# 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′ = Λ1/2w. We then apply ISTA to solve the problem in terms of w′. The new parameters are a′ = Λ−1/2a, A′=Λ−1/2−1/2, and thresholds λ√τk that are specific to each coefficient. Noting that ΛA is positive-definite if and only if IA′ is positive-definite leads us to L = 2. The iteration wi+1′=Tλ√τ(wi′+(a′−Awi′)) can be rewritten, in terms of the original variable, as wi+1=Tλτ(wi+Λ−1(aAwi)). The latter is nothing but an iteration of SISTA (see Algorithm 1). According to Proposition 1, we have C(Λ−1/2wi′)− C(w)≤ ||wi0′−Λ1/2w ||2/(ii0), 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′ = Λ1/2w and apply FISTA to solve the new reconstruction problem. The ISTA step wi+1′ = Tλ√τ(vi′+(a′−Avi′)) is equivalent to a SISTA step in terms of the original variable wi+1 = Tλτ(vi+Λ−1(aAvi)). The convergence results of FISTA [20, Thm. 4.4] applies on the sequence {wi′}, which leads to C(Λ−1/2wi′)− C(w)≤ 4/(i+1)2||w0′−Λ1/2w ||2. In the strongly convex case, we have

 є 2
⎪⎪
⎪⎪
wiw ⎪⎪
⎪⎪
22 ≤  C(wi)− C(w).