Abstract: To eliminate the zigzagging and blocky effects in homogeneous areas in an interpolated image, a variable exponent variational method was proposed for image interpolation. An exponent function with diffusion characteristic of image interpolution was introduced by analyzing the diffusion characteristic of variable exponent variational model. Two parameters in the exponent function act on interpolation: the one controlled the intensity of diffusion which eliminated the width of image edges while the other controlled the intensity of smoothness which retained the fine textures in the image. The new variable exponent variatonal model made the Total Variation (TV) variational diffuse along image contours and the heat diffusion on smooth areas. The numerical experiment results on real images show that image interpolated by the proposed method has better interpolated edges, especially for fine textures. Compared to the method proposed by Chen et al. (CHEN Y M, LEVINE S, RAO M. Variable exponent, linear growth functionals in image restoration. SIAM Journal on Applied Mathematics, 2006, 66(4): 1383-1406) and robust soft-decision interpolation method, the visual improvement is prominent for retaining fine textures, and the Mean Structural SIMilarity (MSSIM) is increased by 0.03 in average. The proposed model is helpful to further study variable exponent variational model for specifical image processing and worthy to practical applications such as image network communication and print.

Key words: variable exponent, Total Variation (TV), heat diffusion, variational method, image interpolation

摘要： 为了消除插值图像在边缘的锯齿现象、在平坦区域的分块现象，提出了一种变指数变分模型的图像插值方法。通过对变指数变分模型扩散特性的研究，引入了一个满足插值扩散特性的指数函数。指数函数中的两个参数实现两方面的功能：一个参数控制扩散强度从而减小图像边缘宽度，另一个控制平滑强度从而保持细小的纹理。这个新的变指数变分模型使总变差（TV）模型沿着图像轮廓方向扩散消除锯齿现象，而热扩散在图像平坦区域起光滑作用消除分块现象。数值实验结果显示，该方法能很好地重建插值图像的边缘。与Chen等的方法（CHEN Y M， LEVINE S， RAO M. Variable exponent， linear growth functionals in image restoration. SIAM Journal on Applied Mathematics， 2006， 66（4）： 1383-1406）以及鲁棒软决策插值方法相比，所提方法对细微纹理的保持有明显的视觉效果改善，平均结构相似度（MSSIM）指标提高0.03左右。该模型对进一步研究符合具体图像处理任务的变指数变分模型具有一定的探索意义，对图像网络传输、打印等具有很强的实际应用价值。

关键词: 变指数, 总变差, 热扩散, 变分方法, 图像插值