We propose the concept of asymptotic projections onto convex sets (APOCS) in general and a wavelet-based, edge-retaining asymptotic POCS (ERAPOCS) algorithm for image interpolation in particular. APOCS differs from POCS in that the projections sequence in one (or more) of the convex sets can enter a desired region rapidly. Like POCS, our proposed algorithm is guaranteed to converge. Our new algorithm alleviates edge blurring by properly amplifying the WT of the image. The amplification in the WT domain biases the projection sequence to the subset of interpolated images that has less edge-blurring. Simulations show that our proposed algorithm performs better in the sense of PSNR and preserves the sharpness of the edges better than do the cubic interpolation and the POCS algorithms. In addition, our algorithm is more computationally efficient than the POCS algorithm since it converges in fewer iterations and has the same computational complexity.