I.6.4: Half-Quadratic Methods

Half-quadratic or HQ methods are used heavily in image restoration and reconstructions problems. They were introduced in two different but related form in Geman and Reynolds [1992] and Geman and Yang [1995].

$$f(x)=\|Ax-z\|_2^2+\beta\sum_{i\in\mathcal{I}}\phi(u_i'x-v_i)$$ potential funcion, regularization function. edge-preserving requires non-quadratic potential function typically $U$ is a discrete version of a differential operator, such as a first or second-order difference matrix $$g(x,y)=\|Ax-z\|_2^2+\beta\sum_{i=1}^r\frac{b_i}{2}\|\mathcal{D}_ix\|_2^2+k(b_i)$$ $$k(b):=\sup_{t\in\mathcal{R}}\left\{-\frac12bt^2+h(t)\right\}.$$ By convex conjugacy $$h(t)=\inf_{b\in\mathcal{R}}\left\{\frac12bt^2+k(b)\right\}$$ and the infimum is attained at $$b=\begin{cases}h''(0^+)&\text{ if }t=0,\\ \frac{h'(t)}{t}&\text{ if }t\not= 0. \end{cases}$$