# Image Magnification Based on Bicubic Approximation with Edge as Constraint
## 1 Introduction
[16, 17] proposed the image super-resolution using deep convolutional networks (SRCNN), which directly learned the end-to-end mapping between HR and LR images. The method improved the quality of magnified images and the amplification speed. [PERSON] et al. [18] achieved remarkable results by establishing a lightweight database and learning adaptive linear regression to map LR image blocks directly into HR blocks. However, there are some problems with these methods. First, the reconstructing effect of these algorithms depends on a trained image library, which limits the application of them. Second, the algorithm requires a lot of time to train the model. Third, the acquired model can only be used for magnification of a fixed multiple; however, the model needs to be retrained to magnify different multiple. In other types of image magnification methods, fractal is an effective tool for describing image texture. Therefore, it is widely used for texture description and classification. In [19], a texture descriptor was proposed, which implicitly combines information from space and frequency domain. Reference [20] proposed a depth upsampling method based on joint bureau partial analysis and boundary consistency analysis. In [21], an image magnification method using a special type of orthogonal fractal coding method is proposed. It can produce better details, but it cannot restore sharper edges. In [22], a texture enhancement method was proposed. Using local fractal analysis to improve image magnification performance, the method can effectively enhance image details. However, it does not provide satisfactory results in random texture regions. Reference [23] proposed a fractal interpolation method applying the fractal analysis method to image interpolation, which can better preserve the edge structure of the image. Firstly, according to image features, the image is divided into textured regions and non-textured regions. Secondly, a rational fractal interpolation model is used in the texture region, and a rational interpolation model is used in the non-texture region. Finally, the HR image is obtained through pixel mapping. This method can recover more satisfactory details than other interpolation methods. Unlike the learning-based method relying on the source of the training patch, this method obtains competitive performance by using LR image patch information. Based on error minimization, iterative back projection can significantly improve the quality of the image. Reference [24] proposed an image restoration and amplification method combining nonlocal self-similarity and global structure sparsity. A group of similar image blocks is reconstructed by adaptive regularization techniques based on weighted kernel norms, and a new strategy is used to maintain the global structure. The strategy decomposes the image into smooth components and sparse residuals. The latter uses L1 norm to regularization, which provides a new method for image restoration and magnification. [PERSON] and [PERSON] [25] continue to degrade the high resolution, to calculate errors, and to optimize the error until the error is less than a certain threshold. The quality of the reconstructed image is significantly enhanced, but there are obvious sawtooth and ringing in the edge area. In [26; 27], a nonlocal iterative back projection method was proposed. By searching for similar blocks in a certain area, the average image of the magnified image is filtered and the sawtooth and ringing phenomena can be corrected to improve the correctness of the error image. This method improves the shortcomings of the method [25] and improves the image quality to a certain extent. If the features of an image can be extracted first and the different edges of the image are segmented [28; 29], then the surface is fitted based on the features and the edge structure; the accuracy and the visual effect of the magnified image can be significantly improved. As the image data points are complex, this paper uses a piecewise polynomial surface to fit the given image.