On the application of Lehmer means in signal and image processing

References

• G. Akram, K. Bibi, K. Rehan and S. Siddiqi, Shape preservation of 4-point interpolating non-stationary subdivision scheme, J. Comput. Appl. Math. 319 (2017), pp. 480–492. doi: 10.1016/j.cam.2017.01.026
   Web of Science ®Google Scholar
• S. Amat, H. Cherif and J.C. Trillo, Denoising using Linear and Nonlinear Multiresolutions, Eng. Comput. 22(7) (2005), pp. 877–891. doi: 10.1108/02644400510619567
   Google Scholar
• S. Amat, K. Dadourian and J. Liandrat, On an nonlinear subdivision scheme avoiding Gibbs oscillations and converging towards Cs functions with s>1, Math. Comput. 80(274) (2010), pp. 959–971. doi: 10.1090/S0025-5718-2010-02434-2
   Google Scholar
• S. Amat, K. Dadourian and J. Liandrat, Analysis of a class of nonlinear subdivision scheme and associated multiresolution transforms, Adv. Comput. Math. 34 (2011), pp. 253–277. doi: 10.1007/s10444-010-9151-6
   Web of Science ®Google Scholar
• S. Amat, K. Dadourian, J. Liandrat and J.C. Trillo, High order nonlinear interpolatory reconstruction operators and associated multiresolution schemes, J. Comput. Appl. Math. 253 (2013), pp. 163–180. doi: 10.1016/j.cam.2013.03.005
   Web of Science ®Google Scholar
• S. Amat, R. Donat, J. Liandrat and J.C. Trillo, Analysis of a new nonlinear subdivision scheme. Applications in image processing, Found. Comput. Math. 6(2) (2006), pp. 193–225. doi: 10.1007/s10208-004-0122-5
   Web of Science ®Google Scholar
• S. Amat and J. Liandrat, On the stability of the PPH nonlinear multiresolution, Appl. Comp. Harm. Anal. 18(2) (2005), pp. 198–206. doi: 10.1016/j.acha.2004.12.002
   Web of Science ®Google Scholar
• S. Amat, J. Liandrat, J. Ruiz and J.C. Trillo, On a nonlinear cell-average multiresolution scheme for image compression, SeMA J. 60 (2012), pp. 75–92. doi: 10.1007/BF03391711
   Google Scholar
• S. Amat, J. Ruiz, J.C. Trillo and D.F. Yáñez, On a stable family of four-point nonlinear subdivision schemes eliminating the Gibbs phenomenon, J. Comput. App. Math. 354 (2018), pp. 310–325. doi: 10.1016/j.cam.2018.04.058
   Web of Science ®Google Scholar
• F. Aràndiga and R. Donat, Nonlinear Multi-scale Decomposition: The Approach of A. Harten, Numer. Algorithms 23 (2000), pp. 175–216. doi: 10.1023/A:1019104118012
   Web of Science ®Google Scholar
• P.S. Bullen, Handbook of Means and Their Inequalities, Mathematics and its Applications, Springer, Dordrecht, 2003.
   Google Scholar
• G. Chaikin, An algorithm for high speed curve generation, Comput. Gr. Image Process. 3 (1974), pp. 346–349. doi: 10.1016/0146-664X(74)90028-8
   Google Scholar
• A. Cohen, I. Daubechies and J.C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. in Pure and Appl. Math. 45 (1992), pp. 485–560. doi: 10.1002/cpa.3160450502
   Web of Science ®Google Scholar
• A. Cohen, N. Dyn and B. Matei, Quasi linear subdivision schemes with applications to ENO interpolation, App. Comput. Harm. Anal. 15 (2003), pp. 89–116. doi: 10.1016/S1063-5203(03)00061-7
   Web of Science ®Google Scholar
• C. Conti, L. Gemignani and L. Romani, From approximating to interpolatory non-stationary subdivision schemes with the same generation properties, Adv. Comput. Math. 35(2–4) (2011), pp. 217–241. doi: 10.1007/s10444-011-9175-6
   Web of Science ®Google Scholar
• G. Deslauriers and S. Dubuc, Symmetric iterative interpolation processes, Const. Approx. 5 (1989), pp. 49–68. doi: 10.1007/BF01889598
   Web of Science ®Google Scholar
• R. Donat, S. López-Ureña and M. Santágueda, A family of non-oscillatory 6-point interpolatory subdivision schemes, Adv. Comput. Math. 43(4) (2017), pp. 849–883. doi: 10.1007/s10444-016-9509-5
   Web of Science ®Google Scholar
• D. Donoho, Denoising by soft thresholding, IEEE Trans. on Inform. Theory 41(3) (1995), pp. 613–627. doi: 10.1109/18.382009
   Web of Science ®Google Scholar
• T. Duchamp, G. Xie and T. Yu, Smoothing nonlinear subdivision schemes by averaging, Numer. Algorithms 77(2) (2018), pp. 361–379. doi: 10.1007/s11075-017-0319-8
   Web of Science ®Google Scholar
• N. Dyn, M.S. Floater and K. Hormann, A C2 four-point subdivision scheme with fourth order accuracy and its extensions, in Methods for Curves and Surfaces, M. Dæhlen, K. Mørken and L.L. Schumaker, eds., Series: Modern Methods in Mathematics, Tromsø, Nashboro Press, 2005, pp. 145–156.
