Abstract
A new nonlinear representation of multiresolution decompositions and new thresholding adapted to the presence of discontinuities are presented and analyzed. They are based on a nonlinear modification of the multiresolution details coming from an initial (linear or nonlinear) scheme and on a data dependent thresholding. Stability results are derived. Numerical advantages are demonstrated on various numerical experiments.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amat, S., Aràndiga, F., Cohen, A., Donat, R.: Tensor product multiresolution analysis with error control for compact image representation. Signal Process. 82(4), 587–608 (2002)
Amat, S., Aràndiga, F., Cohen, A., Donat, R., García G., von Oehsen, M.: Data compression with ENO schemes: a case study. Appl. Comput. Harmon. Anal. 11, 273–288 (2001)
Amat, S., Donat, R., Liandrat, J., Trillo, J.C.: Analysis of a new nonlinear subdivision scheme. Applications in image processing. Found. Comput. Math. 6(2), 193–225 (2006)
Amat, S., Donat, R., Trillo, J.C.: On specific stability bounds for linear multiresolution schemes based on piecewise polynomial Lagrange interpolation. J. Math. Anal. Appl. 358(1), 18–27 (2009)
Amat, S., Liandrat, J.: On the stability of the PPH nonlinear multiresolution. Appl. Comput. Harmon. Anal. 18(2), 198–206 (2005)
Aràndiga, F., Donat, R.: Nonlinear multi-scale decomposition: the approach of A. Harten. Numer. Algorithms 23, 175–216 (2000)
Cohen, A.: Theoretical, applied and computational aspects of nonlinear approximation. (English summary) Multiscale Problems and Methods in Numerical Simulations, pp. 1–29. Lecture Notes in Math., vol. 1825. Springer, Berlin (2003)
Cohen, A., Dyn N., Matei, B.: Quasilinear subdivision schemes with applications to ENO interpolation. Appl. Comput. Harmon. Anal. 15, 89–116 (2003)
Daubechies, I., Runborg, O., Sweldens, W.: Normal multiresolution approximation of curves. Constr. Approx. 20(3), 399–463 (2004)
Dyn, N.: Subdivision schemes in computer-aided geometric design. Advances in Numerical Analysis, II, 36-104 (1992)
Floater, M.: Nonlinear means in geometric modeling. Approximation Theory: In Memory of Varma, A. K., Govil, N.K., Mohapatra, R.N., Nashed, Z., Sharma, A., Szabado, J. (eds.) Monographs and Textbooks in Pure and Applied Mathematics, vol. 212, pp. 197–207. Marcel Dekker Publishers, New York (1998)
Guskov, I., Vidimče, K.K., Sweldens, W., Schröder, P.: Normal meshes. In: Computer Graphics (SIGGRAPH’00 Proceedings), pp. 259–268. ACM Press, New York (2000)
Harizanov, S., Oswald, P.: Stability of nonlinear subdivision and multiscale transforms. Constr. Approx. 31, 359–393 (2010)
Harten, A.: Multiresolution representation of data II. SIAM J. Numer. Anal. 33(3), 1205–1256 (1996)
Matei, B.: Smoothness characterization and stability in nonlinear multiscale framework: theoretical results. Asymptot. Anal. 41(3–4), 277–309 (2005)
Oswald, P.: Smoothness of nonlinear median-interpolation subdivision. Adv. Comput. Math. 20(4), 401–423 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Charles Micchelli.
Research supported in part by the Spanish grants MICINN-FEDER MTM2010-17508 and 08662/PI/08.
Rights and permissions
About this article
Cite this article
Amat, S., Liandrat, J. Nonlinear thresholding of multiresolution decompositions adapted to the presence of discontinuities. Adv Comput Math 38, 133–146 (2013). https://doi.org/10.1007/s10444-011-9231-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-011-9231-2