Poly-scale refinability and subdivision

A stationary subdivision scheme is a two-scale process, where values at the next level of refinement are computed from the values of the current level using a single given mask P = {pk}_k∈ℤd. Under a certain restriction on the mask it can be shown that there exists a distributional solution for the functional equation φ = ∑_k∈ℤd pkφ(2 · -k). It is well known that the limit of a convergent subdivision scheme initialized by data f⁰ = {f⁰ _k}_k∈ℤd can be represented as ∑_k∈ℤd f_k⁰ φ (x - k), where φ is a continuous solution of the functional equation. In this work we generalize this framework in the following sense. The (poly) M-scale subdivision scheme computes the next level of refinement from the M-1 scales of the previous level, using M - 1 given masks, P_m = {p_m,k} _k∈ℤd, m = 1,...,M - 1. With a certain restriction on the masks there exists a distributional solution for the poly-scale functional equation φ = ∑_m=1^M-1 ∑_k∈ℤd p_m,kφ (2^m · -k). We show that a convergent poly-scale subdivision process initialized by data f⁰ = {f⁰_k}_k∈ℤd converges to ∑_k∈ℤd f⁰_kφ(x - k), where φ is a continuous solution of the poly-scale functional equation. In applications, the poly-scale framework allows the design of subdivision schemes with features that are not possible in the standard two-scale case.