摘要： The previous chapter has shown the importance of maintaining the image edges in image interpolation to obtain a natural-looking interpolated image. We have discussed the basic structure and model of image edges. The multi-resolution analysis presented in the chapter on wavelet (Chapter 6) has shown that an ideal step edge obeys the geometric regularity property [43], which refers to the correlation structure of an ideal step edge being independent of the scale of the image. In other words, the correlation structure of the local region that has enclosed an edge feature in the low-resolution image should have a similar correlation structure in the corresponding region in the high-resolution image. The edge-directed interpolated algorithms presented in the last chapter aim to preserve the correlation structures across scales by estimating the unknown pixels with the consideration of the locations and orientations of the image edges. The locations and the orientations of the image edges are specified by explicit edge map obtained through some kind of edge detection techniques. There are several problems associated with such approach. Firstly, the edge detection technique quantized the edge orientation in finite number of cases. As a result, the interpolation results can only preserve limited classes of correlation structures during the interpolation process. In other words, the geometric regularity cannot be fully preserved between the low-resolution image and the interpolated high-resolution image. Secondly, the edge detection algorithm has detection accuracy problems in both the spatial location of the edge pixel and the orientation of the associated edge. As a result, the edge-directed interpolation algorithm might falsely preserve a nonexisting edge structure and results in image artifacts with unnatural appearance and low in peak signal-to-noise ratio (PSNR). To achieve good image interpolation result, the geometric regularity property should be fully exploited. Various algorithms have been proposed in literature that aim to preserve the geometric regularity between the low-resolution image and the interpolated high-resolution image, which include directly considering the geometric regularity in spatial domain [12, 38], reformulating the problem into a set of partial differential equations [6, 47], and constructing a convex set of geometric regularity [54], and proposed a projection-onto-convex-set algorithm to achieve an interpolated image with the best matched correlation structure between the low-resolution image and the interpolated high-resolution image.