Visual sensitivity to small color difference is an important factor for precision color matching. Small color differences can be measured by the line element theory in terms of color distances between a color point and neighborhoods of points in a color space. This theory gives a smooth positive definite symmetric metric tensor which describes threshold of color differences by ellipsoids in three dimensions and ellipses in two dimensions. The metric tensor is also known as the Riemannian metric tensor. In regard to the color differences, there are many color difference formulas and color spaces to predict visual difference between two colors but, it is still challenging due to the nonexistence of a perfect uniform color space. In such case, the Riemannian metric tensor can be used as a tool to study the performance of various color spaces and color difference metrics for measuring the perceptual color differences. It also computes the shortest length or the distance between any two points in a color space. The shortest length is called a geodesic. According to Schrödinger's hypothesis geodesics starting from the neutral point of a surface of constant brightness correspond to the curves of constant hue. The chroma contours are closed curves at constant intervals from the origin measured as the distance along the constant hue geodesics. This hypothesis can be utilized to test the performance of color difference formulas to predict perceptual attributes (hue and chroma) and distribution of color stimulus in any color space. In this research work, a method to formulate line element models of color difference formulas the ΔE*ab, the ΔE*uv, the OSA-UCS ΔEE and infinitesimal approximation of CIEDE2000 (ΔE00) is presented. The Jacobian method is employed to transfer their Riemannian metric tensors in other color spaces. The coefficients of such metric tensors are used to compute ellipses in two dimensions. The performance of these four color difference formulas is evaluated by comparing computed ellipses with experimentally observed ellipses in different chromaticity diagrams. A method is also developed for comparing the similarity between a pair of ellipses. The technique works by calculating the ratio of the area of intersection and the area of union of a pair of ellipses. Similarly, at a fixed value of lightness L*, hue geodesics originating from the achromatic point and their corresponding chroma contours of the above four formulas in the CIELAB color space are computed by solving the Euler-Lagrange equations in association with their Riemannian metrics. They are compared with with the Munsell chromas and hue circles at the Munsell values 3, 5 and 7. The result shows that neither formulas are fully perfect for matching visual color difference data sets. However, Riemannized ΔE00 and the ΔEE formulas measure the visual color differences better than the ΔE*ab and the ΔE*uv formulas at local level. It is interesting to note that the latest color difference formulas like the OSA-UCS ΔEE and the Riemannized ΔE00 do not show better performance to predict hue geodesics and chroma contours than the conventional CIELAB and CIELUV color difference formulas and none of these formulas fit the Munsell data accurately
| Identifer | oai:union.ndltd.org:CCSD/oai:tel.archives-ouvertes.fr:tel-00981484 |
| Date | 17 February 2012 |
| Creators | Pant, Dibakar Raj |
| Publisher | Université Jean Monnet - Saint-Etienne |
| Source Sets | CCSD theses-EN-ligne, France |
| Language | English |
| Detected Language | English |
| Type | PhD thesis |
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