  We propose a model for finding one-dimensional structure in a given measure.
Our approach is based on minimizing an objective functional which combines the
average-distance functional to measure the quality of the approximation and
penalizes the curvature, similarly to the elastica functional. Introducing the
curvature penalization overcomes some of the shortcomings of the
average-distance functional, in particular the lack of regularity of
minimizers. We establish existence, uniqueness and regularity of minimizers of
the proposed functional. In particular we establish $C^{1,1}$ estimates on the
minimizers.
