一种基于L曲线准则的正则化图像复原算法

针对因大气湍流而引起的模糊图像的复原问题，通过对偏差函数添加正则函数以使得要求解的方程组是适定的. 以偏差函数和正则化函数为坐标，得到复原图像关于正则化参数的参数曲线，用三次多项式拟合了该曲线的左端部分，并以该曲线的最大曲率点对应的参数为最优正则化参数. 当图像边界满足周期性条件时，利用傅里叶矩阵可以将分块循环矩阵对角化，进而利用二维离散傅里叶变化和反变换求解线性方程组而得到复原图像. 实验结果表明，当噪声方差较小时，可得到比较满意的复原结果. 

Regularized Image Restoration Algorithm Based on the L-Curve Criterion

For the restoration of blurred image caused by atmospheric turbulence, the regularization function was added to discrepancy function to ensure that the linear system to be solved is well-posed. The parametric curve for restoring blurred image can be obtained, which coordinates are discrepancy function and regularization function, respectively. The left end portion of the curve was fitted with cubic polynomial, and the parameter corresponding to the maximum curvature point in L-curve was selected as optimal regularization parameter. When image boundary satisfy periodic condition, the block circulant matrix could be diagonalized by Fourier matrix, and then the restored image was received by solving linear systems with two-dimensional discrete Fourier transform and its inverse transform. Experimental results show that when the noise variance is small, the satisfactory recovery results could be achieved.