# Parametric Curve Reconstruction from Point Clouds using Minimization Techniques

### Oscar E. Ruiz, C. Cortés, M. Aristizábal, Diego A. Acosta, Carlos A. Vanegas

#### Abstract

Curve reconstruction from noisy point samples is central to surface reconstruction and therefore to reverse engineering, medical imaging, etc. Although Piecewise Linear (PL) curve reconstruction plays an important role, smooth (C^1-, C^2-,...) curves are needed for many applications. In reconstruction of parametric curves from noisy point samples there remain unsolved issues such as (1) high computational expenses, (2) presence of artifacts and outlier curls, (3) erratic behavior of self-intersecting curves, and (4) erratic excursions at sharp corners. Some of these issues are related to non-Nyquist (i.e. sparse) samples. In response to these shortcomings, this article reports the minimization-based fitting of parametric curves for noisy point clouds. Our approach features: (a) Principal Component Analysis (PCA) pre-processing to obtain a topologically correct approximation of the sampled curve. (b) Numerical, instead of algebraic, calculation of roots in point-to-curve distances. (c) Penalties for curve excursions by using point cloud – to - curve and curve – to – point cloud. (d) Objective functions which are economic to minimize. The implemented algorithms successfully deal with self - intersecting and / or non-Nyquist samples. Ongoing research includes self-tuning of the algorithms and decimation of the point cloud and the control polygon.

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#### Paper Citation

#### in Harvard Style

E. Ruiz O., Cortés C., Aristizábal M., A. Acosta D. and A. Vanegas C. (2013). **Parametric Curve Reconstruction from Point Clouds using Minimization Techniques** . In *Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information Visualization Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2013)* ISBN 978-989-8565-46-4, pages 35-48. DOI: 10.5220/0004283900350048

#### in Bibtex Style

@conference{grapp13,

author={Oscar E. Ruiz and C. Cortés and M. Aristizábal and Diego A. Acosta and Carlos A. Vanegas},

title={Parametric Curve Reconstruction from Point Clouds using Minimization Techniques},

booktitle={Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information Visualization Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2013)},

year={2013},

pages={35-48},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004283900350048},

isbn={978-989-8565-46-4},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information Visualization Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2013)

TI - Parametric Curve Reconstruction from Point Clouds using Minimization Techniques

SN - 978-989-8565-46-4

AU - E. Ruiz O.

AU - Cortés C.

AU - Aristizábal M.

AU - A. Acosta D.

AU - A. Vanegas C.

PY - 2013

SP - 35

EP - 48

DO - 10.5220/0004283900350048