  Shearlet systems have so far been only considered as a means to analyze
$L^2$-functions defined on $\R^2$, which exhibit curvilinear singularities.
However, in applications such as image processing or numerical solvers of
partial differential equations the function to be analyzed or efficiently
encoded is typically defined on a non-rectangular shaped bounded domain.
Motivated by these applications, in this paper, we first introduce a novel
model for cartoon-like images defined on a bounded domain. We then prove that
compactly supported shearlet frames satisfying some weak decay and smoothness
conditions, when orthogonally projected onto the bounded domain, do provide
(almost) optimally sparse approximations of elements belonging to this model
class.
