In this paper we give a new method to find a grayscale image from a color image. The idea is that the structure tensors of the grayscale image and the color image should be as equal as possible. This is measured by the energy of the tensor differences. We deduce an Euler-Lagrange equation and a second variational inequality. The second variational inequality is remarkably simple in its form. Our equation does not involve several steps, such as finding a gradient first and then integrating it. We show that if a color image is at least two times continuous differentiable, the resulting grayscale image is not necessarily two times continuous differentiable.
Hans Jakob Rivertz, "On an Euler-Lagrange equation for color to grayscale conversion" in Proc. IS&T 27th Color and Imaging Conf., 2019, pp 95 - 98, https://doi.org/10.2352/issn.2169-2629.2019.27.18