where α and β are two nonnegative tuning parameters which balance the strength of backward diffusion and forward diffusion.
In this paper, we employ the two-step splitting (TSS) method to implement the proposed model (13), which allows for further fine tuning of strength of backward diffusion and forward diffusion.
We are given the function f t,x,y,z) defined on (left [0,Tright ]times {mathcal {R}}^{k}times {mathcal {R}}times {mathcal {R}}^{k}), function (Phi (x), xin {mathcal {R}}^{k}) and k-dimensional diffusion process (forward) begin{array}rcl@ mathrm{d}X_{t}=Sleft vartheta,t,X_{t}right)mathrm{d}t+varepsilon sigma left t,X_{t}right),mathrm{d}W_{t}, quad X_{0}, ; 0leq tleq T. end{array} (9).
For each source, only data at detectors that were more than one transport mean free path away from the source were used in order to avoid singularities in the diffusion approximation (forward model).
where β1 and β2 control the steepness for the min-max transition region of forward diffusion and backward diffusion, respectively.
So, by using forward diffusion (step 1) and backward diffusion (step 2) alternatively, we can remove the noise, simultaneously preserve the contrast.
First, we notice that if we take α = β, then α/s > β|φ ' (s)|/s, i.e., the forward diffusion velocity is larger than the backward diffusion velocity.
In fact, the term α/|∇u| > 0 is forward diffusion velocity, which measures the ability of denoising for diffusion equation (15); and βφ ' (|∇u|)/|∇u| < 0 is backward diffusion velocity, which controls the ability of contrast enhancing for diffusion equation (15).
We adopted the efficient projection algorithm in the dual framework to solve the forward diffusion in the first step, and then employed the simple finite differences scheme to solve the backward diffusion to compensate the loss of contrast occurred in the previous step.
We adopt the efficient projection algorithm in the dual framework to solve the forward diffusion in the first step, and then use the simple finite differences scheme to solve the backward diffusion to compensate the loss of contrast occurred in the previous step.
