Learning a linear filter

Hurlbert and Poggio (1987, 1988) have shown that color constancy may be learnt from examples. Owing to computational constraints, they worked on individual lines of simulated color Mondrians. The color Mondrians were created by artificially illuminating a reflectance image of a color Mondrian with illumination gradients. In their model, the measurement made by the sensor / is proportional to the product of the illumination intensity L and the surface reflectance R. 

the logarithm is applied which turns the product of the reflectance R and the illumination intensity L into a sum. 

A set of training vectors was assembled from this data. Let ns be the number of training samples. Each input vector vc, with i e 1. ns] corresponded to a horizontal line from the illuminated Mondrian. Since they worked with simulated data, the corresponding reflectances vr, were known. Let nx be the width of the Mondrian image. Therefore, each vector vc, and vr, contained nx data samples. The training samples were collected in a matrix of size ns x nx. This resulted in two matrices R and C. They assumed that the transform from input to output, i.e. from measured intensities to reflectances, is linear. 

The solution of this equation, which is optimal in the least-squares sense is given by 

Color Constancy M. Ebner 2007 John Wiley Sons, Ltd 

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