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

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