We study the problem of estimating the illuminant's direction from
images of textured surfaces. Given an isotropic, Gaussian random
surface with constant albedo, Koenderink and Pont [JOSA 03] developed
a theory for recovering the illuminant's azimuthal angle from a single
image of the texture formed under a Lambertian model. In this paper,
we extend the theory to deal with cases of spatially varying albedo.
First, we generalise the theory to explain why their method should work
even for certain types of spatially varying albedo. Our generalisation
also predicts that the coherence of the structure tensor should lie
below 0.8 in such non-constant albedo cases and accurately predicts
the ``deviation'' from the true value observed by Koenderink and Pont
on the Columbia-Utrecht (CUReT) texture database. Next, we extend the
theory to account for arbitrarily varying albedo. We also investigate
local, rather than global, estimates of the direction, and demonstrate
our theory on the CUReT and the Heriot-Watt TextureLab databases where
estimated directions are compared to ground truth.