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 C1,1 estimates on the minimizers.
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Keywords: Average-distance problem, elastica functional, Willmore energy, curve fitting
@article{COCV_2021__27_1_A10_0,
author = {Lu, Xin Yang and Slep\v{c}ev, Dejan},
title = {Average-distance problem with curvature penalization for data parameterization: regularity of minimizers},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021002},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021002/}
}
TY - JOUR AU - Lu, Xin Yang AU - Slepčev, Dejan TI - Average-distance problem with curvature penalization for data parameterization: regularity of minimizers JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021002/ DO - 10.1051/cocv/2021002 LA - en ID - COCV_2021__27_1_A10_0 ER -
%0 Journal Article %A Lu, Xin Yang %A Slepčev, Dejan %T Average-distance problem with curvature penalization for data parameterization: regularity of minimizers %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021002/ %R 10.1051/cocv/2021002 %G en %F COCV_2021__27_1_A10_0
Lu, Xin Yang; Slepčev, Dejan. Average-distance problem with curvature penalization for data parameterization: regularity of minimizers. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 8. doi: 10.1051/cocv/2021002
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