The Relative Illuminant

 

Assume that two images of the same scene, taken under different conditions are given. One of the images is chosen as the reference image and the task is to determine the chromatic relation between the reference and a second, test image. This relation is quantified by the relative illuminant which is postulated as being homogeneous over the entire scene, $L(\lambda )$.Let the spectrum corresponding to a pixel at a location ${\bf x}$ in the reference image be $R(\lambda, {\bf x})$, and the spectrum of the same pixel in the test image be $M(\lambda, {\bf x})$, where $\lambda$ is the radiation wavelength. As will be shown below, there is no need to register the pixels in the two images!

We will also assume that there is no systematic (global) change in the image formation geometry. This assumption can also be relaxed due to the statistical nature of the technique. Then, the above defined quantities are connected by the relation

$$M(\lambda ,{\bf x}) = R(\lambda ,{\bf x})\cdot L(\lambda) \ \ .$$ (1)

Taking logarithms (lower case letters denote the logarithm of the upper case)  

$$m(\lambda ,{\bf x}) = r(\lambda ,{\bf x}) + l(\lambda) \ \ ,$$ (2)

a linear relation is obtained. For $k = 1 \ldots K$ let $b_k(\lambda)$ be the eigenvectors of the log-spectra from the combined NCS/Munsell color appearance system. The Karhunen-Loeve expansions of the three spectra are

$$m(\lambda ,{\bf x}) \approx \sum_{k=1}^K \mu_k({\bf x}) b_k(\lambda) \ \ ,$$

$$r(\lambda ,{\bf x}) \approx \sum_{k=1}^K \rho_k({\bf x}) b_k(\lambda) \ \ ,$$

$$l(\lambda ,{\bf x}) \approx \sum_{k=1}^K \alpha_k b_k(\lambda) \ \ .$$

Which yields from (2)  

$$\mu_k({\bf x}) = \rho_k({\bf x}) + \alpha_k \ \ .$$ (3)

In the log-spectral space, the K-L coefficients describing the spectrum of a pixel in the test image $\mu_k({\bf x})$are related to the K-L coefficients of the pixel in the reference image $\rho_k({\bf x})$by a constant, global shift parameter $\alpha_k$.

The equation (3) connects the colors of a matched pixel pair, and therefore have no practical value. However, when the relation is considered for a large set of pixels randomly chosen from the two images it reveals an important property. The distributions of the log-spectral coefficients of the test and reference images are related by constant shifts. These shift parameters provide the representation of the relative illuminant in the log-spectral space. Note that the relative illuminant can also be used as a global index for a single image. In this case, instead of the distributions derived from the reference image, the ones generated from the complete Munsell/NCS system are used. The relative illuminant provides a description of the image in the color appearance space.

The general algorithm to determine the relative illuminant is then:

1.

For both the reference and test image:

• Randomly select a few thousand of pixels of representative colors.
• For each pixel find its correspondent in the color appearance space, and thus the K-L coefficients in the log-spectral space.
• Build the K-L coefficient distributions.
• Compute for each distribution the value of its mode. When only the global representation of an image is of interest, use instead of the reference image the distributions of the complete Munsell/NCS system.

2.

Determine the shift parameters $\alpha_k$, $k = 1 \ldots K$,by subtracting the mode estimates (3).

3.

Compute the spectrum of the relative illuminant $\hat{L}(\lambda) = \exp{\sum_{k=1}^K \alpha_k\cdot b_k(\lambda)}$.

4.

(Optional) Normalize the test image towards similarity with the reference image.

Several practical problems are to be solved when the algorithm is implemented. They are discussed in the following sections.