Filtered polynomial interpolation for scaling 3D images

Image scaling methods allows us to obtain a given image at a different, higher (upscaling) or lower (downscaling), resolution with the aim of preserving as much as possible the original content and the quality of the image. In this paper, we focus on interpolation methods for scaling three-dimensional grayscale images. Within a unified framework, we introduce two different scaling methods, respectively based on the Lagrange and filtered de la Vall\'ee Poussin type interpolation at the 1st kind's Chebyshev zeros. In both cases, using a non-standard sampling model, we take (via tensor product) the associated trivariate polynomial interpolating the input image. It represents a continuous approximate 3D image to resample at the desired resolution. Using discrete linf and l2 norms, we theoretically estimate the error achieved in output, showing how it depends on the error in input and on the smoothness of the specific image we are processing. Finally, taking the special case of medical images as a case study, we experimentally compare the performances of the proposed methods among them and with the classical multivariate cubic and Lanczos interpolation methods.