# Image Magnification Based on Bicubic Approximation with Edge as Constraint
## 1 Introduction  
Image magnification is one of the important issues in the fields of image processing, computational medicine, computer graphics, computer vision, and virtual reality. It has a wide range of applications in medical image-aided diagnosis, product defect inspection, and visualization. The purpose of image magnification is to increase the number of pixels in the image so as to increase the details of the image and to make its features clearer, helping doctors better understand the details of the lesions. Image magnification methods can be divided into three categories: fitting-based methods, learning-based methods, and other methods. In a fitting-based method, a surface is constructed from a low-resolution image. Then, more sampling points are obtained by the surface to generate a high-resolution image. Early fitting methods include bilinear interpolation [1] and bicubic interpolation [2; 3]. In these methods, the magnified images produced by bicubic interpolation have higher accuracy, while the least square [4] is similar to the two types of interpolation methods. Sometimes, there are obvious sawtooth and ringing, and serious distortion in the magnified image [1; 2; 3; 4]. The common feature of the magnified images by these methods is that the effect is better in flat areas, but it is not ideal in non-flat area (edge and texture). In order to reduce the sawtooth and blur in the magnified image, [PERSON] et al. proposed a softdecision estimation method [5]. Compared with other interpolation methods, this method calculates the pixel values of a group of high-resolution images at one time, but the generated high resolution of image has no significant improvement in visual effect. In [6, 7], an image magnification method based on gradient direction was proposed. The gradient direction of the high-resolution image is estimated by the gradient direction of the low-resolution image. In [8], images are magnified by optimizing the second-order directional derivative of the edge direction. However, in most cases, the effect of the image magnification is not ideal. [PERSON] et al. proposed a method for fitting an image using a bicubic polynomial surface generated by combining quadratic polynomial surfaces [9, 10]. The magnified image has quadratic polynomial precision in the following sense. That is, if a given image is obtained by sampling from a quadratic polynomial function, a quadratic polynomial function is reconstructed by a bicubic polynomial surface, which is generated by combining the quadratic polynomial surfaces. The images by References [9, 10] can reduce the distortion phenomena such as sawtooth and ringing, but the accuracy and visual quality of the magnified image are not ideal. The methods of image amplification based on learning can learn the missing high-frequency information from the training set of HR and LR image pairs. These methods can be roughly divided into two categories: the first category relies on an external dictionary constructed by a group of external training images, and the second category uses LR image itself to replace the external training set. For the former, LR image generates an HR image with the help of a training pair. Common methods include regression-based methods [11], sparse expression-based methods [12, 13], and so on. These methods using the external data sets usually perform well for certain categories of images, but the amplification effect of some images is not ideal. In [14], for each LR patch, a linear function needs to be learned to map it to its HR version. However, it is inevitable to generate errors by searching for magnified image block to find LR-HR patch pairs. Reference [15] proposed a high-resolution image generation method based on self-similarity (SelfExSR). The method expands the internal patch searching space through geometric changes and improves the visual effect. However, if the LR image does not contain enough similar patterns, these methods will produce sharp edges. In addition, [PERSON] et al.