Nikolova Mila, Chan Raymond H
Centre de Mathématiques et de Leurs Applications CNRS-UMR 8536). ENS de Cachan, 94235 Cachan Cedex, France.
IEEE Trans Image Process. 2007 Jun;16(6):1623-7. doi: 10.1109/tip.2007.896622.
A popular way to restore images comprising edges is to minimize a cost function combining a quadratic data-fidelity term and an edge-preserving (possibly nonconvex) regularizalion term. Mainly because of the latter term, the calculation of the solution is slow and cumbersome. Half-quadratic (HQ) minimization (multiplicative form) was pioneered by Geman and Reynolds (1992) in order to alleviate the computational task in the context of image reconstruction with nonconvex regularization. By promoting the idea of locally homogeneous image models with a continuous-valued line process, they reformulated the optimization problem in terms of an augmented cost function which is quadratic with respect to the image and separable with respect to the line process, hence the name "half quadratic." Since then, a large amount of papers were dedicated to HQ minimization and important results--including edge-preservation along with convex regularization and convergence-have been obtained. In this paper, we show that HQ minimization (multiplicative form) is equivalent to the most simple and basic method where the gradient of the cost function is linearized at each iteration step. In fact, both methods give exactly the same iterations. Furthermore, connections of HQ minimization with other methods, such as the quasi-Newton method and the generalized Weiszfeld's method, are straightforward.
一种流行的恢复包含边缘图像的方法是最小化一个代价函数，该函数结合了二次数据保真项和边缘保持（可能是非凸的）正则化项。主要由于后一项，解的计算缓慢且繁琐。半二次（HQ）最小化（乘法形式）由Geman和Reynolds于1992年率先提出，以减轻非凸正则化背景下图像重建中的计算任务。通过用连续值线过程推广局部均匀图像模型的思想，他们根据一个关于图像是二次的且关于线过程可分离的增强代价函数重新表述了优化问题，因此得名“半二次”。从那时起，大量论文致力于HQ最小化，并取得了重要成果，包括沿凸正则化的边缘保持和收敛。在本文中，我们表明HQ最小化（乘法形式）等同于最基本的简单方法，即在每次迭代步骤中对代价函数的梯度进行线性化。事实上，这两种方法给出的迭代完全相同。此外，HQ最小化与其他方法，如拟牛顿法和广义Weiszfeld法的联系很直接。
