  In this paper, a variational, multi-dimensional model for image
reconstruction is proposed, in which the regularization term consists of the
$r$-order (an)-isotropic total variation seminorms $TV^r$, with $r\in \mathbb
R^+$, defined via the Riemann-Liouville fractional derivative. Key properties,
such as the lower semi-continuity and compactness with respect to both the
function $u$ and the order of derivative $r$, are studied. This paper, the
first of our series works on analytical and numerical aspects of the model, as
well as the learning of optimal order $r$ for particular imaging tasks,
provides a comprehensive analysis of the behavior of $TV^r$ in the space of
functions with bounded (fractional order) total variation.
