  This article is the second work in our series of papers dedicated to image
processing models based on the fractional order total variation $TV^r$. In our
first work of this series, we studied key analytic properties of these
semi-norms.
  Here we focus on the more applied aspects of such models: first, in order to
obtain a better reconstructed image, we propose several extensions of the
fractional order total variation. Such generalizations, collectively denoted by
RVL, will be modular, i.e. the parameters therein are mutually independent, and
can be fine tuned to the particular task. Then, we will study the bilevel
training schemes based on RVL, and show that such schemes are well defined,
i.e. they admit minimizers. Finally, we will provide some numerical examples,
showing that training schemes based on RVL are effectively better than those
based on classical regularizer $TV$ and $TGV^2$.
