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

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 $$\alpha||f||^2$$?

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This is an example of the Fidelity Term and Prior Term model.

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:

$$ \frac{1}{2} {\left\| A \boldsymbol{f} - \tilde{\boldsymbol{g}} \right\|}_{2}^{2} $$

The prior terms is usually something else we know about the data in question. It can be something about its distribution, about its derivatives or in the case of Variational Models something about its response to some operator (For example, Derivative).

In 3.12 above they use the Tikhonov Model which I wrote about in your other question.

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