the gradient need, of course, not fall exactly on a discrete pixel location, when the gradient magnitude is considered as a continuous function of position, Edge locations can be estimated to finer than pixel resolution if this continuous function of image position is first estimated from the discrete samples of the magnitude of the brightness gradient at pixels. We can approximate the magnitude of the gradient locally as a power series 2 2 M(x, y) =ax +bxy +cy +dx +ey +f where (x, y) is the displacement from the center of a 3 ×3 region of pixels. Show that the parameters a, b, c, d, e, and f can be estimated from M, Mx, My, Mxx, Mxy and Myy evaluated at the origin i.e. at the center pixel. Mx is the derivative of M (X, Y) along x direction, etc.