(a) Show that the magnitude of the gradient given in Eq. (3.6-11) is an isotropic operation. (See Problem 3.24.)
(3.6-11)
3.24 Show that the Laplacian defined in Eq. (3.6-3) is isotropic (invariant to rotation). You will need the following equations relating coordinates for axis rotation by an angle θ:
where (x,y) are the unrotated and (x′, y′) are the rotated coordinates.
(b) Show that the isotropic property is lost in general if the gradient is computed using Eq. (3.6-12).
(3.6-12)
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