两相图像变分分割凸松弛模型快速算法

Fast Algorithms for Two-Phase Image Variation Segmentation Based on Convex Relaxation Models

主要研究两相图像分割凸模型的三类快速数值算法.首先,分别针对无约束和有约束的图像分割凸模型分别提出相应的具有O(1/k)阶收敛速率的梯度投影算法,并结合快速迭代收缩算法的加速收敛策略,将所提出的梯度投影算法的收敛速率从O(1/k)阶提高到O(1/k2)阶;其次,基于分块协调下降的思想,对无约束的图像分割凸模型采用Newton法求解,该算法不仅是单调下降的,而且具有二阶收敛性;然后,根据交互式迭代算法的思想,在约束模型的Fenchel原始-对偶形式的基础上,提出了一种通过原始变量和对偶变量交互式混合迭代求解的算法,所提出的算法在求解过程中避免了梯度算子和散度算子作用于未知变量,使得迭代形式更简单;最后,仿真实验表明了这3类算法的有效性和在收敛速率上的优势.

This paper focuses on three fast numerical algorithms for two-phase convex variational image segmentation models. At first, two gradient projection algorithms are proposed separately for unconstrained and constrained convex image segmentation models, which converge as O（1/k）. Combining the accelerating convergence strategy of the fast iterative shrinkage /thresholding algorithm, the convergence rate is improved from O（1/k） to O（1/k2）. Next, a Newton method based on block coordinate descent is proposed for the unconstrained model, which is not only monotonically decreasing, but also converging quadratically. And then, a Primal-dual alternating iterative algorithm is applied to the constrained model, based on the Fenchel primal-dual formulation. It alternates between the primal and dual problems, and avoids the gradient operator and the diver- gence operator acting on unknowns. So, the iterative formula is more simple. At last, the validity and the advantages on convergence rate of all algorithms are illustrated by numerical examples.