Resolving the color‐image irradiance equation

The case of orthogonal projection of a smooth and piece-wise uniformly colored lambertian surface with complex color illumination is considered. The algorithm presented here is based on two equations: n·n = 1 (the norm constancy condition) and Φn·n = 0 (the integrability condition, where Φ is a linear differential operator). Both equations hold over the whole image. Starting with the color-image irradiance equation (CHE) for the trichromatic visual system, we infer an algebraic formulae for direct computation of the normal vector field n up to some rotation U, the same one for all the points in the region of color and illuminant constancy. The transition from the image to the normal field is performed in this stage of the algorithm with a symmetric and nonnegative matrix B, which is constant in the region. This property is used for labeling segments of the color image. The second step of the algorithm computes the rotation U mentioned above. This computation is based on CHE and the integrability condition, but the way of finding a solution is quite different. An error-function dependent upon the rotation parameters is developed and some effective optimization algorithm is used for estimating the parameters. As a result, the algorithm can compute: (1) division of the image into regions of constant color and illumination; (2) the matrices U·B (in each segment) of linear transformation to recover the shape n of the surface. Results of computational experiments are given.