图像上梯度相关的优化问题快速算法

针对图像处理中目标函数为对图像梯度的约束,形式为正则项与保真项之和的优化问题,提出了一种对该优化问题的变形形式,并给出了基于交替方向乘子法(alternating direction method of multipliers,ADMM)的优化算法进行求解.在约束条件下采用每个图像中的最小单元上的分段式,使得在每步迭代中的每个子问题可以分化为在每个最小单元上的二元优化问题,从而可直接获得优化问题的最优解.所提出的优化形式与优化算法可以控制每步迭代的时间复杂度在O(N),其中N为优化问题在该图像区域中最小单元的个数,还可进一步根据图像的分割进行并行化.文中给出了2个图像上比较经典的优化问题:L0模优化问题和Poisson图像编辑的优化算法.与现有的基于迭代算法相比,文中算法在达到相似结果的同时,可具有更快计算速度与更小的内存消耗.

Fast Algorithm for Gradient Domain Optimization on Image

In many digital image processing problems,the objective function is a constraint on the image gradient and the objective energy includes the regularization term and fidelity term.In this paper,we propose a new optimization formulation and an alternating direction method of multipliers(ADMM)based method to solve these problems.With this new formulation,the original optimization problem can be decomposed into many small problems,and each sub-problem has closed form solution.The time complexity of the proposed algorithm in each iteration is proportional to the image resolution.Besides,the algorithm can be further parallelized based on segmenting the image.We apply the proposed algorithm to two classic image processing problems:L0 norm based image smoothing and Poisson image editing.Compared with the existing iterative algorithms,our proposed algorithm can achieve faster computation speed and cost less memory while achieving similar results.