Lagrange–Chebyshev Interpolation for image resizing

Image resizing is a basic tool in image processing, and in literature, we have many methods based on different approaches, which are often specialized in only upscaling or downscaling. In this paper, independently of the (reduced or enlarged) size we aim to get, we approach the problem at a continuous scale where the underlying function representing the image is globally approximated by its Lagrange–Chebyshev I kind interpolation polynomial corresponding to suitable (tensor product) grids of first kind Chebyshev zeros. This is a well-known approximation tool widely used in many applicative fields due to the optimal behavior of the related Lebesgue constants. Here we aim to show how Lagrange–Chebyshev interpolation can be fruitfully applied also for resizing any digital image in both downscaling and upscaling at any desired size. The performance of the proposed method has been tested in terms of the standard SSIM (Structured Similarity Index Measurement) and PSNR (Peak Signal to Noise Ratio) metrics. The results indicate that, in upscaling, it is almost comparable with the classical Bicubic resizing method with slightly better metrics, but in downscaling a much higher performance has been observed in comparison with Bicubic and other recent methods too. Moreover, for all downscaling cases with an odd scale factor, we give a theoretical estimate of the MSE (Mean Squared Error) of the output image produced by our method, stating that it is certainly null (hence PSNR equals infinite and SSIM equals one) if the input image's MSE is null.