Land's Retinex Theory

For simplicity, Land assumed all of the information processing regarding color perception takes place in the retina, thus the name ``retinex'' for his theory. Land postulated that the perception of color is a process containing the following operations.

• Ratio signal

  A retinal cell k+1 of one of the three cone types generates a signal proportional to the logarithm of the ratio between the intensities of the incoming light received by this cell and that received by a neighboring cell k (the local spatial comparison):
  

  $$r^{\lambda}_k \stackrel{\triangle}{=} \log\frac{i^{\lambda}_{k+1}}{i^{\lambda}_k} =\log {i^{\lambda}_{k+1}-i^{\lambda}_k}$$
  
  
  This operation is the local spatial comparison.

• Thresholding This signal is then thresholded by some threshold T:
  
  $$\delta log\frac{I^{\lambda}_{k+1}}{I^{\lambda}_k}=\left\{ \be... ...{k+1}}{I^{\lambda}_k} > T$ } \\ 0 & else \end{array} \right.$$
  
  
  where $I^{\lambda}_{k+1}$ and $I^{\lambda}_k$ are respectively the intensities of the incoming light at the cell in question and one of its neighbors. Here $\lambda \in \{ L, M, S \}$ represents one of the three types of cones.

• Propagation The thresholded signal from a cell then propagates radially outward to its neighboring cells where the signal is added to the signals similarly generated locally along the propagation path (assumed to be from cell j to cell i):
  
  $$R^{\lambda}(i,j)=\sum{k=j}^{i-1} \delta log \frac{I_{k+1}}{I_k}$$
  
  
  This operation is the global spatial comparison.

• Averaging over space

  The response of a retinal cone cell i is the average of all such propagating signals from hundreds cells in the visual field arriving at the cell.
  

  $$R^{\lambda}(i)=\frac{1}{N} \sum_{j=1}^N R^{\lambda}(i,j)$$
  
  

The color constancy results from the above computational model for basically two reasons:

• The term log (I_k+1 / I_k)=log I_k+1-log I_k extracts the difference between two neighboring retina cone cells which emphasizes the spatial details of the reflection (such as the boundaries in the Mondrian display);
• The threshold operation suppresses to zero the small differences between neighboring cells which may be caused by nonuniform illumination over the scene.

As we know the retina cone cells do not seem likely to have the computational functions as speculated here, it may be more proper to assume computation like this may take place in somewhere in the visual cortex.