Measuring the accordance of color shades based on the intersection volume of confidence ellipsoids

In many applications the colors of products and parts have to match with respect to defined color parameters or dimensions. Obviously, parts painted in a plant should not be distinguished from parts of the same color painted in another part. Therefore, a measure for the accordance of color shades is needed. Of course, a useful representation for color shades is necessary. The Lab representation provides a common approach for defining colors. It is based on three parameters L, a and b. L represents lightness , a and b stand for the color-opponent dimensions, based on nonlinearly compressed coordinates. For measuring accordance we need two samples with measured color dimensions, where each sample represents a certain producing or painting process. Now, variation within each sample can be represented by the three-dimensional confidence ellipsoid, especially by the volume of this ellipsoid. Obviously, if the ellipsoids of both samples overlap completely, no differences are recognized, at least from a statistial point of view. For determining a measure of accordance we calculate the volume of the intersection of the two ellipsoids. Its percentage share of the sum of the volumes of the two ellipsoids defines the measure of accordance. From a mathematical point of view it is not trivial to calculate this volume of the intersection. But Monte Carlo integration provides a quite simple method with sufficient accuracy. In our talk we want to present a tool realized in SAS-JMP.