Relaxed Mumford-Shah regularization tutorial

[mrava87]

In the tutorial https://github.com/PyLops/pyproximal/blob/dev/tutorials/relaxed_mumford-shah.py we state rMS is a combination of Tikhonov and TV regularization, which may be a bit misleading. We should consider improving this wording.

[mrava87]

@NickLuiken

[tnorth]

I am also wondering if the description of the functional is correct, ie:

$$\text{rMS}(x) = \min (\alpha\Vert x\Vert_2^2, \kappa)$$

It is described in Strekalovskiy et al. as $\text{rMS}(x) = \min (\alpha |x|^2, \kappa)$, as the minimum between $\alpha$ and the absolute value of the gradient vector squared and a constant, not the minimum between $\alpha$ times the squared L2 norm of the gradient and a constant.

Should the optimization problem then be written as something like:
$$\underset{d}{\text{argmin~}} \frac{1}{2} \Vert d - Gm\Vert_2^2 + \Vert \min(\alpha |{\nabla d}|^2, \kappa) \Vert_2$$

?

[mrava87]

@NickLuiken