Image Restoration Using Functional and Anatomical Information Fusion with Application to SPECT-MRI Images

Algorithm 1

NAS-RIF Algorithm.

Compute the filter coefficients 𝑢(𝑥,𝑦,𝑧) using a conjugate gradient optimization routine.

1.          Definitions

           𝛾A positive constant associated with 𝐽₃

           𝛿A constant associated with 𝐽₄

           𝑙The iteration step

           𝑡A positive real number called the speed-gradient algorithm

        𝑢^[𝑙]Vector of filter of dimension 𝑁_𝑥𝑢×𝑁_𝑦𝑢×𝑁_𝑧𝑢 at the lth iteration

        𝑑^[𝑙] Vector of dimension 𝑁_𝑥𝑢×𝑁_𝑦𝑢×𝑁_𝑧𝑢 at the 𝑙th iteration

            𝐽A cost function to be minimized

           ∇𝐽Gradient vector of 𝐽 of dimension 𝑁_𝑥𝑢𝑁_𝑦𝑢𝑁_𝑧𝑢×1

        ⟨⋅,⋅⟩Scalar product

2.          Initialization

  𝛾=0

  𝛿=25

  𝑙=0

  𝑢^[0]=(0⋯1⋯0)^𝑇

  𝑑^[0]=−∇𝐽(𝑢^[0])

  Set speed-gradient algorithm 𝑡>0

3.          FIR filter parameters update

repeat

   𝑓(𝑥,𝑦,𝑧)=𝑔(𝑥,𝑦,𝑧)∗𝑢(𝑥,𝑦,𝑧)

    Calculate the gradient vector of 𝐽 as indicated in  (7)

    if 𝑙=0𝐭𝐡𝐞𝐧

     𝑑^[𝑙]=−∇𝐽(𝑢^[𝑙])

    else

          𝛽^[𝑙−1]=⟨∇𝐽(𝑢^[𝑙])−∇𝐽(𝑢^[𝑙−1]),∇𝐽(𝑢^[𝑙])⟩⟨∇𝐽(𝑢^[𝑙−1]),∇𝐽(𝑢^𝑙−1)⟩

          𝑑^[𝑙]=−∇𝐽(𝑢^[𝑙])+𝛽^[𝑙−1]𝑑^[𝑙−1]

   𝑢^[𝑙+1]=𝑢^[𝑙]+𝑡×𝑑^[𝑙]

   𝑙=𝑙+1;

until a stopping criterion is met;