International Journal of Biomedical Imaging / 2011 / Article / Alg 1

Research Article

Sparse Regularization-Based Reconstruction for Bioluminescence Tomography Using a Multilevel Adaptive Finite Element Method

Algorithm 1

TNIPM algorithm for BLT reconstruction at the π‘˜ th mesh level.
Initialize the parameters: relative tolerance πœ‰ > 0 , 𝑑 = 1 / πœ† , 𝑆 𝑝 π‘˜ = 0 , 𝑒 = 𝟏 = ( 1 , . . . , 1 ) ∈ 𝑅 𝑁 𝑝 ;
repeat
1. Compute the search direction ( Ξ” 𝑆 𝑝 π‘˜ , Ξ” 𝑒 ) as an approximate solution to the Newton system
  (14) by preconditioned conjugate gradient method;
2. Compute the step size a by backtracking line search;
3. Update the iterate by ( 𝑆 𝑝 π‘˜ , 𝑒 ) ∢ = ( 𝑆 𝑝 π‘˜ , 𝑒 ) + π‘Ž ( Ξ” 𝑆 𝑝 π‘˜ , Ξ” 𝑒 ) ;
4. Construct a dual feasible point 𝜈 from (10);
5. Evaluate the duality gap 𝛾 from (11);
6. quit  if 𝛾 / 𝐺 ( 𝑣 ) ≀ πœ‰ ;
7. Update t: 𝑑 { m a x { πœ‡ m i n { 2 𝑁 𝑝 / 𝛾 , 𝑑 } , 𝑑 } , π‘Ž β‰₯ π‘Ž m i n 𝑑 , π‘Ž < π‘Ž m i n , where πœ‡ > 1 and π‘Ž m i n ∈ ( 0 , 1 ] .