Let u(x, y) = [0, 1] represent a two-dimensional black and white image so that each value of (x, y) is mapped to a grayscale value u(x, y) between 0 (black) and 1 (white). Suppose that you obtain a very pixelated image v(x, y) and you want to recreate the full non-pixelated image. Come up with a cost function for u(x, y) that will minimize the difference between u(x, y) and v(x, y), but will also not allow for the grayscale to change to rapidly, i.e. you don't want u(x, y) to have large gradients anywhere either. What are the Euler-Lagrange equations you end up with?