Part I: Image filtering Considering an image denoising problem, let x be a noise-free m m image degraded by an additive white Gaussian noise w. The observed corrupted image y is given by: y = x +w. (1) The Gaussian filter recovers the original image x by a linear filtering that re- places the noisy intensity value yp at each pixel location p with a weighted average of the neighboring pixels q E N(p), such that: xp = f(p, q) ya qEN (p) f(p, q) qEN (p) (2) 1. Write the analytical expression of the Gaussian kernel f with standard deviation of? 2. Show that (2) may be rewritten using convolutions. 3. Assuming the pixels yp are i.i.d. with a standard deviation oy, compute the noise reduction r = where o is the variance of the filtered image Ty 2. 1 4. Give a suitable integer-valued convolution mask f of size (5 x 5) that approximates Gaussian filter f with a standard deviation of = 1.4.