Smoothness Analysis of Nonlinear Subdivision Schemes of Homogeneous and Affine Invariant Type

Nonlinear subdivision schemes arise from, among other applications, nonlinear multiscale signal processing and shape preserving interpolation. For the univariate homogeneous subdivision operator $S:\ell(\bZ) \goto \ell(\bZ)$ we establish a set of commutation/recurrence relations which can be used to analyze the asymptotic decay rate of $\|\Delta^r S^j m\|_{\ell^\infty}$, $j=1,2,\ldots,$ the latter in turn determines the convergence and H\”older regularity of $S$. We apply these results to prove that the critical H\”older regularity exponent of a nonlinear subdivision scheme based on median-interpolation is equal to that of an approximating linear subdivision scheme, resolving a conjecture by Donoho and Yu. We also consider a family of nonlinear but affine invariant subdivision operators based on interpolation-imputation of $p$-mean (of which median corresponds to the special case $p=1$) as well as general continuous $M$-estimators. We propose a linearization principle which, when applied to $p$-mean subdivision operators, leads to a family of linear subdivision schemes. Numerical evidence indicates that in at least many cases the critical smoothness of a $p$-mean subdivision scheme is the same as that of the corresponding linear scheme. This suggests a more coherent view of the result obtained in this paper.