$\text{New}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Objective}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Function}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{(}X\text{)}=\text{Old}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Objective}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Function}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{(}X\text{)}+\beta \left|\right|\nabla X|{|}^{2},$

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 ...

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