The Gray World Assumption

From the theory of color image formation, we have seen that the intensity /, f.v. y) measured by a sensor i with i e r, g, b at position (x, y) on the sensor array can be approximated by 

For the derivation of this equation, we have also assumed ideal sensors that only respond to light of a single wavelength, i.e. the sensor can be described by a delta function. This is an 

Let E[GRj be the expected value of the geometry factor G multiplied by the reflectance R,. Both can be considered as independent random variables, as there is no correlation between the shape and the color of an object. Let us assume that the reflectances are uniformly distributed, i.e. many different colors are present in the scene and each color is equally likely. Therefore, the reflectance can be considered to be a random variable drawn from the range [0, 1]. We obtain (Johnson and Bhattacharyya 2001) 

Instead of assuming that the reflectances are uniformly distributed over the range [0, 1], we can also use the actual distribution of reflectances to compute E[Ri] (Barnard et al. 2002a). In this case, we also need to know the shape of the response curves of the camera to compute the expected value. This value would then depend on the set of reflectances chosen for the surrounding and would also depend on the type of camera used. 

We now see that we can use space average color to estimate the color of the illuminant as 

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The gray world assumption works nicely if we only have a single illuminant. However, if we have multiple illuminants, then the gray world assumption does not work. We can see this in image (b) of Figure 6.4. This is not surprising, as one of the assumptions was that we... 

Figure 6.5 The gray world assumption will fail to produce correct colors if sufficiently large numbers of colors are not present in the scene. A leaf from a banana plant is shown in (a). The image in (b) shows the output image.

The gray world assumption as well as the white patch retinex algorithm are used frequently for automatic white balance. The popular draw utility written by Coffin (2004) scales each color channel using the average as an automatic white-balance option. After rescaling, the white point is set at the 99th percentile. In other words, all channels are scaled equally such that only the top 1% of all pixels are clipped. This assumes that there are only a few highlights. 

Instead of normalizing to white, we can also use the gray world assumption. If we compute the average of the n pixel values after the logarithm has been applied, we obtain... 

Figure 6.12 Results obtained using a simplified version of Horn s algorithm. The gray world assumption was used for normalization.

The influence of the illuminant can be removed by using the gray world assumption. 

Figure 6.26 Comprehensive color normalization iteratively normalizes all pixels and then rescales the color channels using the gray world assumption. Therefore, intensity information is lost. Since it is based on the gray world assumption, it does not work if we have a nonuniform illumination.

Cardei and Funt (1999) suggested to combine the output from multiple color constancy algorithms that estimate the chromaticity of the illuminant. Their approach is called committee-based color constancy. By combining multiple estimates into one, the root mean squared error between the estimated chromaticity and the actual chromaticity is reduced. Cardei and Funt experimented with committees formed using the gray world assumption,... 

Combining White Patch Retinex and the Gray World Assumption... 

Figure 12.3 Even a nonlinear change of the illuminant can be considered to be linear provided the area of support is sufficiently small (a). However, in practice, we use quite large areas of support. Otherwise the gray world assumption would not be correct. In this case, the area of support will probably contain nonlinearities and the estimated local space average color will be incorrect (b).

If we apply the gray world assumption, then the output color o of the sample is given by... 

Thus, the output color is independent of the light that illuminates the sample. The output color will be achromatic for all combinations of illuminants and backgrounds. Thus, the gray world assumption is not in agreement with the results obtained by Helson. 

Comprehensive color normalization, developed by Finlayson et al. (1998), which is described in detail in Section 6.7, interleaves color normalization and normalization based on the gray world assumption. First, the image pixels are normalized by dividing each channel by the sum over all channels. This algorithm is denoted by Aiorm- Next, the image pixels are normalized by dividing each channel independently by the sum over all pixels. 

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