Let $I(x,y)$ be the intensity at position $(x,y)$, $\rho(x,y)$ be the albedo at this point (a constant), $n=(p,q,1)$ be the surface gradient vector and $s_{0},s_{1},s_{2}$ all be vectors from the point to $(x,y)$.

Assume that $I(x,y)=I_{1}$ when $s_{0}=(1,1,0)$ and similarly, $I(x,y)=I_{2}$ when $s_{1}=(1,0,1)$ and $I(x,y)$ = $I_{3}$ when $s_{2}=(0,1,0)$

Lambert's cosine Law is then:

$I(x,y)=\rho(x,y)\dfrac{s_{x}p+s_{y}q+s_{z}}{|n||s|}$

We can calculate the following. $|n|=\sqrt{p^{2}+q^{2}+1}$ and $|s_{0}|=|s_{1}|=|s_{2}|=1$.

All I would like to have are equations for $p$ and $q$ in terms of $I_{1},I_{2},I_{3}$.