Average-distance problem with curvature penalization for data parameterization: regularity of minimizers

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.