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A Closed Form Solution for the Brightness Preserving Colour to Greyscale Image Conversion

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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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Document Type: Research Article

Publication date: January 1, 2012

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  • CIC is the premier annual technical gathering for scientists, technologists, and engineers working in the areas of color science and systems, and their application to color imaging. Participants represent disciplines ranging from psychophysics, optical physics, image processing, color science to graphic arts, systems engineering, and hardware and software development. While a broad mix of professional interests is the hallmark of these conferences, the focus is color. CICs traditionally offer two days of short courses followed by three days of technical sessions that include three keynotes, an evening lecture, a vibrant interactive (poster) papers session, and workshops. An endearing symbol of the meeting is the Cactus Award, given each year to the author(s) of the best interactive paper; there are also Best Paper and Best Student Paper awards.

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