Image Reconstruction

$\text{New}\text{Objective}\text{Function}\text{(}X\text{)}=\text{Old}\text{Objective}\text{Function}\text{(}X\text{)}+\beta \left|\right|\nabla X|{|}^2,$

where β is a user specified controlling parameter. Using the squared norm of the gradient $\left|\right|\nabla X|{|}^2$ as a penalty term can be generalized as using an “energy” function U(X) as penalty term. The energy function U(X) is defined as

$U(X)={\displaystyle \sum _{i,j}{w}_{ij}V(x_j-x_j),}$

where the summation is over a neighborhood (clique), and V is a convex function, which may or may not be quadratic (see Figure 6.19). If V is a quadratic function, this energy function encourages smoothness and penalizes jumps. If the function V increases more slowly than a quadratic function (say, V increases linearly), then it can preserve edges and smooth out the noise. How does the algorithm ...