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Q1: Image Restoration (12 marks) = Suppose an image f(x,y) is first blurred by a point spread function (PSF) h(x, y) and then corrupted by a noise process n(x,y). The degraded image is denoted as g(x,y) = h(x,y) * f(x,y) + n(x, y). It is known that the noise n(x, y) and image f(x,y) are uncorrelated, and n(x,y) is white Gaussian with zero mean and variance o 2. 1) Assume that h(x,y) = 8(x,y) and there are L noisy observations {g}(x,y)}}=1 of f(x,y), i.e., g(x,y) = f(x,y) + ni(x, y), l = 1,...,L, and {n(x, y)}{= are independent realizations of the white Gaussian noise process n(x, y). A denoised image is obtained as f(x,y) = {{-191(x, y). Determine the noise variance in the denoised image f (x,y). (2 marks) 2) Suppose the size of f(x, y) is 216 x 216 (i.e., x, y = 0, ---,215) and ie PSF h(x, y) is given below: h(x,y) 1 = 3 ro 0 1 0 1 0 1 0 0 a) Derive the frequency response of h(x, y). (4 marks) b) Derive the direct inverse filter in the frequency domain to restore f(x,y). Discuss whether there is any frequency where the response of the direct inverse filter is infinite. (3 marks) c) Assume the power spectral density (PSD) of the image (process) is given by Pry(u, v), u, v = 0,---,215. Derive a linear filter in the frequency domain to restore f(x,y) that achieves the minimum mean square error (MMSE). Discuss whether there is any frequency where the frequency response of the derived filter is infinite