A Wavelet-Based Multiresolution Reconstruction Method for Fluorescent Molecular Tomography

Algorithm 2

(1) Set 𝐱₀ to an initial guess;

(2)𝐱=𝐱₀, Compute 𝐲 and 𝐉 at 𝐱 with Algorithm 1;

(3) Perform wavelet transform for 𝐊 and 𝐛 level by level to

    obtain 𝐊_𝑙 and 𝐛_𝑙 for 𝑙=−1,…,−𝐿, with 𝐊_𝑙 and 𝐛_𝑙

    being the wavelet transform at the 𝑙th level

    for 𝐊 and 𝐛;

(4) Set 𝑙=−𝐿 and Δ𝐱⁽⁰⁾_−𝐿=0;

(5) While (𝑙<0)

   {

 Obtain a solution from 𝐊_𝑙Δ𝐱_𝑙=𝐛_𝑙 using the CGD

 method with an initial value Δ𝐱_𝑙⁽⁰⁾;

 Prolongate Δ𝐱_𝑙 through padding zeros to obtain an initial

 guess for Δ𝐱_𝑙+1 at the next higher resolution, that is,

  Δ𝐱⁽⁰⁾_𝑙+1=[Δ𝐱^𝑇_𝑙,𝟎^𝑇]^𝑇;

  𝑙=𝑙+1;

   }

(6) Solve 𝐊Δ𝐱=𝐛 with Δ𝐱⁽⁰⁾=𝐖^T_𝐱Δ𝐱₀⁽⁰⁾ as an initial guess;

(7) 𝐱₀=𝐱₀+Δ𝐱;

(8) If ‖Δ𝐱‖>𝜀 (Define termination criterion 𝜀) go to (2);

    Else 𝐱=𝐱₀, output 𝐱 .