We propose a new procedure for image illumination correction with data-driven parameter choice. This procedure aims at estimating the reflectance image from a corrupted version in which the corruption is due to pointwise multiplicative illumination artifact. The log-illumination artefact consists of "smooth" variations of the intensity which are modelled by a function lying in a finite dimensional space. Then a γ-correction is incorporated. The question of model selection is difficult to solve. We propose an entropy minimization criterion for the selection of both the approximating log-illumination space dimension and the γ-coefficient, so that no parameter tuning is needed. Several experiments are presented using this approach. A comparison to other methods illustrates the relevance of this approach.