A mathematical approach to best luminance maps

An algorithm to calculate the best global mapping from color to grayscale is presented. We assert that the best mapping minimizes the difference between the multi-channel local tensor and the tensor of the resultant mono-chromatic image. To minimize the objective function, we represent the grayscale image as a weighted sum of the RGB channels, three channels and their second-order polynomial and three channels and their root polynomial. The optimization searches for the best weights to combine the linear, polynomial, and root polynomial functions. Our results show that the optimal weights can half the root mean square difference between the color gradients and those achieved by the conventional luminance transformation. Further improvement is achieved by adding the squared and root squared channels to the solution. The improvements are also visually evident.