Gray vector

In principle, the algorithms for color constancy, which are described in the following chapters, can also be applied to the rotated coordinates (Figure 2.20). Land notes that, in practice, the transformation may not be that simple if the algorithm is nonlinear, i.e. contains a thresholding operation. We see later, e.g. in Section 6.6, Section 7.5, or Section 11.2, that some of the algorithms for color constancy are also based on a rotated coordinate system where the gray vector, i.e. the achromatic axis, plays a central role. 

Figure 5.7 HSV color space. The RGB cube is projected along onto a plane that is perpendicular to the gray vector. The result is a hexagonal disk.

Figure 6.23 Color cluster rotation. The cluster of points is first shifted to the origin. Then, the major axis of the cluster is aligned with the gray vector. This is done by rotating the cluster around a vector that is perpendicular to the gray vector and also perpendicular to the major axis of the cluster. Finally, the cluster is shifted to maintain the original average intensity.

Figure 8.3 Algorithm of Pomierski and GroB (1995). (a) A cloud of points inside the RGB color cube. The main axis is also shown, (b) The cloud of points is rotated onto the gray vector, (c) The point cloud is rescaled such that the color space is completely filled.

Figure 11.4 The gray vector passes from [0, 0, 0] to [1,1, 1] directly through the middle of the color cube. If local space average color is located away from the gray vector, we can use a shift perpendicular to the gray vector to move the color back to the center.

If we subtract this vector from the current color, we move local space average color back to the gray vector. This is visualized in Figure 11.6. Let C = [cr, c . cb T be the color of the input pixel. Thus, output colors can be calculated by subtracting the component of local space average color, which is perpendicular to the gray vector. 

Figure 11.5 For these images, output pixels were calculated by subtracting the component of local space average color, which is perpendicular to the gray vector, from the current pixel color.

That is, we project both colors onto the plane with r + g + b = 1. We can calculate the component perpendicular to the gray vector by projecting local space average color onto the gray vector and subtracting the resulting vector from local space average color. 

Figure 11.8 The color of the current pixel c, and local space average color a are projected onto the plane r + g + b = 1. Let c and a be the normalized points. Now, normalized local space average color is projected onto the gray vector w. The projection is subtracted from a, which gives us ax- The component ax is orthogonal to the gray vector w. This component is subtracted from the color of the current pixel that gives us the normalized output color 6. Finally, the output color is scaled back to the intensity of the input pixel.

Figure 11.12 Shifted color gamut (a). The white point lies at the position a. A color correction can be performed by shifting the colors toward the gray vector (b). Now the color gamut is centered around the gray vector. In order to fully use the available color space, we can increase the color gamut as shown in (c).

Color cluster rotation, which is described in Section 6.6, views pixels of the input image as a cloud of points. A principal component analysis is done to determine the main axis of this cloud of points. The main axis is rotated onto the gray vector. For the input data in Helson s experiments, there are only two different colors sensed by the sensor. The two colors line up along the axis ei, which is defined by the illuminant. 

The algorithm described in Section 11.2 subtracts the component of local space average color that is orthogonal to the gray vector from the color of the input pixel. Let cp be the color of the input pixel and let a be the local space average color. Let c = [cr,cg, cj,] be the average value of the components of a pixel, i.e. a denotes the average value of... 

Color cluster rotation Described in Section 6.6. The cloud of points is aligned with the gray vector, positioned at the center of the RGB cube, and rescaled to fill most of the cube using the inverse of the square root of the largest eigenvalue as a scaling factor. 

---