Abstract
This paper deals with two approximation methods for obtaining the optimal deformable model: a discretization scheme by finite differences that generates an algorithm providing the approximating solution for energy-minimizing surfaces and a reconstruction method based on the Chebyshev discrete best approximation which approximates a plane deformable model represented by a finite set of points. Estimates for the approximation-error of these methods and results concerning their convergence or the topological structure of the set of unbounded divergence are presented, too.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Caselles, V., Catte, F., Coll, T., Dibos, F.: A geometric model for active contours. Numer. Math. 66, 1–31 (1993)
Cheney, W., Light, W.: A Course in Approximation Theory. Amer. Math. Soc., Pacific Grove (2009)
Cobzas, S., Muntean, I., Superdense, A.E.: Unbounded divergence of some approximation processes of analysis. Anal. Exch. 25, 501–512 (1999/2000)
Cohen, L.D., Cohen, I.: Finite element methods for active contour models and balloons for 2D and 3D images. IEEE Trans. Pattern Anal. Mach. Intell. 15(11), 1131–1150 (1993)
Curtis, P.C. Jr.: Convergence of approximating polynomials. Proc. Am. Math. Soc. 13, 385–387 (1962)
He, L., Pens, Zh., Guarding, B., Wang, X., Han, C.Y., Weiss, K.L., Wer, W.G.: A comparative study of deformable contour methods on medical image segmentation. Image Comp. Vision 26, 141–163 (2008)
Mc Inerney, T., Terzopoulos, D.: Deformable models in medical image analysis: A survey. Med. Image Anal. 1(2), 91–106 (1996)
Kaas, M., Witkin, A., Terzopoulos, D.: Snakes: Active contour models. Int. J. Comput. Vis. 1, 321–331 (1987)
Malladi, R., Sethian, J., Vemuri, B.: Shape modeling with front propagation: A level set approach. IEEE Trans. Pattern Anal. Mach. Intell. 17(2), 158–175 (1995)
Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)
Szabados, J., Vértesi, P.: Interpolation of Functions. World Scientific, Singapore (1990)
Szegö, G.: Orthogonal Polynomials. Amer. Math. Soc., Providence (1975)
Terzopoulos, D.: On matching deformable models to images. Technical Report Technical Report 60, Schlumberg Palo Alto Research (1986). Reprinter in Topical Meeting on Machine Vision, Technical Digest Series, vol. 12, pp. 160–167. Optical Society of America, Washington (1987)
Terzopoulos, D., Fleischer, K.: Deformable models. Vis. Comput. 4(6), 306–331 (1988)
Trif, D.: Numerical Methods for Differential Equations and Dynamic Systems. Transilvania Press, Romania (1997)
Acknowledgements
These researches are supported by the Project PN2-Partnership, No. 11018 (MoDef).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Basel
About this paper
Cite this paper
Mitrea, P., Gurzău, O.M., Mitrea, A.I. (2012). On the Approximation-Error of Some Numerical Methods for Obtaining the Optimal Deformable Model. In: Bandle, C., Gilányi, A., Losonczi, L., Plum, M. (eds) Inequalities and Applications 2010. International Series of Numerical Mathematics, vol 161. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0249-9_7
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0249-9_7
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0248-2
Online ISBN: 978-3-0348-0249-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)