The Sobolev Regularity of Refinable Functions

By Amos Ron, Zuowei Shen. 2000

72 citations recorded by the source corpus.

Refinable functions underlie the theory and constructions of wavelet systems on the one hand and the theory and convergence analysis of uniform subdivision algorithms on the other. The regularity of such functions dictates, in the context of wavelets, the smoothness of the derived wavelet system and, in the subdivision context, the smoothness of the limiting surface of the iterative process. Since the refinable function is, in many circumstances, not known analytically, the analysis of its regularity must be based on the explicitly known mask. We establish in this paper a formula that computes, for isotropic dilation and in any number of variables, the sharp L"2-regularity of the refinable function @f in terms of the spectral radius of the restriction of the associated transfer operator to a specific invariant subspace. For a compactly supported refinable function @f, the relevant invariant space is proved to be finite dimensional and is completely characterized in terms of the dependence relations among the shifts of @f together with the polynomials that these shifts reproduce. The previously known formula for this compact support case requires the further assumptions that the mask is finitely supported and that the shifts of @f are stable. Adopting a stability assumption (but without assuming the finiteness of the mask), we derive that known formula from our general one. Moreover, we show that in the absence of stability, the lower bound provided by that previously known formula may be abysmal. Our characterization is further extended to the FSI (i.e., vector) case, to the unisotropic dilation matrix case, and to even snore general setups. We also establish corresponding results for refinable distributions.

Subfields: Biomedical Engineering, Computational Mechanics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/b9a5fa299a422cc8f062f9185e33195dbaa46b2e

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Citations in the corpus

Subdivision schemes in geometric modelling (2002, unknown): “The partial results in this direction include the construction of schemes with bounded curvatures, in Loop (2001), and the special patch construction by Prautzsch and Umlauf (1998).”
Dual Wavelet Frames and Riesz Bases in Sobolev Spaces (2009, background): “The interested reader should consult [11, 20, 21, 27, 30, 38, 49, 50, 51] and the references therein for more details.”
PSEUDO-SPLINES, WAVELETS AND FRAMELETS (2007, methodology): “The regularity has to be checked one by one numerically using methods given in [9, 15, 28, 34, 52], although the regularity of a refinable function seems to increase as the order of Strang-Fix condition increases by numerical tests.”