(BV, L2) Multiscale Hierarchical Decomposition: Modes and Rates of Convergence

Tadmor, Nezzar and Vese [Eitan Tadmor, Suzanne Nezzar, and Luminita Vese. A multiscale image representation using hierarchical (BV, L2) decompositions. Multiscale Model. Simul., 2(4):554–579, 2004.] developed a total variation based multiscale method for decomposing a function f ∈ BV into a countable set of features {uk : k = 0, 1, 2 . . .} associated to a sequence of dyadic scales {λk = λ02−k : k = 0, 1, 2, . . .} such that for each k, [uk+1, vk+1] = arg min{λk|Du| + ‖v‖L2 : u + v = f }. They showed that f = ∑∞ k=0 uk in L 2(Ω) and strongly in W−1,∞(Ω). In this paper, we study the convergence of the series ∞ ∑ k=0 uk in the weak*, strict and normed topologies of the space of functions with bounded variation. We show that in general, the convergence of the series f = ∞ ∑ k=0 uk in any of the three topologies of BV is conditioned by its rate of convergence in L2, and prove that the convergence in L2 is geometric.