CAP5416: Assignment #2, Due Date:Oct. 8, 2019 Make sure that you writing is legible, or else, type your answers using your favourite text formatter. You should turn in your hwk. on the due date in class before the class begins. Please staple your answer sheets prior to submission. Its your responsibility to staple them together otherwise they may be lost! Use the definition of Fourier transform with the 2pi in the exponent for the first two problems. 1. Let Π(x, y) denote the rectangle function defined as, Π(x, y) = 1 for |x| < 1/2, |y| < 1/2 = 0 Otherwise Derive the Fourier transform of Π(x−bc , y). Note that the FT of Π(x, y) = Π(x)Π(y) is sinc(x, y) = sinc(x)sinc(y), and sinc(x) = sin(pix)pix , for x 6= 0. 2. Prove the rotation theorem for Fourier transforms i.e., prove that f(xcosθ − ysinθ, xsinθ + ycosθ)↔ F (ucosθ − vsinθ, usinθ + vcosθ) The double arrow denotes the Fourier transform pair. 3. Consider the discrete approximation of the Laplacian given by the following convolution mask: 1 62 1 4 1 4 −20 4 1 4 1 Where, is the spacing between pixels. Write down this weighting scheme (mask) as a sum of nine impulse functions. Find the Fourier transform of this mask and evaluate it as → 0. 4. Compute the following convolutions and write the solution in matrix form. 1 4 6 4 1 4 16 24 16 4 6 24 36 24 6 4 16 24 16 4 1 4 6 4 1 ∗ [ 1 −1 ] 3 2 32 3 2 1 2 3 ∗ [ 1 0 0 −1 ] 5. One way to find edges in an image, I(x, y), is to find the zero-crossings in the Laplacian of the image function. It however suffers from a poor signal to noise performance. One might instead use the second derivative of brightness in the direction of the brightness gradient. (a) Given a unit vector in the direction of the brightness gradient, find the sine and cosine of the angle θ between thsi vector and the x-axis. Hint: depeding on your approach to this problem, you might need the trignometric identities (1 + tan2θ) = sec2θ and cos2θ + sin2θ = 1. (b) What is the first directional derivative of brightness in the direction of the brightness gradient? How is this related to the magnitude of the brightness gradient? (c) Express the second directional derivative, I ′′(x, y) (in any direction v) in terms of the first and second partial derivatives with respect of x and y.
