Emerging display technologies are able to produce images with a much wider color gamut than those of conventional distribution gamuts for cinema and TV, creating an opportunity for the development of gamut extension algorithms (GEAs) that exploit the full color potential of these new systems. In this paper, we present a novel GEA, implemented as a PDE-based optimization procedure related to visual perception models, that performs gamut extension (GE) by taking into account the analysis of distortions in hue, chroma, and saturation. User studies performed using a digital cinema projector under cinematic (low ambient light, large screen) conditions show that the proposed algorithm outperforms the state of the art, producing gamut extended images that are perceptually more faithful to the wide-gamut ground truth, as well as free of color artifacts and hue shifts. We also show how currently available image quality metrics, when applied to the GE problem, provide results that do not correlate with users' choices.