Most scenes are illuminated by several light sources, where the traditional assumption of uniform illumination is invalid. This issue is ignored in most color constancy methods, primarily due to the complex spatial impact of multiple light sources on the image. Moreover, most existing multi-illuminant methods fail to preserve the smooth change of illumination, which stems from spatial dependencies in natural images. Motivated by this, we propose a novel multi-illuminant color constancy method, by learning pixel-wise illumination maps caused by multiple light sources. The proposed method enforces smoothness within neighboring pixels, by regularizing the training with the total variation loss. Moreover, a bilateral filter is provisioned further to enhance the natural appearance of the estimated images, while preserving the edges. Additionally, we propose a label-smoothing technique that enables the model to generalize well despite the uncertainties in ground truth. Quantitative and qualitative experiments demonstrate that the proposed method outperforms the state-of-the-art.