In this paper, we take interest in the process of assigning a given color distribution to a given image. Two examples of such an image modification are histogram equalization (or specification) and color transfer, in which the color palette of a style image is assigned to a source image. Classical methods for gray level specification, as well as more recent methods for color transfer, can be defined as optimal transportation problems. The corresponding image modifications are known to produce visually unpleasing effects such as the removal of details and texture, as well as the enhancement of noise or compression patterns. In this paper, a new method is proposed for the suppression of these artifacts. The method relies on a non local regularization of the transportation map, defined as the difference between the original image and the modified one. The interest of using this method is demonstrated on the aforementioned applications: contrast adjustment and color transfer.