Normal Multi-scale Transforms for Curves
Extending upon Daubechies et al. (Constr. Approx. 20:399–463, 2004) and Runborg (Multiscale Methods in Science and Engineering, pp. 205–224, 2005), we provide the theoretical analysis of normal multi-scale transforms for curves with general linear predictor S, and a more flexible choice of normal directions. The main parameters influencing the asymptotic properties (convergence, decay estimates for detail coefficients, smoothness of normal re-parametrization) of this transform are the smoothness of the curve, the smoothness of S, and its order of exact polynomial reproduction. Our results give another indication why approximating S may not be the first choice in compression applications of normal multi-scale transforms.
KeywordsNonlinear geometric multi-scale transforms Approximating subdivision schemes Lipschitz smoothness Curve representation Detail decay estimate
Mathematics Subject Classification (2000)65D17 65D15 65T60 26A16
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- 3.P. Binev, N. Dyn, R.A. DeVore, N. Dyn, Adaptive approximation of curves, in Approximation Theory (Acad. Publ. House, Sofia, 2004), pp. 43–57. Google Scholar
- 4.A.S. Cavaretta, W. Dahmen, C.A. Micchelli, Stationary Subdivision, Memoirs AMS, vol. 93 (Am. Math. Soc., Providence, 1991). Google Scholar
- 6.N. Dyn, M.S. Floater, K. Hormann, A C 2 four-point subdivision scheme with fourth order accuracy and its extensions, in Mathematical Methods for Curves and Surfaces, ed. by M. Dæhlen, K. Mørken, L.L. Schumaker (Nashboro Press, Brentwood, 2005), pp. 145–156. Google Scholar
- 11.I. Guskov, K. Vidimce, W. Sweldens, P. Schröder, Normal meshes, in Computer Graphics (SIGGRAPH’00: Proceedings), ed. by K. Akeley (ACM, New York, 2000), pp. 95–102. Google Scholar
- 12.A. Khodakovsky, I. Guskov, Compression of normal meshes, in Geometric Modeling for Scientific Visualization (Springer, Berlin, 2003), pp. 189–207. Google Scholar
- 14.S. Lavu, H. Choi, R. Baraniuk, Geometry compression of normal meshes using rate-distortion algorithms, in Proceedings of the Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, ed. by L. Kobbelt, P. Schröder, H. Hoppe (Eurographics Association, Aire-la-Ville, 2003), pp. 52–61. Google Scholar