When producing a mosaic of multiple multi-spectral images one needs to harmonize the colours so that the tone transition is smooth from one image to the other. Given two images Im(a) and Im(b), a transform T is sought to map Im(b) to an image that is harmonious in multi-spectral appearance to Im(a). We give the above problem of tonal harmonization an analytical framework, in which both ideal and practical solutions of the problem are studied. Using a physically motivated image formation model, we prove that a perfect tonal harmonizing operator cannot in general be found, but that whenever such an operator exists it is linear. In the latter case, finding the optimal harmonizing transformation can be cast as a linear programme (LP), which is a type of problem that can be efficiently solved using known techniques. Finally, strong empirical evidence is provided for the efficacy of the proposed solution.