数字图像放大分数阶偏微分方程方法

Method of Image Magnification Based on Fractional Order Partial Differential Equations

图像放大过程中会导致图像的边缘和纹理等细节模糊化,采用分数阶偏微分方程可有效解决这一问题。将Riesz导数化成Hadamard有限积分部分,用分段二次插值多项式对其逼近,从而得到该分数阶导数的一种收敛阶为3-α的差分格式,然后应用该差分格式对图像放大结果进行边缘信息增强。由于该差分格式对非零的Dirchlet边界条件有效,因此相比一般的高阶方法其更适合图像处理。实验结果表明:该方法与现有方法相比,能更有效地还原图像的边缘纹理等细节。

The edges and textures of the image are blurred in the process of image magnification,and it can be solved effectively by using fractional order partial differential equations. The Riesz fractional derivative may be expressed in terms of Hadamard finite part integral,and a 3-α oder approximation scheme to the fractional derivative is introduced by approximating the Hadamard finitepart integral with the piecewise quadratic interpolation polynomials. The result of image magnification is revised by the difference scheme. Since the difference scheme is valid for nonzero Dirchlet boundary conditions,it is more suitable for image processing than general high order methods. Experimental results show that the proposed method which is more effectively can restore edges,textures and other details than existing method.