# Variational Regularization Method in Image Processing

I would like to understand better variational regularisation methods in image processing. In particular, the formulas in this image:

Why formula (3.13)? In the notes that I read I cannot find anything about its background, why is the solution of (3.12) given by (3.13). Why do we need this part $$\alpha||f||^2$$?

In many Inverse Problems we assume some model on the additive noise. This part is modeled by the Fidelity Term ($$\mathcal{D} \left(A \boldsymbol{f}, \tilde{\boldsymbol{g}} \right)$$ in your example). For Gaussian Noise it is given by Least Squares Term:
$$\frac{1}{2} {\left\| A \boldsymbol{f} - \tilde{\boldsymbol{g}} \right\|}_{2}^{2}$$
In 3.12 above they use the Tikhonov Model which I wrote about in your other question.