  Even though the statistical theory of linear inverse problems is a
well-studied topic, certain relevant cases remain open. Among these is the
estimation of functions of bounded variation ($BV$), meaning $L^1$ functions on
a $d$-dimensional domain whose weak first derivatives are finite Radon
measures. The estimation of $BV$ functions is relevant in many applications,
since it involves minimal smoothness assumptions and gives simplified,
interpretable cartoonized reconstructions. In this paper we propose a novel
technique for estimating $BV$ functions in an inverse problem setting, and
provide theoretical guaranties by showing that the proposed estimator is
minimax optimal up to logarithms with respect to the $L^q$-risk, for any
$q\in[1,\infty)$. This is to the best of our knowledge the first convergence
result for $BV$ functions in inverse problems in dimension $d\geq 2$, and it
extends the results by Donoho (Appl. Comput. Harmon. Anal., 2(2):101--126,
1995) in $d=1$. Furthermore, our analysis unravels a novel regime for large $q$
in which the minimax rate is slower than $n^{-1/(d+2\beta+2)}$, where $\beta$
is the degree of ill-posedness: our analysis shows that this slower rate arises
from the low smoothness of $BV$ functions. The proposed estimator combines
variational regularization techniques with the wavelet-vaguelette decomposition
of operators.
