  Reconstructing the missing parts of a curve has been the subject of much
computational research, with applications in image inpainting, object
synthesis, etc. Different approaches for solving that problem are typically
based on processes that seek visually pleasing or perceptually plausible
completions. In this work we focus on reconstructing the underlying physically
likely shape by utilizing the global statistics of natural curves. More
specifically, we develop a reconstruction model that seeks the mean physical
curve for a given inducer configuration. This simple model is both
straightforward to compute and it is receptive to diverse additional
information, but it requires enough samples for all curve configurations, a
practical requirement that limits its effective utilization. To address this
practical issue we explore and exploit statistical geometrical properties of
natural curves, and in particular, we show that in many cases the mean curve is
scale invariant and oftentimes it is extensible. This, in turn, allows to boost
the number of examples and thus the robustness of the statistics and its
applicability. The reconstruction results are not only more physically
plausible but they also lead to important insights on the reconstruction
problem, including an elegant explanation why certain inducer configurations
are more likely to yield consistent perceptual completions than others.
