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Volume: 20 | Article ID: art00014
A Closed Form Solution for the Brightness Preserving Colour to Greyscale Image Conversion
  DOI :  10.2352/CIC.2012.20.1.art00014  Published OnlineJanuary 2012

There are many methods for converting a colour image to a grey scale counterpart. The luminance image can be calculated as a weighted sum of R, G and B. However, when equiluminant edges appear in images, they disappear in the greyscale reproduction. Alternate greyscale computations attempt to mitigate this problem by finding the best solution according to an optimisation criterion. Optimisations include best representing the colour difference in grey scale or maximising the variance of the greyscale reproduction. A promising previous approach proposed maximising the contrast of a greyscale reproduction subject to the constraint that the brightness was preserved (i.e. the grey scale reproduction would have the same brightness as the colour original). The required greyscale was found using a quadratic programming optimisation. While this made the algorithm simple to describe it limited its practical utility (e.g. it is unlikely to get QP implemented in a digital camera). The main result of this paper is to show that there exists a closed form solution for finding the maximum contrast and brightness preserving greyscale.As in the previous work, we define that a greyscale is a weighted sum of R, G and B, and that resulting greyscale has the same average as the colour original. We propose that the individual weights should be between 0 and 1 and their sum is equal to 1 (this constraint appeals to our notion of reasonableness and ensures white is preserved). These constraints coupled with our requirement that brightness is preserved is interpreted geometrically. We show that the vector of 3 weighting factors must lie on a line segment and that the best solution is always at one of the endpoints. It is straightforward to directly solve for these endpoints and so directly solve the maximum contrast brightness preserving greyscale problem.

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Graham D. Finlayson, Toby N. Matheson, "A Closed Form Solution for the Brightness Preserving Colour to Greyscale Image Conversionin Proc. IS&T 20th Color and Imaging Conf.,  2012,  pp 76 - 81,

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