Resampling of warped images has been a topic of research for a long time but only seldomly has focused on theoretically exact resampling. We present a resampling method for minification, applied on the texture mapping function of a 3D graphics pipeline, that is derived from sampling theory without making any approximations. Our method supports freely selectable 2D integratable prefilter (anti-aliasing) functions and uses a 2D box reconstruction filter. We have implemented our method both for CPU and GPU (OpenGL) using multiple prefilter functions defined by piece-wise polynomials. The correctness of our exact resampling method has been made plausible by comparing texture mapping results of our method with those of extreme supersampling. We additionally show how the prefilter of our method can also be applied for high quality polygon edge anti-aliasing. Since our proposed method does not use any approximations, up to numerical precision, it can be used as a reference for approximate texture mapping methods.