Wavelet Compressive Sampling Signal Reconstruction Using Upside-Down Tree Structure

Algorithm 1

Upside-down tree orthogonal matching pursuit.

Inputs: CS matrix D, measurement 𝐲, upward extending coefficient l, weighting value c, and

iterative number K, wavelet transform level J, 𝐒₀ = {1}

Outputs: approximated signal 𝑥

Initialize: 𝜶₀ = {0}, 𝐫₀ = {0}, k = 0, Λ₀ = {1,2,…, 𝑁_𝐽 (= N/2^J)}

𝑟₀=𝑦−𝑃_span{Λ0}𝑦

While halting criterion false do

   (1) k = k +1

   (2) 𝜽^𝑘_𝑖=𝐒_𝑘−1(𝑖)⋅|⟨𝐫_𝑘−1,𝐝_𝑖⟩|

   (3) 𝜆^𝑘=argmax_{1≤𝑖≤𝑁}{𝜽^𝑘_𝑖}

   (4) construct upside-down tree 𝑇_𝑘

     𝐒_𝑘(i) = {𝑐∶𝑖∈𝑇_𝑘}

   (5) Λ_𝑘=Λ_𝑘−1∪𝜆^𝑘

   (6) 𝐔_𝑘=𝑃_GS(Λ_𝑘)

   (7) 𝐫_𝑘=𝐲−𝑃_span{Λk}𝐲

end while

𝜶=𝐔_𝑘𝐲

return 𝐱=ΦΨ𝜶