# Image Magnification Based on Bicubic Approximation with Edge as Constraint
## 6 Conclusions  
It is an effective method to construct a surface to approximate the image and then sampling the surface to generate the magnified image, especially when magnifying medical images. The key problem is how to construct the approximation surface. Based on the complexity of image shape, it is an effective strategy to construct local surfaces and to generate a whole sampling surface by the weighted combination of the local surfaces. In this paper, the second-order difference quotient is used to describe the characteristics of the pixels at the edge, which better distinguishes the characteristics difference between the pixels at the edge and the pixels at the non-edge. In the process of local surface construction, we use the edge features of the image as constraints to construct a quadratic polynomial surface patch on the adjacent area of each pixel. Four adjacent quadratic polynomial patches are weighted to form a bicubic surface. All the bicubic surfaces are put together to form the whole sampling surface. The constructed bicubic surface has quadratic polynomial precision. It is inevitable that there are errors in the magnified image obtained from the whole sampling surface. The correction surface can be formed by the approximation error, so as to correct the errors and to improve the approximation accuracy of the whole sampling surface. Due to the use of edge feature constraints to construct local surfaces, each local surface can better maintain the edge features or texture features near the pixels. Experiment results showed that it is an effective method to magnify the image by constructing local surface patches with edge feature constraints, especially for medical image. Although the magnified image keeps information such as the edge texture of the image well, there is still a lack of edge and texture information. Moreover, the bicubic surface should have bicubic polynomial precision, but the method in this paper has only quadratic polynomial precision. Therefore, our future research work is to improve the retention of edge and texture information and the interpolation accuracy, so that the magnified image has higher image quality and better visual effect. In addition, Table 2 and Figure 8 show that, for the aliasing images in AIS, the eight methods are not ideal for enlarging the images. It is necessary to study a new image magnification method according to the characteristics of the aliasing images. Because in the aliasing images, there are many edges and the difference of pixel values at the edge is large. One idea to consider is to magnify the aliasing image by the combination of segmentation and fitting. Firstly, the image is segmented, and then, the piecewise fitting patches are constructed based on the segmentation results so as to reduce the oscillation of the patchs as much as possible. This is our second research work we will do next. Conceptualization, A.C.; methodology, H.H. ; software, L.J. ; validation, R.Z. ; formal analysis, A.C., H.H., L.J., and R.Z. ; investigation, L.J. ; resources, H.H. ; data curation, R.Z. ; writing-original draft preparation, L.J. ; writing-review and editing, A.C.; visualization, L.J. and R.Z. ; supervision, H.H. ; project administration, A.C.; funding acquisition, H.H. All authors have read and agreed to the published version of the manuscript. This work is supported by NSFC Joint Fund with Zhejiang Integration of Informatization and Industrialization under Key Project (U1609218) and by the Science and Technology Innovation Program for Distributed Young Talents of Shandong Province Higher Education Institutions under Grant 2019 KJN042. The authors declare no conflict of interest. The following abbreviations are used in this manuscript: \begin{tabular}{l l} PSNR & Peak Signal to Noise Ratio \\ SSIM & Structural Similarity \\ HR & High Resolution \\ LR & Low Resolution \\ AIS & Aliasing Image Set \\ DCCI & Directional Cubic Convolution Interpolation \\ ICBI & Iterative Curvature-Based Interpolation \\ \end{tabular}