   Google Scholar
• N. Dyn and D. Levin, Subdivision schemes in geometric modelling, Acta Numer. 11 (2002), pp. 73–144. doi: 10.1017/S0962492902000028
   Google Scholar
• R. Gonzalez and R.E. Woods, Digital Image Processing, Chapter 5: Image Restoration, 2nd ed., Prentice Hall, Upper Saddle River, NJ, 2008.
   Google Scholar
• A. Guessab, M. Moncayo and G. Schmeisser, A class of nonlinear four-point subdivision schemes, Adv. Comput. Math. 37(2) (2012), pp. 151–190. doi: 10.1007/s10444-011-9199-y
   Web of Science ®Google Scholar
• R. Hameed and G. Mustafa, Family of a-point b-ary subdivision schemes with bell-shaped mask, Appl. Math. Comput. 309 (2017), pp. 289–302.
   Web of Science ®Google Scholar
• A. Harten, Multiresolution representation of data II, SIAM J. Numer. Anal. 33(3) (1996), pp. 1205–1256. doi: 10.1137/0733060
   Web of Science ®Google Scholar
• F. Kuijt, Convexity preserving interpolation, Nonlinear Subdivision and Splines. PhD thesis, University of Twente, 1998.
   Google Scholar
• D.H. Lehmer, On the compounding of certain means, J. Math. Anal. Appl. 36(1) (1971), pp. 183–200. doi: 10.1016/0022-247X(71)90029-1
   Web of Science ®Google Scholar
• X. Li and Y. Chang, Non-uniform interpolatory subdivision surface, Appl. Math. Comput. 324 (2018), pp. 239–253. doi: 10.1016/j.cam.2018.01.031
   Web of Science ®Google Scholar
• X. Li and J. Zheng, An alternative method for constructing interpolatory subdivision from approximating subdivision, Comput. Aided Geom. Design 29(7) (2012), pp. 474–484. doi: 10.1016/j.cagd.2012.03.008
   Web of Science ®Google Scholar
• Z. Liu, Remark on inequalities between Hölder and Lehmer means, J. Math. Anal. Appl. 247(1) (2000), pp. 309–313. doi: 10.1006/jmaa.2000.6834
   Web of Science ®Google Scholar
• Z. Luo and W. Qi, On interpolatory subdivision from approximating subdivision scheme, Appl. Math. Comput. 220 (2013), pp. 339–349.
   Web of Science ®Google Scholar
• J.L. Merrien and T. Sauer, From Hermite to stationary subdivision schemes in one and several variables, Adv. Comput. Math. 36(4) (2012), pp. 547–579. doi: 10.1007/s10444-011-9190-7
   Web of Science ®Google Scholar
• P. Oswald, Smoothness of Nonlinear Median-Interpolation Subdivision, Adv. Comput. Math. 20(4) (2004), pp. 401–423. doi: 10.1023/A:1027315032100
   Web of Science ®Google Scholar
• K. Rehan and S. S. Siddiqi, A family of ternary subdivision schemes for curves, Appl. Math. Comput. 270 (2015), pp. 114–123.
   Web of Science ®Google Scholar
• X. Si, J. Baccou and J. Liandrat, On four-point penalized Lagrange subdivision schemes, Appl. Math. Comput. 281 (2016), pp. 278–299.
   Web of Science ®Google Scholar
• S. Siddiqi and K. Rehan, Ternary 2N-point Lagrange subdivision schemes, Appl. Math. Comput. 249 (2014), pp. 444–452.
   Web of Science ®Google Scholar
• J. Tan, G. Tong, L. Zhang and J. Xie, Four point interpolatory-corner cutting subdivision, Appl. Math. Comput. 265 (2015), pp. 819–825.
   Web of Science ®Google Scholar
• Y. Wang and Z. Li, A family of convexity-preserving subdivision schemes, J. Math. Res. Appl. 37(4) (2017), pp. 489–495.
   Google Scholar
• J. Wang, H. Zheng, F. Xu and D. Liu, Fractal properties of the generalized Chaikin corner-cutting subdivision scheme, Comput. Math. Appl. 61(8) (2011), pp. 2197–2200. doi: 10.1016/j.camwa.2010.09.014
   Web of Science ®Google Scholar