Interpolating m-refinable functions with compact support: The second generation class

Abstract

We present an algorithm for the construction of a new class of compactly supported interpolating refinable functions that we call the second generation class since, contrary to the existing class, is associated to subdivision schemes with an even-symmetric mask that does not contain the submask {0..., 0, 1, 0, ...0}.As application examples of the proposed algorithm we present interpolating 4-refinable functions that are generated by parameter-dependent, even-symmetric quaternary schemes never considered in the literature so far.

Recursive Process for Constructing the Refinement Rules of New Combined Subdivision Schemes and Its Extended Form

“…Dual binary subdivision schemes on the other hand are the schemes that create two new points at the old edges and discard the old points. Most of the dual schemes are approximating subdivision schemes; however, recently Romani [19,20] introduced interpolatory subdivision schemes that are dual in nature. Detailed information about the primal and dual subdivision schemes can be found in [16].…”

“…Now, firstly we put α 0 � α and α 1 � 1 in ( 19) and then we use ( 17) and ( 21) in (19). Hence, we get the following 4-point relaxed subdivision scheme:…”

Dual Univariate Interpolatory Subdivision of Every Arity: Algebraic Characterization and Construction

“…For the matrix M we have to consider two different cases: M odd and M even , respectively for ( 21) and (26). Starting from the case m ∈ 2N + 1, the right-hand side of ( 21) leads to…”

“…Beside the use of primal interpolatory schemes, there exist other approaches to interpolate points via subdivision, such as those that apply an approximating scheme after suitably preprocessing the data to be interpolated (see, e.g., [13,27,29]). However, before [26], none of the existing approaches took into account the possibility of constructing a native dual interpolatory scheme which does not have the property of retaining the initial data at each iteration, but achieves the interpolation in the sense that the initial data are still preserved in the limit function. All dual subdivision schemes we can find in literature are indeed not interpolatory [9], except for the family of dual quaternary schemes introduced in [26] and for the class of dual (2n)-point subdivision schemes proposed in [12], which is shown to possess the interpolation property only when n tends to infinity, i.e., when the subdivision mask has infinite length.…”

The Family of Multiparameter Quaternary Subdivision Schemes

“…Conti and Romani [11] presented an algebraic technique for the generation of m-ary subdivision schemes. Romani [12] presented an algorithm for the generation of dual interpolating m-ary subdivision schemes. Romani and Viscardi [13] presented a new class of univariate stationary interpolating subdivision schemes of arity m. Recently, Mustafa et al [14] presented a family of integer-point ternary parametric subdivision schemes.…”