Stability Through Synchronization in Nonlinear Multiscale Transformations
Francesc Aràndiga, Rosa Donat · SIAM Journal on Scientific Computing · 2007
Nonlinear analogues of classical wavelet transformations have been designed within spatially oriented multiresolution frameworks such as Harten’s framework [J. Appl. Numer. Math., 12 (1993), pp. 153–193] and the lifting framework [R. L. Claypoole et al., IEEE Trans. Image Process., 12 (2003), pp. 1449–1459]. In designing and using nonlinear multiscale transformations for data compression purposes, the main issue is that of ensuring stability. In [J. Appl. Numer. Math., 12 (1993), pp. 153–193], Harten ensures stability by designing a special error‐control mechanism that provides estimates for the compression error. In [R. L. Claypoole et al., IEEE Trans. Image Process., 12 (2003), pp. 1449–1459] no theoretical results are provided, but the authors argue that stability is a consequence of synchronization between the nonlinear decisions made by the prediction operators in the direct and inverse transformations. By interpreting the nonlinear multiscale transformation of R. L. Claypoole et al. within Harten’s cell‐average framework, we are able to provide stability bounds for the scheme of R. L. Claypoole et al. and establish a theoretical basis for comparison between both nonlinear multiresolution schemes.