Minimization of length and curvature on planar curves

In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional integral root 1 + K-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 find non-existence of minimizers on various functional spaces in which the problem is naturally formulated. In this case, minimizing sequences of trajectories can converge to curves with angles. We instead prove existence of minimizers for the "time-reparameterized" functional integral parallel to(gamma)over dot(t)parallel to root 1 + K-gamma(2) dt for all boundary conditions if initial and final directions are considered regardless to orientation.