Appendix B  Efficient Wavelet-Based Reconstruction

B.1  Proof of Proposition 1

Proof. We rewrite the cost function (2) with the change of variable w′ = Λ^1/2w. We then apply ISTA to solve the problem in terms of w′. The new parameters are a′ = Λ^−1/2a, A′=Λ^−1/2AΛ^−1/2, and thresholds λ√τ_k that are specific to each coefficient. Noting that Λ−A is positive-definite if and only if I −A′ is positive-definite leads us to L = 2. The iteration w_i+1′=T_λ√τ(w_i′+(a′−A′w_i′)) can be rewritten, in terms of the original variable, as w_i+1=T_λτ(w_i+Λ⁻¹(a−Aw_i)). The latter is nothing but an iteration of SISTA (see Algorithm 1). According to Proposition 1, we have C(Λ^−1/2w_i′)− C(w^⋆)≤ ||w_i0′−Λ^1/2w^⋆ ||²/(i−i₀), which translates directly into the proposed result.

B.2  Proof of Proposition 2

Proof. In the spirit of the proof of Proposition 1, we consider the change of variable w′ = Λ^1/2w and apply FISTA to solve the new reconstruction problem. The ISTA step w_i+1′ = T_λ√τ(v_i′+(a′−A′v_i′)) is equivalent to a SISTA step in terms of the original variable w_i+1 = T_λτ(v_i+Λ⁻¹(a−Av_i)). The convergence results of FISTA [20, Thm. 4.4] applies on the sequence {w_i′}, which leads to C(Λ^−1/2w_i′)− C(w^⋆)≤ 4/(i+1)²||w₀′−Λ^1/2w^⋆ ||². In the strongly convex case, we have