Gamut Boundaries expressed with Zernike polynomials

In cross-media color image reproduction, gamut mapping is needed due to gamut difference among different media. In order to implement gamut mapping, gamut boundaries of each medium involved should be first determined. It may be expected that an analytical expression for a boundary is preferred than a set of discrete data , since it would take less storage space and make the determination of the intersection point between a boundary and a "mapping line" easier and faster. In this article, a form of Zernike polynomial expression is suggested to be used as the expression of gamut boundary surface. For instance, if C1E1976L∗a∗b∗ is adopted as the color space for gamut mapping, then each color(point) on the boundary can be expressed as L∗=L∗(a∗,b) and the boundary can be expanded into a series of Zernike polynomials with an appropriate coefficient for each of which. These coefficients can be obtained with sufficient experimental data of boundary points and existing algorithms. Experiments have been executed for a color printer with(R,GB) as its input. The 6 boundaries in ROB space would consist of (O,B),(R,O,B),(R,GO),(255,GB),(R,255,B) and (R,G255) where each of R,B varies from 0 to 255. Then 6 corresponding sets of Zernike coefficients are calculated, based on about half of the measured L∗a∗b∗'s for each boundary. A comparison between original measured data and the data predicted by Zernike polynomials shows that, not only for the data that have been used to calculate the coefficients, but also for those not used, the differences are acceptably small even negligible with only a few exceptions.