Computation of Lightness and Color

A threshold operator is used to separate these two components. Let (x) be a threshold operator 

The expression -V2log(x2 + y2) is equivalent to the delta function 5(x, y). This gives us 

After that, we exponentiate and obtain an estimate of the reflectances. 

Unfortunately, it is not possible to completely recover the original reflectances. For instance, scaling and a constant factor cannot be recovered. Any solution to the equation V/(x, y) = 0 can be added to the input without changing the result of the algorithm. One has to define suitable boundary conditions. Because of this, Horn calls the result lightness 

After thresholding, one needs to undo the Laplacian. In the continuous case, we convolved the output with the function + log/-2. In the discrete case, we have to solve the discretized version of 

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