  In this paper we consider the problem of reconstructing a curve that is
partially hidden or corrupted by minimizing the functional $\int
\sqrt{1+K_\gamma^2} ds$, depending both on length and curvature $K$. We fix
starting and ending points as well as initial and final directions.
  For this functional we discuss the problem of existence of minimizers on
various functional spaces. We find non-existence of minimizers in cases in
which initial and final directions are considered with orientation. In this
case, minimizing sequences of trajectories can converge to curves with angles.
  We instead prove existence of minimizers for the "time-reparameterized"
functional $$\int \| \dot\gamma(t) \|\sqrt{1+K_\ga^2} dt$$ for all boundary
conditions if initial and final directions are considered regardless to
orientation. In this case, minimizers can present cusps (at most two) but not
angles.
