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The Laplace operator is a second-order differential operator in the "n" -dimensional Euclidean space, defined as the divergence of the gradient (Vf). Thus if f is a twice-differentiable real-valued function; then the Laplacian of f is defined by: Af = V2f = V . Vf where the latter notations derive from formally writing: 623' Dx Now, consider a 2D image I(z,y) and its Laplacian, given by Al = I II +Iyy: Here the second partial derivatives are taken with respect to the directions of the variables €, y associated with the image grid for convenience. Show that the Laplacian is in fact rotation invariant_ In other words, show that AI Irr + Iyv' , where r and are any two orthogonal directions Hint: Start by using polar coordinates to describe a chosen location (x, Y) Then use the chain rule_

          The Laplace operator is a second-order differential operator in the "n" -dimensional Euclidean space, defined as the divergence of the gradient (Vf). Thus if f is a twice-differentiable real-valued function; then the Laplacian of f is defined by:
Af = V2f = V . Vf
where the latter notations derive from formally writing: 623' Dx Now, consider a 2D image I(z,y) and its Laplacian, given by Al = I II +Iyy: Here the second partial derivatives are taken with respect to the directions of the variables €, y associated with the image grid for convenience.  Show that the Laplacian is in fact rotation invariant_ In other words, show that AI Irr + Iyv' , where r and are any two orthogonal directions Hint: Start by using polar coordinates to describe a chosen location (x, Y) Then use the chain rule_

the laplace operator is a second order differential operator in the n dimensional euclidean space defined as the divergence of the gradient vf thus if f is a twice differentiable real valued 07089

Added by Pamela V.

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