1. Poisson equation can be solved by sine transformation or cosine transformation. Explain in detail how the Poisson equation can be solved by taking advantage of the fourier transformation and depending on the cosine transformation.
2. Gray-scale photographs want to be colored using the Poisson equation. In this context, describe in detail how to colorize gray-level photos, provided that the colored scratches received as input from the user are used at the boundary values of the solution of the poisson equation.