Poster on Colour constancy via horizontal cells, page 05

feedback algorithm

To make the algorithm more universal an amplification factor K for the subtracted average was introduced in step (2). Otherwise, without additional amplification, it is impossible to achieve good colour constancy.

X_i^R = S^R ·r_i^R

(1) log X_i^R = log S^R + log r_i^R

(2) z_i^R = log X_i^R – K · y^R

(3) y^R = ¹/_N ·S z_i^R – C^R = log S^R + ¹/_N ·S log r_i^R – K · y^R – C^R

Again, the constant C^R have to be adjusted to compensate the second term in (3):
C^R = ¹/_N ·S log r_i^R

Then y^R = log S^R– K · y^R and a steady-state value for the average, y^R, amounts only a fraction of an illuminant:
By substituting y^R in (2) for the latter expression and in view of (1), we get:
If K goes to infinity the first term in the latter expression tends to zero and we get: z_i^R = log r_i^R

(4)
Z_i^R = exp( z_i^R ) = r_i^R

X_i^R = S^R ·r_i^R
(1)	log X_i^R = log S^R + log r_i^R
(2)	z_i^R = log X_i^R – K · y^R
(3)	y^R = ¹/_N ·S z_i^R – C^R = log S^R + ¹/_N ·S log r_i^R – K · y^R – C^R
Again, the constant C^R have to be adjusted to compensate the second term in (3): C^R = ¹/_N ·S log r_i^R
Then y^R = log S^R– K · y^R and a steady-state value for the average, y^R, amounts only a fraction of an illuminant: By substituting y^R in (2) for the latter expression and in view of (1), we get: If K goes to infinity the first term in the latter expression tends to zero and we get: z_i^R = log r_i^R
(4)	Z_i^R = exp( z_i^R ) = r_i^R