In this paper, a novel single image super-resolution (SR) method based on progressive-iterative approximation is proposed. To preserve textures and clear edges, the image SR reconstruction is treated as an image progressive-iterative fitting procedure and achieved by iterative interpolation. Due to different features in different regions, we first employ the nonsubsampled contourlet transform (NSCT) to divide the image into smooth regions, texture regions, and edges. Then, a hybrid interpolation scheme based on curves and surfaces is proposed, which differs from the traditional surface interpolation methods. Specifically, smooth regions are interpolated by the non-uniform rational basis spline (NURBS) surface geometric iteration. To retain textures, control points are increased, and the progressive iterative approximation of the NURBS surface is employed to interpolate the texture regions. By considering edges in an image as curve segments that are connected by pixels with dramatic changes, we use NURBS curve progressive-iterative approximation to interpolate the edges, which sharpens the edges and can maintain the image edge structure without jaggy and block artifacts. The experimental results demonstrate that the proposed method significantly outperforms the state-of-the-art methods in terms of both subjective and objective measures.