Line of constant illumination

Figure 12.4 Nonlinear change of the illuminant. The arrow shows the direction of the gradient. The line of constant illumination is perpendicular to the direction of the gradient.

Figure 12.5 Rotated coordinate system. The directions front/back point along the direction of the gradient. The directions left/right point along the line of constant illumination.

Let a(front), a(back), a(left), and a(right) be the interpolated colors of the rotated coordinate system. Let c(x, y) be the color at the current element. Now we can calculate local space average color a, for color band i by averaging the colors obtained from the left and the right along the line of constant illumination ... 

If we iterate long enough, we obtain local space average color, which is computed along the line of constant illumination. 

Figure 12.8 Care must be taken at the border elements. The pixel values outside of the grid of processing elements are unknown. No assumption should be made about these pixel values. Therefore, we only average the data of the current element and the interpolated value along the line of constant illumination.

Figure 12.10 Output images for interpolation method (b). The first two images (a) and (b) are very similar to the input images shown in Figure 12.9. The third image (c) differs because here the line of constant illumination is curved. For this image, the values obtained along the line of constant illumination are too low. The spotlight is blurred and comes out too dark.

Figure 12.11 If the line of constant illumination is curved, then the values, which are obtained in a direction perpendicular to the gradient, are not correct.

The solution to this problem is also clear. We need to calculate the local curvature of the illuminant for each processing element. Then, we can average the values along the line of constant illumination even if the shape of the change of the illuminant is curved. The curvature K of a point (x, y) on a surface F(x, y) is defined as (Bronstein et al. 2001)... 

The sign of the curvature K tells us on which side of the curve the center of the curvature lies. If K > 0, then the center of the curvature lies on the positive side of the curve normal. It lies on the negative side of the curve normal if K < 0. If K = 0, then the line of constant illumination is really a straight line. 

Figure 12.12 The values along the line of constant illumination can be obtained by calculating the intersection between the unit circle and the circle that describes the curvature at the current processing element. We have to obtain pixel values at positions Pi and l if the center of the curvature lies at position (0, r ).

If the center of the curvature does not lie on the X-axis, we can simply perform an appropriate rotation of the coordinate system. Now that we know the points of intersection, we can obtain pixel values at these positions using one of the methods of interpolation as described earlier. Let a, be the previous estimate of local space average color along the line of constant illumination. If only the two values along the line of constant illumination are averaged, then we compute... 

Table 12.2 Root mean squared error between the input image and the image with the computed local space average color. Local space average color was computed by averaging the data obtained along the line of constant illumination. The line of constant illumination was computed from the curvature at each pixel.

Figure 12.13 Output images for interpolation method (h) when the local curvature is calculated in order to obtain the correct values along the line of constant illumination. The center of the spotlight is now much brighter.

Figure 12.16 Anisotropic local space average color that was computed along the lines of constant illumination.

Computation of anisotropic local space average color along the line of iso-illumination was discussed in Chapter 12. Since only two different colors are present in the input image, local space average color a computed along the line of constant illumination will again be given by... 

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