目的 对图像纹理区域的细节保持一直以来是图像插值技术的一个难题，为此提出了一种梯度优化的有理函数图像插值算法。方法 首先，构造了一种新的含有可调参数的双变量有理插值函数，随着参数的不同取值，该函数具有不同的表达形式，它是多项式模型和有理模型的有机统一体；其次，根据图像的区域特征，利用等值线方法将图像自适应地划分为纹理区域和平滑区域，纹理区域采用有理模型插值，平滑区域采用多项式模型插值；最后，根据各向同性Sobel算子计算插值单元的图像梯度，确定纹理方向，不同纹理方向的插值单元用相应的权重对中心点进行优化。结果 从客观数据、主观效果、时间复杂度3个方面对重建图像进行评价，客观数据包括峰值信噪比（PSNR）和结构相似性（SSIM），从实验结果可以看出，本文算法的PSNR平均提高了0.14~1.50 dB，SSIM平均提高了0.005~0.097。从主观效果来看，本文算法的重建图像的纹理细节更加丰富，边缘结构更加清晰，从时间复杂度来看，本文算法的平均运行时间是3.77 s，分别比DFDF（directional filtering and data fusion）、NEDI（new edge-directed interpolation）、RSAI（robust soft-decision adaptive interpolation）、Lee's、NARM（nonlocal autoregressive model）算法快了3.28倍、5.26倍、53.28倍、43.53倍、418.54倍。特别地，对于Baboon、Barbara、Metal这类纹理细节丰富的图像，本文算法在峰值信噪比和结构相似性上较对比算法有突出优势，主观效果有明显提高。结论 基于构造的双变量有理插值模型，本文提出了一个梯度优化的有理函数图像插值算法，实验结果表明，该算法在图像纹理细节和边缘结构保持方面具有良好的视觉效果，有效提高了插值图像质量，且时间复杂度较低。

Objective Image interpolation has become an active area of research in image processing, which can be easily extended to diverse applications ranging from medical imaging, remote sensing, aviation, animation production, and multimedia entertainment industries. A large number of image interpolation methods have been proposed by researchers. Generally, the interpolation methods can be divided into discrete and continuous methods. The adaptive interpolation methods based on discrete ideas can preserve the image structure of the edge. However, its performance in maintaining image details is less than satisfactory. The image cannot be amplified at any multiple by using discrete methods. And such methods are considerably time consuming. The interpolation methods based on continuous ideas can obtain rich image detail information but cannot maintain the image edge structure well. A new method of rational function image interpolation based on gradient optimization that has the advantages of the discrete and continuous methods is proposed. Method First, a novel bivariate rational interpolation function is constructed. With varying shape parameters, the function has different forms of expression, i.e., an organic unity of polynomial and rational models. The constructed C² continuous rational function interpolation model has the advantages of the continuous method, in which the appearance of the jagged edge is reduced to some extent and becomes smooth. Second, according to the regional characteristics, the image is divided into the texture and smooth regions automatically using the isoline method. If the interpolation unit has at least an isoline, then the unit belongs to the texture region. If the interpolation unit does not have isolines, then the unit belongs to the smooth region. The smooth region is interpolated by the polynomial model, and the texture region is interpolated by the rational model. Finally, according to the isotropic Sobel operator, the image gradient of the interpolation unit is calculated and the direction of the texture region is determined. According to the image gradient and texture direction, the weight of the influencing factor of every interpolation unit is obtained. Then, the center of the image patch with different directions is optimized by convoluting with the weight matrix. Result A rational function image interpolation algorithm based on gradient optimization is proposed. The proposed algorithm is tested in three different aspects, namely, objective data, visual effect, and time complexity. Compared with the state-of-the-art interpolation algorithms, the average peak signal-to-noise ratio of the proposed method is 1.5, 0.36, 0.14, 0.28, 1.11, and 0.95 dB higher than that of the bicubic, RSAI, DFDF, NARM, NEDI, and Lee's algorithms, respectively. The average structural similarity of the proposed method is 0.0968, 0.007 2, 0.007 6, 0.005 2, 0.014 1, and 0.023 7 higher than that of the bicubic, RSAI, DFDF, NARM, NEDI, and Lee's algorithms, respectively. The image reconstructed by the proposed method has richer texture detail and sharper edge structure than that by the bicubic, RSAI, DFDF, NARM, NEDI, and Lee's algorithms. The average runtime of the proposed method is 7 s, which is 3.28, 5.26, 53.28, 43.53, and 418.54 times faster than that of DFDF, NEDI, RSAI, Lee's, and NARM algorithms. For texture images such as Baboon, Barbara, and Metal, the proposed method is highly competitive not only in objective data but also in visual effect. Conclusion We construct a bivariate rational interpolation function in this study. On the basis of this model, an image interpolation algorithm based on gradient optimization is presented, which is not only able to scale the reconstructed image indefinitely but also has a low time complexity. Experimental results show that the proposed algorithm preserves the image details and structures of the edge effectively and generates a high-quality interpolation image.