Pentland (
1982) developed one of the first image-based methods for estimating lighting direction in an image. Precluding any direct estimation of the 3D geometry of the surface, but assuming the surface normals are isotropically distributed, he showed that the slant and tilt of the lighting direction can be estimated from the local shading derivatives. Lee and Rosenfeld (
1985) extended and improved the method, deriving the following estimate of lighting tilt
θ:
where
E(
I _{ y }) is the maximum-likelihood estimate of the image derivative along the
y direction and
E(
I _{ x }) is the maximum-likelihood estimate of the image derivative along the
x direction. The slant
φ of the lighting direction is
where
λ is the illumination brightness,
ρ is the surface albedo, and
E(
I ^{2}) is the expectation of
I ^{2} taken along the tilt direction
θ. The slant and tilt estimates determine the lighting direction as follows:
Thus, we can solve for the lighting direction using the 2D content of the image and an estimate of albedo and illumination brightness, provided that the global shape of the surface is convex. Because this approach is based on the 2D image information, we refer to it as the image-based method.