多网格法解总变分问题及在医学图像增强中的应用

传统的各向同性平滑方法，如拉普拉斯平滑方法，虽然能去掉图像的噪声，但同时也可能使图像的边缘信息模糊，甚至丢失。针对这种情况，基于总变分的平滑方法得到重视，因为该方法可以在去除噪声的同时，对边缘的信息进行增强，但是由于基于总变分的平滑方法计算量大，且用松弛法迭代的收敛速度比较慢，因此引入了多网格预处理的共轭梯度算法来解总变分问题。计算结果表明，共轭梯度法的收敛速度明显高于松弛法，而采用多网格法收敛速度还可以得到进一步提高。为说明该方法的优点，最后对用这两种方法处理的超声医学图像的收敛曲线和平滑结果进行了比较。

Total Variation Regularization Solved by Multi-grid Method and Applied in Image Denoising

The isotropic diffusion method for image denoising such as those based on the Laplace regularization can smooth out the noise in image, but it may simultaneously blur the edge or boundary of the objects. In order to overcome this problem, recently many researchers pay attention to the smooth method based on the total variation (TV) regularization because it can reserve or even enhance the information of edge when smoothing the noise. However, since equation system deduced by TV method is a strongly nonlinear system, the convergence rate is very slow when solving TV equations using relaxation method. So in this paper, we introduce the multi-grid algorithm and conjugate gradient (CG) algorithm to solve this system. By smoothing out the noise in the echocardiograpgic images, numerical results indicate that the convergence rate of CG is fast, the algorithm of multi-grid has more efficiency and the image can be recovered with satisfied result even contamination of strong noise. As a result, the multi-grid algorithm is a good alternative method for solving the TV questions.