The linear Monge-Kantorovitch linear colour mapping for example-based colour transfer

A common task in image editing is to change the colours of a picture to match the desired colour grade of another picture. Finding the correct colour mapping is tricky because it involves numerous interrelated operations, like balancing the colours, mixing the colour channels or adjusting the contrast. Recently, a number of automated tools have been proposed to find an adequate one-to-one colour mapping. The focus in this paper is on finding the best linear colour transformation. Linear transformations have been proposed in the literature but independently. The aim of this paper is thus to establish a common mathematical background to all these methods. Also, this paper proposes a novel transformation, which is derived from the Monge-Kantorovitch theory of mass transportation. The proposed solution is optimal in the sense that it minimises the amount of changes in the picture colours. It favourably compares theoretically and experimentally with other techniques for various images and under various colour spaces.