3D Perturbation Algorithm

(i) Assume that each ROI consists of n slices, and each slice represents a connected region. Let slices of ROI be ROIz, where z = 0, 1, …, n;(ii) Pick slice 0 as the starting slice, and calculate the average gradient using the following equations:(a) Let ${(x_z,y_z)}_r$ be rth voxel on L of slice z, find the voxel ${(x_{z+1},y_{z+1})}_r$ on slice $z+1$ such that $\parallel x_z-x_{z+1}{\parallel }^2+{\parallel y_z-y_{z+1}\parallel }^2$ is the smallest; (b) For each voxel r, we repeat the previous step to build a list of voxels as follows: $\mathbf{\Phi }(r)=[{(x_0,y_0)}_r,{(x_1,y_1)}_r,{(x_2,y_2)}_r,\text{…},{(x_n,y_n)}_r]$ (c) For each voxel r, calculate the average gradient as $$\overline{\mathrm{avg}\_\mathrm{gra}{d}_r}=\frac{1}{\left|\mathbf{\Phi }(r)\right|}\sum _{z=0}^n\text{APTARAHARPOON}\widehat{\mathrm{Grad}}{\left(x_z,y_z\right)}_r.$$ Pick voxel s 0 such that $\overline{avg\_gra{d}_r}$ at $r=s_0$is the smallest in magnitude and use this s 0 in the TG function (6).