Improved colour to greyscale via integrability correction

The classical approach to converting colour to greyscale is to code the luminance signal as a grey value image. However, the problem with this approach is that the detail at equiluminant edges vanishes, and in the worst case the greyscale reproduction of an equiluminant image is a single uniform grey value. The solution to this problem, adopted by all algorithms in the field, is to try to code colour difference (or contrast) in the greyscale image. In this paper we reconsider the Socolinsky and Wolff algorithm for colour to greyscale conversion. This algorithm, which is the most mathematically elegant, often scores well in preference experiments but can introduce artefacts which spoil the appearance of the final image. These artefacts are intrinsic to the method and stem from the underlying approach which computes a greyscale image by a) calculating approximate luminance-type derivatives for the colour image and b) re-integrating these to obtain a greyscale image. Unfortunately, the sign of the derivative vector is sometimes unknown on an equiluminant edge and, in the current theory, is set arbitrarily. However, choosing the wrong sign can lead to unnatural contrast gradients (not apparent in the colour original). Our contribution is to show how this sign problem can be ameliorated using a generalised definition of luminance and a Markov relaxation.