(variational-image-processing)=
Variational Image Processing
===

In the last chapter we have observed, that the maximum a-posteriori
estimation yields an optimization problem consisting of the sum of a
data term and a regularization term including prior information. In this
chapter we are analyzing such a class of problems using variational
calculus. We want to minimize a family of functionals
$\operatorname{arg\,min}_{u\in X} E(u;f)$ written as

```{math}
:label: eq:variational_model

    E(u; f) = \frac{1}{2} \int _{\Omega} \left( f(x) - (Au)(x) \right)^2 \, dx + \alpha R(u)\,,
```

where $\alpha > 0$ is a regularization weight for the regularizer
$R:X\to \R$, and $A:X\to X$ denotes a linear operator modelling the
'forward' problem (e.g. the convolution, for denoising tasks we choose
$A = \operatorname{Id}$).\
We start with some examples for a choice of regularizer.

````{prf:example} Variational Regularizer
:label: ex:variational_regularizer

1.  **Tikhonov regularization** penalizes large $u$ in the least squares
    sense
    ```{math}
    R(u) = \frac{1}{2}\|u\|_2^2\,.
    ```

2.  $\mathbf{l^1}$**-regularization** (LASSO) is sparsity enducing
    ```{math}
    R(u) = \|u\|_1\,.
    ```

3.  **Gradient regularization** penalizes the rate of change of a
    function and makes it smoother
    ```{math}
    R(u) = \frac{1}{2}\|\nabla u\|_2^2\,.
    ```

4.  **Total variation regularization** penalizes the gradient in the
    $l^1$ sense. It induces discontinuouities (jumps) and piecewise
    constant functions.
    ```{math}
    R(u) = \|\nabla u\|_1\,.
    ```

````