基于变分分数阶偏微分方程的图像放大算法

Fractional-order variational partial differential equations for image zooming

为了提高放大后的图像质量,将分数阶偏微分方程和基于全变差的图像放大思想相结合,建立了一个新的基于分数阶梯度的全变差,该全变差的最小解即是放大后的图像,然后采用交替方向最小化方法,给出该放大模型的分数阶投影算法及其收敛性证明。得益于分数阶偏微分方程处理图像的优势,新模型不仅能够较好地保持图像边缘和纹理细节,还能非线性地保留图像平滑区的纹理特征。数值模拟实验表明,新模型可得到高质量的放大图像。

In order to improve the quality of an enlarged image, based on fractional-order partial differential equations and en ergy functional minimization, a new model for image zooming is proposed. Firstly a new energy functional for image zooming is built, which is based on fractional order gradient. The minimal solution of the energy functional is the enlarged im age. Then the alternating direction minimization method is used to solve the minimization problem. Finally a fractional-order projection method of the amplification model is proposed and the convergence of the algorithm is proved. Compared with Chambolle model, the new model not only maintains better the image edge and texture detail, but also preserves the texture features of image smoothing zone nonlinearly because of the advantages of the fractional-order PDEs. Numerical simulations show that the new model can obtain high quality zooming images.