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Error indicators and adaptive refinement of the discrete thin plate spline smoother
Author(s) -
Lishan Fang
Publication year - 2019
Publication title -
australian and new zealand industrial and applied mathematics journal. electronic supplement
Language(s) - English
Resource type - Journals
ISSN - 1445-8810
DOI - 10.21914/anziamj.v60i0.14061
Subject(s) - spline (mechanical) , grid , finite element method , smoothing , computer science , algorithm , piecewise , thin plate spline , smoothing spline , polynomial , mathematics , mathematical optimization , spline interpolation , geometry , mathematical analysis , physics , structural engineering , bilinear interpolation , engineering , computer vision , thermodynamics
The discrete thin plate spline is a data fitting and smoothing technique for large datasets. Current research only uses uniform grids for this discrete smoother, which may require a fine grid to achieve a certain accuracy. This leads to a large system of equations and high computational costs. Adaptive refinement adapts the precision of the solution to reduce computational costs by refining only in sensitive regions. The error indicator is an essential part of the adaptive refinement as it identifies whether certain regions should be refined. Error indicators are well researched in the finite element method, but they might not work for the discrete smoother as data may be perturbed by noise and not uniformly distributed. Two error indicators are presented: one computes errors by solving an auxiliary problem and the other uses the bounds of the finite element error. Their performances are evaluated and compared with 2D model problems. ReferencesH. Chui and A. Rangarajan. A new point matching algorithm for non-rigid registration. Comput. Vis. Image Und., 89 (23): 114141, 2003. doi:10.1016/S1077-3142(03)00009-2.W. F. Mitchell. A comparison of adaptive refinement techniques for elliptic problems. ACM T. Math. Software, 15 (4): 326347, 1989. doi:10.1145/76909.76912.S. Roberts, M. Hegland, and I. Altas. Approximation of a thin plate spline smoother using continuous piecewise polynomial functions. SIAM J. Numer. Anal., 41(1):208234, 2003. doi:10.1137/S0036142901383296.G. Sewell. Analysis of a finite element method. Springer-Verlag, 1985. doi:10.1007/978-1-4684-6331-6.R. Sprengel, K. Rohr, and H. S. Stiehl. Thin-plate spline approximation for image registration. In P. IEEE EMBS, volume 3, pages 11901191. IEEE, 1996. doi:10.1109/IEMBS.1996.652767.L. Stals. Efficient solution techniques for a finite element thin plate spline formulation. J. Sci. Comput., 63(2):374409, 2015. doi:10.1007/s10915-014-9898-x.G. Wahba. Spline models for observational data, volume 59 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, 1990. doi:10.1137/1.9781611970128.

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