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ds6 - ECE 178 Digital Image Processing Discussion Session#6...

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ECE 178 Digital Image Processing Discussion Session #6 Anindya Sarkar & Mehmet Emre Sargin { anindya,msargin } @ece.ucsb.edu February 16, 2007 Problem 3-25 Gonzalez & Woods: The Laplacian operator is defined as: 2 f = 2 f ∂x 2 + 2 f ∂y 2 (1) for the unrotated coordinates and as 2 f = 2 f ∂x 0 2 + 2 f ∂y 0 2 (2) for rotated coordinates. It is given that x = x 0 cos θ - y 0 sin θ and x = x 0 sin θ + y 0 cos θ (3) where θ is the angle of rotation. We want to show that the right sides of the first two equations are equal. We start with: ∂f ∂x 0 = ∂f ∂x ∂x ∂x 0 + ∂f ∂y ∂y ∂x 0 (4) = ∂f ∂x cos θ + ∂f ∂y sin θ (5) Taking the partial derivative of this expression again with respect to x 0 yields 2 f ∂x 0 2 = 2 f ∂x 2 cos 2 θ + ∂x ∂f ∂y sin θ cos θ + ∂y ∂f ∂x cos θ sin θ + 2 f ∂y 2 sin 2 θ (6) Next, we compute ∂f ∂y 0 = ∂f ∂x ∂x ∂y 0 + ∂f ∂y ∂y ∂y 0 (7) = - ∂f ∂x sin θ + ∂f ∂y cos θ (8) Taking the derivative of this expression again with respect to y 0 gives
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